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Mechanical Measurements
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Contents
1
The Process of Measurement: An Overview
2
Standards and Dimensional Units of Measurement
3
Assessing and Presenting Experimental Data
4
The Analog Measurand: Time-Dependent Characteristics
5
The Response of Measuring Systems
6
Sensors
Thomas G. Beckwith/Roy D. Marangoni/John H. Lienhard V
Thomas G. Beckwith/Roy D. Marangoni/John H. Lienhard V
Thomas G. Beckwith/Roy D. Marangoni/John H.Lienhard V
Thomas G. Beckwilh/Roy D. Marangoni/John H. Lienhard V
Thomas G. Beckwith/Roy D. Marangoni/John H. Lienhard V
Thomas G. Beckwith/Roy D. Marangoni/John H. Lienhard V
7
Signal Conditioning
Thomas G. Beckwith/Roy D. Marangoni/John H. Lienhard V
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Digital Techniques in Mechanic al Measurements
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Strain and Stress: Measurement and Analysis
Thomas G. Beckwith/Roy D. Marangoni/John H. Lienhard V . • .. . . . . . . . . . . . . . . .. . . . .... . . . . ... . . . . . . . . . . . 347
10
Measurement of Pressure
11
Measurement of Fluid Flow
Thomas G. Beckwilh/Roy D. Marangoni/John H. Lienhard V
Thomas G. Beckwith/Roy D. Marangoni/John H. Lienhard V
12
Temperature Measurements
G. Beckwith/Roy D. Marangoni/John H. Lienhard V
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Measurement of Motion
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Marangoni/John H. Lienhard V
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Appendix: Standards and Conversion Equations
Thomas G. Beckwith/Roy D.
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15
Appendix: Theoretical Basis for Fourier Analysis
16
Appendix: Number Systems
Thomas G. Beckwith/Roy D. Mara11go11VJoh11 H. Lienhard
Thomas G. Beckwith/Roy D. Marongo11VJoh11 H. Lienhard
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17
Appendix: Some Useful Data
18
Appendix: Stress and Strain Relationships
19
Appendix: Statistical Tests of Least Squares Fits
Thomas
G.
Beckwith/Roy D. Marangoni/John H. Lienhard V
Thomas G. Beckwith/Roy D. Mara11gonVJohn
H. Lienhard V
Thomas G. Beckwith/Roy D. MarangonVJohn H.
Index
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603
609
615
621
637
641
The Process of Measu rement:
An Overview
INTRODUCTION
2
THE SIGNIACANCE OF M ECHANICAL M EASUREMENT
3
FUNDAMENTAL M ETHODS OF MEASUREM ENT
4
THE GENERALIZED MEASURING SYSTEM
s
6
7
8
9
10
TYPES OF INPUT QUANTmEs
M EASUREMENT STANDARDS
CALIBRATION
UNCERTAINTY: ACCURACY OF RESULTS
REPORTING RESULTS
FINAL REMARKS
INTRODUCTION
It has been said, "Whatever exists, exists in some amount." The detennination of the amount
is what measurement is all about. If those things that exist are related 10 the practice of
mechanical engineering, then the de!ermination of their amounts conslilutes the subject of
mechanical measurements.1
The process or the act of measurement consists of obtaining a quantitative comparison
between a predefined standard and a measurand. The word measurand is used to designate
the particular physical parameter being observed and quanlified; that is, the input quantity
to the measuring process. The act of measurement produces a result (see Fig. I ).
The standard of comparison must be of the same character as the measuranid, and
usually, bu! not always, is prescribed and defined by a legal or recognized agency or
organization-for example, the National Institute of Standards and Technology (NIST),
fonnerly called the National Bureau of Standards (NBS), the International Organization for
Standardization (ISO), or the American Naiional Standards Institute (ANSI). The me:ter, for
example, is a clearly defined siandard of length.
Such quantities as temperature, strain, and the parameters associated with fluid flow,
acoustics, and motion, in addiiion to the fundamental quantities of mass, length, time, and
so on. arc typical of those within the scope of mechanical measurements. Unavoidably,
The
1 Meclumical meantremtnU are not neeessarily accomplished by mechanical means: rather, ii is to the measured
quantity itself that the term mtchankal is directed.
phrase mtasunmtnl of mechanical quantit1its, or of
paramettrJ, would perhaps express more oomp�ly the naning intended. In the interest of brevity, however,
the subject is simply called mechanical mecuurY11Un1s.
The Process of Measurement An Overview
Measurand
(input)
Processor
comparison
(measurement)
Result
(readout)
FIGURE I: Fundamental measuring process.
the measurement of mechanical quantities also involves consideration of things electrical,
since it is often convenient or necessary to change, or transduce, a mechanical measurand
into a corresponding electrical quantity.
2
THE SIGNIFICANCE OF MECHANICAL MEASUREMENT
Measurement provides quantitative information on the actual state of physical variables and
processes that otherwise could only be estimated. As such, measurement is both the vehicle
for new understanding of the physical world and the ultimate test of any theory or design.
Measurement is the fundamental basis for all research, design, and development, and its
role is prominent in many engineering activities.
All mechanical design of any complexity involves three clements: experience, the
rational element, and the experimental element. The element of experience is based on
previous exposure to similar systems and on an engineer's common sense. The rational
element relies on quantitative engineering principles, the laws of physics, and so on. The
experimental element is based on measurement-that is, on measurement of the various
quantities pertaining to the operation and performance of the device or process being devel­
oped. Measurement provides a comparison between what was intended and what was
actually achieved.
Measurement is also a fundamental element of any control process. The concept
of control requires the measured discrepancy between the actual and the desired perfor­
mances. The controlling portion of the system must know the magnitude and direction of
the difference in order to react intelligently.
In addition, many daily operations require measurement for proper performance. An
example is in the central power station. Temperatures, flows, pressures, and vibrational
amplitudes must be constantly monitored by measurement to ensure proper performance
of the system. Moreover, measurement is vital to commerce. Costs are established on the
basis of amounts of materials, power, expenditure of time and labor, and other constraints.
To be useful, measurement must be reliable. Having incorrect information is poten­
tially more damaging than having no information. The situation, of course, raises the
question of the accuracy or uncertainty of a measurement. Arnold 0. Beckman, founder of
Beckman Instruments, once stated, "One thing you learn in science is that there is no perfect
3
The Process of Measurement: An Overview
answer, no perfect measure. "2 It is quite important that engineers interpreting the results of
measurement have some basis for evaluating the likely uncertainty. Engineers should never
simply
read a scale or printout and blindly accept the numbers. They must carefully place
realistic tolerances on each of the measured values, and not only should have a doubting
mind but also should attempt to quantify their doubts. We will discuss uncertainty in more
detail in Section
3
8.
FUNDAMENTAL METHODS OF MEASUREMENT
( I ) direct comparison with either a primary
indirect comparison through the use of a calibrated system.
There are two basic methods of measurement:
or a secondary standard and (2)
3.1
Direct
Comparison
How would you measure the length of a bar of steel? If you were to
measurement to within, let us say,
be satisfied with
a
k in. (approximately 3 mm), you would probably use a
steel tape measure. You would compare the length of the bar with a standard and would find
that the
bar is so many inches long because that many inch-units on your standard are the
same length as the bar. Thus you would have determined the length by direct comparison.
The standard that you have
used is called a secondary standard. No doubt you could trace
its ancestry back through no more than four generations to the primary length standard,
which is related to the speed of light.
Although to measure by direct comparison is to strip the measurement process to its
barest essentials, the method is not always adequate.
The human senses are not equipped
In many cases they arc not
to make direct comparisons of all quantities with equal facility.
sensitive enough. We can make direct comparisons of small distances using a steel rule,
with a precision of about
1 mm (approximately 0.04 i n.) .
Often we require greater accuracy.
Then we must call for additional assistance from some more complex form of
system. Measurement by direct comparison is thus less common th a n
is
measuring
measurement by
indirect comparison.
3.2
Using a Calibrated System
Indirect comparison makes use of some form of transducing device coupled to a chain of
connecting apparatus, which we shall call, in toto, the measuring system. This chain of
devices converts the basic form of input into an analogous form, which it then processes
and presents al the output as a known function of the original input. Such a conversion is
often necessary so that the desired information will be intelligible. The human senses arc
simply not designed to detect the strain in a machine member, for instance. Assistance is
required from a system that senses, converts, and finally presents an analogous output in
the form of a displacement on a scale or chart or as a digital readout.
Processing of the analogous signal may take many forms. Often it is necessary to
an amplitude or a power through some form of amplification. Or in another case
it may, be necessary to extract the de.�ired information from a mass of extraneous input
increase
by a process of filtering. A remote reading or recording may be needed, such as ground
recording of a temperature or pressure within a rocket in flight. ln this case the pressure or
2Emphasis added by lhe authon.
4
The Process of Measurement: An Overview
temperature measurement must be combined with a radio-frequency signal for transmission
to the ground.
In each of the various cases requiring amplification, or filtering, or remote record­
ing, electrical methods suggest themselves. In fact, the majority of transducers in use,
particularly for dynmrnc mechanical measurements, conven the mechanical input into an
analogous electrical fonn for processing.
4
THE GENERALIZED MEASURING SYSTEM
Most measuring systems fall within lhc framework of a general arrangement consisting of
three phases or stages:
Stage 1. A detection-transduction, or sensor-transducer, stage
Stage 2. An intermediate stage, which we shall call the signal-conditioning stage
Stage 3• . A tenninating, or readou1-recording, stage
Each stage consists of a distinct component or group of components that perfonns
required and definite steps in the measuremenL These are called basic elements; their
scope is detennined by their function rather than by their construction. Figure 2 and Table I
outline the significance of each of these stages.
4.1
First. or Sensor-Transducer, Stage
The primary function of the first stage is lo detect or to sense the measurand. Al the same
time, ideally, this stage should be insensitive to every other possible input For instance, if
il is a pressure pickup, it should be insensitive to, say, acceleration; if it is a strain gage, it
should be insensitive to temperature; if a linear accelerometer, it should be insensitive lo
angular acceleration; and so on. Unfonunately, ii is rare indeed lo find a detecting device
that is completely selective. Unwanted sensitivity is a measuring error, called noise when
ii varies rapidly and drift when ii varies very slowly.
Frequently one finds more than a single transduction (change in signal character) in
the first stage, particularly if the first-stage output is electrical.
4.2
Second, or Signal-Conditioning, Stage
The purpose of the second stage of the general system is to modify the transduced infonna­
tion so that it is acceptable to the third, or tenninating, stage. In addition, it may perform
one or more basic operations, such as selective filtering to remove noise, integration, difCatibration
Auxfliary power
in�t
(not always required)
''
''
Measurand "-
�
·�
Sensor-
transducer
....-
Auxiliary power
(usually required )
signal
Transduced
(analogous
lo Input)
Signal
conditioner
-
driving
Recorder
Computer
Analogous
signal
Indicator
�
-
FIGURE 2: Block diagram of the generalized measuring system.
Processor
Controller
The Process of Measurement: An Overview
TABLE 1: Stages of the General Measurement System
Stage 1: Sensor-Transducer
Senses desired input to exclusion
of all others and provides
analogous output
Stage 2: Signal Conditioning
Modifies transduced signal into
Provides an indication or
fonn usable by
recording in form that can be
final stage.
Usually increases amplitude
and/or power, depending on
requirement. May also selectively
into p ul sed fonn
filter unwanted components or
Types and Examples
Mechanical:
Con tacting spindle,
spring-mass, elastic devices (e.g ..
for force), gyro
Bourdon tube for pressure,
proving ring
·
Stage 3: Readout-Recording
convert signal
Types and Exampks
Mechanical: Gearing,
cranks,
slides, connecting links. cams,
etc.
evaluated by an unaided human
sense or by a control ler. Records
data digitally on a computer
l)pes and Examples
Moving pointer and scale, moving
lndiclJ/ors (dispkicement ''pe):
scale and
index, light beam and
beam and sale
scale, electron
(oscilloscope), liquid column
Hydraulic-pneunuuic: Buoyant
Hydraulic-pneumatic: Piping.
float, orifice, venturi. vane,
valving, dashpots, plenum
propeller
chambers
alphanumeric readout
Optical: Photographic film.
Optical: Mirrors, lenses, optical
Recorders:
photoelectric diodes and
filters, optical fibers. spatial filters
(pinhole, slit)
pen
transistors, phOlomultiplier tubes.
holographic plates
Electrical:
Contacts, resistance.
capacitance, inductance,
piezoelectric crystals and
polymers. thermocouple.
semiconductor junction
Electrical: Amplifying or
filters, te lemetering systems.
various special-purpose
in tegrated-circ uit devices
attenuating systems, bridges,
lndicalors (digital type):
IDirect
Digital printing, inked
and chart, direct phot1>graphy.
magnetic
recording (hard disk)
Processors and compUle,,r:
Various types or computing
systems. either special-pu rpose or
general . used to feed
readout/recording devices and/or
controlling systems
Controllers: AU types
ferentiation, or telemetering, as may be required.
Probably the most common function of the second stage is to increase either a11Dpli tude
or power of the s ign al , or both, to the level required to drive the final terminating. device.
In addition, the second stage must be designed for proper matching characteristics loetween
the first and second and between the second and third stages.
4.3
Third , or Readout-Recording, Stage
The third stage provides the information sought in a form comprehensible to one of the
human senses or to a controller. If the output is intended for immediate human recc•gnition,
it is, with rare excepti o n , presented in one of the following forms:
1. As a re/alive displacement, such as movement of an indicating hand or displacement
of oscilloscope trace
The Process of Measurement: An Overview
Stem
Compression
spring
Piston
Cylinder
(a)
- - ----- - - '
'
'
'
'
:
'
'
'
: (None)
'
'
'
Sensor-transducer
Piston/cylinder
Input
pressure
(Pressure lo
lorce)
Spring
(Fon:a lo
displacement)
-
-
•
Signal
concfiUoning
·-
- - - - - - .. - - .!
Readout
(Scale and
index)
(b)
FIGURE 3: (a) Gage for measuring pressure in automobile tires. (b) Block diagram of tire-gage
functions. In this example, the spring serves as a secondary transducer.
2. In digital form, as presented by a counter such as an automobile odometer, or by
a liquid crystal display (LCD) or light-emitting diode (L_ED) display as on a digital
voltmeter
To illustrate a very simple measuring system, let us consider the familiar tire gage
used for checking automobile tire pressure. Such a device is shown in Fig. 3(a). It consists
of a cylinder and piston, a spring resisting the piston movement, and a stem with scale
divisions. As the air pressure bears against the piston, the resulting force compresses the
spring until the spring and air forces balance. The calibrated stem, which remains in place
after the spring returns the piston, indicates the applied pressure.
The piston-cylinder combination constitutes a force-summing apparatus, sensing and
transducing pressure to force. As a secondary transducer, the spring converts the force to
a displacement. Finally, the transduced input is transferred without signal conditioning to
the scale and index for readout [see Fig. 3(b)].
The Process of Measurement: An Overview
As an example of a more complex system, let us say that a velocity is to be measured,
as shown in Fig. 4. The first-stage device, the accelerometer, provides a voltage analogous
to acceleration. 3 In addition to a voltage amplifier, the
second
stage may also include a
filter that selectively attenuates unwanted high-frequency noise components. It may also
integrate the analog signal with respect to time, thereby p roviding a velocity-time relation,
rather than an acceleration-time signal. Finally, the signal voltage will probably need lo
be increased to the level necessary to be sensed by the
stage,
third,
or
recording and readout,
which may consist of a data-acquisition computer and printer. The final record will
then be in the form of a computer-generated graph; with the proper c;ilibration. an accurate
velocity-versus-time measurement should be the result.
5
5.1
TYPES OF INPUT QUANTITIES
Time Dependence
Mechanical quantities, in addition to their inherent defining characteristics, also have dis­
tinctive time-amplitude properties, which may be classified as follows :
1.
Static-<:<>nstant in time
2. Dynamic-varying in time
(a)
(b)
Steady-state periodic
Nonrepetitive or transient
i.
ii.
Single pulse or aperiodic
Continuing or random
Of course. the unchanging, static mcasurand is the most easily measured. If the system
is terminated by some form of meter-type indicator. the meter's pointer has no difficulty
in eventually reaching a definite indication. The rapidly changing. dynamic measurand
presents the real measurement challenge.
Two general forms of dynamic input are possible: steady-stale periodic input and
transient input. The steady-stale periodic quantity is one whose magnitude has a definite
repeating time cycle, whereas the time variation of a transient quantity docs not repeat .
"Sixty-cycle" line voltage is an example of a steady-slate periodic signal. So also are many
mechanical vibrations, after a balance has been reached between a constant input exciting
energy and energy dissipated by damping.
An example of a pulsed transient quantity is the acceleration-lime relationship accom ­
panying an isolated mechanical impact. The magnitude is temporary. being completed in
a matter of milliseconds, with the portions of interest existing perhaps for only a few
microseconds. The presence of extremely high rates of change, or wavefronts. can place
seven: demands on the measuring system.
5.2
Analog and Digital Signals
Most measurands of interest vary with time in a continuous manner over a range of magni­
tudes. For instance, the speed of an automobile. as it starts from rest, has some magnitude
l Although lhe accelerometer may be susceptible to an analysis of "stages" within i1self, we shall forgo such
an analysis in this ex.ample.
Stage 2
Signal-conditioning system
Stage 1
Sensor-lransducer
I
i
Stage
3
i Recording-readout system
I
I
Integrating
circuit
Fitter
Voltage output
from accelerometer
with unwanted "noise"
Signal
with noisa
removed
Ampllller
I
I
I
I Data-acquisition
computer
Time-integrated
'IOltage analogus
tovelocily
Tlme,s
Computer graph
FIGURE 4:
Block diagram of a relatively complex measuring system.
·Printer
The Process of Measurement: An Overview
at every instant during its motion. A sensor that responds to velocity will produce: an output
signal having a time variation analogous to the time change in the auto's speed. We refer to
such a signal as an analog signal because it is analogous to a continuous physical process.
An analog signal has a value at every instant in time, and it usually varies smoothly in
magnitude.
Some quantities, however, may change in a stepwise manner between two distinct
magnitudes: a high and low voltage« on and off, for instance. The revolutions of a shaft
could be counted with a cam-actuated electrical switch that is open or closed, depending on
the position of the cam. If the switch controls current from a battery, current either flows
with a given magnitude or does not flow. The current flow varies discretely bet ween two
values, which we could represent as single digits: I (flowing) and 0 (not Hewing). The
amplitude of such a signal may thus be called digital.
Many electronic circuits store numbers as sets of digits-strings of ls and Os-with
each string held in a separate memory register. When digital circuits, such as those in
computers, are used to record an analog signal, they do so only at discrete points in time
. because they have only a fixed number of memory registers. The analog signal, which has
a value at every instant of time, becomes a digital signal A digital signal is a set of discrete
numbers, each corresponding to the value of the analog signal at a single specilfic instant
of time. Clearly, the digital signal contains no information about the value of tine analog
signal at times other than sample times.
.
Mechanical quantities-such as temperatures, fluid-flow rates, pressure, stress, and
strain-normally behave timewise in an analog manner. However, distinct advantages
are often obtained in converting an analog signal to an equivalent digital signal for the
purposes of signal conditioning and/or readout. Noise problems are reduced or s1Jmetimes
eliminated altogether, and data transmission is simpler. Computers are designed to process
digital information, and direct numerical display or recording is more easily accomplished
by manipulating digital quantities.
6
M EASUREMENT STANDARDS
As stated earlier, measurement is a process of comparison. Therefore, regardless of our
measurement method, we must employ a basis of comparison-standardized units. The
standards must be precisely defined, and, because different systems of units exist, the method
of conversion from system to system must be mutually agreed upon.
Most importantly, a relationship between the standards and the readout scale of each
measuring system must be established through a process known as calibration.
7
CALIBRATION
At some point during the preparation of a measuring system, known magnitudes of the input
quantity must be fed into the sensor-transducer, and the system's output behavior· must be
observed. Such a comparison allows the magnitude of the output to be correctly interpreted
in terms of the magnitude of the input. This calibration procedure establishes the correct
output scale for the measuring system.
By performing such a test on an instrument, we both calibrate its scale and prove its
ability to measure reliably. In this sense, we sometimes speak ofproving an instrument. Of
10
The Process of Measurement: An Overview
course, if the calibration is to be meaningful, the known input must i tsel f be derived from
a defined standard.
If the
output is exactly proportional to the input (output =
cons tant
x input ), then
a single simultaneous observation of input and output will suffice to fix the constant of
proponionality. This is called
calibrations are used,
single-point calibration.
M ore often, however,
wherein a number of different input values
are applied.
multipoint
Multipoint
calibration works when the output is not simply proportional, and, more generally, improves
the accu racy of the calibration.
If a
meas ur ing system will be used to detect a time-varying input, then the calibration
should ideally
be
made using a time-dependent input standard. Such
dynamic
cal i bration
can be di fficult, however, and astatic calibration, using a constant input signal, is frequently
substituted. Naturally, this procedure.is not optimal; the more nearly the calibration standard
corresponds to the measurand in all its characteristics, the better the resulting measurements.
Occasionally, the nature of the system or one of its components makes the introduc­
tion of a sample of the basic input quantity di ffi cul t OI' impossible. One of the important
characteristics of the bonded resistance-type strain gage is the fact that, through quality con­
trol at the time of manufacture, spot calibration may be appl ied to a complete lot of gages.
As a result, an indirect calibration of a strain-measuring system may be provided through
the gage factor s up p l ied by the manufacturer. Instead of attempting to apply a known unit
strain to the gage installed on the test s tructure-which , if possible, would often result in an
ambiguous situation-a resistance change is substituted. Through the predetermined gage
factor, the system's suain response may thereby be obtained.
8
UNCE RTAINTY: ACCURACY OF RESULTS
Error may be defined as the difference between the
measured result and
the
true
value of
the quantity being measured. While we do know the measured value, we do not know the
true value, and so we do not know the error either. If we estimate a likely upper bound on
the magnitude of the error, that bound is called the uncertainty: We estimate, with some
level of odds, that the error will be no larger than the uncertainty.
To estimate the size of errors, we must have some understanding of their causes and
classifications. Errors can be of two basic types:
or
bias,
or
systematic, error and precisio11,
random, error.
Should an unscrupulous butcher place a ball of putty under the scale pan, the scale
readouts would be consistently in error. The scale would indicate a weight of product too
zero offset represents one type of sys tematic error.
are used to make patterns for the casting of metals. Cast steel shrinks in
great by the weight of the putty. This
Shrink rules
cooling by about 2%; hence the patterns used for preparing the molds are oversized by the
proper percentage amounts. The pattern maker uses a shrink rule on which the dimensional
units are increased by that amount. Should a pattern maker's shrink rule for cast steel be
inadvenently used for ord inary length measurements, the readouts would be consistently
undersized by
s\i in one (that is, by 2%). This is an example of scale error.
In each of the foregoing examples the errors are cons tant and of a syste matic nature.
Such
errors cannot be uncovered by statistical analysis.
An inexpensive frequency counter may us e the 60-Hz power- l ine frequency as a com­
parison standard. Power-line frequency is held very close to the 60-Hz standard. Although
it does wander slowly above and below the average value, over a period of time-say,
The Process of Measurement: An Overview
day-the average is very close to 60 Hz. The wandering is random and the moment­
precision, or
random, error.
Randomness may also be introduced by variations in the measurand itself. If a
a
to-moment error in the frequency meter readout (from this source) is called
number of hardness readings is made on a given sample of steel, a range of readings will
be obtained. An average hardness may be calculated and presented as the actual hardness.
Single readings will deviate from the average, some higher and some lower. Of course,
the primary reason for the variations in this case is the nonhomogeneity of the crystalline
structure of the test specimen. The deviations will be random and are due to variations in
the measurand. Random error may be estimated by statistical methods.
9
REPORTING RESULTS
When experimental setups
are made and time and effort are expended to obtain results, it
normally follows that some form of written record or report is to be made. The purpose
of such a reconl will determine its form. In fact, in some cases, several versions will be
prepared. Reports may be categorized as follows:
1. Executive summary
2. Laboratory note or tech nical memo
3. Progress report
4. Full technical report
S. Technical paper
Very briefly, an executive summary is directed at a busy overseer who wants only
the key features of the work: what was done and what was concluded, outlined in a few
paragraphs. A laboratory note is written to be read by someone thoroughly familiar with the
project, such as an immediate supervisor or the experimentalist himself or herself. A full
report tells the complete story to one who is interested in the subject but who has not been
in direct touch with the specific work-perhaps top officials of a large company or a review
committee of a sponsoring agency. A progress report is just that-one of possibly several
interim reports describing 1he current status of an ongoing project, which will eventually be
incorporated in a full report. Ordinarily, a technical paper is a brief summary of a project,
the extent of which must be tailored to fit either a time allotmenl at a meeting or space in a
publication.
Several factors are common to all the various forms. With each type, the first priority
is to make sure that the problem or project that has been tackled is clearly stated. There is
nothing quite so frustrating as reading details in a technical report while never being certain
of the raison d'etre. It is extremely importanl 10 make certain 1hat the reader is quickly
clued in on the
before one attempts to explain the
and the results. A clearly stated
objective can be considered the most important part of the report. The.entire report should
be written in simple language. A rule stated by Samuel Clemens is not inappropriate: "Omit
unnecessary adverbs and adjectives."
why
9.1
how
Laboratory Note or Technical Memo
The laboratory nole is written for a very limited audience, possibly even only as a memory
jogger for the experimenter or, perhaps more often, for the information of an immediate
supervisor who is thoroughly familiar with the work. In some cases, a single page may
12
The Process of Measurement: An Overview
be sufficient, including a sentence or two statin g the problem . a block diagram of the
experimental setup, and some data presented either in tabular form or as a plotted diagram.
Any pertinent observations not directly evident from the data should also be included.
Sufficient information should be included so that the experimenter can mentally reconstruct
the situation and results I year or even years hence. A date and signature should always be
included and, if there is a possibility of important developments stemming from the work,
such as a patent, a second witnessing signature should be includL'<I and dated.
5
9.2
Full Report
93
Technical Paper
10
FINAL REMARKS
The full report must relate all the facts pertinent to the p roj ect . II is even more important in
this case to make the purpose of the project comp le tel y clear, for the report will be read by
persons not closely assoc iated with the work. The full report should also include enough
detail to allow another pro fessional lo repeal the measurements and calculations.
One format that has much merit is to make the report proper-<he main body-short
and to the point, releg ati ng to appendices the supporting materials, su ch as data, detailed
descriptions of equipm ent, review of literature, sample calculations, and so on. Frequent
reference to these mate rial s can be made throughout the report proper, but the option to
peruse the de tai ls is left to the reader. This scheme also provides a good basis for the
technical paper, should it be planned.
A primary purpose of a technical paper is to make known (to advertise) the work of the
writer. For this reason, two particularly important portions of the writing are the problem
statement and the results. Adequately done, these two item s will attract the attenti on of
other workers interested in the particular field , who can then make direct contact with the
writer{s) for additional details and discussion.
Space, number of words, limits on illustrat ions, and perhaps lime are all factors
m aking the preparation of a technical paper particularly challe n gin g . Once the problem
statement and the p ri mary results have been adeq uately established, the rem aining available
space may be used to summarize procedures, test setups, and the like.
An attempt has been
made in th i s chapter to provide an overall preview of the problems
of mechan ical measurement. In conformance with Section 9, we have tried to state the
problem as fu lly as possible in only a few pages.
PROBLEMS
I.
2.
3.
Write an executive summary or this chapter.
Consider a mercury-in-glass thermome ter as
the various stages of this
a
temperature-measuring system. Discuss
measuring system in detail .
For the thermometer of Problem 2,
specify how practical single point calibration may be
obtained.
4.
Set up test procedures you would use to estim at e, with the aid only of your present judgment
and experience, the magnitudes of the common quant it ies list ed.
( a ) Distanc e between the centerlines or two holes in a machined part
13
The Process of Measurement: An Overview
( b) Weighl of two small objects of differenl densities
( c ) nme intervals
( d) Temperature of waler
( e) Frequency of pure tones
5.
Consider the impac1 frame shown in Figure 5. Mass M, which travels along gui�le rails, is
raised to an initial height H and released from resl Discuss how you would me:asure the
mass velocity just prior to impacl wilh lhe leSI ilem in order to accounl for friction between
mass M and lhe guide rails.
M
I
H
Test
Item
FIGURE 5: Impact test frame for Problem
5.
Sta ndards and Di mensional
U n its of M easurement
2
3
4
5
6
7
B
9
10
1
INTRODUCTION
HISTORICAL BACKGROUND OF MEASUREMENT IN THE UNITED STATES
THE SI SYSTEM
THE STANDARD OF LENGTH
THE STANDARD OF MASS
TIME AND FREQUENcY STANDARDS
TEMPERATURE STANDARDS
ELECTRICAL STANDARDS
CONVERSIONS BETWEEN SYSTEMS OF UNITS
SUMMARY
INTRODUCTION
The basis of measurement is the comparison between a measurand and a suitable standard.
In this article, we will take a clos er look at the establishment of standards.
The term dimension connotes the defini ng characteristics of an entity (measurand),
and the dimensional unit is the basis for quantification of the entity. For example,
a dimension, whereas centimeter is a unit of length;
a unit of time.
A
time is a dimension,
length
is
and the second is
dimension is unique; however, a partic ular dim ension--;ay, length-may
be measured in various units, such as feet, meters, inches, or miles. Systems of units must
be established and agreed to; that is, the systems must be standardized. Because there are
various systems, there must also be agreement on the basis for conversion from system to
system. It is clear, then, that standards of measurement apply to units, to systems of units,
and to unitary conversion between such systems.
In general terms, standards are ubiquitous. There are standards governing food prepa­
ration, m arketing , professional behavior, and so on. Many
are
established and governed
by either federal or state laws. So that we may avoid chaos, it is especially important that
the basic measurement standards carry the authority of not only federal, but also interna­
tional, laws.
In the following sections, we will disc uss those standards, systems of units, and
2
problems of conversion that are fundamental to mechanical measurement.
HISTORICAL BACKGROUND OF MEASUREMENT IN THE UNITED STATES
The legal au thority to control measurement standards in the United States was assigned
by the U.S. Constitution.
Quoting from Article I, Section 8, Paragraph S, of the U.S.
Prom Mechmiical Measunments, Sixth Edition, Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V.
Copyright 0 2007 by Peanon Education. Inc. Published by Prentice HaU. All rights reserved.
15
Standards and Dimensional Units of Measurement
Constitution: "1be congress shall have power to . . . fix the standard or weights and mea­
sures." Although Congress was given the power, considerable time elapsed before anything
was done about it. In 1 832, the Treasury Department introduced a uniform system or
weights and measures to assist the customs service; in 1 836, these standards were approved
by Congress [ 1 ) . In 1 866, the Revised Statutes or the United States, Section 35 69 , added
the stipulation that "It shall be lawful throughout the United States of America to employ
the weights and measures of the metric system." This simply makes it clear that the metric
system may be used. In addition, this act established the following (and now obsolete)
relation for conversion:
1 meter = 39.37 inches
An international convention help in Paris in 1 875 resulted in an agreement signed for
the United States by the U.S. ambassador to France. The following is quoted therefrom:
"The high contracting parties engage to establish and maintain, at their expense, a scien tific
and permanent international bureau of weights and measures , the location of which shall
be Paris" (2,3 ]. Although this established a central bureau of standards, which set up at
Sevres, a suburb of Paris, i t did n ot, of course, bind the United States to make use of or
adopt such standards.
On April 5, 1893, in all absence of further congressional action, Superintendent
Mendenhall of the Coast and Geodetic Survey issued the following order [2,4J:
The Office of Weights and Measures with the approval of the Secretary of the Treasury, will in
the future regard the international prototype meter and the kilogram as fundamental standards.
and the customary unils, the yard and pound, will be derived therefrom in accordan�-c with the
Act of July 28, 1 866.
The Mendenhall Order turned out to be a very important action. First, it recogn i zed
the meter and the kilogram as bei ng fundamental units on which all other units of length
and mass should be based. Second, it tied togeth er the metric and English systems of le n g t h
and mass in a definite relationship, thereby making possible international exchange on an
exact basis.
In response to requests from scientific and industrial sources, and to a great degree
influenced by the establishment or like institutions in Great Britain and Germany, 1 Congress
on March 3, 1901 passed an act providing that "The Office of Standard Weights and Measures
shall hereafter be known as 'The National Bureau of Standards"' [5). Expanded functions
of the new bureau were set forth and included development or standards, research basic to
standards, and the calibration of standards and devices. The National Bureau of Standards
(NBS) was formally established in July 1 9 1 0, and its functi ons were considerably expanded
by an amendme nt passed in 1 950. In 1 988, Congress changed the name of the bureau to
"The National Institute of Standards and Technology" (NIST) [6].
Commercial standards are largely regulated by state laws; to mainta in uniformity,
regular meetings (National Conferences on Weights and Measures) are held by officials
of NIST and officers of state governments. Essent i ally all state standards of weighL• and
measures are in accordance w i th t he Conference's standards and codes. International uni­
fomtity is maintained through regularly scheduled meetings (held at about 6-ycar intervals),
1 The N11ional Physical laboratory. Teddington. Middlesex and Physikalisch-TL-chnische Reichsanstalt. 8raun­
schweig.
16
Standards and Dimensional Units of Measurement
called the GeMral Conference on Weights and Measures and attended by representatives
from most of the industrial countries of the world. In addition, numerous interim meetings
are held to consider solutions to more specific problems, for later action by lhe General
Conference.
3
3.1
THE SI SYSTEM
Establishment of the SI System
The International System of Units, or SI System, has its origins in the Decimal Metric System
that was introduced at the time of the French Revolution. During the next two centuries,
metric systems of measurement continued to evolve, and they came to encompass both
mechanical and electrical dimensions. Finally, in 1 960, the Eleventh General Conference
on Weights and Measures formally established the SI System, consisting of dimensional
standards for length, mass, time, electric curren t, thermodynamic temperature, and luminous
inten sity. The Fourteenth General Conference on Weights and Measures ( 1 97 1 ) added the
mole as the unit for amount of substance, completing the seven dimensional system in use
today (7] .
The seven base units of the SI S ystem are listed in Table 1. Other dimensions can be
derived from these base units by multiplying or dividing them. A few such derived units
are assigned special names; others are not For example, the unit of force, the newton , is
obtained from the kilogram, the meter, and the second as
2
I newton = I kg/m . s
In contrast, area is simply meters squared (m2 ) . Work and energy are expressed in joules
(kg · m 2/s). The term hertz is used for frequency (s- 1 ) and the term pascal is used for
pressure (N/m 2 ). Some derived units carrying special names are listed in Table 2. and some
without special names are given in Table 3. Note 1hat, whereas those assigned special names
that originate from proper names are not capitalized, the corresponding abbreviations are
capitalized.
It should be clear that all the various derived units can be expressed in terms ofbase
units. In certain instances when a unit balance is attempted for a given equation. it may be
desirable, or necessary, to convert all vari ab les to base uni1s.
TABLE 1: Base Units in the SI System
Name
Quantity
length
mass
time
elcc1ric current
temperature
amount of substance
luminous intensity
17
Unit
meter
kilogram
second
ampere
kelvin
mole
candela
Symbol
m
kg
s
A
K
mol
cd
Standards and Dimensional Units of Measurement
TABLE 2: SI-Derived Unils wilh Special Names and Symbols
Q uantity
Expressed in
Other Unit1a
Unit
Symbol
plane angle
radian
rad
solid angle
steradian
sr
frequency
hertz
Hz
force
newton
N
pressure, stress
pascal
Pa
energy
joule
power
watt
electric charge
coulomb
A · s
electric potential difference
volt
w
c
v
W/A
electric capacitance
farad
electric resistance
ohm
F
0
VIA
magnetic ftux
weber
Wb
magnetic flux density
tesla
T
Wblm2
H
Wb/A
inductance
henry
m · m-1
2
2
m . m-
s-•
kg · mls2
N/m2
N · m
J
J/s
CN
v. s
To accommodate the writing of very large or very small values, lhe SI S:ystem defines
lhe multiplying prefixes shown in Table 4. For example,
2,500,000 Hz may be
written as
2 .5 MHz (megahertz), and 0.000 000 000 005 farad as 5 pF (picofarad). Only one prefix
should be used with a given dimension; thus, it would be incorrect to write 2.5 lckHz in place
of 2 .5 MHz. Likewise, for units of mass, I 000 kg might be written as I Mg (megagram).
In lhe following seclions, we discuss lhe SI standards of length, mass, time, and
current in greater detail. The standards of luminous intensity and amount of subslance are
described in reference [7).
TABLE 3: Some SI-Derived Units
Derived Quantity
Symbol
area
m2
acceleration
mJs2
angular acceleration
rad/s 2
angular velocity
rad/s
density
dynamic viscosity
heat flux
3
kg/m
Pa . s
W/m2
moment of force
N · m
specific heat capacity
J/kg . K
velocity
volume
18
mis
m3
Standards and Dimensional Units of Measurement
TABLE 4: Multiplying Factors
Mu ltip le
10 1
1 <>2
1<>3
1 1>6
109
1 0 12
101s
10 1 8
1 02 1
1 024
3.2
Prefi x
dcka
hecto
kilo
mega
giga
tera
peta
exa
zetta
yotta
Symbol
Multiple
Prefix
Symbol
da
h
k
M
G
T
10- 1
10- 2
10-3
10- 6
10-9
10- 1 2
10- I S
10- 18
10- 21
10- 24
deci
centi
milli
micro
nano
pico .
fem to
atto
zepto
yocto
d
c
m
p
E
z
y
µ,
n
p
f
a
z
y
Metric Conversion in the United States
In May 1 965, the United States announced its intention of adopting the SI system. In 1968,
the passage of Public Law 90-472 authorized the Secretacy of Commerce to make a "U.S.
Metric Study" to be reported by August 1 97 1 . After prolonged debates, studies, and public
pronouncements of I0-year conversion plans, on December 23, 1 975, the 94th Congress
approved Public Law 94- 168, called the Metric Conversion Act of 1975. Its stated purpose
wa� as follows: ''To declare a national policy of coordinating the increasing use of the metric
system in the United Slates, and to establish.a United States Metric Board to coordinate the
voluntary conversion to the metric system." Note especially that the conversion was lo be
volu111ary· and that no time limit was set. The Act made clear that, in using the term metric,
the SI System of units was intended.
In 1 98 1 , the U.S. Metric Board reported lo Congress that it lacked the clear Congres­
sional mandate needed to effectively bring about national conversion lo the metric system;
funding for the Board was eliminated after fiscal year 1 982 [8].
Congress subsequently amended the Metric Conversion Act with the Omnibus Trade
and Competitiveness Act of 1988, Public Law 100-4 1 8 (6) . These amendments provided
strong incentives for industrial conversion to SI units. The amended act declares that the
metric system is "the preferred system of weights and measures for United States trade
and commerce." II further requires that all federal agencies use the metric system i n
procurement, grants, and business-related activities; this requirement w as t o be met by
the end of fiscal year 1992, except in cases where conversion would harm international
competitiveness.
Metrication in the United States has progressed, especially in the automotive industry
and certain parts of the food and drink industries. Classroom use has increased to the point
that most engineer ing courses rely primarily on SI units. Throughout this book, we shall
use both the SI system and the English Engineering system, 2 with the hope of encouraging
the complete conversion to SI units. 3
2 This term may appear to be incongruous given that the United Kingdom has adopted the SI System. llowe>er.
this usage is so well established that the term has outlived its origins.
J Attention is dilo<ted to ref..,,nce (9(. which is an excellent guide for applying the metric system.
19
Standards and Dimensional Units of Measurement
4
THE STANDARD OF LENGTH
The meter was originally intended to be one ten-millionth of the earth's quadran t. ln
1 889, the First General Conference on Weights and Measures defined the meter as the
length of the International Prototype Meter, the distance between two finely scribed lines
on a platinum-iridium bar when subject to certain specified conditions. On October 14,
1 960, the Eleventh General Conference on Weights and Measures adopted a new de fi n i tio n
of the meter as 1 ,650,763.73 wavelengths in vacuum of the rad i ati on correspondi ng to
the transition between the levels 2p1o and Sds of the kry pton - R6 atom. The Na ti onal
Bureau of Standards of the United States also adopted this standard, and the inch hccame
4 1 ,929.398 54 wavelengths of the krypton light.
As it turned out, the wavelength of krypton light could only be determined to about
4 parts per billion, limiting the accuracy of the meter to a similar level. During the 1 960s
and early 1970s, laser-based measurements of frequency and wavelength evolved to such
accuracy that the uncertainly in the meter became the limiti ng u ncertai nty in determining
the speed of light ( 1 0, 1 1 ). This limitation was of serious concern in both atomic and
cosmological physics, and on October 20, 1 983, the Seventeenth General Conference on
Weights and Measures redefined the meter directly in terms of the speed of l i gh t :
The meter is the length of the path
11299,792,458 of a second
traveUed by light
in vacuum during a ti me interval of
This definition has the profound effect of defi11ing the speed of light to be 299, 792,458 mis,
which had been the accepted experimental value since 1 975 [ 12).
4.1
Relationship of the Meter to the Inch
The 1 866 U.S. Statute had specified t hat 1 m
=
I in. = 2.540 005 08 cm
39.37 inches, resulting in the relationship
(approximately)
In 1959, the National Bureau of Standards m ade a small adjustment to th is re l ationship to
ensure international agreement on the definition of the inch [ 1 3 ] :
I in. = 2.54 c m
(exac t ly )
This simpler relationship had already been used as an appro xi m at i on by e ng i neers for years .
The difference between these two standards may be written as
2.54 005 08/2.54
�
-
I = 0.000 002
or 0.0002% , which is about in. per mile.
We gain a sense of the significance of the difference by con sidering the fo l l owing
situation. In 1959, the work of the United States National Geodetic Survey was based on the
39.37 inJm relationship and a coordinate system with its ori g in located in Kansas. Ch ang i ng
the relationship from 39.37 inJm (exactly) to 2.54 in .fem (exactly) would have caused
discrepancies of almost 1 6 ft at a distance of 1 500 miles. One can only i m ag i ne the confusion
over property lines if such a change had been made! This prob lem was resolved by defin i ng
separately the U.S. surveyfoot ( 1 2139.37 m) and the i11ternat(o11a/foot ( 1 2 x 2 .54 cm). The
survey foot is still used with U.S. geodetic data and U.S. statute miles [ 14).
20
Standards and Dimensional Units
5
of
Measurement
THE STANDARD OF MASS
The kilogram is defined as the mass of the International Prototype Kilogram, a platinum­
iridium weight kept at the International Bureau of Weights and Measures near Paris. Of
the basic standards, this remains the only one established by a prototype (by which is
meant
the
original model or pattern, the unique example, to which all others are referred
for comparison).
Various National Prototype Kilogram masses have been calibrated by
n
comparison to the International Prototype Kilogram. These 1 asses
are in tum used by the
standards agencies of various countries to calibrate other standard masses, and so on, until
one reaches masses or weights of day-to-day goods and services.
Apart from the inconvenience of maintaining this chain of calibration, the definition
of the kilogram by an international prototype leads to several
very fundamental
problems:
The prototype can be damaged or destroyed; the mass of the prototype fluctuates by about
one part in 108 owing to gas absorption and cleaning; and the prototype ages i n an unknown
manner, perhaps having resulted in 50 µ.g of variation during the past century [ 1 5].
In recent years, considerable effort has been given to developing a new mass slandard
that can be reproduced in any suitably equipped lab, without the use of a prototype. One
approach being considered is to precisely detennine Avogadro 's number by mass and density
measurements of silicon crystals. This value of Avogadro's number could then be used with
an atomic unit of mass to define the kilogram as the mass of a specific number of atoms
[ 1 5 , l 6). An alternative approach, which promises somewhatbetter accuracy, uses a "moving
coil wall balance" to compare the mechanical and electrical power exerted on a current­
carrying conductor that moves against gravity in a magnetic field. This technique leads to
a definition of the kilogram in terms of fundamental physical quantities [ 1 7, 1 8) .
The pound was defined in terms of the kilogram by the Mendenhall Order o f 1 893.
In 1 959, the definition was slightly adjusted ( 1 3 ], g iving the relationship still in use today :
I pound avoirdupois
6
=
0.453 592 37 kilogram
TIME AND FREQUENCY STANDARDS
Until 1 956, the second was defined as 1186,400 of the a
verage period of revolution of the
earth on its axis. Although this seems to be a relatively simple and straightforward definition,
problems remained. There is a gradual slowing of the earth's rotation (about 0.001 second
per century) ( 1 9), and, in addition, the rotation is irregular.
Therefore, in 1 956, an improved standard was agreed on; the second was defined as
1 13 1 ,556,925.9747 of the time required by the earth to orbit the sun in the year 1 900. This is
called the ephemeris second. Although the unit is defined with a high degree of exactness,
implementation of the definition was dependent on astronomical observation, which was
incapable of realizing the implied precision.
In the 1 950s, atomic research led to the observation that the frequency of electro­
magnetic radiation associated with certain atomic transitions may be measured with great
21
Standards and Dimensional Units of Measurement
repeatabil ity. One-the hyperfine transition of the cesium atom-was related to t he ephem­
eris second with an estimated accuracy of two parts i n 109. On October 13, 1967, in Paris,
the Thirteenth General Co n ference on Weights and Measures offic ially adopted the follow­
i ng definition or the second as the un it ofti me in the SJ System (7):
The second is the duration or 9 . 1 92,63 1 .770 periods or lhe radia1ion com:sponding 10 1he
lransition belween lhc lwo hyperline levels of lhe ground stale or the ce siu m 1 33 alom.
Alomic apparatuses, commonly c a lled "ato mic cloc ks ," are used to produce the fre­
quency of the tra n s i tion (20). In a fountain clock (2 1 ), a gas of cesium atoms is introduced
into a vacuum chamber, where a set of laser beams is used to slow the molecular motion,
push i ng a gro up of atoms into a ball and coo li ng them to a temperature near absolute zero.
Another laser is then used to toss the ball of atoms upward into a microwave cavi ty, where
some of the atoms are excited to higher energy levels. When the ball falls again, yet another
laser is used to force the emi ss i o n of radi atio n . This radiation is detected, yielding the
desired frequency. The be s t cesium standards reprod uce the second to an accuracy better
than one part in 1 0 1
s.
7
TEMPERATURE STANDARDS
The basic unit of temperature, the kelvin (K), is defi ned as the fraction .11273 . 1 6 of the
thermod ynamic temperature of the triple point of water, the temperature at which the solid,
liquid , and vapor phases of water coexist in equilibrium. The degree Celsius (0C) is defined
by the re l ation ship
I = T - 273. 1 5
where I a nd T represe n t temperatures in degrees Celsius and in ke l vi n s, respective l y.
In real i ty, lwo te mpe ratu re scales are defined, a thennodyrlamic scale and a practical
scale . The latter is 1he . u su al basis for measurement. The thermodynamic te m perature
scale is defi ned in terms of e n tropy and the properties of heat e n gi nes [ 22). It can be
i mplemented directly only with specialized thermometers that use media havi ng a prec ise ly
known equation of state (a c o nstan t volume, ideal gas the rmometer, for ex amp le) . Such
thermometers are di fficul t and time consumi ng to use if accuracy is desired, and, as a result, a
correspo nding scale w hich is more eas ily realizable is needed [7]. Th us , the 11hennodynamic
sc a le is normally approx i m ated using a so-called practical' scale.
A prac t ic al scale has two components. The first i s a set of fixed re ference 1emper­
atures, defined by specific states of matter. The second is a procedure for i nterpo lat i ng
between those reference poi n ts , for e x am ple, by measuring a temperatu re-dependen t elec­
trical resistance. Using the i n terpolation formulae and fixed points of the practical scale,
one can calibrate any other temperature measu r i ng dev ice .
The fixed re ference temperalures must correspond to th ermody namic slates that are
very accu rate l y reproduci ble. Zero degrees Celsius is the temperature of equilibrium
between pure ice and air-salurated pure wate r at normal at mospheric pressure. However, a
more precise dal um , i ndepe nde n t of bo1h ambient pressure and poss ible c ontaminants, is
lhe triple point te mperatu re of water. As noled above, the value 273. 16 K (or O.O I OO°C)
is assig ned to this temperature. Relatively si m pl e apparatus can be used to reproduce 1his
temperature fixed point (23).
Standards and Dimensional Units of Measurement
In 1 927, 1he national laboratories of lhe United S1a1es, Great Britain, and Germany
proposed a practical temperature slandard lhat became known as lhe
International Tem­
perature Scale (ITS-27). This standard, adopted by 31 nations, conformed as closely as
possible to the thermodynamic scale lhat had been proposed by Lord Kelvin in 1 854. It was
based on six fixed-temperature points dependent on physical properties of cenain maleri­
als, including lhe ice and sleam poinls of water. Several revisions have since been made,
nolably in 1948, 1968, and 1 990. The praclical lemperature scale currently in effecl is lhe
International Temperature Scale of 1990 (ITS-90), adopted by the International Committee
on Weights and Measures and aulhorized by lhe Eighteenlh General Conference (24).
The ITS-90 defines a number of fixed reference temperalures points, some of which
are shown in Fig. 1 and Table S. Between these fixed points, elaborate inlerpolation equa­
tions are specified by ITS-90 for use wilh lhe various interpolation slandards. From 0.6S K
to S.O K. 1he standard is based on measurement of lhe vapor pressure of helium and use of
1 400
K
'C
'F
1200
.!31L31
_1 �I·�-
1 234.93
_1?_�.g_o_
660.323
933.473
1 220.581
� 12.�2z
692 .677
g3_! �!
2�·.!!7�
0.010
_g11:!�
.!�..:.1!
1 000
-��.:?!
800
600
400
200
0
-200
C!«>.!.d.P��-=1:.91!.'P.!'!�"!..o.!_ �li�ri_!!� �� _ _ _ _
soUd and
liquid gold
Sliver polnt-TemperalUre of
equilibrium between-
- - - - - - Soiid-a'-il;QuidSi1V"er - - - - - -
-
--
Aluminum point-TemperalUre of equilibrium
- - - - - - - - betWeen-sOiid &nd 1iquklaluminum - -
_7!7.:.1 !9_ �� .e�.!-:.T!llJJ!'!!�e_ot_esu.!!'�'!!."!.b.!'?�i:!.. _ _ _ _
solid IW1d liquid zinc
_4i9.:.4?.0_ Tin point-Temperature of equilibrium between
-----
32.018
Soi'"idaii!Aquid �- - - - - - - - - - - - -
Triple point of watar-TemperalUre of equilibrium
- - - - - - - - - - c.rsOiid. ikiuid. and" ViiPO<phaS8S -
of equlibrtum
-11!�� .!3..:.8.!!5� -308.81 96 Triple
- - -point
- - -of-argon-TemperalUre
- - - - fsOl
-273. 1 5
_ ..9J L
FIGURE I :
-�52-!7_ Absolute zero
---
o
id.li<iUid,and 'VaPor phBses-
-------- - --- -------- - --
Some of 1hc fixed-point temperatures esiablished by the ITS-90.
23
Standards and Dimensional U nits of Measurement
TABLE 5: Defining Fixed Points of the ITS-90
Assigned Values of Temperature
Vapor pressure of helium•
Triple pointt o f hydrogen
Vapor pressure of hydrogen
Vapor pressure o f hydrogen
Triple point of neon
Triple point of oxygen
Triple point of argon
Triple point of mercury
Triple point of water
Melting point* of gallium
Freezing point* of indium
Freezing point of tin
Freezing point of zinc
Freezing point of aluminum
Freez ing point of silver
Freezing point of gold
Freezing point of copper
oc
K
Equilibrium State
3 to 5
-270. 1 5 to - 268. 1 5
1 3 . 8033
"" 1 7
"" 20.3
24.556 1
54.3584
83.8058
234.3 1 56
273 . 1 6
-259.3467
"" -256. 1 5
"" -252.85
- 248.5939
- 2 1 8.79 1 6
- 1 89.3442
-38.8344
0.0 1
302.9 1 46
29.7646
429.7485
505.078
692 . 67 7
933.473
1234.93
1 337 .33
1357 .77
156.5985
23 1 .928
4 1 9.527
660.323
96 1 .78
1064. 1 8
1084.62
•Temperature is calculaled by substituting measured vapor pressure
into an equation or state.
t Equilibrium among solid. liquid, and vapor phases.
:j: Melting and freezing point tempera!� com:spond to standard
111nOspheric pn:ssure ( 1 01 ,325 Nim ).
equations describing the vapor-pressure-versus-temperature relationship of helium. From
3 .0 K to 24.556 1 K (the triple point of neon), a constant-volume helium gas thermometer
is used. Ovec the broad range from the triple point of hydrogen ( 1 3.8033 K) to the nor­
mal freezing point of silver ( 1 234.93 K), the standard is defined by means of a platinum
resistance thermometer. Complex equations expressing platinum's resistance as a func­
tion of temperature are prescribed, along with calibration procedures for each of several
temperature subranges. To calibrate, the resistance of a platinum sensor is measured al
several fixed-point temperatures within a given subrange, and these measurements are used
to determine unknown constants in the temperature-resistance equations.
Finally, above the melting point of silver, temperatures are determined by measure­
ment of the thennal radiation em i tted by a black-body cavity in vacuum and the Planck
radiation law:
EA (T)
exp(C2/ATrer) - I
EA (T,.r) = exp(C2/AT) - I
24
Standards and Dimensional Units of Measurement
·
where
the radiant energy emitted by the black body per unil lime, per
unit area, and per unit wavelength al a wavelenglh >., and at a
lemperature T or Tref, respeclively
The freezing point lemperature of either sil ver ( 1 234.93 K),
gold ( 1 337.33 K), or copper ( 1 357.77 Kl
0.0 14388 m · K
The radiant energy is typically measured by oplical pyromcuy. The unknown tempcralure
is lhen calculaled by comparing the emission of a source al lhe unknown 1emperature to
lhal from a source al lhe reference temperature.
The International Temperature Scale of 1990 thus establishes means of de1ermining
any lemperature from 0.65 K to more lhan 4000 K. In actual applications, lhe slandardiz.ed
pyrometer, the standardiz.ed resistance thermometer, or the standardized gas thermometers
are used as secondary standards for calibralion of working instruments. Apart from any
uncertainties introduced in the calibration procedures, lhe major uncertainlies in ITS-90
arise in realizing lhe fixed points. Al a lu level the uncertainties in lhe fixed-point temper­
alures are ±0.5 to 1.5 mK for temperatures up 10 lhe mel1ing point of gallium, increasing
to ±60 mK at the freezing point of copper (25 ].
The temperature units of the English Engineering Sys1em are defined in 1erms of lhe
kelvin. Absolute tempera1ure takes units of degrees Rankine (0R). which differ from lhe
kelvin by a fac1or of 1 .8:
T(0R) = 1 .8
x
T(K)
The degree Fahrenheit (0F) is defined b y sub1rac1ing 459.67 from 1hc 1empcra1Urc in degrees
Rankine:
8
ELECTRICAL STANDARDS
T(°F) = T (0 R )
-
459.67
In the SI System, all electrical unils originale from 1he defini1ion of the ampere. One
ampere is defined as lhe currenl 1ha1 produces a magnelic force of 2 x I o- 7 Nim on a pair
of lhin parallel wires carrying 1hat currenl and separalcd by one meler. The force on an
appropriate pair of conduc1ors can be measured direc1ly, using a so-called currenl balance
(26). The CUlTCnl may be then calculaled from 1he relalions of eleclromagnetic lheory. The
remaining electrical units, such as volts and ohms, can all be derived from the value of
lhe ampere and lhe mechanical unils of mass, l englh, and time, again usi ng the results of
elec1romagne1ic 1heory.
The meas11remen1 of currenl from the SI defini1ion of lhe ampere is cumbersome, j usl
as is the measuremenl of lemperature from the lhermodynamic 1emperature scale. Obtaining
the volt and the ohm from the SI definition is also difficult. Consequently, practical standards
are normally used in place of the SI definitions in order to obtain the voll and the ohm.
Tradilionally, lhe praclical realizalion of the volt was a so-called standard cell, an
electrochemical cell of relatively high stabilily. National standards laboralories mainlained
25
Standards and Dimensional Units of Measurement
such cells as their voltage standards. By the early 1970s, however, the: superconducting
Josephson junction effect, discovered in 1962, had displaced the standaid cell. Josephson
junctions create voltages that are repeatable to about four puts in 1 0 10 , wh1ereas the voltages
of standard cells had shown greater drift (27 ,28). The ultimate accuracy of the Josephson
junction voltage is limited pri marily by the accuracy of the time standard.
The traditional practical standard for the ohm was the standard resistor, typically" a
specially alloyed wire held in an oil bath to stabilize its temperature. These: standard resistors
have an accuracy of a few parts in I 07 , but were subject to aging and lo 1resistor-to-resistor
variations. The quantum Hall effect, discovered in 1 980, quickly becmne an alternative
resistance standard. The quantized Hall resistance allows the ohm to be determined directly
from fundainental physical constants to a repeatability of about 2 puts in 1 09 (28).
The Eighteenth General Conference on Weights and Measures declared that, from
1990 onward, the Josephson junction and lhe quantum Hall effect would be the practical
standards for the volt and the ohm, respectively. The Conference alsc> standardized the
values of the physical constants that characterize the Josephson junctiotn and the quantum
Hall effec t, so that the saine volt and ohm would be obtained in any !al> using these devices.
The resulting realizations of the volt and the ohm are believed to agree with the formal S I
definitions to within a few p arts in 107 (7).
9
CONVERSIONS BETWEEN SYSTEMS OF UNITS
Over 1he centuries, various systems of units have evolved. Five systems a.re listed in Table 6.
To be acceptabl e , each system of units must be compatible with the physical laws of the
universe. If compatible with the laws of nature, the values expressed in one system must
be con vcni ble to equally legitimate values in any of the other systems.
Systems of units differ in their use of defined and derived units. This becomes most
apparent in the English Engineering System's treatment of weight and mass. To see how
this issue arises, let us consider Newton's second law:
A particle acted upon by an external force will be accelerated in proportion to the force magni­
tude and in inverse proportion to the mass or the particle; the direction ol' the a=leration will
coincide with the line-of-action or the foroe.
Algebraically,
F = ma
{I)
where
F = the magnitude of the applied force,
m = the mass of particle,4 and
a = the resulting acceleration
4Cdutioo: Particular note should be made or the use lhrou1hout this text of the synnbols ··m·· and ··w." The
synnbol m is used to ieprcscnt the magnitude or the dimension mass, and it canies u.nits of kilog""" (kg) or
po•nd-mass (lbm). Weigh� w. which is a force, carries the units pountls·/on:e (lbO or ••ewtons (N). Note should
also be made of the use of the symbol "m" to denote the unit of length. meter. Context should always make clear
1he in1ent.
26
TABLE 6: Systems of U nits . Four di me ns i ons are considered for each system, with derived units u nderscored. The English En gineering System
assigns all fou r dimensions, res u lting in the need for a dimensional constanl not equal to unity.
System
N
.....
E ng lish
Engineering
(mass, length, time) (mass, length, time) (force, mass,
SI
CGS
Force
meter (m)
second (s)
kilogram (kg)
newton (N)
centimeter (cm)
second (s)
gram (g)
dyne
Dimensional
constant, &c
1 -N · s2
Quantity
Length
Time
Mass
kg · m
1
g · cm
dyne · s2
--
foot (ft)
length,
second (s)
pound-mass (lbm)
pound-force (lbf)
· ft
32 . 1 7 lbm
lbf . 52
Absolute
Technical
foot (fl)
foot (ft)
second (s)
slug
pound-force (lbf)
English
English
lime) (mass, length, time) (force, length, time)
second (s)
pound-mass (lbm)
poundal
I
lbm · ft
poundal s2
·
slug · ft
1 tbf · s2
Standards and Dimensional Units of Measurement
From experiment (29), we know that near the earth's surface a body ac!ed on solely
by gravitational attraction accelerates at a race of about 32.2 ftls2 (9.8 1 m/s2 ). 5 In this
situation, the acting force is weighl, w, which may be expressed in pounds-force (lbf),
newtons, dynes, and so on, depending on the particular system of units that is used, and the
magnitude of mass may be expressed variously as slugs, pounds-mass (lbm), kilograms,
and so on. In any case, whichever system is used, a consistent, compatible balance of units
must be maintained. Newton's inertial law is of particular interest in this regard because it
demands a careful distinction between the units of force and mass. In the United States, it
has long been the habit to use the abbreviation lb as the unit for both mass and force, except
when a distinction is absolutely required; then the abbreviations lbm and lbf are used.
This distinction shows how the English Engineering System differs from the SI System
in its treatment of the unit of force. The SI System assigns the units of kilograms, meters,
and seconds to the dimensions mass, length, and time, respectively. The unit of force.
the newton, is a derived unit, equal to I kg · rn/s 2 . In contrast, the English Engineering
System assigns the units pounds-mass, feet, seconds, and pounds-force. so that force has
an assigned, rather than derived, unit.
When the assigned units are applied 10 Eq. ( I), we must introduce a factor called the
dimensional constant gc. which converts units of mass, length, and time 10 units of force.
Equation (I) is modified as follows:
(2)
If we select the English System as an example and assume that I lbf acts on
we know results in an acceleration of 32.2 ftls2 , then we fi nd that
2
Kc = ( I l bm)( 3 2 . 2 fi/s ) / ( I lbf)
= 32.2 (lbm
·
I lbm, which
fl /lbf · s 2 )
For the SI System, force is a derived unit: I N is defined as the force required lo accelerate
I kg at I m 2/s. Hence,
gc = ( I kg) ( I m/s2 ) / ( I N)
= I (kg · m/N · s2 ) = I
As a result, the factor 8c has the value unity in the SI System.
The other systems in Table 6 use derived units of either force or ma�s. with the result
that gc is also unity in those systems. Such systems are called "consistent.'' In the Technical
English System, for example, the unit of mass is the slug. which is defined a� the mass that
a force of I lbf accelerates al I ft/s 2 . In the CGS System (centimeter-gram-second system),
the derived unit is that for force: the dyne, which is equal lo I g · cmls 2 . In the Absolute
English System, the derived unit is again that for force, the poundal, <.'q Ual 10 I lbm · flls 2 .
In the past, physicists have bee n partial lo the CGS System. whereas engineers have
mainly used the English Engineering System and the Technical English System. Throughout
this book, we shall use both the SI and the English Engineering Systems. All systems other
than SI, however, are regarded as obsolete.
5The standard gravi1ational body fon:e ("acce tcracion due 10 gravi1y"l is lakcn as 9.80665
Of course, the actual value depends on the SP<-'Cific locali1y.
32. 174 flls2 .
28
m2/s (exaclly) or
Standards and Dimensional Units of Measurement
EXAMPLE
1
Detennine the conversion factor between pounds-force and newtons.
Solution The c onversio n factors between inches and meters and between pounds-mass
and kilograms were given in Sections 4 and 5.
I lbf =
3 2 . 1 74 lbm . ft/s2
= 3 2 . 1 74 (0.453 592 37 kg) ( 1 2
x 0.0254
m)/s2
= 4.4482 kg · m/s2
= 4 .4482 N
EXAMPLE Z
D.
Water of densi ty P and dynamic viscosity µ, Hows with velocity V through a pipe of diameter
Calculate the Reynolds
number, Re , from the data supplied, using (a) the English Engi­
ri
nee ng S ystem of units and (b) the SI System. Before making the numerical calculations,
check the balance of uni ts.
Re = P
a nd its value is unitless and hence is independent of the system of
u nits u sed : th us, we shou ld obtain the same umerical answers for both parts (a) and (b).
n
Data
VDIµ
D = 8.00 in. = 8/ 1 2 ft = 0.203 m
p = 62 .3 lbm/ft 3 = 998 kg/m 3
= 4.00 ft/s = 1 .22 m/s
V
µ = 2 . 02 x 10- 5 lbf
= 9. 67 x 1 0- 4
Solulion
N
·
·
s/ ft 2
s/m 2
1 . If we en ter the u n i ts for each of 1he separate quantities appearing in the equation for
R e, we have
(lbm/ft 3 )(fl/s)(ft)(f1 2 /lbf s)( I / gc)
·
or, entering the units for gc ,
(lbm /ft 3 ) ( ft/s)(ft)(fl 2 /lbf · s)(lbf · s 2 /lbm · fl)
We see Iha! th e vari ous
number is unilless.
I n magnitude,
Re =
units cancel, confinning the s1a1emen1
( 62 . 3 ) (4.00) (8/ 1 2)/(2.02
x
that the Reynolds
5
1 0 - ) (32.2) "" 255,000
Standards and Dimensional Units of Measurement
2. In tcnns of SI units, we have
( kg/m 3 ) (m/s)(m) (m2 / N · s)( l /gc)
or, when the units for 8c are entered,
(kg/m3 )(m/s) (m)(m2 / N · s)(N · s2 /kg · m)
Again, we see that the units cancel. The same result would be obtained by omitting
8c entirely and replacing the newton by its the definition (this is the usual practice in
the SI System).
In magnitude. using SI units,
Re = (998)( 1 .22)(0.203)/(9.67
x
1 0 -4 )( 1 )
�
256,000
Note: The lack of exact numerical agreement in the final numbers results from inexact
conversions
10
and
rounding
of the numbers.
SUMMARY
All measurements are based on defined standards, most of which have been established
by internalional agreement and U.S. laws. A measurement consists of quantifying the
dimensional magnitude of an unknown relative to that established by the standard.
I. The SI System of units employs seven base units (meter, kilogram., second, ampere,
kelvin, mole, and candela), a number of derived units, and various miultiplying factors
(Section 3).
2. The standard of length is the meter, defined
as the distance travelled by light in
l n99.792,458 of a second (Section 4).
3. The standard of mass is the kilogram, defined by the International Prototype Kilogram
(Section 5).
4. The standard of time is the second, defined as 9, 1 92,63 1 ,770 periods of hyperfine·
transition radiation from a cesium atom (Section 6).
5. The operational (or "practical" ) temperature standard is the lntematiional Temperature
Scale of 1990, defined with respect to the thennodynamic temperatures of specific
states of mailer (Section 7).
6. SI electrical units are derived from the ampere. The ampere is defin·ed by the mechan·
ical force present in a particular type of electrical circuit. Practical standards are
normally used to obtain the volt and the ohm (Section 8).
7.
Although SI is the preferred system of measurement, other systems are still commonly
used. The most important of these is the English Engineering System (Section 9).
SUGGESTED READINGS
Many of the following publications may be downloaded from Web pages of the National Institute of
Standards and Technology.
30
Standards and Dimensional Units of Measurement
Butcher, T., L. Crown, R. Suiter, and J. Williams (eds.). General tables or units or measurement.
Appendix C or NIST Handbook 44: SpecificaJions, Tolerances, and Other Technical RequiremelllS
for W<ighing and Measuring Devices. Gaithersburg, Md.: National nstitute or Standards
I
Technology, 2005.
and
Judson, L. V. Weights and Measures Standanls of the United States: A Brief History, Gaithersburg,
Md.: National Bureau of Standards, Special Publication 447. March 1 976.
Gaithersburg, Md. : Office or Weights
and Measures/Metric Program. National Institute or Standards and Technology, Letter Circular
The United States and the Metric System: A Capsule History.
1 1 36, October 1997.
Preston-Thomas, H. The International Temperature Scale of 1990 (ITS-90). Metrologia, 27:3- 10 and
107. 1990.
Taylor, B. N. Guide/or the Use ofthe lntemalional System of Units (SI). Gaithersburg, Md.: National
Institute or Standards and Technology, Special Publication 8 1 1 , 1 995.
Taylor, B. N. (ed.) The
Standards and Technol ogy, Special Publication
/nlemalibnal System of Units. Gaithersburg,
Md.: National Institute of
330. 200 1 .
PROBLEMS
1.
Determine the speed of light in vacuum
in
(a ) miles/hour
( b ) feet/s
2.
Convert the fol lowing temperatures to equivalent temperatures i n K:
( a ) I OO"C
(b) lOO"F
( c ) 595°R
3.
Detennine the
following temperatures in °R:
(a ) Freezing poi nt of tin
( b ) Freezing point of aluminum
( c ) Freezing point of copper
4.
5.
Calculate the
force (lbf) necessary to accelerate a weight of 0.5 lbf at 5 ft/s2 .
Determine the force (N)
necessary to accelerate a mass of 200 g at 25 cmls2• What is the
lbf?
magnitude of this force in units of
6.
Prepare a l ist of the best secondary standards that are available to you at present as cali­
bration sources for
( a ) Length
( b ) Time
( c ) Mass
(d)
(e)
Temperature
Pressure
31
Standards and Dimensional Units of Measurement
7.
Deicrmine by calculation the rela1ionship for 1he following conversions:
( a ) Pressure in lbf/in. 2 unilS IO N/m2 unilS
( b) Viscosity in lbf . slft2 unilS 10 kg/m · s uniis
( c ) Specific heal in lcJ/kg · K u ni ls 10 B lu/lbm · ° F
8.
9.
( d ) Dyna mic viscosity in poise ( I dy ne · s/cm2 ) units to lbm/h · ft
( e ) Heal ftux in W/cm2 unilS lo B lu/h · fl2
Deicrmine the faclor for convening volume ftow ralC in cm3/s unilS 10 gaVmin.
Express the universal gas constant of I 545 fl · lbf/lbm · mol · 0R in SI uniis.
REFERENCES
[ 1 ) Judson, L. V. Weights
tory, Gaithersburg,
March 1976.
and Measures Standards of tile United States: A Brief His·
National Bureau of Standards, Special Publication 447,
Md.:
(2) Units a/Weights and Measure. Gaithersburg, Md.: National Bureau of Standards, Misc .
Publication 2 1 4, July 1 955.
(3) Terrien, J. Scientific metrology on the international plane and lhe Bureau International
des Poids et Mesures. Metrologia 1 (2): 1 5, January 1 965.
(4) U.S. Coast and Geodetic Survey, Bull. 26, April 1893.
[SJ Cochrane, R. D.
Measuresfor Progress, A History ofthe National Bureau ofStandards.
Washington, D.C.: U.S. Dept. of Commerce, 1966, p. 47.
(6) United States Congress, Omnibus Trade and Competitiveness Act of 1988, Public Law
100-4 1 8 , Section 5 164.
(7) Taylor, B. N. (ed.) Tiie International System of Units. Gai1hcrsburg, Md. : National
Institute of Standards and Technology, Special Publication 330, 200 1 .
[8] The United States and the Metric System: A Capsule History. Gaithersburg, Md. :
Office of Weights and Measures/Metric Program, National Institute of Standards and
Technology, Letter Circular 1 1 36, October 1 997 .
[9] Taylor, B. N. Guidefor the Use ofthe lnternationul System of Units (SI). Gaithersburg,
Md.: National Institute of Standards and Technology, Special Publication 8 1 1 , 1 995.
(10] Terrien, J. lnternnlional agreement on the velocity of light. Metrologia, 1 0:9, 1974.
( 1 1) Svenson, K. M., et al. Speed of light from direct frequency and wavelength measure­
ments of the methane-stabilized laser. Phys. Rev. Letters, 29( 1 9): 1 346-49, 1 972.
( 12) Documents concerning the new definition of the metre. Metro/ogia, 19: 1 63-1 77, 1 984.
[ 13) Refinement of values for 1hc yard and 1he pound, Federal Register, Docu men t 59-
5442, June 25, 1 959.
( 14) Units and Systems of Weights and Measures: Their Origin. Development, and Present
Status. Gaithersburg, Md.: National Bureau of Standards, Letter Circular 1035, Novem­
ber 1 985.
32
Standards and Dimensional Units of Measurement
( 15) Seyfried, P., and P. Becker. The role of NA in the SI: an atomic path to the kilogram.
Metrologia, 3 1 : 1 67- 1 72, 1 994.
[ 16) Quinn, T. J. Conclusions of the International Workshop on the Avogadro Constant and
the Representation of the Silicon Mole. Metrologia, 3 1 :275-276, 1994.
N. Determining the Avogadro constant from electrical measurements.
Metrologia, 3 1 : 1 8 1 - 1 94, 1 994.
(17) Taylor, B.
[18) Taylor, B.
65, 1 999.
N ., and
P. J. Mohr. On the redefinition of the kilogram. Metrologia, 36:64-
( 19) Clemence, G. M . Time and its measurement. Am. Scientist, 40(2):260, April 1952.
[20) Sullivan, D. B., et al. Primary atomic frequency standards at NIST. J. Res. Natl. Inst.
Stand. Technol , 1 06( 1 ):47�3. 2001 .
(21) http:/ftf.nist.gov/cesiumlfountain.htm. This National Institute of Standards and Tech­
nology Web page provides clear description of the NIST-Fl fountain clock, including
animation.
(22) Bejan, A. Advanced Engineering Thermodynamics.
New York: John Wiley,
1 988.
[23) Mangum, B. W., and G. T. Furukawa. Guidelines for reali4ing the International Tem­
pera/Ure Scale of 1990 (ITS-90). Gaithersburg, Md.: National Institute of Standards
and Technology, Technical Note 1 965, August 1 990.
(24) Preston-Thomas, H. The International Temperature Scale of 1 990 (ITS-90). Metrolo­
gia, 27:3- 1 0, 1 990 (with corrections in Metrologia, 27: 1 07 . 1 990).
(25) Rusby, R. L., et al. Thermodynamic basis of the ITS-90. Metrologia, 28:9- 1 8 , 199 1 .
(26) Driscoll, R . L., and R . D . Cutkoskey. Measurement o fcurrent with the National Bureau
of Standards current balance. Natl. Bur. Stand. J. Res. . 60, April 1 958.
(27) Taylor, B. N. New measurements standards for 1 990. Phys. Today, 23--2 6, August
1989.
(28) Petley, B. W. Electrical units-the last thirty years. Metrologia, 32:495-502, 1994/95.
(29) Cook, A. H. The absolute determination of the acceleration due to gravity. Metrologia,
1 (3):84, 1 965.
ANSWERS TO SELECTED PROBLEMS
2
3
s
8
(b) 3 1 0.93 K
(b) I , 680.23°R
().Q I 1 24 lbf
I gal/min
=
63.09 cm 3 /s
Assessi n g a nd Presenti ng
Experi menta l Data
INTRODUCTION
2
COMMON TYPES OF ERROR
3
INTRODUCTION TO UNCERTAINTY
4
ESTIMATION OF PRECISION U N CERTAINTY
S
THEORY BASED ON THE POPULATION
6
7
THEORY BASED ON THE SAMPLE
B
STATISTICAL ANALYSIS BY COMPUTER
9
BIAS AND SINGLE-SAMPLE UNCERTAINTY
10
PROPAGATION OF UNCERTAI NTY
11
EXAMPLES OF UNCERTAINTY ANALYSIS
THE CHI-SQUARE
(x 2 ) DISTRIBUTION
12
M I NIMIZING ERROR IN DESIGNING EXPERI M E NTS
13
GRAPHICAL PRESENTATIO N O F DATA
14
L I N E ATTI NG AND T H E M ETHOD OF LEAST SQUARES
1S
SUMMARY
INTRODUCTION
"How good are the data?" is the first qu e st i o n put to any experi mentalist w h o draws a
conclusion from a set of measurements. The da1a may beco me the foundation of a new
theory or the undoing of an existing one.
They may form a critical test o f a structural
an ai rc ra ft wing that must never fail dur i n g operation. Before a data set can be
used in an engi n eeri ng or sc ie nti fic appl icat ion. its qualily must be establ ished.
member in
The answer to the question revolves around the mean i ng we assi gn to the word good.
Our first templation may
be
to call t he data "good" if they agree well with a theoretically
derived result. Theory, however, is simply a model intended to m i m ic the behavior of the
real system being studied; there is
no guarantee that it actually doe.� represent the physical
system well. The accuracy of eve n the m ost fu nda me n t a l theory, such as Newton's laws. is
limited both by the accuracy of the dala from which the theory was developed and by the
the data and assump1ions used when calcula1 ing with it. Thus, measurements
should not be compared to a t heory in order to assess their quality. What we are really
after is the actual value of the physical q u a n tiiy being measured. and that is the standard
agai n st which data should be tested. The error of a measurement is thus d e fi ned as the
difference between the measured value and the true phys i c al value of the quantity. The
orig i n a l quest io n could be more c l earl y phrased as. "What is the e rror of the data'?"
accuracy of
36
Assessing and Presenting Experimental Data
The definition of error is helpful, but it suffers from one major ftaw: The error
cannot be calculated exactly unless we know the true value of the quantity being measured!
Obviously, we can never know the true value of a physical quantity without first measuring
it, and, because some error is present in every measurement, the true value is something we
can never know exactly. Hence, we can never know the error exactly, either.
The definition of error is not as circular as ii seems, however, because we can usually
estimate lhe likelihood that the error exceeds some specific value. For example, 95% of
the readings from one particular ftowmeter will have an error of less than l Us. Thus, we
can say with 95% confidence ( 19 times out of 20) that a reading raken from that meter has
an error of I Us or less, or, equivalently, rhat the reading has an uncertainry of 1 Us at
confidence level of 95%. A theoretical result that disagrees with lhe reading by more than
I Us shows a measurable inaccuracy; a theory within I Lis is supported by lhe reading at
that level of confidence.
Error or uncertainty may be estimated with statistical tools when a large number of
measurements are laken. However, the experi mentalist must also bri ng lo bear his or her
own knowledge of how the instruments perform and of how well they are calibrated in
older to establish the possible errors and their probable magnitudes. This chapter describes
how to estimate the uncertainty in a measurement and how to present the corresponding
experimental data in an easily interpreted way.
2
COMMON TYPES OF ERROR
We have defined the error in measuring a quantity x
value, Xm , and the true value, XtJue :
Error = £
as the difference between the measured
= Xm - Xuue
( I)
A primary objective in designing and executing an experiment is to minimize the error.
However, after the experiment i s completed, we must tum our attention to estimating a
bound on E with some level of confidence. This bo1111d is typically of the form
-11
:::
t:
::: + 11
(n
:
I)
(2 )
where 11 is the 1111certainty estimated at odds of 11 : I . In other words, only one measurement
in n will have an error whose magnirude is greater rhan 11. This bound is equivalent to saying
that
Xm - 11 ::0 Xtrue ::0 Xm + u
We would, of course,
(n : I )
(3)
give higher odds that the lrue value would lie wirhin a wider inrerval
and lower odds !hat ii would lie wirhin a narrower interval.
The first step in bounding a measurement's error is lo idenlify its possible causes.
The specific causes of error will vary from experiment lo experiment, and even a single
experiment may include a dozen sources of inaccuracy. But, in spite of !his diversily, most
errors
can be placed into one of two general classes [ I ]: bias errors and preciJion errors.
Bias errors. also referred to as systematic errors, are !hose that occu r lhe same way
each rime a measuremenl is made. For example, if lhe scale on an instrument consisrently
reads 5% high, rhen the entire set of measuremcnls will be biased by +5% above lhe true
value. Alternatively, the scale may have a fixed offset error, so that the indicated value for
every reading of x is higher rhan rhe true value by an amount Xoffset ·
37
Assessing and Presenting Experimental Data
Precision errors, also called random errors, are different for each suc:cessive measu re­
ment hut have an average value of zero. For example, mechanical friction or vibration may
cause the reading of a measu ri ng mechanism to ftuctuate about the true ·value, somet imes
reading high and sometimes reading low. This lack of mechanical precision will cause
sequential readings of the same quantity to differ sl ightly, creating a disb:ibution of values
surrounding the true value.
If enough readings are taken, the distribution of precision errors will ll>ecome apparent.
The successive re ad ings will generally cluster about a central value and will extend over
a limited interval surroundi ng that central value. In this situation, we may use statistical
ana ly sis to esti mate the l i kely size of the error or, equ i vale ntly, the likely rainge of x i n which
the true value l ies.
In co ntrast, bias errors cann ot be treated using statistical techniques, because such
errors are filled and do not show a distribution. However, bias error can be estimated by
comparison of the instrument lo a more accurate standard, from our knowledge of how the
instrument was c al i brated , or from our ellperience with instruments of that particular type.
In prac tice , bias and precision errors occur simultaneously. The combined effect on
repeated measurements of x is shown in Figs. l (a) and (b). In Fig. l(a), the bias error
is larger than the typical precision error. In Fig. l (b), the typical precision error exceeds
the bias error. In other situations, bias and precision errors may be of the same size. The
total error in a particular measured Xm is the sum of the bias and prec is ion errors for that
measurement.
2.1
Classification of Errors
A fu l l classification of all possible errors as e i ther bias or prec i sion erro1: }YOu ld be conve­
nie nt but is nearly impossible to make, since categories of error overl ap and arc at times
ambiguous. Some errors behave as bias error in one situation and as precision error in
other situations; some errors do not fit neatly i nto either category. How•ever, for purposes
of discussion, typical errors m ay be roughly sorted as follows:
1. Bias or systematic error
(a) Calibration errors
(b) Certai n consistently recu rri n g human errors
(c) Certain errors caused by defective equipment
(d) Loading errors
(e) Li mi tations of system resolution
2. Precision or random error
(a) Errors caused by disturbances to the equipment
(b) Errors c aused by ftuctuating experimental conditions
(c) Errors derived from insufficient measuring-system sensitivil.y
3. Illegitimate error
(a) B l unders and mistakes d uri ng an e x periment
(b) Computational errors after an e xperi men t
38
Assessing and Presenting Experimental Data
Bias error
I
I
0
Precision error
Measured value, Xm
(a)
Total error
Measured value, Xm
(b)
FIGURE I : B ias and precision errors: (a) bias error larger than the typical pn.-cision error,
(b) typical precision error
larger than t he bias error.
39
Assessing and Presenting Experimental Data
Actual response ---,
Input, x,..,
FIGURE 2: Calibration errors. For ideal response, Xmeasun:d = xuue. Actual response may
include zero-offset error (Xoffsea) and scale error (/J 'F I) so that Xmeasurcd = fJxuue + xoffse•·
4. Errors that are sometimes bias error and sometimes precision error
(a) Errors from instrument backlash, friction, and hysteresis
(b) Errors from calibration drift and variation in test or environmental conditions
(c) Errors resulting from variations in procedure or definition among
ex peri men ters
The most common form of bias error is error in calibration. These errors occ ur when
an instrument's scale has not been adjusted to read the measured value properly. Typical
calibration errors may be zero-offset errors, which cause all readings to be offset by a
constant amount Xoffsea. or scale errors in the slope of the output relative to the input which
cause all readings to err by a fixed percentage. Figure 2 illustrates these ty pes of error.
Calibration procedures normally attempt to identify and eliminate these errors by
"proving" the measuring system's readout scales through a comparison with a standard.
Of course, the standards themselves also have uncertainties, albeit smal ler ones. 1 The
impreciseness of any calibration procedure guarantees that some calibration-related bias
error is present in all measuring systems.
standard (the International PnMotype Kilogram) has by definition a
1The uniqueness or cenain prinwy srandanls makes them exa:plions to this swement. In panicular, the mass
mass or exactly
I kg. In the sense or practical
applications, however, uncertainty will nevcnheless occu r. Even primary standards require the use of ancillary
apparatus. whk:h necessarily inlroduces some unccnai nty.
40
Assessing and Presenting Experimental Data
Human errors may well be sys te matic , as when an i ndiv idual experimenter consis­
te nt ly tends to read high or to ·�ump the g u n" when synchronized readings are to be taken.
The equipment
itsel f may introduce hui lt-in errors resulting from incorrect design,
fabrication, or mai nten ance .
Such errors resu l t from defective mechanical or electrical
components, incorrect scale graduat ions , and so forth. Errors of this type are often con­
sistent in sign and mag n i tude , and because of their consistency they may sometimes be
corrected by calibration. When the input is time varying, however, introducing a correction
is more com plicated . For example, distortion caused by poor frequency response cannot
be corrected by the u sual "static cal i brati on ," one based on a signal that is constant in time.
Such frequency-response errors arise i n con ne<:lion with seismic motion detectors.
Loading error is of particular importance. It refers to the influence of the measure­
ment procedure on the system be i ng tested. The measuring process inevitably alters the
characteristics ofboth the source of the measured quantity and the measuring system itself;
thus, the measured value will always differ by some amount from the quantity whose mea­
suremenr is sor1ght. For example, the sound-pressure level sensed by a microphone is not
the same as the sound-pressure level that would exist at that location if the microphone were
not pn:sent. Minimizing the influence of the measuring instrument on a measured
variable
is a major objective in designing any experiment
Precision errors are also of several typical forms. The experimenter may
be inconsis­
tent in estimating successive read i ngs from his or her instruments. Precision errors in the
instrumentation itself may arise from outside disturbances to the measuring system, such as
temperature variations or mechanical vibrations. The meas uring system may also include
poorly controlled processes that lead 10 ran do m variations in the system output.
Vari at ions i n the actual q uanti t y being measured may also appear as precision error
i n the results. Sometimes these variations are a result of poor experimental design, as
when a sy stem designed 10 run at a constant speed instead has a varying speed . Sometimes
the variati o n s are an inherent feature of the process under study, as when manufactur­
i ng variations create a distribution in t he operating lifetimes of a group of l igh t bul bs.
S trictly speaking, variation in the measured quantity is not a measurement "error''; how­
ever, it is poss ib le to apply the same stati st ica l techn iques to variations in the measured
variable and lo treat them as if they were errors. In particular, if you wish lo find the
mean value of the measured quantity. its variations may be averaged (together with the
precision errors of the equipment), and the mean value may
be calculated along with its
uncertainty.
Illegitimate errors are errors that would not be expected. These i ncl ude outrigh t mis­
takes (which can be eliminated through exercise of care or repetition of the measurement),
such as incorrectly writing down a number, failing to tum on an instrument, or miscalcu­
lati ng during data reduction. Sometimes a stat ist ical analysis will reveal such data as bei ng
extremely unlikely to ha ve arisen from prec i s ion error.
Backlash and mechanical friction arc i m portant sources of variation in measuri ng
systems. For example, friction may cause a mechanical element, such as a gal vanometer
needle, to lag behind advances in its inlcndcd position, thus reading low while lhc mea­
sured variable is increased and reading h igh while the measured variable is decreased.
Such
hysteresis i s illustrated i n Fig. 3. Since this error depends on how a sequence of measure­
ments is taken, it may be have as either a bias error or a precision error. One means of
detecti ng-and often correcting-this type of error is to make measurements while first
41
�/
Assessing and Presenting Experimental Data
Ideal response
FIGURE 3: Hysteresis error.
increasing and then decreasing the measured quantity,
method of symmetry.
an
approach sometimes called the
Drift of an i nstrument's calibration may occur if the response of an instrument varies
in time. Often drift results from the sensitivity of an instrument to ternpernture or hu midity
fluctuations. If changes in environmental conditions occur between the time an instrument
is calibrated and the time it is used, a bias error may appear in the readings. Conversely, if
the test duration is long, environmental cond i ti on s may fluctuate throu.ghou t the test, causing
different calibration errors for each suceessive measurement. In this case, the nuctuations
create a precision error.
When an experiment is repeated using .different eq u ipm ent or by different experi­
menters, the bias errors of successive experiments are unrelated. If en ough different exper­
iments are performed, the bias errors are effectively randomized, andl they become another
form of precision error in the set of all experiments. For example, the speed of light in
vacuum has bee n measured by many experimenters, each using different techniques and
apparatus to obtain what is supposedly a unique quantity. Each experiment included its
own bias and preci sion errors; but taken together their results show a d ist ri but io n about a
mean value, which may be estimated statistically (F i g. 4 ).
In contrast, the randomness of precision error may sometime:s work against itself.
For example, computer signal-processing techniques can extract des ired information from
very noisy signal, as when photographs from satellites are enhanced to reveal planetary
topography. In such cases, the systematic nature of the desired infonnation enables it to be
separated from the completely random overlying noise.
a
42
Assessing and Presenting Experimental Data
299.7800
299.7900
Meaiued speed , thousands ol kilometers per second
299 .8000
FIGURE 4: Measured values of the speed of light, 1947-1967 (Data from Froome and Essen (2]).
Z.2
Terms Used In Rating Instrument Performance
The following terms are often employed to describe the qualily of an instrument's readings.
They are related to the expected errors of the instrument.
•
Accuracy. The difference between the measured and true values. Typically, a manu­
facturer will specify a maximum error as the accuracy; manufacturers often neglect
lo report the odds that an error will not exceed this maximum value.
•
Precision. The difference between the instrument's reported values during repeated
measurements of the same quantity. Typically, this value is determined by statistical
analysis of repeated measurements.
•
Resolution. The smallest increment of change in the measured value that can be
determined from the instrument's readout scale. The resolution is often on the same
order as the precision; sometimes it is smaller.
•
Sensitivity. The change of an instrument or transducer's output per unit change in
the measured quantity. A more sensitive instrument's reading changes significantly
in response to smaller changes in the measured quantity. Typically, an instrument
with higher sensitivity will also have finer resolution, better precision, and higher
accuracy.
Reading error refers to error introduced when reading a number from the display scale
of an instrument. This type of error may sometimes be a bias error caused by truncation or
rounding of the actual value to one within the resolution of the display. Reading error will
also include error from inadequate instrument sensitivity if the instrument does not respond
to the smallest fluctuations of the measured quantity. For example, a digital display may
truncate an actual value of 10.4 to a displayed value or I 0. The reading error of the digital
Assessing and Presenting Experimental Data
10.4
!
display is thus ± of the last digit read. 2 This is a bias error in the sense that
will
always be displayed as 1 0. When many different values are to be read, the error may be
thought of as a precision error if the many values have no particular relation to one another.
For a needle display on a galvanometer or the scale on a micrometer one may be
able lo estimate to ± or even
of the finest graduation. Depending on the particular
experimenter, such error may be either bias or p reci s ion error, as discussed previously.
!
3
±!
INTRODUCTION TO UNCERTAINTY
When estimating uncertainty, we are usua l l y concerned with two types of error, precision
and bias error, and with two classes of experiments. single-sample experiments and repeal­
sample experiments.
A sample, in this sense, refers 10 an individual measurement of a specific quantity.
When we measure the s1rain in a structural member several times under identical loading
conditions, we have repeatedly sampled that particular strain. With such repeat sampling,
we can, for example, statistically estimate the distribution of precision errors in the strain
measurement. If we measure that strain only once, we have instead a single sample of the
strain, and our result does not reveal the distribution of precision error. In that case, we
must reson to other means for estimating the precision error in our result.
Much of the remainder of this chapter is devoted to methods of estimating bias and
precision error. Procedures for statistical analysis of precision error in repeal-sampled data
are described in Sections 4-7. The estimation of bias uncertainty is covered in Section 9.
The estimation of precision uncertainty for single-sample experiments is also considered in
Section 9. Section 1 0 describes how uncertainty in measured variables leads to uncertainly
in results calculated from those variables. Examples of uncertainty analysis are given in
Section 1 1 .
After determining the individual bias and precision uncertain i n a measurement of
x, we must combine them to obtain the total uncertainly in our result for x. If the bias
uncenainty is Bx and the precision uncertainly is P, , 3 then the two may be combined in a
root-mean-square sense as
(4 )
to yield the total uncectainty, Ux .
Th e j usti fi cation for combining the two uncertainty· estimates this way i s largely
empirical [ I ). However, the underlying assumption is that Bx and Px are associated with
independent sources of error, so lhal the errors are unlikely to have their maximum values
simultaneously. When B., and [', are each estimates for 95% confidence, then Ux is also a
95% confidence estimate; under the same conditions. ii turns out [ I ] lhal simply adding Bx
and Px algebraically yields an uncertainly that roughly covers 99% of the data.
21t may be shown that 9S% or the ""'dinp ( 1 9 out or 20)
±O.S( l9/20)
where t
=
±0.47S "'
will differ rrom the value displayed by less than
±0.S. The uncertainty is ±0.5 at 1 <1 : l odds.
lThe precision uncertainty is evaluated in Section
6.
In tenns defined later it has the value Px = r S, / .fii,
is the l·SWis1ic. Si is lhe sample sundard devialion, and n is the sample size.
44
Assessing and Presenting Experimental Data
EXAMPLE
1
brass rod is repeated I y loaded to a fixed tensile load and the axial strain in the rod is
determi ned usi ng a strain gage. Thirty results are obtained under fixed test conditions,
yielding an average strain of e = 520 µ-strain (520 ppm). Statistical analysis of the
distribution of measurements gives a preci sion uncertainty of P, = 21 µ-strain at a 95%
confidence level. The bias uncertaint y is estimated to be B, = 29 µ-strain with odds
of 19 : I (95% confidence). What i s the total uncertainty of the strain?
A
Solution The t ota l uncertai nty
U, =
(
for 95% coverage
s; + Pl)
112
=
is
36 µ-strain
(95%)
In other words, at a confidence leve l of 95% the true strain lies in the i nterval 520 ± 36 µ­
strain:
484 µ-strain !:: 6 !:: 556 µ-strain
4
ESTIMATION OF PRECISION UNCERTAINTY
Two fundamental concepts form the basis for analyzing precision errors. The first is that of
a distribmion of error. The distribution characterizes the probability that an error of a given
size will occ ur. The second concept is that of a population from which samples are drawn.
Usually, we have only a limited set of observations, our sample, from which to infer the
characteristics of the larger population.
error usually assumes a model for the distribution of errors
or nonnal, distribution. Usi n g this assumed dis­
tribution, we may estimate the probable difference between, say, the average value of a
small sample and the true mean value of the larger population. This probab le difference, or
co1ifidence inten•a/, provides an estimate of the precision uncertainty associated with our
Statistical analysis of
in
a populat i on, generally the Gaussian,
measured sample.
This sect ion and the next four examine some basic probability distributions, the char­
of a populalion that satisfies a Gaussian di s tributio n , and the accuracy with
which the statistics of samp l es represent an underlying population. The I-distribution is
i n troduced for treating small sam pl es . and the x -distribu tion is introduced for other statis­
tical purposes. These sections should provide y ou with sufficient bac kg rou nd to estimate
the preci si o n u ncerta i nty in eleme ntary engineering experiments.
acteristics
4. 1
2
Sample versus Population
Manu facturing variations in a prod uct i on
lot of marbles will create a distribution of diam­
To est i m ate the mean diameter, we may take a handful of marbles, measure them,
and average the resull (fig. 5). The handful is our sample, drawn from the production lot,
which is our population.
No two samples from the same populat ion w il l yield precisel y the same average
value; however, eac h s ho u ld approxi mate the average of the population to some level of
eters.
45
Assessing and Presenting Experimental Data
Random sample
Bag ol maitlles
FIGURE 5: Sample taken from a population.
uncertain ty. The difference between the sample characteristics and those of the
popula t i on
will dec.rease as the sample is made larger.
Because our handful of marbles inc lud es a number o f members of the population, it
may be regarded as a repeated sample. By contrast, if we had drawn onl :y one marble, we
would obtain a single sample , which would give no direct evidence of dl.e d ist ri bu tion of
marble diameters.
Experimental errors can a ls o be viewed in terms of population and sample. If we
measure me d ia meter of a si ng le marble repeatedly, the set of measured d i ameters gi ves a
sample of me precision error in measu ri ng the diameter. In this case, we could measure
me diameter as many ti mes as we liked, and each measurement would i nclude a slightly
different preci s ion error. Thus, the population of precision errors is t heciretically infinite.
(Note, however, that this particular repeat sampl i n g of the prec i s io n error is performed on
a s ingl e sample from the marble population.)
The discussion of this and s ubseq uent sections applies to both of llhe following two
classes of sampling:
1. A sample of si ze n is drawn from a finite popu lat io n of size p.
used to
The sample is
estimate properties of the population. Additional data cannot be added to
the population; for example, we assume that no more marbles can1 be added to the
p arti cu l ar produc tion lot from the same source. Further, the sa mplt: size is assumed
to be small compared to the population size: n « p.
Assessing and Presenting Experimental Data
2. Ajinite number ofitems, n, is randomly drawn from what is assumed to be a populalion
of indefinite size. The properties of the assumed populalion are inferred from the
sample.
An importanl qualification underlies this discussion: The sample mus1 be randomly
selected from lhe populalion. If we selecl only the largest marbles from lhe bag, our sample
will nol acc urately represenl the whole population of marbles.
4.2
'Probabllity Distributions
Probability is an expression of lhe likelihood of a particular even! laking place. measured
wilh reference IO all possible events. Specifically, suppose thal one of n equally likely cases
will occur and lhat m of these cases correspond lo an evenl A. The probabilily lhal event A
will occu r is m/n.
A penny is lossed. The IOlal number of possible outcomes is tw�eads and lllils. If
we choose heads (or !ails) as event A, then the probability of A is I in 2, or 50%.
A slightly more complex example is that of throwing a pair of dice. One possible
outcome yields a sum of 2, six outcomes yield a sum of 7, and lhe remai n ing outcomes are
as distributed in Fig. 6. The bar chart used here is termed a histogram. If we divide the
ordinate of lhe chart by the total number of possible outcomes (36). we obtain a graph of
lhe probability distribution. For example, !he probability of rolling a 7 i s 6 in 36, or 1 6.6'l&.
Olher distributions lhat will be considered in lhe following sect io ns are these:
1. The Gaussian, or normal, probability distribution. When examining experimental
dala, lhis dislribulion is undoubledly lhe first !hat is considered. The Gaussian dis­
lribution describes the popula1ion of possible errors i n a meas u reme n l when many
independent sources of error contribute simultaneously 10 1he 101al precision error in
a measurement. These sources of error must be unrelated, random, and of roughly lhe
2
3
4
5
6
7
8 9
Sum of spolS for a pa ir
ol lassed clce
10
11
12
FIGURE 6: Distribulion of resulls for a pair of lhrown dice.
Assessing and Presenting Experimental Data
same size. 4 Although we will emphasize this particular distribution, you must keep
in mind that data do not always abide by the normal distribution. For tabulation and
calculation, the Gaussian distribution is recast in a standard form, sometimes called
the z-distribution [see Eq. ( 1 1 ) and Table 2].
2. Student's t-distribUlion. This distribution is used in predicting the mean value of a
Gaussian population when only a small sample of data is available [see Eq. (2 1 ) and
Tuble 5).
3. The x 2 -distribution. This distribution he l ps in predicting the width or scatter of a
population's distribution, in compari ng the uniformity of samples, and in checking
the goodness offit for assumed distributions (see Section 7 and Table 6).
5
THEORY BASED ON THE POPULATION
From a practical standpoint, we are l imi ted to samples from which to extract statistical
information. In most cases, it is either impractical or impossible to manipulate the entire
population. Nevertheless, some useful and important results can be established at the outset
by considering the properties of the entire population.
Consider an infinite population of data, each datum representing a measurement of
a single quantity, and assume each datum, x, differs in magnitude from the rest only as
a result of precision error. Effectively, each time a different member of the population is
randomly selected and measured, a value of x is determined and, a different precision error
occurs. Some measurements are larger and some are smaller. The probability of obtaining a
specific value of x depends the magnitude of .t, and the probability distribution of x-values
is described by a probability densityfunction, / (x ) (Fig. 7).
' '
I
'
'
I
I
I
I
1
1
0'2&
FIGURE 7: Norma l distribution curve. More precise data are represented by the dashed
curve than by the solid curve.
4
Note 1ha1 the disuibution of pr«is;on errors accompanying experimen tal data and the distribution or dimtm·
sional variations in manufacturing optfrllion.s are very similar. For example. a drawing may specify tolerances
within which the diameter or a shaft is allowed to vary about a nominal value. 1be variations that actually exist
w it hin a production lot are often found to be distribu&ed in nonnal, or Gaussian . manner. Quality conuol of
machining is based on essentially lhc same 1heories as applied 10 dislribution or experimental error.
Assessing and Presenting Experimental Data
n te an
bea ee easuri
Since the population is infinite, the probability density function, or PDF, is a con
tinuous curve, unlike the previous bar-graph histogram describing rolled represents dice.
Consequently the ord i a of the PDF must be carefully i nterpreted: It represents the prob­
ability of occurrence per unit ch ge of x. The probability of measuring a given x is not
f(x) i tsel f; instead, the prob bi li ty of m
ng an x in the i n r al tu = xi
xi is the
area under the PDF curve tw n xt and x2
Probability(x, -x?)
p a
­
te v
=
ix,
-
(5)
f(x) dx
XI
In any measurement, some value of x will be observed, so the total area under the PDF'
curve is unity (i.e., the rob bi l i ty of measuring some x value is I00% ).
PDFs come in a variety of shapes, which are determined by the nature of the data
considered. Precision error in experimental data is often distributed according to the familiar
bel l shaped curve given by the Gaussian, or normal, distribution (Fig. 7). Most of the
remaining statistical discussion in this chapter is based on that premise. For an infinite
population the mathematical e p sion for the Gaussian probability density function is
-
,
whe
re
x res
/(x ) =
I
--
u ..fiii
[
exp -
(x
- µ)2
2u 2
---
]
(6)
x = the magnitude of a
µ.
u
particular measurement,
the mean value of the entire population, and
= the standard devi ation of the entire popu l at ion
ear po ti
=
The mean value. µ., is that which would be obtained if every x in the pu la on could
averaged together; for an infinite population, such averaging is cl l y impossible, and
thus µ remains unknown. Since we assume that the various x 's differ as a result of precision
error, µ. also represents the true value of the quantity we are attempting to measure, and the
average sought amounts to averaging out all the precision errors.
If a large number of easuremen are taken with equal care, then the arithmetic
average of these n measurements,
be
m
.t
ts
=
Xt + x2 +
n
mv a e re
d x-
· · ·
+ Xn
(7)
can be shown to be the most probable single 1•al11e for the q11antity, µ..
sample allows us to estimate the true al u .
The amount
by which a single
e
su
=
me nt
µ.
Averaging a large
is in error is termed the deviation d:
(8)
Assessing and Presenting Experimental Data
TABLE 1 : Summary of Probability Estimates Based on the Nonna! Distribution
Common Name for
"Error" Level
% Confidence
That Deviation
of x from Mean
is Smaller
Error Level
in Terms of CT
Odds That
Deviation of
x is Greater
abt. I i n 3
Standard deviation
±CT
68.3
Two-si gma error
± 1 .960"
95.0
I in 20
99.7
I in 370
Three-s igma error
±3a
The mean squared deviation, a 2 , is approximated by averagi ng the squared deviations of a
very large sample5 :
(9)
a,
The quantity,
is called the standard deviation of the population; it characterizes both
the typical deviation of measurements from the mean value and the: width of Gaussian
distribution (Fig. 7). When a is smaller, the data are more prec i se. llue standard deviation
is a very important parameter in characterizing both population s a nd sam p les .
The actual deviation or error for a particular measurement is, of cou rse, never know n.
However, the likely size of the error, or uncenainty, can be es ti mated with various levels
of con fidence by using our knowledge of the distribution of t he popu l at io n . For example,
if a population has a Gaussian distribution, then the probabi li ty that a single measurement
will have a deviation greater than ±a is 3 1 .7%, or about one chance in three. For a single
measurement, then, we can be 68% confident tha t the deviation is less than ±" . A deviati on
greater than ± l .96a has a probability of5.0% ( I in 20); greatcr than ± :lo . about 0.27% ( I in
370); and greater than ±40' , about 0.0063% ( I in 16,000) . The most popular uncertainty
estimate is that for 95% con fidence , ± I . 96a . Table 1 summarizes the various le ve ls of
probability for the normal distribution.
One common criterion for discarding a data point as illeg itimate is that the data poin t
exceeds the 3a level : since the probability of an error larger than tln is is I in 370. such
values are unlikely in modest-sized data sets. Such data are som etime:s called outliers.
For purposes of tabulation, the Gaussian PDF may be trans fonmld by i ntroduc i ng the
variable z :
x -
µ.
z = -­
a
Eq uation (6) is now
a2
/(z )
=
_1_ e-z'12
a
J�::<x
x or (z -
µ)2.
-
µl' f(x ) Jx.
This amounts
( 1 1)
..fii
5 Fonnally. the mean and lhe standard deviation may be calculaled asintegrals oflhe PDF: µ
=
( 1 0)
=
f�:: xf(x) Jx:
lo sununing the probable contribution to the obse""'d value of each
Assessing and Presenting Experimental Data
Probability (area)
listed in table
z
FIGURE 8: Standard nonnal distribution curve. Note that a and b are inftection points.
which is the standard curve shown at the top of Table 2 and in Fig. 8. The table lists
the areas under the curve between 0 and various val ues of z. Since the curve is symmetric
about zero, the tabulation lists values for only half the curve. Bear in mind that the total area
beneath the curve is equal to unity. This tabulation is s ometimes called the z-distribution.
The following examples illustrate the nature and use of the tabulated data.
EXAMPLE 2
(a) What is the area under the curve between z
(b) What is the si gnificance of this area?
Solulion
=
- 1 .43 and z
(a) From Table 2, read 0.4236. This represents half the
area is 2 x 0.4236 = 0.8472.
area
=
1 .43?
sought. Therefore, the tota l
(b) The significance is that for data following the normal distribution, 84.72% of the
populatio n lies within the range - 1 .43 < z < 1 .43.
EXAMPLE
3
What range of x will contain 90% of the data?
Solulion We need to find z such that 90%12 = 45% of the data lie between zero and +z;
the other 45% will lie between -z and zero. Reading from Table 2, we find zo.4 S "" 1 .645
Assessing and Presenting Experimental Data
TABLE 2: Areas under the Standard Nonnal Curve
0
z
0.0
0.1
0.2
0.3
0.4
0.5
0.8
0.7
0.8
0.9
1.0
1.1
1 .2
1 .3
1 .4
1 .5
1 .8
1 .7
1 .8
1 .9
0.00
0.0000
0.0398
0.0793
0. 1 1 79
0. 1 554
0. 1 9 1 5
0.225 7
0.2580
0.1881
0.3 1 59
0.34 1 3
0.3643
0. 3849
0 .4032
0.4 1 92
0.4332
0.4452
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.119
0.0040
0.0438
0.0832
0.0080
0.0478
0.0 120
0 .0 1 60
0.0 1 99
0.0596
0.0987
0. 1 368
0.1 736
0.0239
0.0636
0. 1026
0. 1406
0. 1 772
0.0279
0.0675
0. 1064
0. 1443
0. 1 808
0.03 19
0.07 1 4
0. 1 103
0.1480
0. 1 844
0.0359
0.2123
0.2454
0.2 157
0.1190
0.25 17
0.3023
0.30.5 1
0.33 15
0.2486
0.2794
0.3078
0.3340
0.1823
0.3106
0.3365
0.3577
0.3790
0.3980
0.4 1 47
0.4192
0.3599
0.3810
0.3997
0.4 1 62
0.4306
0.3611
0.3830
0.40 1 5
0.4177
0.43 1 9
0.4406
0.45 1 5
0.4608
0.4686
0.4 750
0.44 1 8
0.4525
0.4616
0.4693
0.4429
0.4535
0.4625
0.4699
0.4761
0.4803
0.4808
0.4 850
0.444 1
0.4545
0.4633
0.4706
0.4767
0. 1591
0.0871
0. 1 25 5
0. 1 62 8
0 . 1 950
0. 1 985
0.12 17
0.2291
0.261 1
0.2910
0.3 1 86
0.20 1 9
0.235 7
0. 2704
0.2939
0. 31 1 2
0.2673
0. 2967
0.3238
0.2088
0.2422
0.2734
0.1995
0.3164
0.3189
0.3508
0 .3729
0. 3925
0.2642
0.2054
0.2389
0.4207
0.4 2 22
0.42 36
0.4099
0.425 1
0.4345
0.4357
0.4370
0.4382
0.4394
0.4495
0.459 1
0.4505
0.4463
0.4938
0.4953
0.4965
0.4974
0.4981
0.4940
3.0
3.1
3.2
3.3
3.4
0.4987
0.4987
0.4990
0.4991
0.4993
0.4995
0.4997
3.7
3.8
3.9
0.2324
0 . 3888
2.5
2.6
2.7
2.8
2.9
0.4998
0.4998
0.4999
0.4999
o.5000'
0. 1 700
0.3531
0.3749
0.3944
0.4 1 1 5
0.4265
0.4778
0.4816
3.5
3.6
0.0557
0.094 8
0. 1 33 1
0.3485
0.3708
0.3907
0.4772
0.4821
0.4861
0.4893
0.49 1 8
0.4995
0.4997
0.09 1 0
0. 1 293
0.1664
0.3461
0.4564
0.4649
0.4993
0.05 1 7
0.3438
0.3665
0.3869
0.4049
0.4554
0.464 1
0.47 1 3
2.0
2.1
2.2
2.3
2.4
z
Second Decimal Place In z
0.47 19
0.4864
0.4896
0.4920
0.4955
0.4966
0.4975
0.4982
0.4998
0.4998
0.4999
0 .4999
0. 3686
0.4066
0.4082
0 .4474
0.4484
0.4573
0.4656
0.4726
0.4664
0 .467 1
0.4732
0.4783
0.4788
0.4738
0.4793
0.487 1
0.490 1
0 .487 5
0.4830
0.4868
0.4898
0.4922
0.494 1
0.4956
0.4582
0.4834
0.4925
0.4943
0.4957
0.4967
0.4976
0.4982
0.4968
0.4977
0.4983
0.4987
0.499 1
0.4988
0.4991
0.4994
0.4995
0.4994
0.4996
0.4997
0.4997
0.4998
0.4998
0.4999
0.4999
0.4999
0.4999
0.4999
0.4999
0 .4838
0.4904
0 .492 7
0.4945
0.4959
0.4969
0.497 7
0.4984
0.4988
0. 4992
0. 4994
0.4996
0.4997
0.4998
0.4999
0.4999
0. 4999
• For z :!: 3.90. the areas are 0. 5000 10 foor decimal places.
52
0.4599
0.4678
0. 4 744
0.4798
0.4842
0.4878
0.4906
0.4 929
0.2764
0.3154
0.3770
0.3962
0.4 1 3 1
0.4279
0.4846
0.4881
0.4909
0.4931
0.4756
0.4884
0.491 1
0.4932
0.4946
0.4960
0.4970
0.4978
0.4984
0.4948
0.4961
0.4971
0.4979
0.4985
0.4971
0.4979
0.4985
0.4989
0.4992
0.4989
0.4992
0.4994
0.4996
0.4997
0.4989
0.4992
0.4995
0.4996
0.4997
0.4998
0.4999
0.4994
0.4996
0.4997
0.4999
0.4999
0.4998
0.4999
0.4999
0.4 999
0.4999
0.4949
0.4961
0.4 8 12
0.4 854
0.4887
0.49 1 3
0.4934
0.4951
0.4963
0.4973
0.4980
0.4986
0.4990
0.4993
0.0753
0.1 14 1
0. 1 5 1 7
0. 1 879
0.2224
0.2549
0.1851
0.3 133
0.3389
0.4 8 1 7
0.4857
0.4890
0.4916
0.4936
0.4952
0.4964
0.4974
0.498 1
0.4 986
0.4990
0.4993
0.4995
0.4995
0.4996
0.499 7
0.4997
0.4998
0.4999
0.4999
0.4999
0.4999
0.4998
0.4998
0.4998
0.4999
0.4999
0.4999
0.4999
Assessing and Presenting Experimental Data
(by interpolation). Hence, since z =
the range
(µ, - zo.4scr )
or
6
(x - µ.)fer , 90% of the population should fall within
<
(µ. - l .645er ) <
x < (µ, + Z0.4ser )
( 12)
x < (µ. + l.645er )
THEORY BASED ON THE SAMPLE
In any real-life situation, we deal with samples from a population and not the population
itself. Typically, our objective is to use average values from the sample to estimate the mean
or standard deviation of the population. Thus, we would calculate the sample mean
•
x;
L.,, x = "°'
n
_
=
xi + x2 +
+ x.
· · ·
-----
n
Isl
( 1 3)
as an approximation to the popu lation mean µ. and the sample standard deviation
S,
=
/ (x1 - .i) 2 + (x2 - .i)2 +
n- l
V
· · ·
+
(x. - .i)2 = (L:'=t xt) - n.i2
n-l
( 1 4)
as an approx i mation to the population standard deviation er. Here n is the number of data
in the sample. The denominator of the standard deviation, (n - I), is called the number of
degrees of/reedom. 6
The difference between population and sample values is emphasized by the use of
symbols for each:
different
For Population
For Sample
Mean
er or er,
Standard deviation
Na1urally, we'd like to have some assurance that the sample mean and standard devia1ion accurately approximate the corresponding values for the population; more specifically,
we'd like to have an estimate of the uncertainty in approximating µ. and er by .i and S, .
A second objective i s often to infer the probability distribution of the population from
that of !he sample. As it turns out, these two objectives can be accomplished independently
for large samples. Conversely, for small samples (n <
knowledge of the distribution
is assumed in est imatin g 1he uncertainty of .i .
30),
6Thc basis for the number of degrees of freedom is the number of independent discrete dala lhat arc being
cvaluaicd. In computing the sample average, i, all • dala arc independent However, the Slandard deviation ascs
lhc rcsull or the previous caJculalion ror i, which is not independent or the remaining data. Thus. the number of
s, : We divide by (• - t) rather than n. In other inslances, lhe
I); Ille nlllllber
of dcgn:cs of freedom is then reduced by two or three. Thc number of degrees of freedom is often denoted as •·
dala may have been used to calculate two or llm:e necessary quantities (sec Section 7 and Appendix
degrees ol freedom is reduced by one in calculating
53
Assessing and Presenting Experimental Data
TABLE
3:
Results of a 1 2-Hour Pressure Test
Pressure, p, In MPa
Number of Results, m
I
3
12
25
33
17
6
2
3.970
3.980
3.990
4.000
4.0 1 0
4.020
4.030
4.040
4.050
6.1
An
Example of Sampling
During a 12-hour test of a steam generator, the inlet pressure is to be held constant at
4.00 MPa. For proper performance, the pressure should not deviate from this value by more
than about 1 %. The inlet pressure was measured 100 times during the te:st. Various factors
caused the readings to Huctuate, and the resulting data are listed in Tabh: 3. The resolution
of the digital pressure gauge used wa� 0.001 MPa. The number of resulw, m, is the number
o f readings falling in an interval of ±0.005 MPa centered about the listt:d pressure.
A first step in assessing the dispersion of the pressure readings m ight be to plot a
histogram of the readings, as in Fig. 9.7 A clear central tendency is ;apparent, as is the
approximate width of the distribution. To quantify these values, we can compute p and SP
(Table 4) to obtain
p = 4.008 M Pa
Sp
=
0.0 1 4 MPa
Is the distribution of readings Gaussian"! A simple test is just to s ubstitu te p and Sp
for µ. and a in Eq. (6) and plot the resultant curve over the histogram. as in Fig. 9. (The
vertical scale for the distribution has bee n arbitrarily increased, for purpose of comparison.)
An eyeball comparison indicates an approximate fit, albeit not a perfect one. How good
must the fit be in order that we can claim a Gaussian distribution and apply Gaussian results?
Goodness of fit is a legitimate concern, which is addressed further in Section 7. I .
Assuming that the population o f pressure readings i s in fact Gaussi.an distributed, with
p � µ. and Sp � a , the results of the previous section can be used to estimate the interval
containing 95% of the pressure readings: µ. ± l .96a � p ± l .96Sp = 4.008 ± 0.027 MPa
One objective of the pressure test was to verify that the pressure did not deviate from
4.00 MPa by more than I % = 0.04 MPa. In terms of the standard deviation, 0.04 MPa
� 2.86a . For a Gaussian distribution, the probability of a 2.86a Huc11aation is about one
chance in 240.
7The choice of bin widlh is, on the one hand, aibill1lr}'. bul on lhc Olherhand, ii can grcJ>lly affecl lhe appearance
of the resulting graph. An empirical rule for the nwnberofintervals io plOI, N, is lhc Stu"'is rut.: N = I + J . J log"·
for n lhe IOlal number of points.
Assessing and Presenting Experimental Data
I
I
30
,., ....
I
I
I
I
I
I
z
\
\
\
\
\
� 10
3.96
3.98
4.00
4.02
Pressure. MPa
4.04
4.06
FIGURE 9: Histogram of the pressure data.
Two final comments should be made. First, these data do not separate measurement
pressure; however, the actual pressure fluctuations are
unlikely to be larger than the combined variation from measurement error and real fluctua­
tions. Second, the analysis tells us nothing about possible hias errors in the readings.
error from actual variations in the
TABLE 4: Calculation of Sample Mean and Standard Deviation
Pressure, p,
ln MPa
Number of
Results, m
3.970
3.980
3.990
4.000
4 .0 1 0
4.020
4.030
4.040
4.050
I
3
12
25
33
17
6
2
L. p = 400.770 n = L. m = IOO
p = 400.77/100 = 4.008 MPa
dz
Deviation, d
-0.038
-0.028
-O.o J 8
-0 .008
0.002
0.0 1 2
0.022
0.032
0.042
L, d2
Sp = J 1 858
x
1 44.4
78.4
32.4
6.4
0.4
14.4
48.4
1 02.4
1 76.4
x
= 1 858.0
x
1 0- S
1 0- s
1 0- s /99 = 0.0 1 4 MPa
Assessing and Presenting Experimental Data
6.2
Confidence Intervals for Large Samples
The example in Section 6. 1 showed how we can assess the dispersion of sample values about
the mean value of the sample. However, it did not yield an estimate for the uncenainty in
using i as an approximation to the true mean µ,. In fact, we obtained from that sample
only a single estimate for the mean value. If the pressure test were repeated and another
I 00 points acquired, the new mean value would differ somewhat from the first mean value.
If we repeated the test many times, we would obtain a set of samples for the mean pressure.
These samples of the mean would also show a dispersion about a central value. A
profound theorem of statistics shows that if 11 for each sampl e is very l arge the distribution
of the mean values is Gaussian and that Gaussian distribution has a standard deviation
Uj =
( 15)
u/,/n
This theorem (the Central Limit Theorem) applies for very large samples even if the distri­
bution of the underlying population is 11ot Gaussian [3).
Armed with this important result, we can attack the uncenainty in our estimate that
Since i is Gaussian distributed, with standard deviation Uj, it follows that
i "" µ,.
( 16)
where z is the same z-distribution gi ven in Table 2. Hence, following Example 3, we can
assert that c% of all readings of .t will lie in the interval
µ. ± lc/2 .;n
u
In other words, with c% confidence, the true m ean value,
about any si n gle reading of i :
_
.t -
a
lc/2 .,/ii
<
( 1 7)
µ,,
u
µ, < X + Zc/2 ,/ii
_
lies in the following interval
( 18)
This interval is termed a c% co11fidence illterval.
Th e confidence interval suffers from only one limitation: a is usually unknown.
However, a reasonable approximation to u is Sx when n is large. Thus, for l arge samples,
standard practice is to set u "" S, i n estimating the confidence interval for i :
( 19)
Often, the statrdard deviatio11 of the sample meQll , Sx . is introduced in this context:
(20)
56
Assessing and Presenting Experimental Data
EXAMPLE 4
Determine a 99% confidence interval for the mean pressure calculated in Sec tion 6. 1 .
Solution
First evaluate Zc/2
P
or
µP =
4.008
-
= Z0.495 = 2.575
�
Z0.49S ../fi <
µP
from Table 2 . Then, wi th u "" Sp,
�
< P + Z0.4YS ../fi
14
MPa = 4.008 ± 0.0036 M Pa
± 2.575 0.0r.;v:.
" 100
(99%)
Note that this interval is much narrower than the dispersion of the data itself. because we
are focusing on the accuracy of the estimate of the popula l io n mean.
Equation ( 19) is appropriate when we want the uncertainty in using the sample mean,
i, as an estimate for the population mean, µ. In contrast. the approach of Table I a nd
Section 6. 1 is appropriate when we desire an estimate of the width of the population distri­
bution or the likelihood of observing a value that deviates from the mean by some particular
amounl For instance, in Section 6. 1 we used Table I 10 esti m ate the interval containing
95% of all individual readings of p; in the last example, we estimated an interval con tain i ng
99% of all measurements of p. The c% co n fidence in1erval for the mean value is narrower
than that of the data by a factor of I/ .,/Ti, because 11 observations h ave been used 10 average
out the random deviations of individual measurements.
6.3
Confidence Intervals for Small Samples
Equations ( 1 8) and ( 1 9) provide confidence intervals for the sa mp le mean, i . when o is
known or can be approximated by S. , The condition u "' S, will generall y apply when
the sample is "large," which, from a practical viewpoint, means n � 30. U n fort u natel y. in
many engi neeri ng experiments, n is substantially less t han 30, and the preceding intervals
are really not much he lp.
An amateur statistician, writing under the pseudonym Student, addressed this mailer
by considering the distribution of a quantity t ,
·
i-µ
(2 1 )
t = --
S, / .,/fi
which replaces u i n Eq. ( 1 6) by S, . Student calculated the probability disiribution of
the t-statistic under the assumplion that the underlying populatio11 satisfies the Gaiwi011
distribution. This PDF, /(t ) , is shown in Fig. 10. Note that the distribu1ion depends on !he
number of samples taken, through the degrees offreedom, v = 11 I .
Th e t·distribution i s qualitatively similar to the z d is t ri butio n . Th e PDF is symmetric
about t = 0, and the lolal area beneath the distribution is again un i ty. Moreover, since the
t-dislribulion is a PDF, the probability that t will lie i n a g iven i nt erval 12 ti is equal to
-
-
-
57
Assessing and Presenting Experimental Data
f(t)
FIGURE
10: Probabi l i ty distribution
the area be neat h the c u rve between 12 and I t . As
approaches the standard Gaussian PDF; for
n
for the t-statistic.
(or v) becomes large, the I-distribution
n > 30, the two distributions are identical.
By analogy to the previous treatment of the z-distribution, the area beneath the 1-
5. However, in this case, the area a between t and I -+ oo
is listed; selected areas are given for a range of sample si:res. Thus the airea a corresponds
to the probabi lity, for a sample si:re of n = v + I, that t will have a valu e greater than that
given in the table [Fig l l (a)] . Since the distribution is symmetric, a is also the probability
that t has a value less than the negative of the tabulated value [Fi g l l (b)]. Conversely, we
can assert with a confidence of c% = ( I - a) that the actual value of t does not fall in the
shaded area (i.e., if a = 0.05, then c = 0.95 = 95%).
distribution is tabulated in Table
Often, we want a two-sided confidence interval for the mean x of a small sample. so
This interval fol.lows directly from
that both upper and lower bounds on the mean are stated.
the preceding intervals; with a confidence of c%, the true mean value li1:s in the interval
-
X - la/2. •
�
..[ii
< f.L <
-
X
�
+ 1af2 .• ..[ii
(c%)
(22)
= I - c and v = n - I [Fig. I l (c)]. This equation is the smaJll-samplc analog of
Sometimes, a is referred to as the leve l of significance.
This confidence i n terval defines the precision uncertainty in the value x. From
(22), the precision uncertainty in x is
where a
Eq
Eq.
( 1 9).
(23)
at a confidence level of c%. This precision uncertainty is that needed for Eq. (4).
Assessing and Presenting Experimental Data
Confidence Interval
0
t..•
(a)
0
FIGURE
1 1:
(b)
-1.12.•.
(c)
1.12.•
Confidence intervals for the I-statistic: (a) one-sided, right, (b) one-sided,
l eft, (c) two-sided. With
interval.
Confidence Interval
-t. .•
a
con fidence of c%
=
(I
-
a), the value of t lies in
the unshaded
Assessing and Presenting Experimental Data
TABLE 5: Studenl's 1-Dislribution (Values of la, • )
0
•
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
00
lo.015, •
3.078
lo.OS,•
3 . 1 82
1.476
2.353
2. 1 32
2.015
1 .440
1 .4 1 5
1 .943
1 .895
1.397
1 .383
1 .372
lo.to,.
1 .886
1 .638
1 .533
6.3 14 1 2 706
4.303
2.920
.
2.776
2 57 1
t.,
lo.OI,•
3 1 .82 1
6.965
4.541
lo.oos, .
63.657
9.925
5.84 1
3.747
3 . 365
4.604
4.032
1 .833
1 .8 1 2
2.447
2.365
2.306
2.262
2.228
3. 143
2.998
2.896
2 .82 1
2.764
3.707
3.499
3.355
3 .250
3 . 1 69
1 . 363
1 .356
1 .350
1 .796
1 .782
1 .77 1
1 . 76 1
1 . 75 3
2.20 1
2. 1 79
2 . 1 60
2. 145
2. 1 3 1
2.7 1 8
2.68 1
2.650
2.624
2.602
3 . 1 06
3.055
1 .337
1 .333
1 .330
1 .328
1 .325
1 .746
1 .740
2. 120
2. 1 IO
2. 1 0 1
2.093
2.086
2.583
2 . 567
2.552
2.539
2.528
2.92 1
2 . 898
2.878
2.86 1
2.845
1 .323
1 .32 1
1 .3 1 9
1 .3 1 8
1 .72 1
1 .7 1 7
2.080
2.074
2.069
2.064
2.060
2.5 1 8
2.508
2.500
2.492
2.485
2.8 19
2.807
2.797
2.787
2.056
2.052
2.04 8
2.045
1 .960
2.479
2.473
2.467
1 .345
1 .34 1
1 .3 16
I .860
1 .734
1 .729
1 . 72 5
1 .7 14
1 71 1
.
1 .798
1 .3 1 5
1 .3 14
1 .3 1 3
1 .706
1 .703
1.701
1 .3 1 1
1 .282
1 .699
1 .645
.
60
2.462
2.326
3.0 1 2
2.977
2.947
2.83 1
2.779
2.77 1
2 .7 6 3
2.756
2.576
•
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
00
Assessing and Presenting Experimental Data
EXAMPLE 5
Twelve values in a sample have an average of .i and a standard deviat ion of Si. What is the
95% confidence i n terval for the true mean value µ.?
,
Solution The required level of significance is a = 1 - 0. 95 = 0.05 and the degree o f
freed om i s 11 = 1 1 . The necessary value of t is la/2. v = lo.02s . 1 1 = 2.201 (from Table 5).
Hence, the 95% confidence i nterval i s
x
-
�
2.20 1 .;n < µ. < x + 2.20 1
-
�
.;n
(95%)
EXAMPLE 6
!-.
A si mple postal scale of the equal-arm balance type, is s u ppl ied with
1 -, 2-, and 4-oz
machined brass weights. For a quality check, the manufacturer randomly selects a sample
of 14 of the I-oz weights and weighs them on a prec ision scale. The results, in ounces, are
as follows:
1 .08
1 .0 1
0.97
1 .03
0.96 0.95
1 .04
0.98
0.99
1 .08
1 .00
0.98
1 .05
1 .0 1
Question: Based on this sample and lhe assu mpt io n that the parent population is
normally d is tribu ted , what is the 95% confide nce interval for the population mean?
Us i ng the t-test, we first ca lc u l ate the sam ple mean and standard deviation, with
n = 14. They are, respec1ively,
Solution
.i = 1 . 009 oz
and
S,
= 0.04 1 78
= 2. 1 60.
From Tab le 5, for 11 = n - I = 1 3, we find
Cal cu lati ng the two-sided confidence limits, we have
± to.02s. 13 S,
11 1 / 2
=
to.oi s. 1 3
±2. 1 60(0 .04 1 78)
( 1 4) 1 /2
= ±0 .024 1 2
Hence we m;iy wri te µ. = 1 .009 ± 0.024 oz, with a confidence of 95%.
6.4
Hypothesis Testing for a Single Mean for a S mall Sample Size (n ::: JO)
We often find i t necessary to employ statistics in order to make certai n decisions regarding
a measurand. For ex am p le consider the pressure mcasuremenls listed in Table 3. These
measurements re prese nt a si n gl e sample con1aining I 00 pressure measu reme n1s ob1ained
during a 1 2 ho u r period which produce a mean pressure o f 400.77 MPa. If additional
s amples were 1aken, and the mean pressure of each sam ple obtained, the mean pressure
ca n be assumed to be n ormally distributed based on lhe Central Limit Theorem. Often
we are concerned whether the press ures observed are representative of a set poin t pressure
of 4.00 MPa. One of the most commonly used methods for making such deci s ions is
h ypothesi s testi ng Typically there are two hypotheses in a h ypo1hes i s test One h y pothesi s
is called t he n111/ hypothesis and the other is often referred lo as the altemate hypothesis.
,
-
.
.
61
Assessing and Presenting Experimental Data
The first step in setting up a hypoth esis test is too choose the null hypothesis. S ince the
hy pothesi s testing often refers to a si mple mean (although it can be used for any parameter)
it takes the form
(24)
Ho : µ = µo
the
where µo is some constant specified valu e.
The second step in h ypot hes is testing involves speci fy ing an afternate hypothesis.
The choice of
alternate hypothesis should reHect on what we are attempting to show.
There are three possibilities for the choice of the al ternative hypothesis .
1. Two-tailed test: If we are p ri m ari ly concerned with determining whether a popu lati on
mean, µ, is di ffere nt from a specific value, µo, then the alternate hypothesis should
read as
(25)
2. Left-tailed test: If we are primarily concerned with determining whether a popu lation
mean, µ, is less than a specific value, µo, it should read as
(26)
Ha : µ < µo
Finally if we want to determine whether a pop•ulation mean. µ, is
greater than µo, it should read as
3. Right-tailed test:
(27)
We can construct useful graphi ca l representations of h ypothes is testi ng :as sh ow n in Fig. 12.
Reject :
Ho :
---.
.
-t..n
Do not
reject Ho
0
Two telled
: Reject
:
Ho
'
t.i:i
Reject
:
Do not reject
Do ncll reject Ho : Reject
Ho
:
�
L �.
Ho
:
-t.
0
0
Left tailed
Pl lght tailed
Confidence level = 1 - a
11 • degrees of freedom • n - 1
t"" s
i - P-o
s.1rn
/Joo = value specified in
Ho
FIGURE 1 2: Criteria for the rej ection of the null h ypothesis .
62
I,,
Assessing and Presenting Experimental Data
For lhe applicalion of hypothesis testing, we must perform the following steps:
I. Define bolh the null and alternate hypothesis.
2. Define a level of confidence, c%.
3. Calculate the value of lex p from the data.
4. From Table 5 determine the proper value of lex, • using lhe degrees of freedom v.
S. If '•• P falls in the reject Ho region, we reject Ho and accepl the alternate hypolhesis
H• .
6. I f '•• P falls i n lhe d o not reject Ho region, w e conclude that we d o nol have suffi­
cient dala to reject Ho at the level of confidence specified. (The strongest statistical
scaccmenc occurs when we can reject Ho and accept Ha .)
EXAMPLE 7
Using che data of Example 6, determine if the sample of 14 of the I -oz weights comes
from a population of weights whose true mean weight is greater than 1 .00 oz, assuming a
confidence level of 99%.
Solution
Ho :
Ha :
I
up
/.L
/.L
= l .OO oz
� l .OO oz
i
-
v = ( 14
la. •
/.LO
- Sx/Jn
=
-
-
1 .009 1 .000 0 806
0.04178/v'f4 -
.
1) = 1 3
lo.01 , CJ = 2 . 650
Since chis is a right tailed test and le•p falls in the do not reject Ho region, we conclude
lhat the 99% confidence level lhat the population mean wa� not significantly differenl lhan
l .OO oz.
6.5
Hypothesis Testing for a Single Mean for a Large Sample Size (n :::: 30)
For a l arge sample size, the six steps listed in Section 6.4 are followed exactly excepl lhat
la. . . is replaced by la and 10,p is replaced by
i - µo
=
Sx/ Jn
Zexp
(28)
Figure 1 2 can now be interpreted as the similar criteria for rejecting or not rejecting the null
hypothesis when th e la . .. limits are replaced by Za ·
Assess ing and Presenting Experimental Data
EXAMPLE 8
From the pressures in Table
3, can we conclude that the experimental data obtained indicate
a target press ure of 4.00 MPa at a confidence level of 99%?
Solution
: µ. 4.00 MPa
: µ. f. 4.00 MPa
i - 4.00 MPa 4.008 - 4.000
=
Sx /..fo
0.0 14/,/100
Ho
=
Ha
Since n
�
=
Zexp
30 we use ::l:Za/2 as our limits.
From Table 2.
Zar.z s Zo.cm = 2.575
I
I
I
I
I
I
-2.575
2.575
Reject Ho :
Since l ex p is in the reject
=
5.714
zo.oos = 2.575.
: Reject Ho
5.714
Ho region, we reject Ho and accept H0 •
indicate that the target pressure of 4.00 MPa is not being accurately
Thus, the data
con trol led based on a
confidence level of 99%.
6.6
The t-Test Comparison of Sample Means
If we wish to compare two samples solely on the basis of their me a ns , we can u se the
following
form of F.q. (2 1 ) (4):
(29)
in which i t . Si . n t and
i2 . S2 , 11 2 are the means, standard deviations and sizes of the two
respective samples. This value of t is compared to the i n ter va l
which a is for an arbitrarily chosen confidence level ( I
be approximated by the fol low i ng e x pressio n :
II =
where 11 is rounded down to the
Section
mea ns .
::l:t.. 12.u fou nd in Table 5, i n
- a). Thc degn.."C of freedom 11 m ay
((sf /11 1 ) + (Si/112)) 2
2
(sf f n i ) 2 (sif 11 2 )
--- + --n1 - I
112
nearest i n tege r [4].
6.4 can be used here to determi ne specific
64
(29a)
-I
test ing procedures of
conditions regarding the two samp le
The hypot hesis
Assessing and Presenting Experimental Data
EXAMPLE 9
An apartment manager wishes to determine if the lifetimes arc different, under similar
conditions, for two major brands of l ig ht bulbs. In the following s am p l e data, the lifetime
is i n months.
Bulb A
Bulb B
Solulion
7.2, 7.6, 6.9, 8.2, 7.3, 7.8, 6.6, 6.9, 5.5, 7.4, 5.7, 6 . 2
7.5, 8.7, 7.7, 7.5, 6.7, 1 1 .2, 7.0, 10.7, 7.0. 8.6. 6. 1 . 6.3.
7.8, 8.7, 6. 1
ta ard deviations of each sample, we find
When we calculate the means and s nd
Bulb A
Bulb B
iA = 6.94 mo is = 7.84 mo
SA = 0.82 mo Ss 1 .53 mo
nA = 12
ns = 15
=
Th e hypothesis t o be tested i s
Ho : iA
Ha : iA
= is
1' is
This results i n a two-tailed test as illustrated in Figure
12.
From Eq.
(29a) we detennine
the degrees of freedom
[(0.82)2/ 1 2 + ( 1 .53) 2 / 15] 2
[<0.82)2/ 12] 2 [(t .53) 2 /15] 2
+
12 - 1
15 - 1
"" 22 (rounded down)
v=
(29) we calculate.the test statistic
6.94 - 7.84
= J(0.82) / 1 2 + ( l .53)l/ t5 = I . 954
fc:xp
2
For a confidence level ( I
a) = 0.95, we find the critical values of I from Table 5 to be
and from Eq.
-
-
±10.os12.n = ±10. 02s.22 =
±2.074
e ec t Ho reg ion . we conclude that there is not
a s ign i fic a nt difference in the l i feti mes of bu l bs A and B at a 95% confidence level. [For
c om paring large sample means (v > 30), we simply replace I in Eq. (29) by z . I
Since the val ue of '••P falls within the do not r j
7
THE CHI-SQUARE (x2) DISTRIBUTION
A variable is considered to have a chi-square distribution if its distribution has the shape of
a right-skewed curve as shown in Fig. 13. The tota l area under the curve is 1 .0: and as the
degrees of freedom , v, becomes large, the chi-square d i stri but i on approaches a symmetric
distribution which resembles the normal distribution. Table 6 presen ts the values of X 2 for
various values of a and v.
65
Assessing and Presenting Experimental Data
FIGURE
13: Probabi l i ty distributions for the x 2 -statist ic .
The chi-square goodness-of-lit-test statistic is defined as
k
x = L.,
i=L
2
where
" (01
-
E;
E; )
2
(30)
= observed frequency
= e x pected frequency
k = total number of variables being compared
O;
E;
v=k- 1
Since the chi-square statistic x 2 , is by de fi n i ti on a positive value, a right-tailed test
is applied for hypothesis testi ng (see Fig. 14). Note that when the observed frequencies
approach the expected frequencies the value of x 2 approach es zero.
,
Do nol reJecl Ho ' AejeCI
---1---
FIGURE
14: x 2 Criteria
Ho
for rejecting th e null
hypothes is .
Assessing and Presenting Experimental Data
TABLE
6: x 2 -Distribution (values xJ.• >
a
•
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
xJ.99s
0.000
0.0 1 0
0.072
0.207
0.4 1 2
0.676
0.989
xJ.99
xJ.91S xJ.9s xJ.os xlois
xJ.0 1
6.635
xJ.oos
7 .378
9.2 1 0
10.597
9.488
9.348
1 1 . 1 43
1 1 .345
1 3.2TI
1 1 .070
12.832
1 5.086
12 .592
16. 8 1 2
1 8.548
14.067
14.449
1 6.0 1 3
1 8.475
20.278
20.090
2 1 .666
23.209
2 1 .955
0.00 1
0.004
3 . 84 1
5.024
0.05 1
0. 1 03
5 . 99 1
0. 1 15
0.297
0.2 1 6
0.352
7.815
0.484
0.554
0.83 1
0.7 1 1
1 . 1 45
0.872
1 .239
1 .237
1 .635
1 .690
2. 1 67
0.000
0.020
1 .646
7.879
1 2.838
14.860
16.750
2. 1 80
2.733
1 .735
2.088
2.700
3.325
15.507
16.9 1 9
1 7.535
1 9.023
2. 1 56
2.558
3.247
3.940
1 8 . 307
20.483
2.603
3.074
3.053
3.57 1
4. 1 07
3.8 1 6
4.404
4.575
19.675
2 1 .920
23.337
24.736
24.725
26.757
5.226
26. 2 1 7
27.688
28.300
29.8 1 9
29. 1 4 1
30.578
3 1 .3 1 9
32.80 1
1 .344
23.589
25. 1 88
4.660
5.009
5.892
2 1 .026
22.362
5.229
5.629
6.262
6.57 1
7.26 1
23.685
24.996
26. 1 1 9
27.488
5 . 1 42
5.8 1 2
6.908
7.962
26.296
28.845
32.000
34.267
5.697
6.265
6.408
7.0 1 5
7.633
7.564
8.672
27.587
30. 1 9 1
33.409
35.7 1 8
8.23 1
8.907
9.380
10. 1 1 7
28.869
37. 1 58
36. 1 9 1
38.582
8.260
8.59 1
1 0.85 1
3 1 .526
32.852
34. 1 70
34.805
30. 144
3 1 .4 1 0
37.566
39.997
8.034
8.897
1 0.283
1 1 .59 1
32.67 1
35.479
38.932
4 1 .40 1
8.643
9.542
10.982
1 2.338
36.78 1
40.289
42.796
9.260
1 0. 196
1 0.856
1 1 .689
12.401
1 3.09 1
1 3.848
33.924
35. 1 72
36.4 1 5
87.652
38.076
39.364
40.646
4 1 .638
42.980
44. 1 8 1
45.558
44.3 14
46.928
38.885
40. 1 1 3
4 1 .923
43. 1 94
45.642
48.290
46.963
49.645
3.565
4.075
4.60 1
6.844
7.434
9.886
1 0.520
1 1 .524
13. 1 20
14.6 1 1
1 1 . 1 60
1 1 .808
1 2. 198
13.844
1 4.573
1 5.379
1 2 .46 1
1 3. 565
15.308
16.928
4 1 . 337
44.46 1
48.278
50.993
13.121
14.256
1 4.953
16.047
1 7.708
42.557
18 .493
43.773
49.588
50.892
52.336
16.79 1
45.722
46.979
1 3 .787
12.879
16. 1 5 1
53.672
•
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Assessing and Presenting Experimental Data
EXAMPLE 1 0
A pair o f dice are tested t o determi ne if they are "true." W hen lhe dice ar e lossed, the
face-up sum total i s read. After 360 such tosses. the results shown in lhe following table
were obtained:
Face-Up Sum
Frequency
2
3
8
25
29
42
53
55
4
5
6
7
8
9
JO
46
11
12
22
39
29
12
Determine if the d ice are tru e a t the 99 % confidence level.
Solution
In this case, the null hypothesis and alternate hy pothes is are
Ho : Frequencies are the same
Ha : Frequencies are different
We can form the following table using the expec ted distribution shown in Fig. 6:
Face-Up
Sum
Expected
Observed
Frequency Frequency
10
20
30
8
2
3
4
5
25
29
42
53
55
46
39
29
22
12
6
7
8
9
JO
11
12
40
50
60
50
40
30
20
10
Th e experimental x 2 -statistic is
2
Xexp =
(8 - J 0) 2
(25 - 20) 2
( 0; - E; ) 2
�
L.,
+
_
•=•
= 3.358
E-'
JO
20
+
...
( 1 2 - J 0)2
+
JO
Assessing and Presenting Experimental Data
For a 99% confidence level, a = 1 -
I = 10.
is v = k -
Thus, since
7.1
0. 99
=
0.01 ;
and the number of degrees of freedom
Therefore the appropriate x 2 -statistic is
xlo1.io = 23.209
x1xp < xJ.o1. to• the dice arc true at the 99% confidence level.
Goodness of Flt Based on the Gaussian Distribution
As staled previously, distribution of experimental data often abides by the Gaussian form
expressed by Eq.
(6).
One must keep in mind. however. that this approximation is not
always justified. For example, fatigue strength data for some metals approximate the so­
called Weibull distribution; other distributions are described in the Suggested Readings
at the end of the article.
Since a given set of data may or may not follow the assumed
distribution and since, at best, the degree of adherence can be only approximate, some
estimale of goodness offit should be made before critical decisions are
based on statistical
error calculations. In the following paragraphs, we discuss tests of fit that may be applied
to the common Gaussian distribution, Eq. (6).
Al
the outset we advise the reader that there is no absolute check in the sense of
producing some perfect and indisputable figure of merit. At best, a qualifying
level
confidence
must be applied, with the final acceptance or rejection left to the judgment of the
experimenter.
The simplest method is simply to plot a histogram and to "eyeball" the result:
Yes, the
distribution appears 10 approximate a bell shaped one; or no, it does not. This approach can
easily result in misleading conclusions. The appearance of the histogram can sometimes
be altered radically, simply by readjusting the number of class intervals.
A
second method, which is relatively ea�y and much more effective is to make a
graphical check using a 11ormal probability plot. This technique requires a special graph
paper8 available from most boo kstores that deal in technical supplies. One axis of the
graph represents the cumulative probability (in percent) of the summed data frequencies.
The other must be scaled to accommodate the range of data values in the sample. The more
nearly the data plots as a straight line and the more nearly the mean corresponds to the
50% point, the belier the fit to normal distribution. '!be fi nal detennination is subjective; it
depends on the judgment of the experimenter. Considerable deviation from a straight line
should raise serious doubts as to the value
of any Gaussian-based calculations, particularly
the significance of the calculated standard deviation. The following example demonstrates
the procedure.
EXAMPLE 1 1
We will illustrate the graphical method by using the data of pressures given i n Table
Solution
In treating these data we will arbitrarily center our class intervals, or bins, on
the mean and will assume eight intervals. each
0.0 1 0
M Pa in width. Using these ground
rules, we prepare Table 7.
8 Scc
3.
re (erenc..-e (5), page 25, for directions for preparing one's own normal probability paper.
Assessing and Presenting Experimental
Data
TABLE 7: Pressure Data or Table 3 Arranged ror a Nonna! Prob.ability Plot
A
3.965
3.975
3.983
3.995
4.005
4.0 1 5
4.025
4.035
4.045
B
c
D
I
I
1 .0 1
3
4
25
16
41
12
33
17
6
2
r
4.04
1 6. 1 6
74
91
97
4 1 .4 1
74.75
9 1 .92
97.98
99
100
99.00
1'00.00
A = l..imics o n class intervals, arbiuarily Lake n as 0.010 MPa.
II = Number or daca items lall in& within respective class inwvals.
c = Cumulalive number or data items.
D � Cumulalive number of data items ia percenL
•A rule or 1humb often wed is to discard lllbittarily any dala !ailing oulSidc " ±3Sx
limit. Theon:tically. di.scanling OUl-ol-tolerance items could make a ieadjusunent or
lhe mean and lhe standard devillion necessary. In lllis panicular case, Ille changes
would be so slight as IO make lhe additional wort unprofillble.
The ordinate of the graph is in terms or the upper limit or each interval. This quantity
correlates with the cumulative values, which are plotted as the abscissa. Data from column
A are plotted versus the percentages in column D, yielding Fig. 15.
To plot either 0 or 100% is impossible.
For this reason, and also because either
absence or presence of even one extra data point in the extreme intervals unduly distorts the
plot, the two endpoints are generally given little consideration in making a final judgment.
On the basis of Fig. I 5, we can
distribution. Figure
16
say that the pressure data show a reasonalbly good Gaussian
IM• discerned from a
illustrates the general discrepancies that may
non-straight-line normal probability plot and their c.auses.
2
Another common test for goodness of fit is based on the x -distrihution. Implemen­
tation of the method requires considerable data manipulation and, as
with other methods, a
judgment on the pan of the experimenter. ln addition, the method does not lend itself well to
small samples of data. The usual practice is to divide the test data into a reasonable number
of class intervals, or bins, to determine the number of observations O; in each interval, and
expected number E; of data items. The expected
a "standard," the source of which depends on the objective of the
then to compare these numbers with an
numbers are based on
test. If the test is to determine the normalcy of test data then the standard is the normal
probability distribution. On the other hand, the standard may simply be ;inother set of data
that, in terms of an objective, is considered satisfactory; for example, how well do test data
fit a standard norm?
Definite limitations apply :
I. The origi nal. experimentally determined values of 01 must be numerical counts. They
arc integer freq11encies; fractional events do not occur.
70
Assessing and Presenting Experimental Data
4.04
4.02
3.98
-
3.96
-
-
-
-
-
0%
._........
..
....
...
._...
..
...... ___..__..__.__._...._._
..
.._..__..__.___._.._..___.._.__.
0.01
0. 1
5
10
3 0 so 70
90
CUllUative % data items
98
99.8
99.99
FIGURE 1 5 : Normal probability plot of data listed in Table 7.
Symmetrjc:al bu t w Hh too
great a proportion of
points dlslributed
near the mean
1%
50%
99%
FIGURE 16: Graphical effects of data skew and offset as displayed on a nonnal probability
plot
Assessing and Presenting Experimental Data
TABLE B:
Pressure Data of Table 3 Arranged for a x 2-Test
Observed
Frequency,
Pressure,
O;
in MPa
3.965
Expected
Frequency,
E1
0.80
3.975
4. 14
3.985
12
12.57
25
20.06
33
31 .47
17
19.63
6
8.54
2
2.27
3.995
4.005
4.015
4.025
4.035
4.045
2.
Frequency values for O; in each bin should be equal to or greater than to unity. There
no unoccupied bins.
The use of x2 is usually questioned if20% of the values in either the O; or the E; cells
or bins have counts less than 5. Often the cells or b i ns can be combined to eliminate
the problem.
should be
3.
For the data in Table 7 the "observed values," O; , are those in column B. To determine the
"expected values," E; , we must define a Gauss ian distribution having a mean pressure, p,
of 4.008 MPa and a standard deviation, of 0.014 MPa, which are the mean pressure and
the standard deviation of the data. Now using the values of p, s,. and n with Table 2 we
can compute E; shown in Table 8. Finally, because several of the 8 bins have fewer than 5
members, we combine the first two and the last two before computing x;.p . so that k = 6.
Applying Eq. (30) and assuming a = 0.05 and v = k 3 = 3,
Sp.
-
2
Xexp
_
-
=
� ( 0;
L.,
i•I
4. 1 5
x; = xJ.ou
.•
-
E;
=
E; ) 2
_
-
(2 - 4.94)2
+
4.94
+
.
. . + (8 - 10.8 1 )2
I0.8 1
7.8 15
Since x;.p < x: with reference to Fig.
distributed at the 95% confidence level.
.•.
1 2.57)2
1 2.57
(12 -
14, we conclude that the pressure data are normally
72
Assessing and Presenting Experimental Data
Of special importance is the fact that v = k - 3. This is because the number of bins,
the mean, and the standard deviation were chosen from the experimental dala.
8
STATISTICAL ANALYSIS BY COMPUTER
Slatistical analysis can often involve very large sets of dala or require the application of
a broad range of statistical tests. Consequently, a number of statistical software packages
have been developed to assist in data analysis. Commercially marketed versions include
MINITAB9 and SPSS. 1 0 Packages such as these are available for use on machines ranging
from mainframes lo personal computers.
The slatistical methods of the preceding sections were developed in the last century
to reduce the laborious calculations that would otherwise be required in drawing statistical
conclusions from samples. These methods are possible largely because the population's
probability distribution has been assumed known. However, the digital computer makes
detailed slatistical computations easier. As a result, some current statistical research is
directed toward using computer methods to relax lhe assumptions associated with classi­
cal slatistics. For example, can we determine small-sample confidence intervals without
assuming that the population is Gaussian distributed?
9
BIAS AND SINGLE-SAMPLE UNCERTAINTY
Precision error in repeat-sampled data reveals its own distribution, enabling us to bound its
magnitude using slalistical me1hods. Bias error. by virtue of its syslematic nature, provides
no direct evidence of either its magnilude or ils presence. The only direct method for
uncovering bias error in a measurement is by comparison with mea�ufCments made using
a separate, and presumably more accurate, apparatus.
Unfortunately, a second set of apparatus is seldom used owing to cost and time
conslraints. lns1ead, we rely on knowledge of our own equipment to make estimales for
the likely sizes of bias errors. Es1imation of bias uncertainly relies heavily on experience
and on an underslanding of calibration accuracy and dimensional tolerances. Even wilh
such experience and underslanding. unexpected sources of bias error can be overlooked.
Diligence, persisten ce, and careful examination of onc·s results are essential in identifying
and eliminating such errors.
Estimates of bias uncertainty should be accompanied by odds or a confidence level (6).
Unlike slatistical confidence levels. odds for bias uncertainty cannot be rigorously deter­
mined. The level of confidence assigned is a product of our knowledge of the system,
reftec1ing our assessment of the fraction of bias errors likely to land within the uncertainty
interval. Sometimes lhe lerrn coverage is used in place of confidence to reflect lhe empirical
na1ure of these estimales in contrast 10 those derived using s1a1is1ical methods.
We have previously discussed general sources of bias error. Lei's look at a few in
more detail.
Dala reduction oflen requires knowledge of physical properlies , sys1em dimensions,
or electrical characleristics. For example, a Oowme1er mcasuremenl may depend on the
density of waler and a tube diameter; an amplifier gain may depend on l he value of a resistor.
Differences between !he assumed values of these components and !heir actual values can
systematically shift all data taken, creating a bias error.
9 MlNITAB is registcn:d to Minitab. lnc .. 308 1 Enlerprise Drive, State College, PA. t680 1 .
I OsPSS is a n:gistercd tradematk of SPSS. Inc . . 444 N. Michigan Avenue. Chicago. IL. 606 1 1 .
73
Assessing and Presenting Experimental Data
Reference standard
(may involve a number of
Identifiable uncartanties)
RGURE
17:
Proper matching
System being
I ndicaled readout
(commonly involves
ancillary apparatus)
calibrated
to be compared with
Uia 'known' Input
Block diagram showing calibration proc:edure.
To find water's density, the temperature is specified and a tabulated handbook value is
taken, giving. for example, 998 kg/ml . Inaccuracy or ambiguity in the specified temperature
may cause a bias uncertainty in density of 2 kg/ml , for instance. This uncenainty may cover
95% of the temperature range we expect is possible for the system. This inaccuracy will
:rysrematically affect all data reduced using the particular value of density.
Similarly, the nominal diameter of a pipe may differ from lhe production diameter by
a percent or so: and a manufacturer's rated value for a resistor can vary substantially from the
resistor's actual resistance. For carbon resistors, the manufacturing lo.lerance may be ±5%
or ± I 0%; for higher-quality, metal-film resistors, the tolerance may be: ±0.01%. Typically,
these tolerances might represent 95% coverage-that is, the variation of95% of all resistors.
Potential sources of error such as these remain unchanged for each meaisuremenl made with
the system.
lf lhe uncenainty related to a manufacturing tolerance is unacceptably large, taking our
own measurement of the specific pan can usually reduce the uncertain1:y substantially. The
uncenainty in a resistor's value can be reduced to the accuracy of the ohmmeter measuring
it; or a pipe's diameter can be measured to the accur.icy of a set of calipers. Physical
propeny data can also
measured, if need be, although it is more common to trust the
carefully determined handbook values.
be
Calibration uncertainty is another very common source of bias u ncenainly. Calibra­
tion requires a reference or standard against which system response can be compared. The
reference may be fixed or one-valued, such as the triple point of waler or the other triple
points and melting points used to define the practical temperature scak,. Alternatively, the
standard may be capable of supplying a range of inputs comparable to the range of the
system, as do various commercially available voltage references. Naturally, the uncenainty
of the standard should be considerably less than that of the system being calibrated. A rule
of thumb is that the uncertainty of the standard should be no more than one-tenth that of
the system being calibrated.
Figure 17 shows a typical calibration arrangement. Normally, the i ndicated readout is
compared to the reference standard and a relation between the two is detennined. Sometimes
the readout scale can be adjusted until agree ment with the standard is .ob·tained; sometimes
a line lit is used lo relate the readout to the standard's value. In eith1�r case, additional
uncenainty appears in the comparison and adjustment process.
Instrument manufacturers often supply calibration data with their products, which
can assist in estimating the uncenainty of the instrument. For example, a particular position
transducer is rated al 0.8 V output per millimeter of sensor displacement. The manufacturer
has not specified the calibration uncenainty directly. However, we might assume that the
uncenainty is roughly 0.05 V/mm, since this is the apparent resolution of the calibration.
The coverage is also unknown, but our experience using the device may :suggest that 90 to
74
Assessing and Presenting Experimental Data
95% coverage is a reasonable assumption. If necessary, we could reduce the uncertainly
by conducting our own calibration.
Examples of estimating bias uncertainty are given in Section I I .
9.1
Single-Sample Precision Uncertainty
When only one, two, or three repeal observations are made, the confidence intervals calcu­
lated stalislically can be quite large. In that circumstance, you may delennine a narrower
range for !he mean value by ucaling precision errors like bias errors and es1imaling a
slandard deviation based on your knowledge of the instruments. For example, random
varialions in lest conditions may cause a digital mullimeler (DMM) reading 10· ftuc1ua1e;
bul if lhe reading is made only once, the random variation simply produces an overall range
of uncertainly for the !rue value of !his variable. The uncertainly (at 19 : 1 odds) is twice
1he standard deviation of lhe lest condition. In olher words, ± 1 .960' :::::. ±2u will cover
95% (or 19 oul of 20) of !he readings made. If, on the other hand, lhe same measuremen1
is made several limes, !he random variations can be averaged oul, and stalislics can be used
10 place a narrower bound on the mean value.
We can conslruct the single-sample estimate a little more fonnally. We begin by
estimating u as a value (u,, say) that is based on our knowledge of lhe experimental system.
Thus, we are assuming that u of the population is known, and, with Eq. ( 1 8), the precision
uncertainty can be estimated as
(3 1 )
With a single reading, n = I , so !hat no averaging is perfonned to reduce the uncertainly.
·
Taking a 95% confidence level, Z0.95/2 = 1 .96 and
Px
:::::.
1 .960',
(95%)
(32)
The polential precision error underlying a single measuremenl of a random variable
can usually be eslimated from your knowledge of how finely an insirumenl will resolve,
of how precisely an instrumenl may repeal a reading, or of how much !he lesl condition
ftuctuales. Often, these estimates can be made in advance of perfonning the experiment, in
order to gauge !he expecled uncertainly in lhe result
Section I I includes an example of single-sample uncertainty analysis.
10
PROPAGATION OF UNCERTAINTY
Often several quanlilies are measured, and the results of those measurements are used
to calculate a desired result. For example, experimental values of density are usually
determined by dividing the measured mass of a sample by the measured volume of that
sample. Each measurement includes some uncertainty, and these uncertain1ies will create
an uncertainly in the calculated result. What is that uncertainly?
Finding the uncertainty in a resull due to uncertainties in the independenl variables is
called finding the propagation ofuncertainty. For uncenain1ies in !he independent variables,
the procedure resls on a statistical theorem that is exact for a linear function y of several
independent variables x; with standard deviations u; ; lhe theorem states that the standard
75
Assessing and Presenting Experimental Data
2 + . . (�an) 2
2
�a
)
(�a1
(
)
axn
ax2 2
ax,
deviuioo of y is
O'y =
...
. +
+
(33)
y
Likewise, a calculated result is a func1ion of several independent measured variables,
{xi . x1, , Xn ; for example, density is a function of mass and volume. Each measured
wlue has some uncenainty,
and these uncenainties lead lo an uncenainty
in y, which we call
• To estimate
1 , we assume that each uncertainty is small enough that
a first-order Taylor expansion of y(x1 ,
. , Xn ) provides a reasonable approximation:
y(1 1 + U J ,
)
{u1 , u2 , . . . , Un )
u
xi,
.12 + u2 , . . . , Xn + Un)
"" y(x 1 . x2, . . .
..
uy
, Xn ) +
ay
ax1
-u 1
+
ay
ay
axn
ax2 u 2 + · · · + -Un
(34)
Under Ibis approximation, y is a linear function of the independent variables. Now we can
apply the theorem, assuming that uncenainties will behave much like standard deviations:
Uy =
( �u 1 ) 2
ax1
+
2
(�u
ax2 2 )
+
· · ·
( )2
+ �
ax. un
(n : l )
(35)
Here, all uncertainties must have the same odds and must be independent of each other.
This approach was established by S. J. Kline a nd F. A. McClintock in 1 95 3 [6).
The uncertain t ies u;, may be either bias uncenainties or precision uncertainties.
Nonnalty, the bias uncenainties and precision uncenainties in y are propagated separately.
The overall uncenainty, U1 , is then calculated by combining 81 and Py using Eq. (4).
Section 1 1 .2 illustrates this proced u re.
Equation (35) can be simplified w he n y has ccnain common functional forms, as
shown in lhe following examples.
,
EXAMPLE 12
Suppose that y has the form
y = Ax1 +
Bx2
arul lhai lhe uncertainties in xi and .<2 are known with odds of n : I . What is the uncenainty
in y7
Solution
Using Eq. (35),
(n : I )
(36)
For additive functions, the absolute uncertainties in each term are combined in root-mean­
square (rms) sense.
76
Assessing and Presenting Experimental Data
EXAMPLE
13
Suppose that y has th e form
X
2
y = A -1_
m.ln
and that the uncertainties in
uncertainty in y ?
x�
x 1 , x2 . and x3 are
k no w n
with odds of
n
I. What is 1he
Solution
Using Eq. (35),
Uy =
(n : I )
so that, for this
case,
�=
)'
For
( )2 ( ) + (k -U3 ) 2
Ut
m-
"2
n-
+
-<2
.q
2
X3
(n
:
I)
multiplicative functions, 1he fractional uncertain1ies are combined in an rms
weighting factors, m, n, and k. in Eq. (37) and !heir sources .
(37)
sense.
Note carefull y the
Normall y, each source of error is independenl of !he
01her sources. The errors will
1heir maximum values simultaneously.
For that reason, Eq. (35) combines 1he uncertain1ies in a root-mean-square sense. In so me
situations, however, various sources of u nce rt ai nty are not independent. Dependent errors
should be added together, before combining them in 1he rool-mean-square sense with other
independent sources of error.
no! all be of the same sign, nor will !hey
11
1 1 .1
all
lake on
EXAMPLES OF UNCERTAINTY ANALYSIS
Rating Resistors
EXAMPLE
14
with color-cod�'<! bands t hat specify their nomi nal resistance.
actual resistance of each resistor varies randomly aboul the nominal value, owi ng to
manufac tu ring varialions. The percentage variation in the resistance of the population of
Carbon resis1ors ar e painted
The
77
Assessing and Presenting Experimental Data
resistors is referred to as the tolerance of the resistors. For commercial carbon resistors,
this variation is 5, I 0, or 20%.
A lab technician has just received a box of2000 resistors. As a result of a production
error, the color-coded bands have not bee n painted on this lot. To determine the nominal
resistance and tolerance, the technician selects I 0 resistors and measures their resistances
with a digital multimeter. His results are as tabulated.
Resistance (kS2)
Number
I
18. 1 2
1 7.95
18. 1 7
1 8.45
16.24
17.82
16.28
16.32
1 7.9 1
1 5.98
2
3
4
5
6
7
8
9
10
Whal is the nominal value of the resistors? What is the uncertainty in that value?
Consider both precision and bias uncertainty. Can you estimate the tolerance?
Solution The precision error in the resistors can be averaged lo find a 95% confidence
inte..Val. From the tabulated data,
R
= 17.32 k!:2
SR = 0.982 kQ
The mean resistance, R = 17.32 kO, is clearly the apparent nominal value of the resistors.
To find the uncertainty in this mean value, both the precision and the bias uncer­
tainties must be estimated. Consider the precision uncertainty first. From Table 5 with
11 = IO - I = 9 and a = ( I - 0.95)/2 = 0.025,
Applying
ta .• = to.025. 9
= 2.262
Eq. (22), the (unbiased) population mean, µ R , is in the range
M =
R
-
± la, v .,(ii
SR
= 1 7.32 ± 2.262
0.982
r.;;
" 10
= 17.32 ± 0.70 kO
kO
(95%)
However, this answer accounts for only precision error, specifically, PR = 0.70 kO. What
is the bias uncertainty in lhis result?
The manual for the DMM describes its calibration; possible bias uncertainty (from
temperature drift, connecting-lead resistances, and other sources) is rated as
±(0.5% of reading + 0.05% of full scale + 0.2 0)
78
Assessing and Presenting Experimental Data
The confidence is not given, but we shall assume it to be
The full-scale readi ng of the
meter is 20 kn, and after evaluating the tenns and summing, the meter's bias uncertainty
can be estimated as
95%.
BR =
±96 . 8 0
= ±0.10
(95%)
0.005
kO
Notice that the reading error in the DMM scale is only
kO, which is much lower than
the actual uncertainty in the DMM reading! This DMM has relativel y high resolution and
precision but much lower accuracy.
The total uncertai n ty in the mean of the population is, from Eq.
(4),
UR = (BR + PR)
] 1/2
[(0.10)2 + (0 70)
0.7 1 (95%)
UR = 0.7 1 (95%),
2
=
=
t/2
2
.
2
kO
kO
The uncertainty of the nominal value is
kO
or about
The precision uncertainty in the mean is the major source of uncertainty. On the other
hand, if a sample of 1000 resistors were used , the precision uncertainty would be reduced
by a factor of ten (why?), and the bias uncertainty would be dominant.
The tolerance of the resistors remains to be found. What we'd like is an estimate of the
percentage deviation from the nomi nal value which includes, say,
of the resistors. One
approach is to note
of a Gaussian population lies within
of the population
mean µ (see Table 1 ). On that basis, we could approximate O'R "'
and
"' R, so that
that 95%
Tolerance % =
±4%.
95%96a
±l.
SR /LR
l.96aR "' l.9�SR 1.96 · 0·982 0.111
17.32
/L R
10%,
R
=
=
that is, a tolerance of % (or about
si nce that's the nearest production tolerance).
In a manufacturing situation, e ng i neers are usually more interested in estimating an
interval that is
certain to contain at least some percentage b of the population. For
example, the man ufactu rer might wish to report, with
confidence, that b
of resistors will have resistances within some specific range of resi stances . As it turns out,
the approach used in the preceding paragraph is a very poor way to estimate such tolerance
limits, because it ignores the inaccuracy of SR an d R as estimates of the population's O'R and
Although
of a Gaussian population lies in the interval
±
that is not
true of the interval R I
For example, our estimate of the mean has a
pr s ion
uncertainty; this means that the interval likely to contain
of the population should
be broadened by something like an additional
of R beyond R ± I
A proper
estimate of tolerance must allow for this uncertainty as well as that in O'R . More advanced
statistical methods
show that, at a confidence of
the
tolerance interval or the
population is almost twice as large as th at estimated previously (i.e., the interval that is
certain to contain at least µ ±
turns out to be R
After this extended disc uss ion, it may interest you to learn that the resistors actually
tested were nominally
kO with a tolera nce of
11
c%
c = 95%
/LR·
95% ± .96SR.
[3)
l.96a
18
= 95%
/J.R l.96aR,4% eci
±4% 95%
.96SR.
95%, 95%
95%
± 3.532SR).
10%.
Assessing and Presenting Experimental Data
1 1 .2
Expected Uncertainty for Rowmeter Calibration
EXAMPLE 1 5
Obstruction meters such as venturis and orifice plates are commonly u sed to measure the
steady How rates of fluids. Tables of emp irical calibration coefficients, K, are avail able for
use in theoreti cally based relationships.
(38)
This tec hniq ue provides a means of measur i ng llow rate in te rms of the pressure drop across
the obstruction. The pu bli shed coe ffi cie nts w i l l yie ld approxi mate How rate s , but accurate
measurement requires careful experimental determination of the coefficient for each specific
i nstall ation
Figure 1 8 shows a proposed arrangement for calibration of a thi n plate orifice meter.
In this case, calibration consists of experimentally determ inin g the coeffici en t K in Eq. (38)
by collecting the flowing fluid (water, in this ca se) in a weigh tank for some period of time.
During the c al i bration period, the flow rate is held as c onstant as possi ble and the pressure
di ffere nce, 6.P = Pi
Pi. is recorded. The flow rate, Q, is the measured weight, W,
d ivided by the liquid dens i ty p, and lhe elapsed time, t (Q = W/pt); the area, A 2 , is
1f D 2 /4, for D the orifice diameter.
Substituting these values into Eq. (38), w e may solve fo r th e ca li bration constant,
.
-
,
-
,
K:
K
=
4 \V
;r
fl""
02 1 y 2PiJ'
(39)
Orifice meter
--�
Source of flow
P,
Weigh lllnk
FIGURE 1 8 : Se t u p for calibrating an orifice.
scale
-+- Plalform·lype
Assessing and Presenting Experimental Data
By inserting the observed values of W, t, and 6 P, along with a measured value of D and
tabulated data for p, the experimental value of K is obtained.
Before undertaking this experiment, we'd like to estimate the expected accuracy of
the result. We could, of course, make the following calculations after the experiment, but
by doing it ahead of time, we can identify those parts of the experiment that contribute most
of the uncertainty and, if necessary, improve them.
Solulion
Both bias and precision error should be considered. However, since we have no
samples for statistical analysis, we can only estimate the expected size of potential precision
errors, using estimates for the standard deviations. Thus this analysis is a single-sample
uncertainty estimate. To proceed, we lirst estimate the single-sample precision and bias
uncertainly in each measured variable and then propagate these uncertainties into K.
The weight, W, is measured with a platform scale. What bias uncertainly exists in the
scale measurement? Has the scale's calibration been checked? How recently and against
what standard? Presumably, we have made some sort of check, at a minimum several point
calibrations using reliable proof weights. If not, the user's manual should include such data.
Let us say that, in our judgment, an uncertainty of
is justified, with a confidence of
about
In practice, the bias uncertainty is undoubtedly dependent to some degree on
the magnitude of the weight measured relative to the scale range; this effect could also be
taken into account if warranted.
Precision uncertainty in the weight measurement will be caused by reading error in
the scale, and possibly by hysteresis, friction, or backlash in the scale mechanism. The size,
or standard deviation, of the reading error will depend on how linely the scale is graduated;
perhaps 0. 1 % error covers one standard deviation in the scale reading (so that ±0. 1 covers
68% of the reading errors). Hysteresis, friction, and backlash should be negligible if the
scale is in good condition; however, if these effects are observed, their errors should also
be taken into account.
The diameter of the orifice must be determined. Assume that it is a sharp-edged
orifice and that we have checked it with an inside micrometer. Did we check the micrometer
against gauge blocks, or are we accepting its scale as is? How experienced are we in using a
micrometer"! We i ntend to measure the orilice only once, using this value in all subsequent
applications of the meter, so the diameter's uncertainty all appears as bias in the results.
After these considerations. we estimate a 95% bias uncertainty of ±0.008 cm in the nominal
4-cm diameter (±0.2% uncertainty).
How accurate is the determination of the
period, t ? If a hand-operated diverter
is used, precisely when did the How start and stop? That is, how well is the time period
defined? If a stopwatch is used, how good is the synchronization between the diverter and
watch actions'!
±I%
95%.
%
time
Note that these time uncertainties are essentially precision errors, which are likely
lo vary from run to run. If we carefully make a series of repeated runs of this particular
procedure, we could accumulate enough data to perform a precision uncertainty analysis,
were such accuracy required. In the present Clllie , we simply estimate the likely standard
deviation to
s; t he total time period is 5 min, the standard deviation of t is ±
Bias error in the time determination may arise if the watch is fast or slow (probably a
very small error in 5 min) or if we systematically stop or start the watch too soon. Without
other information, we' ll assume that bias in the time reading is negligible compared to the
precision error already discussed.
be ±3 if
I%.
81
Asse ssing and Presenting Experimental Data
The density will be read from a handbook table at.the tempera1ture of the experiments.
Between o•c and
the density of water decreases by about Cl.
Temperature may
vary slightly between each experiment, leading to a precision en-or if only one value of
density is used . However, if we use a thermocouple to measure the correct temperature for
each experiment, precision error will still arise from the reading error of the temperature
measurement. If standard deviation in temperature is ±0. 1 °C, then the corres ponding
density variation is only
38°C,
7%.
0.002%.
On the other hand, the bias error in the temperature reading may be higher. perhaps
, and the resulting bias uncertainty in the density used would then be
(again
). If we don't bother to measure the temperature, the bia� uncertainty would be larger
C
95%
± I°
at
still, maybe
±0.02%
0.2% if we just assume a standard room temperature of 20°C.
The value of t;. p is measured using a manometer. The dominant uncenainty is that
in reading the difference in the heights of the manometer columns , essentially a precision
uncenainty resulting from reading error. This uncertainty has relatively constant size,
independent of the magnitude of t;.
so that a percentage uncertainty is a bit misleading.
At small /';.
it may be
but al high /';. it may only he
To keep the uncenainty
analysis simple, let's take a representative value of
for the standard deviation in the
pressure. Bias uncertainty in the manometer turns out to be substantially smaller, about
at about
confidence.
0.1 %
P, P
3%,
P,
0.1 %.
± 1%
95%
Summarizing our results gives the following:
Blas Uncertainty,
Standard Deviation,
1%
0.2%
"" o
0.02%. ' 0.2% t
0.1%
0.1%
Weight, W
Diameter, D
lime, t
Density, p
Pressure, t;.
•
Ux/;r
Bx/X (95%)
Variable
P
�
°'
1 .0%
0.002%
1%
measured.
is not measured.
I C ternpellltu,. is
t tr temperatur<
K
2 + (- - ) ]
(-) ( lld )l + ( -"') 2 ( -2' Up)
p
2
[
Now we can apply Eq.
U
=
K
(37) to
U '°
2
W
K:
t
+ 2D
BK - [(Bw)2
'
-
+
t;. p
U t;P
2 �
(2 - ) ( -8' ) 2 ( -1 - ) + ( - ) ]
t
2
t;.P
1/2
(1)2 (2 0 2) 2 (0)2 + G 0.02) G 0. 1 ) ]
First., we calculate the bias uncenainty i n K :
- K
=
=
W
[1.08%
+
8, 2
+
D
+
+
Bp 2
p
2
2
x
.
+
x
(95%)
82
2 l /l
-
l 86 P
2
+
x
Assessing and Presenting Experimental Data
Likewise, the standard deviation of
K
-l.96uK
is
Our estimate for the single-sample precision uncertainty in
PK
K
=
K- =
From Eq. (4), the total uncertainly in
K
is, from Eq.
(32).
=
1.96(0.0 112) 2.20% (95%)
K
is
2
i; [ ( �) \ ( � r
[<o.010s)2 + (0.0220>2] 1 '2
2.45% (95%)
=
=
r
=
Inspection of these results quickly reveals the parameters having the greatest con­
tribution to the uncertainty. Improvement of the timing and weighing procedures would
improve the results the most. Improvement of the pressure and diameter measurements
would contribute significantly less improvement.
Most of the total uncenainty is caused by precision uncertainty. We can reduce
that uncenainty considerably by repeating the calibration experiment several times and
averaging the results. Since PK will decrease as .jn, taking n = experiments will reduce
PK to about 1 .0%.
Note that the density contributes almost nothing to the total uncenainty. Even if we
don't bother to measure the temperature, the contribution of density uncenainty remains
4
0.2) 2
negligible; that is, (i ·
«
We conclude that careful temperature measurement,
in this case, would be a waste of effon.
Are the various uncenainty estimates simply good guesses? To a degree, they are,
but dismissing them as nothing but guesses would be flippant. The specific considerations
leading to each estimate were not arbitrary; when properly made, such "guesses" have a
strong foundation in the actual perfonnance of the equipment and the method of taking the
data. Even the estimated confidence percentages (usually
are a quantitative assessment
of our expectation for the variability of the data, although they are essentially just educated
guesses. But if we admit to guessing, why not simply guess the overall uncertainty and skip
all the intermediate steps? In answer, the detailed analysis provides a means for evaluating
the relative effect of each identifiable source of error, thereby separating the more imponant
ones from the less imponant ones. Funhennore, one can evaluate the uncertainties of each
of the individual variables with considerably more assurance than one could judge the total.
(1)2•
95%)
Assessing and Presenting Experimental Data
12
MINIMIZING ERROR IN DESIGNING EXPERIMENTS
·
The best ti me to minimize experimental error is in the design stage, when an experimen tal
procedure i s being deve l oped . First and foremost, one should perfo rm a si ngle-sample
uncertai n ty analysis of the proposed experimental arrangement prior lo beginning con­
structi on , in order to determine whether the expected uncertainly i s acceptably smal l and
lo iden tify the major sources of uncertainty. Some general precautions 10 observe when
desi gning an experi me nt are as follows:
I.
Avoid approaches that require two large numbers to be measured in order lo determine
the small difference between them. For example, large uncertainly is l i ke ly when
meas uring 8 = (x 1 - .t2 ) i f 8 « .t t . unless .t1 can be measured with great accuracy.
2. Design experi me nts or sensors that amplify the signal strength in order to improve
sens it ivity. For examp le , a thermopile uses several thermocouples to resolve a s i ngle
temperature, and a strain gage uses many loops of wire lo measu re a single slraiIL
3. Build "null des igns ." in· which the output is measured as a change from zero rather
than as a change in a non zero value. This reduces both bias and precision error.
Such designs often make the output proportional to the difference of two sensors. An
excellent exam pl e of this approach is the W heats tone bridge circuit.
4. Avoid experiments i n which large "correction factors" must be applied as part of the
data-reduction procedure .
S. Allempt to minimize the inHuencc of the measuring system on the measured variable.
6. Cal ibrate entire systems. rather than ind ividual components, in order lo minimize
calibra1ion-rela1ed bias errors.
13
GRAPHICAL PRESENTATION O F DATA
According to 1he American Standards Association (7),
used 10 present facts, interpretations of facts, or t heoret ical relation­
usually serves to communicate knowledge from the author to his
readers, and lo help them visualize the features that he considers important .
When
s hips . a graph
A graph should be u sed when it will convey information and portray s ig nifican t features
more effici eml y than words or tabulations.
A graph is nearly always the most effective formal for convey i ng the interrelation
invaluable in co ns truc ti ng acceptable curve fits of
experimental data and in iden ti fy ing outliers or erroneous measurements. Graphs are also
use fu l in testing theoretical calculations against real ex perimental results and i n ide ntify i ng
the conditions under which a theoretical model fails.
The c lari ty i mparted by graph i ng data can be substantial. Table 9 shows the
atmosphe ri c pressure measured in Cam bri dge, Massachusetts, during Hurricane Bob on
August 19, 199 1 . In tabular form the trend is unclear. If, on the other hand, the data are
graphed (Fig. 19) the progress of the storm is apparent. At mospheric pressure declined
steadily until about 4:00 P.M . and then began 10 increase. The 1 1 :30 A.M. read ing lies well
below the trend delined by the other data, and it can probably be assumed 10 be in error
(some checking of the data reductions verified this conclusion). A faired curve has bee n
of experi men ta l variables. Graphs are
84
Assessing and Presenting Experimental Data
TABLE 9: Atmospheric Pressure during Hurricane Bob
Time of Day
IO:OO A.M.
1 1:30 A.M.
1:00 P.M.
2:15 P.M.
3:40 P.M.
4:40 P.M.
5:40 P.M.
Pressure (mbar)
1009.0
984.2
999.8
989.0
977.I
981.2
990.0
1 010
1 000
Pressure.
p (mbar)
990
980
9:00 A M.
1 1 :00
1 :00 �M
3:00
5:00
Time of day
FIG URE
19:
Atmospheric pressure during Hurricane Bob.
empirical ly sketched through the remaining data. 1 1 This curve may be used to estimate the
pressure at limes other than those measured.
1 3.1
General Rules for Making Graphs
By observing the following guidelines, you can help to ensure that your graph will be easy
for your readers to understand. Figures 20(a) and (b) illustrate some of these points 171.
1. The graph should be designed to require minimum effon from the reader in under­
standin8 and interpreting the information il conveys.
1 1 fuirrd in lhis sense means smooth and without
im:gulari1y: a draftsman's curve 1hrough the data.
Assessing and Presenting Experimental Data
2.0
1 .5
g
0
...
�
esl�te�m�p!:;e::;ra:.1u_. re
�h�
H�ig
__�
1 .0
__J
0.5
0
1910
1920
1930
1940
(a)
1950
nme
1960
1970
1980
1 990
100
S2"
�
i
�
f
t
F
e
50
20
10
5
2
1900
1
(b)
o : Highest tem�l8t'llture
- : Fitted
0
line
1 920
1980
1960
1 940
Year of discovery
superconductor
2000
FIGURE 20: (a) A poor graph; (b) graph of Fig. 20(a) improved by following graphing
guideline numbers 2, 3, 5, 8, I
0, and 12.
2. The axes should have clear labels that name the quantity plotted,
its units, and its
·
symbol if one is in use.
3. Axes should be clearly numbered and should have tick marks for significant numerical
divisions. "JYpically, ticks should appear in increments of I , 2, or unils of measure­
ment mu l tip lied or divided by factors of
. . . ). Not every lick needs, to
be numbered; in fact, using too many numbers will just clutter the axes. Tick marks
should be directed toward the interior of the figure.
5
JO ( 1 ,10,100,
4. Use scientific notalion to avoid placing too many digits on the graph. For example,
use
x HY rather than 50,000. A particular power of I
need appear on l y once
50
0
Assessing and Presenting Experimental Data
along each axis. Avoid co nfusi ng labels such as "Press ure, Pa x
use units such
as MPa instead.
S. When ploning on semilog or full-log coordinates, use real logarithmic axes; do not
Logarithmic scales should have
plot the logarith m itself (e.g., plot
not
I
tick marks at powe rs of
and intermediate values, such as
6. The axes should usually incl ude zero; if you w ish 10 focus on a smaller range of data,
include zero and break the axi s, as sh ow n i n Fig. 4.
7. The choice of scales and proponions should be commensurate with the relative impor­
tance of the variations shown in the results. If variations by increments of
arc
si gn ifica nt, the graph should not be scaled to e mphasi ze variations by increments
of I .
8 . U se symbols such as Q , 0 , A , and 0 for data poi nts . D o not use dots O for data.
Open symbols should be used before filled symbols. You may place a legend defining
symbols on the graph (if space permits) or in the fi gure capti on.
106 ;"
50, 1.70). 12
10
IO, 20, 50, 00, 200, . . . .
10
9. Place error bars on data points to indicate the estimated uncertai n ty ofthe measurement
or else use symbols that arc the same size as the range of uncertainty.
10. When several cu rves are ploued on one graph, different lines (solid, dashed, dash­
dot, . . . ) should be used for each if the curves are closely spaced. The graph should
i nc lude labels or a legend identi fy i ng each c urve . Avoid usi n g colors to di fferen tiate
curves, since colors are usually lost when the graph is photocopied . Theoretical curves
should be plotted as lines, without showing calculated poi n ts. Curves fitted 10 data
do not need to pass through every measurement l i ke a do1-to-do1 cartoon ; however, if
a data point lies far from the lined curve, a discrepancy m ay be indicated [as for the
lirst and the last three points in Fig. 20(b)).
1 1. Lettering o n the graph should be held to the minimum necessary for clarity. Too much
1ex1 (or too much data) creates crowdi ng and confusion.
12. Labels on the axes and curves should be orien ted 10 be read from the bouom or from
the right. Avoid forcing the reader 10 rotate the figure in order to read it.
13. The graph shou ld have a descriptive but concise title. The t itle should appear as a
caption to the ligure rather than on the graph itself.
Good graphing software can hel p produce graphs that adhere to these gu ide l i n es.
However, some graphing packages violate even the simplest of these rules. Discretion is
advised !
13.2
Choosing Coordinates and Producing Straight Lines
The lirst step in making any graph is to decide which variables 10 p lot and on what scale
to plot them. Four basic graphical scales occ u r frequently in engineering work (Fig.
).
li11 ear coordi nates have a l i near variation of both the x - and y-scales. If a variable changes
by several orders of magnitude or is ex ponen tial ly related 10 anot her variable, then a loga­
rithmically scaled axis may be preferab le . Graphs havi n g one logari thmical ly scaled axis
and one linearly scaled axis are called semilogarithmic (or semi.log). Those for which both
axes are logarithmic arc calledfall logarithmic (fall log, or log-log). When a quantity varies
with an angle, polar coordinates prov ide a physica ll y suggest i ve formal for the data.
21
1 2An exceplion is made when lhc unit tkci/Mls is
plotted.
Assess ing and Presenting Experimental Data
10
8
2
1 .0
0.5
Displacement (mm)
(a)
1 .00
<l' 0. 98
�
;s
§
0.96
.,
E!'
i
O
.6.
0
0 94
.
ct
D = I in.
0 = 2 in.
D • 6 in.
D = B in.
- Faired curve
0.92
1a4
FIGURE 2 1 :
1�
ReynoklS number, Re0
(b)
1�
1 07
continued
(a) Linear coordinates, (b) semilogarithmic coordinates.
The choice of wh ich scaling to use is nonnally guided by your expectations for the
physical behavior of the system being studied. You may also attempt to deduce the right
scaling by studying a test graph made on linear coordinates. Often, the objective in selecting
88
Assessing and Presenting Experimental Data
100
10
0.1
0.01
�-----....__
_
_____._______.______�
5
10
20
Frequency, I (Hz)
50
(c)
1 00
continued
FIGURE 2 1 : (c) Full logarithmic coordinates.
a scaling is to find coordinates i n which the plotted data fal l on a straight line, because straight
lines are the easiest curves to fit .
Fig ure 22(a) shows a set of data that represent the cooling of a warm metal slug
suddenly immersed in cold liquid. The difference between the slug's temperature and the
liquid temperature was
recorded at several tiOM:s artcr the slug was submerged. The graph
has the fonn of an eitponential decay; indeed, heat transfer theory suggests that the cooling
curve shou Id have the form
(40)
Assessing and Presenting Experimental Data
90•
'----'----"l.L....---'--'"""'---.W. ----'-----' O"
30
2D
1D
10
D
Temperature rise,
20
30
40
•c
(d)
FIGURE 2 1 : (d) Polar coordinates.
where 6 T is the measured tcmperarure difference at any time, 6 :ro is the temperature
1 .25
1 .00
t::.,
:
Measurements
Fitted curve
0.75
(:�)
0.50
0.25
0.00
�--��--�---�---� ---�
0
40
80
120
Time,
l (s)
1 60
200
(a)
FIGURE 22: Cooling data. (a) Linear coordinates.
240
Assessing and Presenting Experimental Data
1 .0
0.5
(:lo)
0.2
1 20
Tine, t (s)
(b)
FIGURE 22: Cooling data. (b) Semilogarithmic coordinates. ( Note logarithmic variation
of ll.T/ll.To.)
difference before the slug is immersed, and T is a time constant for the cooling. You may
desire to find an experimental value for T, so that you can use Eq. (40) to estimate A T at
values of t where you have no measurements. That task is not straightforward using the
linear scaling of Fig. 22(a).
Instead, you could plot log(ll. T/ti. To) as a function of t . Then the relationship between
ll. T and t is
log
( )
ll. T
ti. To
0.434 3 t
= - ---
T
(4 1 )
which is the equation of a line with slope - 0 .4343/T and intercept zero. 13 The graph is
most easily made using semilogarithmic coordinates [Figure 22(b)], and t can be calculated
from the slope of the line:
_ 0 .4343/T
= log [ll. T (ti ) / ti. Toi - log [ ti. T(12 )/ ll. To ]
t i - ti
(42 )
resulting in T = 98 s. Note that while ll. T/ ti. To is plotted on the logarithmic coordinates.
log(ll. T/ ll. To) must be used in calculating the slope.
llease
10 logarithms are slandanl in graphical work; log 1 0 • = 0.4343.
Assessing and Presenting Experimental Data
TABLE 10:
Straight-Line Transformations [8): y = /(JC) -+ Y = A + BX
Variables to Be Plotted
/(:r)
y = a + b/JC
y = l /(a + b:r)
or
l / y = a + bx
y = JC/(a + b:r)
or
:r/y = a + bJC
y = ab'
y = acbx
Y = a:r b
y = a + bx" ,
if n is
known
y
Straight-Une
y
x
I /JC
Intercept, A.
a
b
l/y
JC
a
b
x/y
x
a
b
log y
log y
logy
JC
log JC
y
x"
log a
log a
log a
x
Slope, B
logb
blogc
b
b
a
Semilog paper, full-log paper, and polar graph paper are available from most univer­
sity bookstores or drafting suppliers. Many computer spreadsheet and plotting programs
can also generate these coordinate systems. Moreover, plotting software can expedite exper­
imentation with different scalings and coordinates, so that the most informative ones
quickly identified.
Logarithmic scaling is only one approach to creating straight-line representations of
data. For example, the function
b
(43)
y=a+­
x
can
be
does not give a straight-line variation of y with x [Fig. 23(a)). However, it does give a
straight-line variation of y with I/ x. The solution is to plot y a function of I/ x rather
than of x itself: then a can determined as the intercept of resultant line and b as its slope
[Fig. 23(b)).
Table 0 offers a guide to straight-line transformations in which a function y = f (x)
is transformed to
as
be
I
Y = A + BX
by ploning an appropriate pair of modified variables, Y and X.
92
(44 )
Assessing and Presenting Experimental Data
x
(a )
1/x
(b)
FIGURE 23: Plol of y = 1 .0 + (2.5/x ) as (a) y versus x and (b) y versus ( l /x).
93
Assessing and Presenting Experimental Data
y
,,.P'
,,,9... ""0
0
o""6
,,. ...
Ybias
0
0
....,_
True line
'---- -- 1C
FIGURE 24: Bias and prec ision error in line fi lling.
14
1 4.1
LINE FllTING AND THE M ETHOD OF LEAST SQUARES
Once the data are in a straight- l ine form, the correc t line m ust be passed 1hrough them and
its slope and in tercep t determined. The s i mple st approac h is just to draw w hat appears 10 be
a good straight line through the data. Whe n this approac h is used, the probable 1cndcncy is
to draw a line that minimizes the total deviation of all points from the l i ne. Results obtained
this way are often acceptably accurate, particularly ir the data set includes only a few po i nt s
or if both x and y have significant error.
The data plotted may include both bias and p recis ion errors. Bi.as errors will tend to
shift the entire data set away from the true line or, perhaps, to ch ange its slope. Precision
errors will cause the data to scatter about the true l ine . E ither y or x may inc l ude both
precisi on and bias error (Fig. 24) . The objective of a curve fit is to a••erage out the preci·
sion errors by drawi ng a curve that follows the apparent central tend1incy of the scauered
data. Curve fitting, like statistical analysi s of preci sion error, does m:>th ing to uncover or
reduce b i as error. Bias errors in fitted curves must be identified by other methods, such as
compari son to independent data sets.
Least Squares for Line Fits
When the prec ision error i n )' is substantially greater than that in x, the method of least
squares, or linear regression, enables us to calculate a line, )' = " + bx, th roug h the
Assessing and Presenting Experimental Data
data (3). The method iden1ifies the slope b and intercept a that minimize the sum of the
squared deviations of the data from the fitted line, s2 :
si = E [y, - y(x, ))2
n
(45)
i=l
Here, for the various measured values of x;, YI is the experimentally detennined ordinate
and y(x; ) = a + bx; is the corresponding value calculaied from the filled line; n is the
number of experimental observations used. The result is
(46)
Most scientific calculators incorporate programs for calculating leas! squares lines, a
great convenience to the experimentalisL Consequently, least squares has become increas­
ingly popular as the methad for fitting lines. But to some degree this predominance has
also promoted misuse of the method. The leas! squares method addresses only the precision
error in y; ; poor results are obtained if x; also includes large precision error. Least squares
assumes, in cffecl, that the experimemal x; are errorfree.
To indicale the reliability of the lit, most pocket calculators and software packages
return the correlation coefficient, r, along wilh the least squares results:
r
where
2 = explained squared variation about
Ym
(47)
10.tal squared variation about Ym
Ym is the mean of the measured y; :
Ym
=
I
-
n
n
L Yi
(48)
ial
Th e explained squared variation results from the straight-line change o f y wi1h x :
II
L [ y(x; ) - Yml 2
i=l
The total squared variation also includes the precision error:
II
ft
i=I
i== l
"
L.._., [y(X; ) - Ym JL.._., ()"; - ym ) 2 = · · · = S2 + "
Thus,
>
(49)
Assessing and Presenting Experimental Data
If the sum of squared deviations S2 is assumed to result only from precision error, then a
"perfect fit" occurs when S2 -+ O and r 2 -+ 1 . Hence, the nearer r is to ± 1 , the "better"
the fit.
Unfortunately, when the data look basically linear, one usually obtains lrl > 0.9; the
correlation coefficient is n ot a very sensitive indicator of the precision of the data. It turns
out that ( I - r 2 ) I /l is a better indicator of the fit's quality; the closer it is to zero, the lower
the precision error in the data. The quantity ( I - r 2 ) t/l is roughly the ratio of the vertical
standard deviation of the data about the line to the total vertical variation of the data.
When both y and x have significant precision error, least squares should not be used;
this case is usually identifiable by the highly scattered appearance of the data. In addition to
eyeball estimates. various other semiempirical line-fitting procedures are available in that
situation, as discussed in reference [ 1 0) . Conversely, if both y and x have precision errors,
but these errors are small relative to the overall variation of the data, then least squares
results may still be acceptably accurate. In this situation, the data will still appear to fall on
a straight line.
14.2
Uncertainty in Line Fits
Sometimes the real issue is to estimate an uncenainty for the slope o r intercept of a fitted
line. Eyeball estimates of the uncertainty are often acceptably accurate; for e xample, you
may be able to vary the slope of a hand-fitted line to determine what range of slopes will still
fit the data with 95% confidence. Similar estimates can be applied to finding the intercept.
This approach is best either when the sample is small or when both y and x have large
precision errors. Statistical confidence intervals may also be derived.
Another major concern in line or curve fitting is that of identifying outliers. Points
well beyond the trend of the remaining data can often be identified by eye. When a particular
point is in doubt. exclude it temporarily, and fit a new line through the remaining data. If
that line shows a much belier fit, the point can probably be dropped. Again, within the
restrictive assumptions of least squares, statistical tests can be applied; essentially, these
consist of estimating a 3a band around the fitted curve. Sometimes, however, data deviate
from a filled line because the actual relationship is rwt a straight line. One must always
avoid forcing nonlinear data to fit an assumed straight-line form, since valuable information
can be lost in the process.
14.3
Software for Curve Fitting
Line fitting is actually a special case of the method of least squares; least squares can be
generalized to fit polynomials of any order. Many other methods of curve fitting have been
developed, some considerably more sophisticated than least squares. Good discussions of
such methods can be found in most texts on numerical analysis.
Assessing and Presenting Experimental Data
EXAMPLE 1 6
A c anti l ever beam de flec ts downward when a
o
is a fu n c t i on of the beam stiffness, K
gravitational body force, g = 9.807 mis:
mass is attached to its free end. Th e
(Nim), applied mass, M (kg), and the
(m),
deflection,
the
To determine
of a small cantilevered steel beam, a student places various
masses on the end of the beam and measures the corresponding deflections. The deflections
are measured using a scale (a ruler) marked in I -mm increments. Each mass is measured
in a balance. His results are as follow:
the stiffness
Mll88 (g)
Deflection (mm)
0
0.6
1 .8
3.0
3.6
4.8
6.0
6.2
7.5
0
50. 1 5
99.90
1 50.
05
200.05
250.20
299.95
350.05
401.00
The est i m ated preci si o n uncertainty in the measured mass, largely from reading error,
is ±0.05 g (95%) and the bias uncertainty, largely from calibration uncertainty, is ±0.I g
(95%).
overall uncertainty i n the mass is UM = 0. 1 1 g (95%), corresponding to
about 0.05% of typical load.
the deflection, reading error is the most likely cause of precision uncertainty; the
student est im ates th i s uncertai nty as ±0.2 mm (95%). The bias u ncertai n ty in the ruler,
from
is estimated at ±0. 1 mm (95%). The overall uncertainty in the
deflection is typically 5-10% of the measured value.
What i s the stiffness of the beam, and what is its uncertainty?
T e stiffness can
found by laki ng a least squares fit through the data, usi ng
the deH ect ion as the y-variable, since it has a much greater prec i si on uncertainty than does
the mass. Setting y = and x = M, we calculate the required sums (perhaps us ing a
a
Thus, the
a
For
manufacturing error,
Solution
be
h
o
pocket calculator subroutine that processes the entered dat ):
n =9
L:> =
1801 g
L x2 = 5 . 109 x 1Cl5 g2
L Y = 33.50mm
E i = 1 19. 3 mm2
L XY = 9959 g mm
·
Assessing and P
1 0.0
res t
e n i ng Experimental Data
8.0
f
Least squares line
Ii = 0.0 1 90 M - 0.076
s.o
\
:
,, '
� ..-: .-". .-"';
QL.""'::--- �
� ;io"'
,'
'
,
'
,
ES11mateoj confidence
interval br the line fit
2.0
0
200
1 00
q are
The least s u
Mass, M (g)
FIGURE 25:
400
3()1)
Beam deHection for various masse s.
s results are then
y = a + bx
-0.0755 mm
g
b = 7< = 0.0190 mmlg
= 0.995886
Fig. 25. The experimental stiffness of the beam is
9 807
= ! ;,,, ·
0.0 190 = 516 Nim
a =
r
The data and the line fit are shown in
i aria n
a
b
o
eq i ene
a
t
a
ncert
K
b
From th e figure, these data d o appear t o fall on a straight line. The correlation coeffic ient ,
r , is nearly unity, but a better test is to consider ( I - r2) 1 12 =
This value i ndicates
ut
of
that the ven cal standard deviation of the data (from precision error) is only
the total venical v
t io caused by the straight-line relationship between y and x .
What i s the uncenainty in the stiffness? The answer i s u val : nt t o the unccnainty
in the line's slope. One way to es ti m e this is just lo vary the lined l i
by eye 10 see what
range of slopes fit the data with
vi n
certainty. These bounds are shown in Fig.
a variation in b of bo ut
Thus, the
u
ain t y in K is also about ±
0.0906.
10%.
95%
95%
9%
25 gi g
10%.
Assessing and Presenting Experimental Data
15
SUMMARY
Every measurement includes some level of error, and th is error can never be known exac tly.
However, a pro bable bound on the error can usually be esti mated . This bound is called
11ncertainty. U ncenai n ty should always be acc om pa nied by the odds (or confidence per­
centage) that a particular error will fall within this bound. When presenting data either
graphically or numericall y, the uncertai nty should also be shown.
1. Errors can usually be classified as eit her bias error or precision error. B ias (or sys tem­
atic) errors occ ur same way for each measurement made . Precision (or rando m) errors
vary i n size and sign with a zero average value. B ias u ncerta i nty must be est i m ated
from our know ledge of the measuring equipment or by comparison to other, more
accurate systems. Precision uncertai nty can be estimated stat istical l y (Sect ions L, 2).
2.
The total uncenainty in a measurement includes both bias and precision error:
The bias and precision uncenainty should have the same confidence level, ty pical ly
95% (Sections 3, L L).
3.
4.
Random variables, such as precision error, may be c haracteri zed in terms of a pop­
ulation an d its distribiltion. The most common distribution for precision error is the
Gaussian or normal distribution (Sections 4, 5).
Properties ofa population are estimated by taking a sample from it. 'llte sample mean.
sample standard
are used to est i mate the population mean and
popul atio n standard deviation. The prec isi on u nce rtai n ty in i is
i, and
deviation, S_.,
P,
with
5.
=
ta/2. •
S,
,/ii
(c%)
I - c and 11 = n - L (Sec ti ons 6, 1 1 ) .
-d i stribut ion may be used to compare the d istri buti on of a sample to an expec ted
a =
The x
2
distribution. The accuracy with which a set of data lit the Gaussian distribution can
be checked us i ng either the x 2 "goodness-of-fit" test or a normal probab il i ty plot
(Section
7).
6. B ias uncenainty and single-sample precision u ncertai n ty
k now ledge of t he measuring sy stem (Sections 9, I I ).
7.
t i mated
from our
When ex perimental data are used to compute a final result, the uncertai nty of the data
must be propagated to determine the uncertainty in the result:
Uy =
Here 11;
I I ).
8.
are es
is eit her a
(
ay
- 11 1
a.q
bias u ncertainty,
)2
+... +
(
ay
iJx,,
;--- 11,,
)i
B; , or a prec i s i on uncertainty,
P;
( S ect io ns I O,
The accuracy ofex peri ments can be improved before they are conducted by ide nt ify i ng
and e l i m i nating major sources of u ncertai n ty ( Sect i ons I 1 .2. 1 2).
Assessing and Presenting Experimental Data
9. Graphical presenlation of data is often the most effective way to convey your results
a n d conclusions. Applying a few s im ple techniques when making yo u r graphs can
he l p ensure that your readers will fully and easi l y understand your work (Section 1 3).
10.
enable your results to lake an an alytica l form. Line fi ts also help to average
out precision errors in data. The method of least squares wi ll yield good line fits when
(a) the y prec is ion error is much larger than the x precision error or (b) both x and y
precision errors are small compared to the overall variation of the dala. Line fits do
not compensate for bias error (Section 1 4).
L i n e fi ts
SUGGESTED READINGS
ANSUASME PTC 1 9. 1 · 1 985. ASME Perrormance Test Codes. Supplement on lnslruments and
Apparatus. Part I, Measurement Uncenainty. New York. 1 985. This source conlains a mu hitude
of valuable references.
Barker, T. B. Quality by Experimental Design. 2nd ed. New York: Marcel Dek ker. 1994.
Coleman. H. W., and W. G. S teele . £.xperi-ntalion mid Uncertainty Analysis for Engineers.
2nd
ed. New York: Wilcy-lnterscicnce, 1 999.
Collection or papers related to engineeri ng measurement uncertainties. Thins. A SME, J. Fluids Engrg.
I 07. June 1 985. This source contains a multilude of valuable references.
Johnson, R., I. R. Miller, and J. E. Freund. Miller and Fn!und's Probability
neers. 7th ed. Englewood Cliffs. NJ.: Prentice Hall, 2004.
and S1a1is1icsfor Enri­
Kline. S. J .• and F. A. McClin1ock. Describing uncenainties in sing le-sam ple experimenls. Mech.
en,. 75: 3-8, January 1 953.
Moffat, R. J. Conlribu1ions 10 t he theory of single-sample uncenainty analysi s . Tra11s. ASMI:.� J. Fluids
£111rg. 1 04. June 1 982.
Turte. E. R. Tlie Vi.sual Display of Qua111i1ari11e lliformalion. Chesire, Conn.: Graphics Press, 1 983.
Weiss. N. A . • and M. J. H assett.
Wesley. 1 999 .
/t11roductory StaliSlics.
PROBLEMS
I odds).
Nole: Ullless Olherwise specified,
( 19 :
I.
5th
ed.
Reading. Mass.: Addison·
al/ 11ncertainties are assumed 10 represem 95% coverage
For a very large set o f data, the measured mean is found to be 200 w i th a standard deviation
20. Assuming the data to be normal l y distributed, determi ne the range within w hich
of
2.
60% of the data are expected Lo fall.
From long-term plant-maintenance data, it is o bserved 1hat pressu re downstream from a
boiler in normal operation has a mean value of 303 psi with a standard dev i at ion of 33 psi .
What is the probability that the pressure will exceed
in normal
Nole:
350 psi during any one measurement
operation"!
The following informalion should be helpful in solving Problems J-8.
For pun! eleclrical eleme111s in series
•
Resistances add din."Ctly.
•
Reciprocals of capacitances add to yield the reciprocal of the overall capacitance.
•
Inductances add directly.
1 00
Assessing and Presenting Experimental Data
For pure electrical elements in parallel
•
3.
Rec i proca ls or resislances add IO yield the reciprocal or the overall resislance.
•
Capaci1ances add directly.
•
,Reciprocals or induclanees add to yield the reciprocal or the overall i ndu ctance .
(a)
68·k0 re sistor is paralleled with a 1 2-kO resistor. Each resistor has a ± 1 ()%
tolerance. Whal will be th e nominal resis1ance and the u nce rtai n t y of the combina·
A
tion?
( b) If the values remain the same except 1ha1 the tolerance on the 68·kQ
dropped 10 ±5%, what will be the uncertainly of the combination?
resistor is
4.
Five 1 00· 0 resisiors, each having a 5% tolerance, are con nec ted in series. W h at is the
overall nominal resistance and tolerance or the combination?
5.
Three 1000- 0 resistors are connected in parallel. Each resistor has a ±5% tolerance.
What is the overall nominal resislancc and what is the besl estimalc or lhe tolerance of the
combination?
6.
A 47·0 resistor is connected in series with a parallel combination or a 1 00-0 resistor and
a 1 80- 0 resistor. What is the overall resistance or the array and what is lhe best estimate
of its iolerancc?
( a ) For individual tolerances equ al 10 1 %
( b ) Fortolerances of the 47-0 resislOrs equal lo 10% and for the 1 80-0 res i s tors equal
I0 5%
7.
A capac i lOr of 0 . 05 µF ± 10% is parallel with a capaciior of O. I µF ± 10%.
(a)
( b)
8.
9.
IO.
What
are the
Whal would
resulting nominal capacilancc and
be
uncertai nly '!
the nominal capacit ance and u ncertai nty if the two clements were
connected in series'!
Two inductances are connected in parallel. Their values are 0.5 mH and 1 .0 mH. Each car·
ries a tolerance or ±20%. Assu mi ng no mutual induc1ance. what is the no mina l inductance
and uncertainty or the combination'!
Power can be measured as / 2 R or V I . Using the dala or Problem 1 7. wh ic h mc1hod wil l
give the most accurate
measuremcm?
The volume of a cylinder is to be delennined from mea surc mems of t he
diameter and
length. If lhe length and diameter are measured at four different locations by means or
a micrometer with an u ncerta i n ty of 0.5% or reading. detenn ine the uncertainty in the
measurement.
Diameter
Length (in.)
3.9920
3.9892
3.996 1
3.9995
4.4940
4.499 1
4.5 1 10
4.522 1
101
Assessing and Presenting Experimental Data
11.
A tube of circular section has a nomi nal length of 52 cm ± O.S cm, a1A outside diameter of
12.
A canti lever beam of circular section has a len gth of 6
20 cm ± 0.04 c m , and an inside diameter of 1 5 cm ± 0.08 cm. Determine the uncenainty
in ca lcu l ated volume.
ft and a di ameter of 2� i n. A
Ibf is applied at the beam end, perpe ndicu lar to the length o f the
beam. If the uncenainty in the length is ± 1.5 i n ., in the diameter is ±0.08 in., and in the
concen trated load of 350
force is ±S lb f, what is the uncenainty in the ca lcu lated maximum bending stress?
13.
If it is determined that the overall uncen ainty in the maximum bending stress for Prob­
lem 12 may be as great as. but no greater than, 6%, what maximum1 uncenainty may be
tolerated in the diameter measurement if the other uncenainties rem:tin unchanged?
14.
It is desired lo compare the design of two bolts based on their tensile st rength c apabi l itie s.
The fo llow i ng lists the resu l ts of the sample testing.
Is there a
15.
16.
Group
Failure Load
Std. Devi ation
Number of Tests
A
B
30 kN
34 kN
2 kN
6 kN
21
9
di fference between the two samples at
lhe 95% confidence level?
From a sample of 1 50 marbles having mean diameter of 10 mm and a standanl dev iation
of ±3.4 mm. how many marbles would you expect to find in the rang1: from 10 to IS mm?
During laboratory testing of a thin· wall pressure vessel. the cylinder di ameter and thickness
were measured at 10 different locations; the resulting data were as follows:
75 = 1 0. 25 in.
'i = 0.25 in.
So
S1
=
=
0 . 25 in.
O.OS in.
If the pressure inside the vessel is measured to be 100 psi with an estimated uncenai nty of
± 10 psi, determine the best estimate of the tangential or hoop stre ss in the vessel (Note:
U9 =
17.
PD/21.)
In ordeno determine the power dis.•ipated across a resistor, the c u rrent How and resistance
values arc measured separate ly. If I = 3.2 A and R = IOOO n are measured values,
determine the uncenainty if the fol lowi ng instru me n ts are u sed :
I n st ru me nt
Vo ltmeter
Re sol u tion
Ohmmeter
l .O mV
1 .o n
Ammeter
0. 1 A
U ncerta inty (% of reading)
0.5%
0. 1 %
0.5%
18.
A total of 1 20 hardness measurements are performed on a large slab of steel. If, us i ng the
19.
In order lo determine whether the use of a rubber backing material li>etween a concrete
co mpre ssion sample and the platen of a testi ng mac hi ne affects the co mpressi ve strength.
six samples with pac k i ng and six wi1hou1 were tested. The strengths are listed in the
fo l lowing table. Determine if the packing material has any effect. US<: a
con dence
Rockwell C scale, the mea n of the measurements is 39 and the standard devi ation is 4.0,
how many of the measurements un be expected to fall between the hardness readings of
35 and 45'!
99% fi
level
1 02
Assessi ng and Presenting Experimental Data
Tensile S1renglh MN/m2
Sample
With Packing
No.
I
2
3
s
4
6
20.
Without Packing
2.48
2.76
2.96
2.72
2.62
2.65
2. 1 8
2.48
2.38
2.00
2. 1 0
2.28
Resuhs from a chemical analysis for 1he carbon conlcnl of two materials are as follows:
Carbon Content, %
Material A
Material B
93.52
92.38
92.8 1
93.2 1
94.32
92.55
93.77
92.05
93.57
92.54
93. 1 2
Determine if there i s a si gn i fi cant difference in carbon content at lhe 99% confidence level.
21.
Figure 9 shows a histogram based on lhe values listed in Table 3. As suggested in Sec­
lion 6. 1 , prepare his1ograms represeming the data, based on (a) seven bins, (b) eight bins,
and (c) ten bins.
22.
The manufacturer of inexpensive outdoor 1hermometers checks a sample of len against a
68°F standard. The fol lowing results were obtained:
68.5
67.5
67
69
68
67
67.5
69
69.
Using S1uden1's t-tcsl, calculate lhe range withi� which the population mean may be
expected to lie with a conlidence level of 95%.
23.
Spacer blocks are manufactured in quantity lo a nominal dimension of 1 25 mm. A sample
of 12 blocks was selec1ed and lhc following measurements were made.
1 .28
1 . 26
1 .24
24.
1 .32
1 .26
1 .23
1 .29
1 .20
1 .26
1 .23 .
1 .29
1 .22
Using Studcnl's /-lest. determine the upper and lower tolerance values within which 1he
popula1ion mean may be expec1ed to fall wi1h a signilicance level of 10%.
In a laborJlory it is suspec1ed 1hat 1he resulls from 1wo different viscometers do not agree.
Ten Ouid samples were tested using apparalus A and corresponding samples were tested
using apparatus B. The resul1s are as follows:
1 03
Assessing and Presenting Experimental Data
Viscosity (Dimensionless)
I
26 .
B
73
45
56
75
53
50
72
54
48
52
72
43
54
75
50
48
73
55
48
50
2
3
4
5
6
7
8
9
10
25.
Using
Apparatus
Using
Apparatus A
Sample No.
Detennine whether there is a significant difference in the two systems al the 99% con fi ­
dence level.
Consider the equation
y
= 1 .0
-
0.2.t +
(0 :::: .r :::: 3)
O.O l.r2
Detennine the maximum uncertainty in y roe :1::2 % uncertainty in the variable .r .
Fo r t he fol lowi ng data determine the equation for y = y (.r) by graphica l analysis.
0.43
1 .54
0
1 .00
y
0.76
3.61
1.21
5.25
2.60
1 0.0
3.5
1 3.50
'1:1.
For the following data, detennine the equation y = y(.r) by gra phi cal ana lys i s .
28.
For the follow ing data. detcnnine the equation y = y(x) by graphi ca l analysis.
29.
"'
I
1.21
1 . 35
1 8.2
0.43
71
2.75
88.0
2.6
26
5. 1
325.0
2.9
19.5
8. 1
800
4.3
1 1 .5
The in fl uence of the size of the test spec i men on the tensile stre ngt h of an epoxy resin
was determi ned by casting seven sam p les of each size and testi ng them accordingly. The
experimental data are as follows:
Spec imen Strengths (kN/m2 )
Sample or Small
Specimens
3475
4326
2262
74 1 5
34 1 8
4404
3788
1 04
Samp le or Large
Specimens
1813
3 1 45
4 1 40
6867
3842
3984
3053
Assessing and Presenting Experimental Data
Determine whether there is a significant difference between the two samples at the
95% confidence level.
30.
31.
In 200 tosses or a coin. 1 1 6 heads and 84 tails were observed. Determine if the coin is
fair using a conlidence level or 95% or a significance level or 5%.
A random number table or 100 d ig its showed the following distribution or the digits
0, I , 2, . . . , 9. Determine if the distribution or the di gi ts differs significantly from the
expec ted distribution at the I % significance level.
Digit
32.
Observed rrequency
0
7
I
12
5
g
6
14
7
12
8
8
9
14
Observed Frequency
8
42
1 07
97
38
8
Are the beari ng diameters normally distributed at the 5% sign i ficance leve l ?
Using the data of Problem 32, construct a normal probability plot. What conclusions can
you determine from this graphical representation regard ing the nonnalcy o r the data?
sample of 100 test specimens of a steel alloy provides the following breaking st rengths.
Detennine whether the data are nonnally d istri buted at the I % sign ifi ca nce level if the
meal'! breaki ng strength is 67.45 ksi and the standard deviation is 2.92 ksi.
A
Breaking
St rength , ksi
59.5-62.5
62.5-65.S
65.5-68.5
68.5-7 1 .5
7 1 .5-74.S
35.
4
6
quality control engineer wants 10 dctennine if the diameters or ball bearings produced
by a machine are nonnally distributed. From a random sample or 300 bearings. he deter·
mines that the sample mean is 1 0.00 mm with a sample standard deviation of ±0. 10 mm.
Moreover, he oblains the following frequency distribution for the diameters.
Under 9.SO
9.80-under 9.90
9.90-under 1 0.00
I 0.00-u nder I 0. 10
1 0. 1 0-under 1 0.20
1 0.20 and over
34.
3
7
A
Diameter. mm
33.
2
12
Observed
Frequency
IS
5
42
27
8
A company subcontracts the mass production of a die c ast i ng or fixed desi gn. Four primary
types of defects have been identified and records have been kept providing a "standard"
against which defect distribution for batches may be judged. For a g i ve n batch of 2243
castings the following data apply. Do t he batch data vary s i gni ficantly from the st andard'!
1 05
Assessing and Presenting Experimental Data
Percen t
Ty pe A
7.2
1 2S
60
7S
4.6
Typc B
1 .9
0.9
Typc C
Typc D
Nonde fective
36.
Results for
Batch 2073
D i stri bution
Defect
Identification
31
8S.4
1 952
1 00.0
2243
A system is calibrated stati ca l l y. The accompanying table lists the results.
Ou1pu1
Output
Input
(Inc reas i ng Input)
(Decreasing Input)
0. 1 2
0. 1 7
0.27
0.32
0.38
0.46
1 .6
2.7
3.7
2.2
2.3
4.2
4.8
S.2
6.S
7.4
4.3
S.6
6.7
7.4
O.S3
0.64
(a ) Plot ou tput versus i npu t.
(b)
3.2
3.9
Ca lcu late the best straigh t- l i ne lit, first for the increasi ng output, then for !he dec reas­
i ng output and finally for the combined data.
( c ) What is the maximum deviation in each case"!
( d ) If it is assumed that 1..ero input should yield zero out pu t, what is the zero offset (b ias)
!hat should be assigned"/
37.
38.
The following data describe the temperature distribution along a length of heated pipe.
Determ ine the be st straight- li ne flt to 1he data.
Temperature, °C
Distance from a Datum, cm
1 00
200
300
400
1 1 .0
19.0
29.0
39.0
so.s
soo
The fon:e-deHection data for a spring are given
in t he fol lowi ng table.
squares flt.
Deflection. in.
0. 10
0.20
0.30
0.40
0.50
0.60
1 06
Force, lbf
9
19
22
40
S2
S9
Determi ne a least
Assessing and Presenting Experimental Data
39.
40.
Solve Problem 38 adding one more data point ---flllmely, zero deRection under zero load .
The data in the accompanying tabu l ation (rrom several sources) provide the resist ivity of
platinum al various temperatures.
( a ) Make a linear plot or the dala points.
( b) De1ermine lhe constants for a linear least squares fit or the entire data set. Plo1 the
fitted line on your graph or lhe data.
( c ) Because 1hc resis1ivity is not a perfoc1ly l inear function or temperalure, a more
accurate fit can be obtained by li m i ting the range of lemperature considered. Obtain
!he cons traints for a li near least fit over the range or O"C to IOOO°C only. Plot the
result on your graph.
Tem perature, •c
Resi stiv ity, n · cm
10.96
10.72
14. 1
1 4. 85
1 7.9
25.4
26.0
40.3
47.0
52.7
58.0
63.0
0
20
1 00
1 00
200
400
400
800
IOOO
1 200
1400
1 600
41.
42.
43.
44.
45.
In constructing a s pring- mass system, a deRcction conslant of 50 lbf/in. is required. Four
springs are available, two having deHection constants o f 25.0 lbf/in. with an uncertainty
(tolerance) of ±2.0 lbr/in. and two having deRection constants of 100 lbf/in. with uncer­
tainties of ±4.0 lbflin. What combinations can be used for a system deRection constant
of 50 lbf/in.'! What w i l l be the uncenainty in each case?
Show th at y = a + bx" will plot as a straight line on linear graph paper when y is ploued
as the ord i nate and x• is plotted as the abscissa. Show that the intercept is equal to a and
the slope is equal to b.
S h ow that i f I / ." versus l /.t is ploued on linear paper, the function x/(ax + b) (which
may al so be written l/y = a + b/x ) will yield a straight line, wit h a as the intercept and
b as the slope.
y = tu:'" wi ll plot as a straight line on linear paper whe n log y is plotted as
the ordinate and x i s plotted as the abscissa and that the inten:ept is equa l to log a and
the slope is equal to b log e. Note that with the slope known, b and c may be found by
Show that
s i mul taneou s solution of the s lope equation and the origi nal equation wriuen for a selected
(x; • Yi ) poi nt .
Se lect a range for .< and make an x versus y plot of y = 1 2x213 on linear graph paper.
Now transform the data to log y and log x and plot on li near paper. The second set of data
should plot as a st r.iight line with a slope of and an intercept of log 1 2.
j
1 07
Assessing and Presenting Experimental Data
46.
From 1960 10 1983, the slandard meter was de fi ned as 1 ,650.763.73 wavelengths of the
light e mi tted during the transition between the 2p10 and the 5ds levels of the kiypton-86
atom. That emission li n.!' is slightly asy mmetric, and its wave leng th has a total uncertainty
of about ±2.4 x 1 0-s A (68%).
By the early 1 970s , laser technology permitted highly-precise determination of
the speed of light, c, by using the relation c = >.f and measured values oflaser wave length,
>., and frequency, /. Laser frequency measurements had at that time reached relative
uncertainties of u t f/ = 6 x 10- 10 (68%).
( a ) Whal was the uncertai nty (95%) in 1he measured speed of light al that time? What
factor limited the accuracy of this measurcmem?
( b) In 1 98 3 , the meter was redefined as "the distance t rave led by light in vacuum during
a lime interval of 11299792458 of a second." How did th i s affect the u ncertainty in
the speed of light (in meters per second)?
REFERENCES
(l) ANSI/ASME 1 9. 1 - 1 985. ASME Performance Test Codes. S u pplement on Instruments
and Apparatus,
(2)
Part l,
Mea.suremenl Uncertainty. New York, 1985.
Froome, K. D., and L. Essen. The Velocity of lighl and Radio Wa�es. New York:
Academic Press , 1 969.
(3) M i l ler, I. R., J. E. Freund, and R. Johnson. Probability a11d Statistics for E11gineers.
4th ed . Englewood Cliffs, NJ.: Prentice Hall. 1 990.
(4) Weiss, N. A., and M. J. Hassett. lntrod11ctory Statisti<"s. 5 t h ed. Reading, Mass.:
Addison-Wesley, 1 999 .
[SJ Sc henk , J., Jr. Theories of Engineering Experime111atio11. 2nd ed. New York: McGraw­
Hill, 1968.
(6) Kline, S. J., and F. A. McClintock. Describing uncertainties in single-sample experi·
ments. Mech. E11gr. 75: 3-8, January 1 953.
(7) Engineering and Scientific Graphs for 1'11blicatio11s, American Standards Association.
New York: American S ociety of Mechanical E ng i nee rs. July 1 947.
(8) Natre lla,
G. Experimental S tat istic s . National 811rea11 of Standards Handbook 9 1 .
Washington, D.C.: U.S. Government Pri n ti ng Office, 1 963.
M.
[9) M<..-C lintock,
F. A. Statistical Estimatio11: linear Regressio11 a11d tire Single Variable,
Research Memo 274, Fatigue and P l ast i city Laboratory. Cambridge: M assac husetts
In$tilUle of Tethnology, February 14, 1 987 .
[ 10) Rabinowicz,
E. Introduction to Experime11tatio11.
Reading. M ass.: Addison-Wesley,
1 970.
[ I I ) Hornbeck, R. W. Numerical Methods. New York: Quantum Publishers. 1 975.
1 08
Assessing and Presenting Experimental Data
ANSWERS TO SELECTED PROBLEMS
1 83.2 to 2 1 6.8
3
(a) 10.2
4
2.24%
7
7.45%
kQ ± 0.0863 kQ
10
1 .256%
12
""' ± 1 0%
IS
">:65 marbles
18
,,.,93
19
Packing is significant
20
Significant difference at 99% confidence level
23
1 .239 < µ. < 1 .279
24
No significant difference
29
No difference
30
Coin not fair
32
Normally distributed at 95% confidi:nce level
35
Significant variation from siandard
38
Force = 28.8
+ 30.7
x Deflection
The Ana log M easu ra n d :
Ti me-Dependent Cha racteristics
I NTRODUCTION
2
SIMPLE HARMONIC RELATIONS
3
CIRCULAR AND CYCLIC FREQUENCY
4
5
AMPLITUDES OF WAVEFORMS
6
7
8
COMPLEX RELATIONS
FREQUENCY SPECTRUM
HARMONIC, OR FOURIER, ANALYSIS
SUMMARY
INTRODUCTION
A parameter common 10 all of measurement is lime: A l l measurands have 1irne-rela1ed
charac1eris1ics. As lime progresses. lhe magnilude of the mcasurand ei lher c han ges or does
nol change. The lime variation of any change is oflen fu l l y as i m p orta nt as is any partic u l ar
amplilude.
In this sec lion we w i l l discuss 1hose q u antiti es necessary to define and describe various
l i me- rel ated charac1eris1ics of m eas u ra nds. We cl assi fy time-relaled mcasurands as either
l.
2.
Slalic-<:onstanl i n time
Dynamic-varying in lime
(a) Steady-slate periodic
(b)
Nonrepelitivc or transient
i. Si ngle-pulse or aperiod ic
ii. Cont i n u i ng or random
2
SIMPLE HARMONIC RE LATIONS
A
functi on is said lo
be a simple lwrmo11ic
function of a
vari able w hen
its second derivat i ve
is proportional lo 1he funct ion but of opposite sign. More o fte n 1han not. the i nde pe nde n t
variable is lime
1.
al t hou g h any 1wo variabl es may be rel ated harmonically.
Some of the mosl common harmonic funclions i n m ech an ic al e n g i nee ring rela1e dis­
placemen t and .lime. I n cleclrical engineeri ng, many of lhe variable quan t i t i es in allernatin g­
current
(ac)
ci rcui l ry are harmon ic functions of time. The harmonic relation is quite basic
From Med1a11ical Mrasun!mems. Sixth Edition. Thomas G. 13eckwith. Koy D. Mamngoni, John H. Lienhard V.
Copyright © 2007 t>y
Pea rson Education. Inc.
Publish•d by
111
Prentice Hall.
A l l rights rcs.rved.
The Analog Measurand: Time-Dependent Characteristics
to dynamic functions, and most quantities that
monically.
In its most elementary
are
time functions may be ex pressed har­
fonn, simple harmonic motion is defined by the relation
(I)
s = so sin cut
where
s = i nstantaneous d isplacement from equilibrium position,
so = amplitude, or max i mum displacement from equilibrium posi t ion .
cu = circular frequency (rad/s),
I = any time interval measured from the instant whe n I =
0s
A small-amplitude pendulum, a mass o n a beam, a weight suspended b y a rubber band-al l
vibrate with simple harmonic motion, or very nearly so.
By differentiation, the following relations may be derived from Eq. ( I ):
11
".' di = sow cos wt
(2)
= sow
(2a)
ds
and
llU
Also,
dv
a = di
= -sow
=
-S(J)2
2.
SID WI
( 3)
(3a)
In addition,
ao =
-sow2
(3 b)
In the preced ing eq uat io ns ,
v = ve loc it y,
11Q
= maximum velocity or velocity am p litu de ,
a = acceleration,
ao = maximum accelerat ion or acceleration ampl it ude
Equation (3a) satisfies the descri pti on of simple harmonic motion given in the first paragraph
of this section: The acceleration a is proportiona l 10 the d i sp laceme nt s. but is of opposite
sign. The proport ion a li ty factor is w2 .
,
112
The Analog Measurand: Time-Oependent Characteristics
3
CIRCULAR AND CYCLIC FREQUENCY
The frequency with which a process repeals ilself is called cyclicfrequency,/, and is lypically
measured in cycles per second, or hem: I Hz I cyclels. However, lhe idea of circular
.frequency, w, is also useful in sludying cyclic relalions. Circular frequency has unils of
radians per second (rad/s). The connec1ion between 1he lwo frequencies is conveniently
illustraled by 1he well-known Scotch-yoke mechanism.
Figure I (a) shows lhe elcmenls of lhe Scolch yoke. consisting of a crank, OA, wilh
a slider block driving lhe yoke-piston combination. IC we mea�ure lhe pislon displacement
from its midstroke posilion, lhe displacement ampliludc will be ± OA . If the crank turns
at w radians per second, lhen the crank angle () may be written as wt . This, of course, is
convenient because it in1roduces lime t inlo lhe relationshi p. which is nol directly apparenl
in 1he term (). Pislon displacemenl may now be written as
=
s
= so sin
wt
which is the same as Eq. ( I ). One cycle lakes place when the crank turns through 21r rad,
and, if f is the frequency in hertz, then
Thus the displacement may
w = 2irf
s
(a)
(4)
i ns lead be expressed in terms of cyclic frequency:
(5 )
= so sin 2irft
(b)
FIGURE I : (a) The Scotch-yoke mechanism provides a simple hannonic molion
pislon; (b) a spring-mass system thal moves wilh s i m p le harmonic molion.
to
the
The Analog Measurand: Time-Dependent Characteristics
FIGURE 2: Motions that are out or phase. The dashed curve lags the s;olid curve by a phase
angle iP.
Either displacement equation showslhat lhe yoke-piston combi nation moves in simple
harmonic motion. Many other mechanical and electrical systems disp1lay simple harmonic
relationships. The spring-mass sysmn shown in Fig. I (b) is such an example. If its
amplitude and natural frequency just happened to match lhe ampli tude and frequency of
the Scotch-yoke mechanism. then the mass and the piston could be made 10 move up and
down in perfect synchronization.
To pul it anOlher way,forel'tly simple l1armo11ic relationship, an analogous Scolch­
yoke mechanism may be devised or ;..,gi11ed. The crank length QA will represent the
vector amplitude. and the angular 1-elocity w of the crank, in radians per second, will
correspond to the circular frequency or the harmonic relation. If the nnass and piston have
the same frequencies and simultaneously reach corresponding extremes of displacement,
their motions are said 10 be in plrase. When they bOlh have the same frequency but do nol
oscillate together, the time diffen:nce (lag or lead) between their motions may be expressed
by an angle referred to as the phase angle, efl (Fig. 2).
4
COMPLEX RELATIONS
Most complex dynamic mechanical signals, steady stale or transien.t, whether they are
functions such as pressure, displacement. strain, or something else, may be expressed as a
combination of simple harmonirn1mpooents. Each component will ha••e ils own amplitude
and frequency and wi II be combined in various phase relations with th•! other components.
A general mathematical statement of this may be written as follows:
)'(I) =
where
Ao. A 0 • and 80
n
=
.�o
T
+
"
.
�)A. cos11wt ± 80 sm 11wt)
11= 1
amplitude-delcnnining constants called lrarmoiiic coefficie111s,
= integers from I to oc, called harmo11ic orders
114
(6)
The Analog Measurand: Time-Depe ndent Cha racteristics
When n is unity, the corresponding sine and cosine terms are said to befundamentals.
2, 3, 4, and so on, the corresponding terms are referred to as second , th ird, fourth
harmonics, and so on. Equation (6) is sometimes called a Fourier series for y(t) .
The variable part or Eq. (6) may be written in tenns of e i ther t h e sine o r the cosine
alone, by introducing a phase angle. Con version is made according to the following rules:
For n
=
Cuse /: For y
= A cos x + B sin x ,
y
=
C cos( -x + tf>i)
(6a)
or
Case II: For )' = A cos x -
C sin(x
+
<t> 1 )
(6b)
C cos(x + tf>i)
(6c)
y = C sin(-x + <t>1 )
(6d)
y=
B sin x,
y=
or
In both cases , C
=
JAi + Bl; and <f>t and tf>2 are positive acute angles, such that
and
and <f>l, A and B are taken as absolu te values.
Although Eq. (6) indicates that all harmonics may be present in defining the signal­
timc relation. such relations usually include only a l imited number or harmonics. In ract,
all measuring systems have some upper and some lower frequency limits beyond which
further harmonics will be anenuatcd. In other words, no measuring system can respond to
an infinite frequency range.
Although it would be utterly impossible to catalog all the many possi ble harmonic
combinations. it is neverthe less useful to consider the effects of a rew variables such as
relative amplitudes, harmonic orders 11, and ph ase relations If>. Therefore, Figs . 3 through
7 are presented for two-component relations, in each case showing the effect of only one
variable on the overall waveform. Figure 3 shows the effect of relative amplitudes; Fig. 4
shows the effect of.relative frequencies; Fig. 5 shows the effect of various phase relations;
Fig. 6 shows the appearance or the waveform for two components having considerably
different frequencies; and Fig. 7 shows the effect of two frequencies that are very nearly
the same.
Note that in calculating <1> 1
The Analog Measurand: Time-Dependent Characteristics
(a)
(b)
�V
'V
\J
V
y1 = 1 0 sin wl + 2 sln 2wl
\J
(\w \)(\ \)
P"�---- \j
p- .� (\ (\
-
y, = 10 sin wt + 4 sin 2wl
�
_
I
y3 = 10 sin wl + 6 sin 2wl
�
�
I
FIGURE 3: Examples of two-component wavefonns with second-harmonic component of
various relative amplitudes.
116
The Ana log Measurand: Time-Dependent Characteristics
y6
�v
=
10 sin wt + 5 sin 2wl
vvv�
Jig = 10 sin wl + S sin Swl
y10 = 10 sin wl + S sin 6wt
FIGURE 4: E11amplc:; of two-component wu"l:forms with second tenn of various relative
frequencies.
The Analog Measurand: Time-Dependent Characteristics
FIGURE 5: Examples of two-component waveforms with second harmonic having various
degrees of phase shift
The Analog Measuran d: Time-Dependent Characteristics
y,6
(b)
=
10 sin wt + 5 sin 1 0wt
Y1 1 = 5 sin wt + 10 sin 1 0wt
FIG URE 6: faamples of wavefonns with lhe lwo components having considerably different
amplitudes.
[V\fr'" "f\f' v0V1Mnv v
VlJIH/
on
IFV
(b)
0 A 8 no • • • n 6 6 0
nrv v v v v v v l[vvlfv v v v
n ••• n n A
vv
(c)
Y20 = 2.S sin 1 0wt + S sin 1 1 wt
7: Examples of wa veforms with lhe two components having frequencies 1ha1 are
nearly the same.
FIGUR E
The Analog Measurand: Time-Dependent Characteristics
FIGURE S:
Pressure-1ime rela1ion: P =
1 00 sin(!101 ) + 50 cos( l 601 - 7r/4).
EXAMPLE 1
As an example of a relation made up of harmonics, lei
pressure-lime funclion con sisting of 1wo harmonic terms:
P = JOO sin 801 + 50 cos
us anal y ze
a rela1ively s i mple
( 1601 - i)
(7)
Solution lnspeclion of the equation shows 1ha1 1he circular frequency of 1he fundamental
has a value of80 rad/s, or 80/27r = 1 2 . 7 Hz. The period for 1 hc pressure variation is lherefore
I / 1 2.7 = 0.0788 s. The second lerm has a frequency 1wicc that of lhc fundamental, as
indicated by its circular frequency of 1 60 racl/s. It also lags the fun d ame nta l by one-eighth
cycle, or :n: /4 rad. In addition, 1he cqua1ion indicales that the am p l i tude of 1he fundamental,
which is 100, is 1wice that of the second harmonic, which is 50. A plot of the relation is
shown in Fig. 8.
EXAMPLE 2
,
As another examp le lei us analyze an acceleration-time rel at i o n that is expressed by the
equalion
{
a = 3800 sin 24501 + 1 750 cos 73 501
- �) + 800 sin ( 36,7501 )
3
(8)
where
a
= an gu l ar acceleration (rad/s2 )
1 = ti me (s)
Solution The relalion consisls of three harmonic eomponenls hav i ng circular frequencies
i n lhe ratio I to 3 lo 1 5 . Hence lhe co m po ne nt s may be referred to as the fundamental, t he
lhird harmonic, and lhe fiflecnth harmonic. Corresponding frequencies are 390. 1 1 70, and
5850 Hz.
1 20
The Analog Measurand: Time-Dependent Characteristics
4. 1
Beat Frequency and Heterodyning
The situation shown in Fig. 7(a) is the basis of an imponant method of frequency measure­
ment. Here two waves of equal amplitude and nearly equal frequency have bee n added. If
one wave has a cyclic frequency of /o and the second wave has a frequency of J1 = Jo + tJ.J ,
!hen the resultant wave is
y=
A sin(2rr/01) + A si n (211" ( Jo + tJ.f)t I
(
= 2A cos 2rr
tJ.J
T
'
) (
·
.
sm
Jo + J1
2rr ---1
2
high-frequency
slowly beating amplilude
)
(9)
wave
This wave undergoes slow "beats" where lhe amplitude rises and falls. Although the cosine
term in the amplitude has cyclic frequency of tJ.J/2, we see that lhe amplitude itself has
two minima per cycle of the cosine. Thus, beats occur at a frequency of tJ.J.
This kind of wave addition happens when a tuning fork is used to tune a musical
instrument. The tuning fork and lhe musical instrument produce nearly equal tones, and,
when lhe two sou nd waves are heard together, a lower beat frequency is also heard. The
instrument is adjusted until the beat frequency is zero, so !hat the instrument's frequency is
identical 10 that of the tuning fork.
When the difference frequency, tJ.J, is much smaller than Jo, addition of waves
allows us lo measure tJ.J with less uncertainly than if we measured Jo and /1 separately
and subtracted them. This technique for frequency measurement is called heterodyning. II
is very important in radio applications and in laser-doppler velocity measurements.
EXAMPLE 3
I012
Helium-neon laser light has a frequency of 473.8 THz (473.8 x
Hz). A helium-neon
laser beam is reflected from a moving large!. This creates a doppler shift in the beam,
which increases its frequency by 3 MHz (3 x I 06 Hz). 1 The reflected beam is "added 10"
an unshifted beam of equal intensity, by using mirrors to bring the beams together. Whal is
lhe resulting signal?
Solution
From Eq. (9), with
Jo = 473.S THz and Ji = Jo + tJ.J = 473.8 THz + 3 MH1�
The amplitude has a cyclic frequency or 1 .5
x
1 06 Hz = 1 .5 M Hz, causing zeros in
amplitude twice per cycle for a beat frequency of
3-MHz. This would be mani fested
1 A doppler shift is an apparcn1 change in 1he fn..oquency or a lighl or sound wave 1ha1 t.X.'t:urs when the wave
sourc� and receiver arc in motion relative 10 one another. One 1ypical cxomplc is the change in pitch or a passing
lrain's whis1le.
1 21
The Analog Measurand: Time-Dependent Characteristics
as a 3-MHz vari atio n between bright and dark at the point where the laser beams were
added-a vari ation which could be detected by a sufficiently fast photodetector, such as a
photo mul tipl ie r tube. N ote that the beat frequency is more than a 1 ()1[) million times smaller
than the freque ncy of the original l i g h t To fi nd this difference betwetm /o and ft d irectly by
subtraction, we wou ld have needed to measu re eac h frequency to an accuracy of better than
part in 100 million ! As a result, the beats prov ide a much easier way to detennine A/.
.
I
4_2
Special Waveforms
A number o f freque n t ly used spec i al wavefonns may be wri tte n as. i nfi n i te trigonometric
series. Several of these are show n in Fig. 9. Table I lists t he co rrespondi ng eq uations
.
Both the square wave and the saw tooth wave are useful in cht:eking the response of
d yn amic measuring systems. In addition, the skewed sawtooth form, Fig . 9(c), gives the
voltage-li me relation necessary for driv i ng the horizontal sweep of a cathode ray oscillo­
scope. A l l these forms may be obtained as voltage-t ime relations from electronic signal,
-
or function, generators.
FOr each case show n i n Fig . 9, all the terms in the in fini te seri.es are necessary if the
prec ise waveform ind icated is to be obtai ned Of course, with incre i n g hannonic order,
their effect on the whole sum becomes smaller and sma ller
.
:as
.
As an example, consider the square wave shown in Fig. 9(a). The complete series
includes all t he terms ind ica ted in the re lat ion
4A
y = 11"
(.
.
I
I .
sm wt + - sm 3wt + - sm 5wt + · · ·
5
3
.
)
By plotti ng only th e tirst three terms, which include the fifth harmonic, the waveform
shown in Fig. O(a) is o btained Fig ure IO(b) shows the resu l ts of p lotti ng terms through
and including t he ninth harmo nic , and Fig I O(c) shows the form for the terms i nc l udi n g the
fifteenth harmonic. As more and more ter ms are added , the waveform gradually approaches
the square wave, which results from the infi n i te series
I
4-3
.
.
Nonperiodic or Transient Waveforms
In the foregoing exam p le s of spec ial waveforms, various combinati•Jns o f harmonic com­
ponents were used . In eac h case the result was a period ic re lat ion repeating i nde fi n i te ly
in every detai l. Many mecha nica l i n puts arc not repct it ivc-- for exa m pl e consider the
acceleration-time relation resu lti ng from an im pact lest [ Fig I l (a)). Although such a rcla·
lio n is tra ns i e n t it may be th ough t of as one cyc le of a periodic relation in which al l other
cyc les are ficl i tiou s [ Fig. l (b)] . On this basis, nonperiodic fu nctio ns may be analyzed in
exactly the same ma n n e r as period ic functions. If the nonperiodic waveform is sampled for
a time period T. the n the fundamental freq ue ncy of t he fi c t it ious period ic wave is f = I I T
(cyclic) or w = 2rr / T ( c irc u lar)
,
.
l
.
1 22
,
The Analog Measurand: Time-Dependent Characteristics
FIGURE 9: Various special waveforms of harmonic nature.
and the abscissa is wt .
1 23
In each case, the ordinate
is )'
The Analog Measurand: Time-Dependent Characteristics
(b)
(c)
FIGURE I 0: Plot of square-wave function: (a) plot of first three terms
only (includes the
fifth harmonic); (b) plot of the first five terms (includes the ninth harmonic); (c) plot of the
first eight terms (includes the fifteenth harmonic).
I�
0
Time
(a )
"'
I \
I
'
I
-,_
I
'I
-T
I \
'
I
-,
I
I
.... .... _
T
0
2T
(b)
FIGURE
11:
(a) Acceleration-time relationship resulting from shock test, (b) co nsideri ng
the nonrepeating function as one real cycle of a peri odic relationship.
1 24
TABLE 1: Equations for Special Periodic Wavefonns Shown in Fig. 9
Figu re
9(a)
4A
9(c)
9(d)
9(e)
I.
I.
I
I
)
)
I.
I
y = - SIR (J)l + - s1R 3wt + - s1R 5wt + · · ·
3
5
,,
9(b)
(.
(
(.
y
y
=
2A
-
"
2A
= -
7r
y =
A
2
-
sin "'' +
SIR WI
4A
- -
8A
(7r )2
(
(
-
(
I
-
2
I
SIR 2wr +
.
'· ·
.
-
3
sin 3wr +
.
···
•· ·
00 [
Equation•
4A
= - 1:
= -
"
9( 1)
y
9(g)
2A
y = ---
=
SA
(7r)
2
2
I
3
4
coswt + cos 3wt + cos 5wt
( 3) 2
(5 ) 2
I
coswt
a (7r - a )
(
oo
L
rr= I
- slR ...,/ + - sm ,JW/ - - s1R 4wt + · · ·
I
v = - sin wi - - sin 3Wl + sin 5wt - · · ·
(5) 2
(1')2
( 3 )2
·
[I
)
7r n=I
2A
+
I
I
2 cos5w1
2 cos 3wr + ' (3)
(5)
.
I
n
=
)
2A
=
.
]
�
- L...
" n=I
00 [
(7r)2 •• I
A
= -
2
.
I
.
]
[(-l)n+ I
]
[ I
n= I
.
--- sm nwt
n
,,
(2n
[
-
1 )2
--- cos(2n -
-
(- 1 )•+ 1
--- sin(2n
(2n - 1 ) 2
- l )wt
I
- L... --- cos n wt
(>r) 2
(211
I )2
•= I
�
SA
.
sm 2a sm 2wt + - sm 3a sm 3wr +
sm a sm wt + (3) 2
(2) 2
.
- l)wr
4A 00
- 2 L
8A
- 1:
=
+--·
,
- SIR nwt
+···
)
)
I
.
-- sm(2n
2n - I
• n as used in lhese equations does not necessarily represent the harmonic order.
· · ·
)
)
]
L
2A
°"
= --a(7r - a ) •=I
[
l )wt
]
.
I .
2 sm na sm nwi
11
]
The Ana l o g Measurand: Time-Dependent Characteristic:>
5
AMPLITUDES OF WAVEFORMS
The magni t ude o f a waveform can be described in several ways . The s i mp lest wave fonn is
a sine or cosine wave:
V (t)
=
Va sin 2n"ft
waveform is v•. The peak-to-peak amplitude i s 2 V0 • On the other
hand, we may want a t i me- average value of this wave. If we sim ply average it over one
period, however, we obtain an uninformative result:
The ampliwde of this
laT
-V = -I
T
o
Va
V0 sin 2Hft dt = - - (cos 27f - cos O) '= 0
2x
area beneath. one period of a sine wave is zero. Thus i t is more' useful to work with
a root-mean-square (rms) value:
The net
Vnns =
I
T
Lo
T
V 2 (t) dt =
.
I {T
. 2 27f/t dt
V02 sm
T lo
=·.·
=
a
..fi.
V
( 10)
For more complex wave form s . the freq ue ncy spectrum (Section 6) provides a c omplete
description of the amplitude of each i ndi v id ual frequency component in the s ig n al . However,
the spectrum can be c u mbersome to use. and a single time-average value is often more
convenient to work with. For this reason, the rms amplitude is ge nerally applied to complex
waveforms as well.
6
FREQUENCY SPECTRUM
Figures 3 through 7 arc plot ted us ing time as the independent variablle. This is the most
common and familiar form. The waveform is displayed as it would appear on t he face of an
ordi nary oscilloscope or on the paper of a strip-chart recorder. A second type of plot is the
frequency spectmm. in which frequency is the i ndependent variable and the ampli tude of
each freque ncy component is displayed as the ordinate. For example, the freq uency spectra
for the plots of Figs. 3(a) and (d) are s how n in Fig. 1 2. Spectra corresponding to F i gs. 4(a)
and (c) are shown in Fig. 1 3 , and the frequ ency spectrum for the square wave is s h ow n in
Fig. 14. Figures 12 th ro ugh 14 use circular freq ue ncy w; cyclic frequ ency f is ofte n used
instead.
The frequency s pec tru m is useful because it allows u s to ide n ti fy at a glance the
freq ue nci es present in a signal. For example, if the waveform resul ts from a vibration
test of a struct ure , we could use the freq uency spectrum to identify th•e structure's natural
freq ue n ci es .
The application of frequency spectrum plots has increased grea.tly since the devel­
opment of the speclrum analyzer and fast Fourier transform. The spectrum an a l yzer is
an electronic device that displays the freq uency spectrum on a cathode - ray screen. The
fast Fourier transform is a co mp u ter algorithm that calculates the freqUt� ncy spectrum from
computer-acquired data (see Secti on 7. I ).
1 26
The Analog Measurand: Time-Dependent Characteristics
AGURE 1 2: (a) Frequency speclrum corresponding to Fig. 3(a), (b} liequency spectrum
correspond ing to Fig. 3(d).
ti
(a)
(b)
��-'-�������
���....�
..
"'
2...
Frequency
Frequency
FIGURE 1 3: (a) Frequency speclrum corresponding 10 Fig . 4(a), (b) frequency spectrum
corresponding 10 Fig. 4(c).
The Analog Measurand: Time-Dependent Characteristics
.,
FIGURE
7
14:
5.,
1 7w
1 3w
9"'
21w
Frequency
Frequency spectrum for lhe square wave shown i n Fig. 9(a).
HARMONIC, OR FOURIER, ANALYSIS
In the preceding sections, we saw how known combinalions of waves could be summed to
produce more complex waveforms. Jn an experiment, the task is reversed: We measure the
complex waveform and seek 10 determine which frequencies are present in it! The process
of determining lhe frequency spectrum of a known wavefonn is called hannonic analysis,
or Fourier analysis. 2
Fourier analysis is a brnnch of classical malhemalics on which entire textbooks have
bee n wriuen. More detailed discussions are available in the Suggested Readings for this
chapter. The key to hannonic analysis is that the hannonic coefficicnls in Eq . (6) are
integrals of the waveform y(I)
An =
B. =
"'- 1 21</"1
- 12n'/w
11:
0
11:
0
W
y(t) cos(w111) d/
II = 0. 1 , 2, . . . .
( 1 1)
y(r) sin(wnt) dt
II = 1 , 2, 3 , . . . .
( 1 2)
.
These relations are reciprocal to Eq. (6). When the harmonic coefficients arc already
known, Eq. (6) can be summed 10 obtain y(t). Conversely, when y(I) is known (as from an
experiment), the integrals can be evaluated to determine lhe harmonic coellicicnts.
Eltperimentally, the waveform is usually measured only for a finite time period T . It
turns out lo be more convenient to write lhe integrals in terms of 1his time period, rather
lhan the fundamental circular frequency, w. Since w = 211: / T. 1hc integrals are just
An = � for y(l)cos(2; n1)
� for ( ; nc)
Bn =
y(t) sin
2
21be term SfMCtrol analysis is also used.
1 28
di
11
di
11
=- 0. 1 . 2 . . . . ,
( 1 3)
= I . 2, 3, . . .
( 1 4)
The Analog Measurand: Time-Dependent Characteristics
Discrete sample
y( t)
Lit
2.1t
· · · · · · · · . . · · ·
..
· · · r .1t
· · · ·
....
· · · · · . . · · ·
NLlt • T
FIGURE I S : Discrete sampling or a continuous analog signal. The value or the signal is
recorded at intervals A I apart for a period T.
Practical harmonic analysis usually falls into o ne o r the following four categories:
y(t) is known mathematical function. In this case, the integrals ( 1 3)
and ( 14) can be evaluated analytically.
1. The waveform
2. The waveform y(t) is an analog signal from a transducer. In this case, the wavefonn
may be processed with an electronic spectrum analyzer to obtain the signal's spectrum.
3. Alternatively, the analog waveform may be recorded by a digital computer. The
computer will store y(I ) only at a series or discrete points in time. Integrals ( 1 3) and
( 14) are replaced by sums and evaluated, as discussed in Section 7. 1 .
4. The waveform i s known graphically, for instance, from a strip-chart recorder or the
screen or an oscilloscope . In this case. y(t) may be read from the graph at a discrete
series of points. and the integrals may again be evaluated as sums. 3
7.1
The Discrete Fourier Transform
The case when y(I) is known only at discrete points in time is very important in practice
because or the wide use or computers and microprocessors for recording signals. Normally,
a computer will read and store signal i nput at time intervals of Al (Fig. I S). The computer
records a total of N points over the time period T = N A t .4 Therefore, in the computer's
memory, the analog signal y(t ) has been reduced to a series of points measured at times
1 = Al, 2 A I , . . . • N A1, spccifically. y ( 6 1 ) . y(261 ) , . . . , y(NAI). We can write this scries
more compactly as y(I, ) by selling 1, = r lit for r = I , 2 . . . . • N .
To perform a Fourier analysis o r a discrete time signal like this, the integrals i n
Eqs. ( 1 3) and ( 1 4) must be replaced b y approximate numerical integration i n the fo rm or
summations. Li kew i se. the continuous time t is replaced by the discrete time t, = r 6t,
poi n t al /
3 An example of this approoch
4 Assume thal lhe
=
is given in 1 1 J. Sci:lion 7.
0 is not rel.'Ordt.-d and Ihat N
1 29
is even.
N
N2l!.-1 �
The Ana log Measurand: Time-Dependent Characteristics
anJ the period T is replaced by
An =
=
(�nrl!.
Nl!.1 1)
l!.t. Making these substitutions in Eq. ( 1 3), we get
L,,
y(t, ) cos
ra l
N
L Y(r l!.l) cos
N r=I
2
l!.I
(2irrn)
N
TI1us, the harmonic coefficients of a discretely sampled wavefonn are
NL
N L=
2
An = -
8,,
N
Y(r l!.t) cos
r= I
=
2
-
N
r
.
Y(r l!. t ) sm
I
N
(2irrn)
n = 0, I, . . , °"2�
-(--n ) n = 1,2, . . . , 2. - I
N
.
2:ir r
N
N
{15)
( 1 6)
for N an even number. The corresponding expression for the discrete wavefonn y(t,) is
y(t, ) =
Ao N/l-I
An cos
+
T
?; [
(2irrn)
N
+ 8., SIR
. (2:irrn)]
N
+
AN 2
-;
/- cos(irr)
( 17)
&juations ( 1 5) and ( 1 6) are called the discrete Fourier transform (DFI} of y(t,) [2]. Equa­
tion ( 17) is called the discrete Fourier series.
In practice, the discrete sample is taken by an a1ialog-10-digital cm1verter connected
to a computer or a microprocessor-driven electronic spectrum analyzer. The computer or
microproces.�or evaluates the sums, Eqs. ( IS) and ( 1 6), often by using the fast Fourier
transfonn algorithm. S The result, which approximates the spectrum of the original analog
signal, is then displayed.
Like the ordinary Fourier series [Eq. (6)], the discrete Fourier se ries expresses y(I)
as a sum of frequency components. In particular,
N
2:irrn
-- =
2ir ( -N t.1 )
n
r
l!.t
= 21r (11 l!.f)t,
for a fundamental frequency (in hertz) of
( 1 8)
;The fast four;., transform. or FFT. algorithm is special factorization or these sums
he
The
that applies
when N is
FFT
a power of 2 (N = 2"', for m an in1eg.er).
number of calcula1ions nonnally required 10 evaluate these sums
log2 N. Thus. the
is proponional to N 2 : w n the FFT algorithm is used. the number is proponional to
N
requires fewer calculations when N is large.
1 30
The Analog Measurand: Time-Oependent Characteristics
and harmonic orders n =
y(t )
=
A°
+
2
I , . . . , N /2. In other words,
N/2- 1
L
[ A,, cos (2ir n !:.ft ) + 8,, sin (2irn t:.ft))
·�·
( 1 9)
Note that the OFT yields only harmonic components up to n
= N /2, whereas the ordinary
Fourier series [Eq. (6)) may have an infinite number of frequency components. This very
important fact is a consequence of the discrete sampling process itself.
7:1.
Frequencies in Discretely Sampled Signals: Aliasing and Frequency Resolution
When an analog waveform is recorded by discrete sampling, some care is needed to ensure
that the wavefonn is accurately recorded. The two sampling parameters that we can control
are rhe sample rate, Is = I/ t:.t, which is the frequency with which samples are recorded, and
the number of points recorded, N. l)'pically, the software controlling the data-acquisition
computer will request values of /, and N as input.
Figure 16 shows two eitamples of sampling a particular wavefonn. In Fig. 16 (a), the
sample rate is low (t:.t is large), and as a result the high frequencies of the original wavefonn
are not well resolved by the discrete samples: The signal seen in the discrete sample (the
dashed curve) does not show the sharp peaks of the original waveform. The total time period
of sampling is also fairly short (N is small), and thus the low frequencies of the signal are
missed as well: Ir isn 'I clear how often rhe signal repeals itself. In Fig. I 6(b), the sample
rate and rhe number of points are each i ncreased, improving the resolurion of both high and
low frequencies. The sample rate and total sampling rime period clearly determine how
well a discrere sample represents the original waveform.
What is the minimum sample rate needed to resolve a particular frequency? Consider
the cases shown in Fig. 1 7 , where a signal of frequency f is sampled at increasing rares. In
(a), when rhe waveform is sampled al a frequency of /,
= f, rhe discrerely sampled signal
appears to be constant! No frequency is seen. In (b). the waveform is sampled at a higher
rate, between f and 2 /; the discrete signal now appears to be a wave, bur it has a frequency
lower than /. In (c). the waveform is sampled at a rate /, = 2/, and the discrete sample
appears to be a wave of the correct frequency, /. Unfortunately, if the sampling begins a
quarter-cycle later at this same rate, as in (d), then the signal again appears lo be constant.
Only when the sample rate is increased above 2/, as in (e), do we always obtain the correct
signal frequency with the discrete sample.
The highest frequency resolved al a given sampling frequency is determined by the
Nyquist freque11cy,
/Nyq
f,
=2=
I
2 t:.t
(20)
Signals with frequencies lower than /Nyq = f,/2 are accurately
frequencies greater than or equal to /Nyq are no/ accurately sampled and
in the discrete
frequencies
sampled. Signals with
appear as lower
sample.
1 31
The Analog Measurand: Time-Oependent Characteristics
(a)
FIGURE 1 6: Effect of sa mple rate and number of sam p les taken: (a) undersamp led , both
100 low; (b) resolution of waveform is improved by
raising the sample rate and number of po in ts .
sample rate and number of points are
The p he nomenon of a d isc retel y samp led signal taking on a different freq uency, as
aliasing. 6 Al iasing occurs whenever the Nyquist frequency falls
below the signal freq ue ncy. Furthermore, the phase ambiguity show n by Figs. l 7(c) and (d)
prohibits sampling al the Nyquist freq ue nc y itself. To pre ve n t these problems, the sampling
frequency should always be chosen lo
more tlran twice a signal's h ighe st frequency, as
i n Fig. 1 7(e).
in Fi g. 1 7(b), is called
be
us h ow 10 correctly resolve the highest freq uenc ies of a
OFT con tains only a finite number of freq ue ncy com ponen ts :
Frequencies higher than the Nyquist are too fast 10 be resolved at the g i ve n sample rate.
In a s i m i l ar fashion, we can de te rm i ne the lowest nonzero frequency found by the
DFr and ex plain the discrete spacing of frequencies. Ir a waveform is to be resolved by
discrete sampling, one or morefull periods of that waveform must be present in the discrete
sampling period, as shown for one wave in Fig. I 8(a). S ince the period of s am p l i ng is
The Nyquist frequency tel ls
signal . It also tells us why the
6
An inco�ctly sampled signal
lakes on a new
identily, or alias.
1 32
The Analog Measurand: Time-Dependent Characteristics
(a) 15 = I
FIGURE 1 7 : EITecl or vary i ng the sample rate, /,. on the apparent signal obtained by
d iscrete sampling. (Con1i1111ed on nexr page)
T = N t.r = NIf, . lhc frequency of th is wave is
fioweSI
I
=T=
I
N 6.1
fs
= N "' !!if
(2 1 )
Thus the fundamental frequency of the DFf, !!if. is also 1ha1 of the lowest-frequency full
wave Iha! fits within the sa m pl i ng period. No lower frequency (other t h an f = 0) is
resolved.
1 33
The Analog Measurand: Time-Dependent Characteristiics
Actual
"'signal
Appa�·h
sJgnaJ
(d) '· • 21
Iv
FIGURE 17: continued
The next-lowest frequency resolved is that for which two full waives lit in the sampling
peri od [Fi g . 18(b)]:
h=r
2
We can
fs
= 2 -;;; = 2 6/
continue adding full waves to show that the only frequencies: resolved by the DFf
are
0, 6/, 2 6/,
· .
. , n 6f,
· .
N
. , z 6f = /Nyq
Note that although the Nyquist frequency itself is present in the DFr, it may not correctly
represent the underlying signal, owing to phase ambiguity.
The frequencies of the DFr are spaced in increments of 6/, and thus 6/ i s sometimes
called thefreq11ency resolution of the DFr. If an analog signal contai1ns a frequency Jo that
lies between two resolved frequencies, say n 6f < Jo < (n + I ) 6/, then this frequency
component will "leak" to the adjacent frequencies of the DFr. The adjacent frequencies
can each show some contribution from Jo. As a result, each frequenc)' component observed
in the DFr has an uncertainty in frequency of approximately ±6//2 (95%) relative to
the frequencies actually present in the original analog signal. We can reduce leakage and
sharpen the peaks in the frequency spectrum by decreasing 6/.
This discussion leads us to the following steps for accurate disi:rete sampling:
1. First, estimate the highest frequency in the signal and choose the Nyquist frequency
to be greater than it. In other words, make the sample rate, f, , gr�aler t han twice the
highest frequency in the signal.
1 34
The Analog Measurand: Time-Dependent Characteristics
.._----- T -------�
(a)
(b)
FIGURE 1 8: Resolving low frequencies: (a) one run wave in the sampling period , (b) two
run waves in the sampling period.
2. If limitations in the sample rate force you
to pick a Nyquist frequency less than the
highest frequency in the signal, then use a low-pass filter to block frequencies greater
than the Nyquist frequency so that those higher frequencies will not be aliased into
your results.
3. After the sample rate is chosen, estimate the lowest frequency in the signal or estimate
the frequency resolution needed to accurately resolve the frequency components in
the signal . Then choose the number of points in the sample, N, to yield the desired
6.f = f, /N at the previously determined value of the sampli ng frequency, /, .
7.3
An Example of Discrete Fourier Analysis
Figure 19 illustrates a simple experiment. Each of three signal generators was set to produce
sine waves and connected to a loudspeaker. Generator A was set to 500 Hz, generator B
to IOOO Hz, and generator C to
Hz. A microphone, which converts sound pressure
to voltage, was used to measure the sound level. The sound level from each speaker was
individually adjusted so that the microphone voltage displayed on the oscilloscope was
50 m V in amplitude. Then all three sources were run simultaneously, so that the three
waveforms were mixed, and the voltage signal produced was recorded by the computer.
1500
The Analog Measurand: Time-Dependent Characteristics
c:J-<Jl))))
L::l-<J l))))
L::l-<J l))))
Signal generators
Oscilloscope
Microphone
1111111
Computer
Loudspeakers
FIGURE 19: Experimental setup in which pure tones from three sound sources are mixed,
detected by a microphone, displayed on an oscilloscope. and recorded by a computer.
not
Note that the signal generators were
in phase. The data acquired are shown in Table 2.
The computer sampled the microphone voltage at a rate of Is = 9000 Hz, corre­
sponding to a Nyquist frequency or
/Nyq
in
=
4500 Hz
The lowest rrequency the signal was known to be 500 Hz, so 18 points were used in the
OFT (N = 18) to yield a frequency
of
resolution
!::./ =
!!_
N
=
500 Hz
Thus, the samples were
at intervals of 0. 1 1 ms (61 = I //,
= 0. 1 1 ms), covering a
·
period of N t:.1 = 2.00 ms .
The test data were analyzed using the computer"s fast Fourier transform program.
The resulting hannonic coefficients are listed in Table 3. Only harmonic components up to
n = N /2 = 9 can be determined with the DFT (or FFT). and the last of these is the Nyquist
frequency component. which suffers from phase ambiguity.
The first, second, and third hannonics were originally set to amplitudes of 50 mV.
Since each wave is phase shifted [recall Eqs. (6a-d)), we consider the amplitude of the sum
of sine and cosine waves at each frequency, C., = ./A� + BJ. This frequency spectrum
.
is shown in Fig. 20. For the first and third harmonic, the calculated amplitudes are within
10% of what we thought they should be. The second harmonic is 40% higher than we
thought. Apart from the lack or precision of the data and any approximation introduced in
the OFT calculation, the test condition itself may have contributed to this discrepancy. The
test was run in a conventional laboratory environment, and sound reHections from the walls
taken
1 36
The Analog Measurand: Time-Dependent Characteristics
TABLE 2: Experimental Data for Microphone Voltage
Time (ms)
Voltage (mV)
0. 1 1
0.22
0.33
0.44
0.56
IO
30
70
65
15
0.67
0.78
0.89
1 .00
1.11
-40
-50
15
100
1 35
90
1 .22
1 .33
1 .44
1 .56
1 .67
- 1 30
- 140
- 1 10
1 .78
1 .89
2.00
-50
- 10
0
-40
TAB LE 3: Calculated Harmonic Coefficients for the Data of Table 2
Harmonic
Order, n
Frequency,
n A/, Hz
0
I
2
3
0
500
4
5
6
7
8
9
IOOO
1 500
2000
2500
3000
3500
4000
4500
1 37
A., mv
B. , mv
c., mv
- 1 .04
-29. 1 1
49.09
-2 1 .96
2.78
1 .97
-0.96
0. 1 5
-0.37
- 1 .08
0.00
46.83
52.70
-47.41
2 .23
-0.47
-0. 5 1
2.93
1 .04
55. 14
72.02
52.25
3.56
2 .03
1 .09
2.93
1.17
1 .08
-I.II
0.00
The Analog Measurand: Time-Dependent Characteristics
80
60
>
.§. 40
o•
20
1 000
0
2000
Frequency, IUSf (Hz)
4000
3000
FIGURE 20: Frequency specll'Um for data of Table 2.
and ceiling may have also contributed to some distortion of the original signals. Ideal l y,
the test should have been run in an anec hoic chamber.
The sum of the measured harmonic components [Eq. ( 1 9)) is plotted together with
the data in Fig. 2 1 . Visually, the computed curve appears to fit the data perfectly. The DFf
calculation reconstructs the original signal to within th e accuracy of the original data.
,"
1 00
>
E
/I \,,I
50
I
4
I
'
; \•
I
II
\I
I
I
t ....
I
4
I
,'"'-
;
I
� \•
I
I
\I
I
I
I
I
•'
' ��.'"+��\ -i._�.,.
� O t-�-i�,...-�,''"+�+\ -+��-,.f'��+;
,
,
,
,
,
,
\ •/ +I p/
1 -50
'
/
\)
.;
./
I
I
I
I
-50
Series
I
I
I
I
I
I
I
I•
,•
p
I
I
I
'
1
•
Data
•,.1
' '
-150 �����....��
..
....��'-...
..
....��·
..
·
-��
�....��
0.00
1 .00
2.00
3.00
4.00
Time, ms
FIGURE 2 1 : Comparison of the calculated Fourier series with the measu red test data.
1 38
The Analog Measurand: Time-Dependent Characteristics
8
SU MMARY
Mechanical and electrical measuring systems often produce time· varying output signals.
We have seen that even fairly complex signals may be broken down and analyzed as a
mixture of hannonic components, each having different frequency and amplitude. Keep in
mind that all dynamic inpulS, those whose magnitudes vary will) time, are in reality only
combinations of simple sinusoidal building blocks.
I. Simple hannonic motion, such as s = so sin cut, is the most basic form of time·
dependent behavior. The frequency of such motion can be described by either the
cyclic frequency, f, or the circular frequency, cu = 27rf. The relationship orthese two
frequencies may be visualized in terms of the Scotch-yoke mechanism (Sections 2, 3).
2. More complex waveforms can be represented by a sum of simple hannonic compo­
nents having different amplitudes and frequencies. Even fairly sharp waveforms, such
as the square wave, can be described by such sums. These sums are called Fourier
series (Sections 4, 4.2).
3. When two waves of nearly equal frequency are added, the resulting waveform under·
goes periodic beats at a frequency of one-half the frequency difference of lhe original
waves (Section 4.1 ).
4. When a nonperiodic waveform is recorded for a time interval T, the recorded por­
tion may be viewed as a periodic waveform of period T and frequency f = I / T
(Section 4.3).
S. The average amplitude of a complex signal is often described using the root-mean­
square, or rms, value (Section 5 ).
6. The frequencies present in a complex waveform may be described using the fre­
quency spectrum, which shows the amplitude of each frequency component present
(Section 6).
7. The process of determining the frequency spectrum of a complex signal is called
hannonic analysis or Fourier analysis. Several methods of hannonic analysis are
available, depending on the nature of the signal being studied (Section 7).
8. When a signal is recorded by a computer, only discrete points are stored. The dis­
crete Fourier transform, or DFf, may be used to find the frequency spectrum of lhe
recorded data (Section 7. I ).
9. Accurate discrete sampling can be ensured by selecting appropriate values of the
sampling frequency, /, , and the number of sample points recorded, N. The correct
values are determined by (a) the Nyquist frequency, /Nyq = /, /2, which must be
greater than lhe highest frequency in the signal; and (b) the frequency resolution,
t..f = Is / N, which is both the lowest frequency seen in the discrete signal and the
spacing of frequencies in lhe signal's DFf (Section 7 .2).
SUGGESTED READINGS
Churchill,
R. V., and J. W. Brown. Fourier Stries and Boundary Value Probltms.
1978.
3rd ed. New York:
McGraw-Hill,
Greenberg, M . D.
Oppenheim, A.
N.J.:
Preniice
Foundations ofApplied Mathematics.
Englewood Cliffs,
V., A. S. Winsky. and S. H. Nawab. Signals
Hall.
1997.
&
Systems.
NJ.: Preniice Hall. 1 978.
2nd ed. Upper Saddle River.
The Analog Measurand: Time-Dependent Characteristics
PROBLEMS
I.
The following expression represenlS thC displace ment of a point as a function of time:
y(t) = 100 + 95 sin 1 5t + 55cos 15t
( a ) What is the fundamental frequency in hertz?
( b) Rewrite lhe equation in terms of cosines only.
2.
Rewrite each of the following expressions in lhe
3.
Construct a frequency
( a ) y = 3. 2 cos (0. 2t - 0.3) + sin(0.21 + 0.4)
( b ) y = 1 2 sin(t - 0.4)
Construct a frequency
spectrum for Fig. 9(c).
S.
Construct a frequency
spectrum for Fig. 9(e).
6.
8.
(6).
spectrum for Fig. 9(a).
4.
7.
fonn of Eq.
Construct a frequency spectrum fo r Fig. 9(0.
Construct a
frequency spectrum for Fig. 9(g).
Figure 22 represenis a trace from
oscilloscope where lhe ordinate is in volts and
the abscissa is in milliseconds. Determine ilS discrete Fourier transform and sketch iis
frequency spectrum. Use a sampling frequency of
Check for periodicity i n
choosing your data window.
an
2000 Hz.
FIGURE 22: Oscilloscope trace for Problem 8.
re1;ord as shown in Fi g. 23. Dctennine its rrequency spectrum.
10.
Consider a pre5$un:-1irne
ll.
Solve Problem 8 using a sampling frequency o f I 000 Hz and compare your results with
those of Problem 8.
9.
If the signal of Problem 9 were 10 be sampled digitally for its discrete Fourier transform.
what sampling frequency would you recommend?
1 40
. The Ana l og Measurand: Time-Dependent Characteristics
Pressure
'®��
1234567
' l(s)
FIGURE 23: Press ure-ti me record for Problem 9.
12.
Using the data files in Table 4, if t is in milliseconds and /(1 ) is in volts, determine lhe
discrete Fourier transfonn for each set of digital data.
(a )
(b)
( c)
( d)
Use /1 (/ ).
Use /2 (t).
Use /3 (1).
Use /4(1).
( e) Use /s (I).
( r) Use /6(1 ).
13.
A 500-Hz sine wave is sampled at a frequency of 4096 Hz. A total of 2048 points are
taken.
( a ) What is the Nyquist frequency?
( b) What is the frequency resolution?
( c ) The student making the measurement suspects that the sampled waveform contains
several harmonics of 500 Hz. Which of these can be accurately measured? What
happens to the others"/
14.
A 150-Hz cosine wave is sampled at a rate of 200 Hz.
( a ) Draw the wave and show the temporal locations al which it is measured.
( b ) What apparent frequency is measured?
( c ) Describe the relation of the measured frequency to aliasing. Give a numerical
justification for your answer.
IS_
( a ) Suppose that a 500-Hz sinusoidal signal is sampled at 750 Hz. Dniw the discrete
time signal found and determine the apparent frequency of the signal.
( b ) If a 200-Hz component were present in the signal of part (a), would it be detected?
Explain.
( c ) If a 375-Hz component were present in the signal of part (a), would i t be detected?
Explain.
141
The Analog Measurand: Time-Dependent CharacteriS'tics
TABLE 4: Data for Problem 1 2
0
l
2
3
4
56
89
11
7
10
12
13
14
65
1918
21
1
1
17
20
22
23
24
25
2627
28
29
30
31
32
33
34
3536
/1 (1)
/2(t )
/3 (1)
4.5.744
0.8
-- ..4
-- 68
- 48
- .4
0.3 08
4.5 74
4.74
0.1 08
- .7
--1..8.68
4
-1.0..428
4.74
. 88
4.19
4.4.6547
4.4.4066
1 0.2
1 0.0
3.01
- 1 .2
2
27
-2.4
1.
1.
1.
2.
2. 7
-2
- 1 .2
-
. 1
.
4
3. 1
- .2
-2.4
2
-2.4
1
-1
- 2.
-2.7
-2
3.0 1
5.4
3 76
3.
3.73
3.6
3.55
89
l.98
3.35
2.
2.36
2.01
2.14
2.24
2 1
2..066
3...556
3. 6
3.88
4.19
4.4.4.456476
4..06
3.3.565
5
2.386
2. 1 6
2 7
13
3
73
3. 7
3 73
3.3
2. 9
9.9.5873
9. 6
8.8.56
8.
7.89
.6
7.. 4
.56
7. 4
8.8.0478
99.8.94847
..6
11.l1.2
.3
I1 I.I.2
10.87
10.0 65
1 .3
2
92
21
7 2
43
73
73
7.
7
. 1
10 3
10
1 0.9
1
11
1 1 .2
11
0.9
10 .
1 .
10.4
0
1 0.2
/4 (1 )
4
2.
0.99
58
4.1.454
10.8.1.0672
fs (t )
/, (/ )
JO.I -9.4
- .528
87. 52 -6.
5.57 2.52
4.1581
3
2.14
0.63 4.17.3
-1.
-3 .4
8.52 -6.
-5 6
2.57
6.66
8.45
8.46 0.2.3646
5.85
-6
3 ·-4.7.6
- 55 -8.3 .5
-1.
0 6 -5.6 ·-4.5
-4.
-1 -2.1 0.06
5 -0.3.1.33555 1.. 83
-5 5.05 11 .
.
-6.7
2
-12-8.5
-2.4 10.6
.5 I0.5 -1
9.34
.3
0
3
-0 .
.9 1
4.
1
1 1 .2
10
3
2 7
.4
.1
2.03
2. 1 1
7. 1 9
7.3
-7.9
-9
-9.9
-JO
JI
0.9
-4
-II
- JO
-9.
4.
-7. J
- .9
3.05
3. 1 1
· - 1 .9
·- 9
·-9.3
·- 9
·-7.9
-3.9
1
- 15
- 1
- 10
49
9. 1 5
I
1
-3.7
6.59
24
0.7
7.94
9.04
9.87
I 0.4
7 1
.29
·- 2 .7
.
-11
·-7
·
2 7
4
10. I
-9 . 7
1
--9.4
The Ana l og Measurand: Time-Dependent Characteristics
16.
An engineer is studying the vibrational spectrum of a large diesel engine. Her modeling
estimates suggest that a strong resonance is likely at 250 Hz, and that weaker frequencies
of up lo
Hz may be exciled also. She has placed an accelerometer on the machine
lo measure the vibration spectrum. She samples the accelerometer output voltage using
her computer's analog-to-digital convener board.
2000
( a ) What is the minimum sample rate she should use?
( b) To reliably test her model of the machine's vibration, she must resolve the peak
resonant frequency 10 ± I Hz. How can she achieve this level of resolution?
17.
18.
A temperature measuring circuit responds fully to frequencies below 8.3 kHz; above
this frequency, the circuit attenuates the signal. This circuit is lo be used 10 measure a
temperature signal with an unknown frequency spectrum. Accuracy of ± I Hz is desired
in the frequency components. If no frequency components above 8.3 kHz are present in
the circuit's output, what sample rate and number of samples should be used to sample
the output?
A cantilever beam of stiffness k supports a large mass m on its free end. The vibrational
frequency of the beam approximately equal lo
! = .:..._
�
21r y ;
In an experiment. this frequency was measured by attaching a strain gage to the beam
to produce a waveform corresponding to the oscillating motion. The waveform was then
discretely sampled using the lab computer, and the frequency of motion was obtained
from an FFT of the waveform.
For one particular case, the mass at the end of the beam was measured to be 60. I 0 g,
to an uncenainty of ±0. 1 1 g (95%). The waveform was sampled at a rate of 128 Hz for
1 28 points, and a peak frequency of IO Hz was returned by the FFT calculation.
( a ) Calculate the beam stiffnes.� and its uncertainty (95% ).
( b ) What is the primary source of uncenainty in this result'! What is the best way
reduce the uncertainty of the result?
lo
( c ) The cxperimenter's disk is nearly full, so he does not wi sh Lo increase lhe number of
points sampled when he repeats the experiment. If the sample rJte can be adjusted
in increments of I Hz, what sample rdle would allow the lowesl uncenainty with
the same number of point�·1 Estima1e the uncenainty in stiffness for that sample
rate.
REFERENCES
( 1) Beckwith, T. G., and R. D. Marangoni . Mechanical Measurements. 4th ed. Read i ng,
Mass.: Addison-Wesley, 1 990.
(2) Oppenheim, A. V., A. S. Willsky, and S. H. Nawab. SignaLv & Systems. 2nd ed. Upper
Saddle River, NJ: Prentice Hall, 1 997.
The Analog Measurand: lime-Dependent Characteristics
ANSWERS TO SELECTED PROBLEMS
8
(a) 15/:br rad; (b) y(t) = 1 00 + 109.77 cos( 15r
-
1 .046)
V (t) � 6.7 cos(2026.8l) + 2.9 sin(3040.3t) - 0.9 cos(4053.7t)
12
(d) V( t) � l O sin wt + 4 cos (l)t - 2 sin 6 "'t
13
16
"' �
27r / .036 radls
(a) /Nyq = 2048 Hz (b) 2 Hz (c) 1 000 Hz, 1500 Hz, 2000 Hz
Is > 4000 Hz, N = 8000 pis. sampled
Th e Response of M easuri ng
Systems
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
INTRODUCTION
AMPLITUDE RESPONSE
FREQUENCY RESPONSE
PHASE RESPONSE
PREDICTING PERFORMANCE FOR COMPLEX WAVEFORMS
DELAY, RISE TIME, ANO SLEW RATE
RESPONSE OF EXPERIMENTAL SYSTEM ELEMENTS
SIMPLIFIED PHYSICAL SYSTEMS
MECHANICAL ELEMENTS
AN EXAMPLE OF A SIMPLE MECHANICAL SYSTEM
THE IMPORTANCE O F DAMPING
DYNAMIC CHARACTERISTICS OF SIMPLIFIED MECHANICAL SYSTEMS
SINGLE-DEGREE-OF-FREEDOM SPRING-MASS-DAMPER SYSTEMS
THE ZERO-ORDER SYSTEM
CHARACTERISTICS O F FIRST-ORDER SYSTEMS
CHARACTERISTICS OF SECOND-ORDER SYSTEMS
ELECTRICAL ELEMENTS
FIRST-ORDER ELECTRICAL SYSTEM
SIMPLE SECOND-ORDER ELECTRICAL SYSTEM
CALIBRATION OF SYSTEM RESPONSE
SUMMARY
INTRODUCTION
Quite s i m p ly, respo11se is a measure of a sy s te m ' s fide li ty to purpose. It may be defined as an
evaluation of the system's abi lity to faithfully sense. transmit, and present all the perti n ent
in formation included in the m cas u ra nd and to exclude all else.
We wou ld like to know if the output i n format ion truly represents the i nput. I f the
i n pu t i n formation is in the form of a sine wave, a sq u are wa ve , or a sawtooth wave, does the
output appear as a sine wave. a square wave, or a sawtooth wave, as the case may be? Is each
of t he harmonic com p o ne nts in a co m p l e x wave treated equally, o r arc s ome attenuated,
completely ignored, or perhaps shifted t imew i se relative to the ot h ers '! These questions are
ans wered by the response characteristics of the partic u l ar system-that is, ( I ) amplitude
response, (2) frequency res po nse , (3) p h ase response, and (4) s lew rate.
1 46
The Response of Measuring Systems
300
o --- o�
200
\
0
\
1 00
0
1<r
1 0"'1
1 0-2
Input, e; {V)
\
FIGURE I : Gain versus i nput vollage for am pli fier section of a commercially available
strain meas uring system for a frequency of I kHz (gain = ou tput voltage/in put voltage).
Z
AMPLITUDE RESPONSE
3
FREQUENCY RESPONSE
Amplitude response is governed by the system's abili ty lo treat all input amplitudes uni­
formly. I f an i nput of 5 units is fed i n to a system and an output of 25 indicator divisions
is obtai n ed , we can ge nerally expect that an input of 10 units will resull in an output of 50
divisions. Although this is the most common case, other special n onl i near responses are
also occ asi onally req uired . Whatever the arrangement, whether it be linear, expo nen tial ,
or some other amplitude fu nction , discrepancy between design expectations in this respect
and actual perfonnance results in poor amplitude response.
Of course no system exists that is capable of respondi ng faithfully over an unlimited
range of ampli tudes . All systems can be overdrive n . Figure I shows the amp l i tude response
of a voltage amplifier s ui table for connecting a strai n -gage bridge to an osc i ll oscope . The
usable range of the amplifier is restricted to the horizontal ponion of the curve. The
plot shows that for i nputs above about 0.01 V the ampli fier becomes overloaded and the
ampl ific ati on ceases to be linear.
Good frequency response is obtained when a system reacts to all freq uency components in
the same way. If a 100-Hz sine wave w ith an i nput amplitude of 5 u n its is fed into a system
and a peak-lo-peak outpu t of 2 ! cm resul ts on an oscilloscope scree n , lo"e can expect that a
i
500-Hz sine-wave i nput of the same amplitude would also resull in a 2 cm peak- to-peak
output. Cha ngi ng the frequency of the i nput si gna l should not al ter the system's output
magnitude so long as the input ampl i tude remains unc h anged.
Yet here again there must be a limit to the range over which good freq ue ncy response
may be expected. This is true for any dynamic system, regardless of its q ual i ty. Figure
illustrates the freq uency response relations for the same voltage amplifier used in Fig.
Frequencies above about 1 0 kHz are atte nuated . Only inputs below this freq uency limit are
ampl i fied in the correct relative proportion.
I.2
1 47
The Response of Measuring Systems
300
Frequency response
1--����-1-���-+-��--1
'
x,
1 o'
'
·�
Frequency, Hz
FIGURE 2:
Frequency response curve for amplifier section of a commen:ially available
strain-measuring system; e1 = 10 mV.
4
PHASE RESPONSE
S
PREDICTING PERFORMANCE FOR COMPLEX WAVEFORMS
Amplilude and frequency respo nses are i m pol'llln l for all types of i n pu l waveforms, whether
simple or complex. Phase respo11se, however. is of primary imponance for lhe complex
wave only.
Time is required for lhe transmission of a signal through any meas uri ng system. Often,
when a simple sine-wave voltage is amplified by a single stage of amplification. lhe output
trails the input by approximately 1 80°, or one-half cycle (see Fig. 20). For two stages, the
phase shift may be about 360°, and so on. The actual shift will not be an exact multiple of
half-wavelengths but will depend on lhe equipment and the frequency. II is the freq uency
dependence that defines phase response.
For a single-sine-wave input, any phase shift would no rmall y be unimponant. The
output produced on the oscilloscope screen would show lhe true waveform, and its proper
parameters could be determined. The fact 1ha1 the shape bei n g shown was actually formed
on lhe screen a few microseconds or a few milliseconds aficr be i ng generated is of no
consequence.
Lei us consider, however, a complex wave made up of numerous hann onics. If the
phase lag is different for each frequency, then each component of lhc complex wave is
delayed by a different amount. The hannonic com po n en ts !hen emerge from the system in
phase relations different from when they entered. The whole waveform and i ts amplitudes
are changed, a result of poor phase response .
Figure 3 illustrates phase response chatacteristics for a typical vohage amplifier.
Response characteristics of an existing system o r a component of a system may be deter­
mined experimentally by injecting as inpul a signal of known form. then detennining lhe
1 48
The Response of Measuring Systems
80
..
..
i!
"'
-3
j..
�
0..
/
/
I
60
/
/
40
20
0
1 0'
Frequency, Hz
FIGURE 3: Phase lag versus frequency (phase response) for the same amplifier used in
Fig. 2.
output, and finally comparing the results. Of course, the most basic test waveform is the
sine wave.
If we know the sine-wave response of a device, can we use this information to predict
how it will respond to a complex input, such as a square wave or one of the various sawtooth
waveforms? The answer is yes, as we dllJllonstrate in the follow i ng example.
EXAMPLE 1
Using a computer program and information given in Figs. 2 and 3. predict the form of
amplifier output to be expected if a perfect square wave is the input. Do this for input
having fundamental frequencies of 1000 and 2000 Hz and amplitude of I m V.
Solution
A square wave is defined by the infinite series
4A
y= -
00
I
L:
--- sin [(2n I)
Jr n ; I
(2n -
1 )2irfl )
(I)
We may modify Eq. ( I ) by introducing frequency and phase distortion factors:
4A
)' = -
r
L:
ir n;I
(
whe e
00
Gn
sin [(2n
2n - I )
-
-
1 )2irft - <fl. I
G n = an amplitude factor based on freq uency.
<fin
=
a phase distortion factor
149
c l a)
The Response of Measuring Systems
!
1
ii
..
,,
'i
..
l
a:
-1
0
-1
21T
0
41T
Dimensionless time, 21Tft
61T
(a)
-
!
61T
- -- Ideal output
-- Actual output
- -,
-1
15.
E
..
�
S!
31
0
[
..
..
a:
I
-1
0
-
_,
21T
41T
Dimensionless time, 21Tft
(b)
61T
81T
- - - Ideal ou1pu1
-- Aclual outpul
FIGURE 4: (a) Compuler-delennined res pon se to a 1 000- Hz square wave by the amplifier
whose characteristics are shown in Fi gs . I through 3, (b) c om puter-determin ed response 10 a
2000- Hz square wave by the amplifier whose characteristics are shown in Figs. I through 3.
(Ideal amplitude = 240 mV.)
Magnitudes for G /1 and <Pn for e ac h harmonic order can be extracted from the respon se
2 and 3. For example, if n = 15, then (2n - I ) / = 29( 1000 Hz) = 29 kHz,
a nd we read G 1 s = 200 from Fig. 2 and tf> i s = 30° from Fig. 3. 1be computer code can
incorporaie tables of such frequency-response and phase-response data taken directly from
the sine-wave response curves. The resulls of us i ng these data in Eq. ( l a) are plotted i n
curves, Figs.
Figs.
4(a) and ( h).
1 50
The Response of Measuring Systems
We can make similar calculations for any wavefonn for which a hanno nic series can
be written. In particular, lo investigate a measuring system's response to a wavefonn of
interest, we can make a Fourier analysis of that wavefonn and investigate the system's
response characteristics using sine-wave test results, as before.
&
DELAY, RISE TIME, AND SLEW RATE
7
RESPONSE OF EXPERIM ENTAL SYSTEM ELEMENTS
Finally, a fourth type of response, which is actually another form of frequency response,
is delay, or rise time. When a stepped or relatively instantaneous input is applied to a
system, the output may lag, as shown in Fig. 5. The time delay after the step is applied,
but before proper output magnitude is reached, is known as rise time. It is a measure of the
system's ability to handle transients. Sometimes rise time is defined specifically as the time,
6.t, required for the system to pass from 10% to 90% of its final response. Alternatively,
transient response may be characterized by the settling time required for the system response
to remain within some small percentage of its final value.
Slew rale is the maximum rate of change that the system can handle. In electrical
terms. it is de/dt, or volts per unit time (e.g., 25 V /µ.s). When the voltages changes rapidly,
the system can respond no faster than the slew rate.
An experimentalist can usually avoid operating a measuring system under conditions when
amplitude response or slew rate are limiting factors. For example, a solid-state amplifier
System input
SetUing lime
FIGURE 5: Response of a typical system to a step input, showing rise time (6.r) and settling
time.
l,__
__ Sys-tem
The Response of Measuring Systems
Input
V1n
·
_
Output
element
____.
v., = V,sln(211'"fl)
f\AA
v...
v... = v. sin(211'"1t + d>
)
FIGURE 6: Response of a system element to a sine-wave input.
that is overloaded ty pically produces a. constant output of ±5 V i rrespecti ve of the input;
such a cond i tion should be fairly obvious, and its result is clearly useless.
Frequency- and phase-response limitation s are not so read il y apparent. One would
prefer zero phase shift and flat frequency response for all frequenc ies of interest. However,
because experimental i nputs often contain a w ide range of freq uencies , poor response
to some subset of freque nc ies may go unnoticed. An evaluation proced u re is general ly
necessary to establish a pan icu lar system's response.
We have seen that a sine-wave test can provide the freque ncy and ph ase res ponse
frequency by freq uency. The sine-wave response of a meas uri n g system e leme n t . such as
a transducer or an amplifier, provides a fou ndation for eval uating the performance of the
overall measurement system.
Each element in the system transfers its input to an output (Fig. 6 ). The output can
differ from the in pu t in both amp l i tude and phase. The ratio of ou t pu t amplitude to i n put
ampl i tude is the gai n (or amplificati on) for that freq uency, and the vari ati on of the gain over
all frequencies is what we h ave called frequ ency response,
o
G ( /) = V
V;
Similarly, we called the variation of th e ph ase shi ft with frequency the phase response,
4>(/). These are, of course, the functions shown in Figs . 2 and 3. t
A complete measuring sys tem consists of a series of elements, from sensors and
transducers to signal condi ti oners to record ing and d isplay devices. For any given frequency
component, the system ' s gain is the produ ct of t he gains of all syste m elements at that
frequency. Likewise, the sys tem ph ase shift is the sum of every stage ' s phase shift. To
obtain an acceptable measurement, every measuring stage must have acceptab l e response
over the frequency range of interest. If any si n gle stage does not respond properly. it will
elision the signal and contaminate the entire measu rem ent .
Figure 7 shows a ty pic al measuring system. A periodic measurnnd is detected by
a sensor. This measurand might be a position, x. that oscillates at a freq ue ncy fx · The
1 The iuder who has studied sySlem dynamics will ttcognizc 1ha1 G(/) and t/l(f) an: 1he magnicude and phase.
=peclivety, of lhc periodic cransrcr runccion. ll(jOJ), or a linear syscem It t.
1 52
The Response of Measuring Systems
Amplifier
Low-pass
filter
Computer
system
FIGURE 7: Measuring system composed of several system cl e men ts; the sig nal i s shown
as il leaves each element.
sensor produces an output voltage in millivolts, which varies with x. However, the sensor
also picks up high-frequency electrical noise from nearby equi pmen t. Thus, the output of
the sensor includes both the low frequency of the signal, fx . and the high frequency of the
noise, J• . The sensor's signal is received by an am plifier (gain = 100), which raises the
signal level to a value convenient for com pu ter recording. Apart from the added noise, both
measuring stages respond faith fu l l y to the input s ig nal .
Certain system elements are chosen spec i fi call y for the l im itations of their frequency­
response characteristics. In the case of Fig. 7, the measurand varies at a fairly l ow frequency,
whereas the u nwan ted electrical n oi se is at a much hi g her frequency. Since the experimen­
talist has evaluated lhe output signal on an oscilloscope, she has a clear idea of which range
of frequencies characterizes th e measurand and which range characterizes the noise. Thus,
she inserted a signal-conditioning low-pass filter after lhe amplifier lo eliminate the noise.
This filler has Hat frequency response al low fr�-quencies (such as fx ) and zero response at
high freq uenci es (such as fn ) , as shown in Fig . 8. The s ig nal that the filler transmits lo the
computer record ing system excludes the noise component.
In contrast to the filter, a well-designed sensor or transducer should respond lo all
frequencies eq ual ly. Unfortunately, most actual sensors and transducers do not. Instead,
such dev ices are characterized by an upper or lower freq ue ncy beyond which response is
attenuated (much like a filter) or by a high or low freq ue ncy al which the sensor resonates
with the input, produc i ng an output that is ridiculously large. Determination of such l i mi ti ng
frequencies is extre me ly important if a dynamic measurand is lo be accurately recorded.
The frequency and phase response of a system element (or of the entire chain of
e le me n ts) can be de termi ned in several ways. For simple elements, we may be able to
construct a phys ica l model of the device that accurately predicts its response. For more
complex systems, we may wish to tes t the response experimentally, for ex am ple, by using
a s ine- wave lest. In other circumstances. wc may re l y on response data provided by the
manufacturer. In any case, once we have determined lhc range of freq uenc ies for which the
system responds accurately, wc will disregard frequencies outside this range , i f they can
1 53
The Response of Measuring Systems
�
-���-'-����- --'-���
o L-
Frequency, f
�
FIGURE 8: Freq uency response for the low-pass filtt:r of Fig. 7.
be identified, and we will take precautions to prevent such freque:ncies from entering the
measuring chain.
The remainder of this chapter is concerned with the identi fication of system response.
Sections !I through 1 9 address the modeling of frequency and ph ase response for simple
phy sica l systems. Section 20 return s to the matter of experim1mtal detennination and
calibration of system response.
8
SIMPLIFIED PHYSICAL SYSTEMS
Whal basic physical factors govern response ? In tenns of practical systems, we are con·
fronted with two fundamental ty pes of construction: mechanical and electrical. The basic
mechanical elements are mass, damping, and some fonn of equilibrium-restoring element,
such as a spri ng . Corresponding electrical elements are i n duc tance , resistance, and capaci­
tance. Although many, if not most, devices and systems involve bod1 electrica l and mechan·
ical elements, for our immediate purposes it is advantageous to con:iider the two separately.
In the next several sections we will discuss some of the mechanical! aspects; and beginning
with Section 17, we will consider the electrical .
9
M ECHANICAL ELEMENTS
A d i sc uss ion of the d yn amic characteristics of an elementary mechamical system necessitates
a short d escription of the elements com posing such a system.
9.1
Mass
It is obvious that in all cases mass will be a factor. Under certain conditions, however, the
masses making up the device or system will not affect its perfonn ance. We will consider
such cases in Sections 14 and 15.
By i t s very nature, mass must be distributed throughout some volume. In many cases,
however, it is not only convenient but also correct, or nearly so, t•o assume that the mass
1 54
The Response of Measuring Systems
of a member is concentrated at a point. Depending on the geometry of the member and its
application, the point of concentration may or may not be the center of gravity. In cenain
cases, the center of percussion may be the location of effective concentration.
9.2
Spring Force
Many mechanical members deHect in direct proportion to the force exerted on them, that is,
t;.F / t;.s = k = a constant, where t;. F is an applied force increment and t;.s is the resulting
deHection increment. Most coil springs, beams, and tension/compression members abide
by this relationship. II may be noted that the force is opposed to the deHection; that is, the
resulting force always allempts to restore equilibrium.
Torsional members commonly adhere to the relationship t;. T/ t;.(J = k, = a constant,
where t;. T is an applied torque increment and t;.(J is the resulting torsional deHection
increment. The constants k and k, are called spring constants, or thflection constants.
Elasticity is not always the source of the restoring force, however. In certain cases,
such as for a beam balance (see Section I 0), the restoring force may be supplied by gravity.
When the motion of a concentrated mass is constrained by an equilibrium-seeking
member [Fig. 9(a)], simple vibration theory shows that the combination will have a natural
frequency
(2)
where
"'•
f
= circular frequency in radians per sec ond
= 21r f,
=
frequency of vibration in hertz,
g0 = the dimensional constant
A system of this sort is said to have a
single degree offreedom; that is, it is assumed to
be constrained in some way to oscillate in a single mode or manner, needing only one
coordinate to fully describe its motion.
m
m
(a)
(b)
FIGURE 9: Elementary spring-mass systems: (a) without damping; (b) with viscous
damping.
The R esponse of Measuring Systems
Time
___
0'191'Shoot
..._
FIGURE 10: lime-displacement relations for damped motion: (a) for damping greater than
critical; (b) for critical damping; (c) for damping less than critical.
9.3
Damping
Another factor important to the usefulness of any system of this type is damping [Fig. 9(b)).
Damping in this connection
is usually thought of as viscous. rather than Coulomb or fric­
tional, and may be obtained by fluids (including gases) or by electrical means.
Viscous damping is a function of velocity, and the force opposing the motion may be
expressed as
(3)
where { = the damping coefficient and ds /dt
= the velocity.
We can see that the damping
coefficient is an evaluation of force per unit velocity. The m:gative sign indicates that the
resulting force opposes the velocity. The effect of viscous damping on a freely vibrating
single-degree-of-freedom system is to reduce the vibrational amplitudes with respect to
time according to an exponential relation.
Damping magnitude is conveniently thought of in terms of critical
damping, which
is the minimum damping that can be used to prevent overshoot when a damped spring­
mass system is deflected from equilibrium and then released (see Section 1 6. 1 for further
discussion). This limiting condition is shown in Fig. 1 0. The value of the critical damping
coefficient, {c. for a simple spring-supponed mass
{c = 2
m is expressed by the relation
f!:
.
(4)
Damping is often specified in terms of the dimensionless damping ratio,
�
=
i.
{c
Many measuring devices or system components involve elements constrained by
gravity or spring force, whose deflection is analogous to the signal input. The ordinary
balance scale is an example, as is the D'Arsonval meter movement. The same is true of
most pressure transducers, elastic-force transducers, and many other measuring devices.
1 56
The Response of Measuring Systems
If lh e syslem is a lranslational one, a spri ng-constrai ned mass may be i nvolved . A tire
gage illuslrates the case in which lhe pislon and slem and a portion of the spring constitute
the mass whose molion is controlled by lhe inleraction of the appl ied pressure and the s pring
force.
These devices depend on equilibrium for correc t indicalion. When equilibrium is
disturbed by a change of inpul, the system requires l ime to rcadjusl to the new equilibrium,
and a number of oscillations may lake pl ace before the new outpul is correctly indicated.
The rale at which the amp l itude of such osci llat i ons decreases is a function of lhe system's
damping. In add i ti on, the freq uency of oscillalion is a function of both damping and
sensitivily.
10
AN EXAMPLE OF A SIMPLE M ECHANICAL SYSTEM
Let us consider a sy mmetrical scale beam without damping (Fig. 1 1 ) . For simpl ificalion ,
we wi ll assume that lhe masses of the scale pans and the weight� bei ng compared are
concentraled al points A and B and lhal they are also included in the mass moment of
inertia /, which is refened lo the main pi vot point 0 . Further, we will assume that a small
difference, l!i. W, exists belwecn the two weights being compared and lhat points A, B, and
0 lie along a straighl l ine. We define sensil ivi ly, 11. as the ratio of the displacement of the
end of the poi nter lo the length of l he pointer, h, divided by l!i. W, or
I
I d
11 = - - = - tan e
6W h
l!i. W
FIGURE 1 1 : Schcmalic diagram o f a be am balance.
1 57
(5 )
The Response of Measuring Systems
The system behaves as a compound pendulum, and it can be shown that the period of
oscillation will be
(6)
where
I
=
the mass moment of inertia,
=
the weight of the
r = the distance between the center of gravity of the beam alone and pivot point 0 ,
wh
beam
With the weights applied,
whi' sin 8
=
�W L cos 8
or
tan 8
L�W
= -­
whi'
(7)
Hence, using Eq. (5), we find that
( I...) 2
(7a)
Combining Eqs. (6) and (7a), we have
'I = !:.
I
211'
· gc =
!:.
I
(-1-)2
211'/
· 8c
(7b)
where / = natural frequency.
Equation (7b) indicates that the sensitivity is a function of T, tht: period ofoscilla!ion
of the balance scale, with increased sensitivity corresponding 10 a long beam and low
moment of inertia. In other words, the more sensitive instrument oscillates more slowly
than the less sensitive instrument. This is an important observation having significant
bearing on the dynamic response of most single-degree-of-freedom iinstruments.
11
THE IMPORTANCE O F DAMPING
The importance of proper damping to dynamic measurement may be understood by assum­
ing that our scale beam in the previous example is part of an instrum•ent that is required lo
come to different equilibria as rapidly as possible.
Suppose that our scale beam has very low damping. When a disturbing force is
applied, the scale will be caused to oscillate, and the oscillation wit I continue for a long
period of time. A final balance will be obtained only by prolonged waiting, and thus the
frequency with which the weighing process may be repeated is limitc:d.
On the other hand, suppose considerable damping is provided--well above critical.
An extreme example of this would be to submerge the entire scale in a c:ontainer of molasses.
Balance would be approached at a very slow rate again, but in this case there would be
no oscillation. Here, again, excessive time would be required before the next weighing
operation could commence.
II appears, therefore, that if we were lo design a beam-type scale for quickly deter­
mining magnitudes of different masses, the final form would necessarily be a result of
1 58
The Response of Measuring Systems
compromise. We would like equilibrium to be reached as quickly as possible in order to get
on with the job. It would seem that there might be an optimum value that should be used.
Although lhis is not exactly the case because of other factors involved, damping of the order
of 60% 10 75% of critical is provided in many instruments of this type (see Section 1 6.2 for
further clarilica1ion of this point).
Although damping will lend to decrease the frequency of oscillalion, it does not change
1he inherenl sensitivily of the device, which is related to the undamped natural frequency.
However, some compromises must still be made in regard to sensitivity. Sensitivity i ncreases
in proportion lo 1he undamped natural period, as shown in the previous section. Because a
high natural peri od (or low natural frequency) usually corresponds lo a lessened frequency
response. scnsi1ivi1ies grea1er than lhose required by lhe application should be avoided in
1hc interest of mainlaining adequale response.
12
DYNAMIC CHARACTERISTICS OF SIMPLIFIED M ECHANICAL SYSTEMS
13
SINGLE-DEGREE-OF-FREEDOM SPRING-MASS-DAMPER SYSTEMS
By making certain simplifying modeling assumptions, we may place lhe dynamic charac­
leristics of most measuring syslems in one of several categories. The basic assumptions are
that any restoring element (such as a spring) is linear, that damping is viscous, and that the
system may be approximaled as a single-degree-of-freedom sysiem.
Figure 1 2 shows a simple single-degree-of-freedom mechanical system. It is single degree
because on ly one coordinale of motion is necessary 10 completely define the molion of lhe
system. We will also assume a general form of cxcita1ion, F (I ), which may or may not be
periodic. Forces acling on the mass will resull from the spring, damping, and lhe exlemal
force, F(t ). Using New1on's second law, we can write
( )
ds m d2s
F(1) - ks - s - = - dt g,. d12
(8)
Nole lhat lhc spring force will always oppose the displacemcnl and that lhe damping is pro­
porlional to vcloci1y and oppusile lhe velocily direction. This relationship can be rearranged
+
s:
T
c::::=::!::;=::=�
F = F(t)
FIGURE 1 2 : Mechanical model of a force-exciled second-order syslem.
The Response of Measuring Systems
lo read
I n bolh equalions,
� [m (�:�)] + { (�) +ks = F(t)
(8a)
s = mass displacemenl from lhe equilibrium posi lion
If we assume F(t) lo be periodic wilh lime, we can s ubst i tu te the appropriale Fourier
series for F(t) or, in general.
f(I)
=
A°
2
+
oo
L c. cos(n Q1 - �. )
(9)
11- 1
where Q = circular for:cing rrequency and
c. =
JA� + B;
B.
Ian � = A.
and
(9a)
We will consider this general case in Seclion 16.3. Firs!, however, we will consider several
special cases.
14
THE ZERO-ORDER SYSTEM
A nearly lrivial case occurs if we remove the spring and damper. The vollage-dividing
polentiomeler is an exampl e . In its si mples ! form lhis device is a single slide wire. Aside
from the mass of lhe slider and any member allached 10 it, there is no apprec iabl e resislance
10 moveme nt . In particular, an equilibrium-seeking for:ce is not present and the oulput is
i ndependent of lime, lhal is,
Output = constant
x
input
Dy n amical ly, the zero-order sysle m requires no further consideration.
15
CHARACTERISTICS OF FIRST-ORDER SYSTEMS
If we ass ume the mass, m, in Fig. 12 and Eq. (Sa) to be zero, we obtain a.first-order system.
S uch syslems have a ba lance between damping for:ces, t; ds/dt. res tori ng for:ces, ks, and
exlemally applied for:ces, F(t). Among i ns trume nls . the mosl common first-order systems
are temperalure-measuring dev ices. A temperature sensor usually responds as a first-order
system because the rale of change of its temperature, d T. /dt, is pro portio nal 10 ils current
temperature, T, . Obviously, a mercu ry- in -g lass thermometer or a thermocouple bears no
physical resemblance lo Fi g . 12. However, the dynamic responses of the three systems are
idenlical .
For the first-order system we can write
{ dt
ds
+ ks = F(t )
1 60
( I O)
The Response of Measuring Systems
15. 1
The Step-Forced First-Order System
Lei
F(t) =
O.
for t < 0
Fo.
for t � O
and
F(t) =
For force equilibrium on the connecting element (which is assumed to be massless),
ds
{ di + ks = Fo
( I I)
where
t
= time,
s = displacement,
{ = the damping coefficient,
k = the deftection constant,
= the amplitude of the constant input force
Fo
' [' ds
L dt = { (Fo ks)
Then
0
from which we obtain
Fo
Fo
The units of � E
{I k
- ks
-
ksA
=
---
e -tr/C =
e-r/r
( 1 2)
time constant.
are seconds, and this quantity is known as the
Equation ( 1 2) can
be written
s = s00[ I
where
·'•
-
e-•l• J
+
SAe-11' = soo
+ [SA - s00)e -'1'
s00 = Fo/ k = the final displacement of the system as I
SA = the initial displacement at t = 0
.....
( 1 3)
oo,
We have assumed that the first-order system represents any dynamic condition wherein
the elements are essentially massless, the displacement constraint is linear, and a significant
viscous rate constraint is present. Generally, Eq.
P = Poo ( I or
where
( 1 3) can be written
e-11' ] + PM-t / r
( 1 4)
P
P00
PA
the magnitude or any
first-order process at time I ,
the limiting magnitude of the process as
the initial magnitude of the process at
1 61
I
.....
t =0
oo,
The Response of Measuring Systems
Although the basic relationship was derived in tenns of a spring-dashpot arrangement,
other processes that behave in an analogous manner include: ( I ) a heated (or cooled) bulk
mass. such as a tem perature sensor subjected to a step temperature change; (2) simple
capacitive-resistive or inductive-resistive circuits; and (3) the decay of a radioactive soun:e
with time.
Figure 13 represents two different process-time conditions for the step-excited first­
order system: (a) a progressive process, wherein the action is an increasing function of
time; and (b) the deca)•ing process, wherein the magnitude decreases with time.
6
P- = 5
- - - - - - - - - - - - - - - - - -
4
3
I
I
I
I
I
I
I
- - - - - -� - - - - - - - - - - - - - - - - - - - - - - - - - - --!
2
0
-:_:-:.:-::;-;:.:-:.:-:.:-�·---
0
I
1 .0
2.0
T
3.0
4.0
llr, Ratio of time to Ume constant
5 .0
(a)
6
�= 5
- - ----------------------------------
4
Process 63.2% completed
3
I
I
I
2
--- -- -l--- -
-
-
- - - - _ _ _::::_::_:-:_�-=-�--·-�----
!
I
'---'--�
T___.__,___...___.___,___, _,___,__�0
0
1 .0
2.0
3.0
4.0
llr, Ratio of time to time constanll
5 .0
(b)
FIGURE 1 3: Characteristics of a first-order system subjected to a step input at t
for a progressive process; (b) for a decaying process.
1 62
== 0: (a )
The Response of Measuring Systems
Significance of the Time Constant,
"C
If we substitute the magnitude of one time constant for 1 in Eq. ( 14 ),
P = Poo + ( P,1. - P00)(0.368)
from which we see that (1 - 0.368), or 63.2%, of the dynamic ponion of the process will
have been completed. TWo time constants yield (I - 0. 1 35) = 86.5%, three yield 95.0%,
four yield 98.2%, fi ve yield 99.3%, and so on. These percentages of completed processes
are imponant because they will always be the same regardless of the process, provided that
the process is described by the conditions of the step-excited first-order system.
It is often assumed thaJ a process is completed during a period offive time constants.
EXAMPLE 2
Assume that a panicular temperature probe approximat� first-order behavior in a panicular
application, that it has a time constant of 6 s, and that it is suddenly subjected to a temperature
step from 75°F to 300°F. What temperature will be indicated 10 s after the process has been
initiated?
Solution
Applying Eq. ( 1 4), we find that
Poo = 300°F,
P,.. = 75°F, I = 10 s,
6
P = 300 + (75 - 300) e- I0/ = 257°F
EXAMPLE 3
Assume the same conditions as those of Example 2, but with a step from 300° F lo 75°F.
Find the indicated temperature after 10 s.
Solution
Poo = 75°F,
P
15.2
=
Pt-. = 300°F,
75 + (300 - 75)e- 10'6
I = 10 s,
=
I 1 7°F
The Harmonically Excited First-Order System
Again referring lo Eq. ( 10), let us now consider the case of
F(t) = Fo cos S'21
or
s
ds
dt
+ ks = Fo cos S'21
( 1 5)
The Response of Measuring Systems
where
Fo = the amplitude of the forcing function,
Q = the circular frequency of the forcing function in radians per second
The solution of Eq. ( 1 5) yields
( 16)
where
At
= a constant whose value depends on the initial conditions,
T = the time co nstan t =
f.
t
</I = the phase lag = tan-
T=
211"
0
Q ("
T
211"
= tan - 1 T r,
( 17)
= the period of excitation cycle in seconds
We see that the first term on the right side of Eq. ( 1 6), the complementary function, is
transient and after a period of several time constanlS becomes very small. The second term
is the steady-state relationship and, except for the short initial period, we can write
s=
or
Fol k
./1
s
s,
and
+ (rQ)2
cos(Qt - </I)
t
cos( O - </I)
Ji + (rQ) 2
Sd
- = --====
s,
./ I + (r Q )2
I
JI + (2n/ T)2
where
and
. (IS)
s4 = the m ax i m u m
amplitude of the
( 19)
periodic dy nam ic displacement
Fo
s, = T
The q uant i ty s, is the static denection that would occur should the force amplitude Fo
be applied as a static force. The ratio s,1 /s., is often called the am plification ratio. For
analogous si tuati o n s. Eq. ( 19) may be written
( 19a)
where P represents the
magnitude
of the applicable proces s .
1 64
The Response of Measuring Systems
100•
eo•
.,.
60"
40•
20"
O"
O.Q1
10.0
1 .0
0.1
I.
1 00
FIGURE 14: Phase lag versus ratio o fexcitation period t o time constant for the hannonically
excited first-order system.
pd
P,
0.6
1------+--
0.4
1-------+--+---,,,_--i
0.0 l-...J..-1.=6.l:::l:=t:::.L...L.J..L__J__LJ_L_L..._L_l.J
1 00
1 0.0
1 .0
0.1
0.01
T
t
FIGURE 15: Amplitude ratio versus ratio of excitation period to time constant for the
harmonically excited first-order system.
Figures 14 and 15 illustrate the relationships of the phase angle and the amplification
ratio described by Eqs. ( 17) and ( 1 9a), respectively.
By calculating the response: of the fim-order system to a sinusoidal input, we have in
fact determined its frequency and phase response. To show this explicitly in the notation of
Section 7, we can rewrite Eq. ( 1 9a) in terms of the cyclic forcing frequency, / = l / T:
G( /) =
PJ
P,
-
I
= -;===::.,
1 65
./1
+ (2rr/ r )2
( 19b)
The Response of Measuring Systems
Likewise, from Eq. ( 1 7), the phase response is
(1 9c)
Note that the ideal response (G -+ 1 and t{> -+ 0) is obtained at frequencies small enough
that 211fr « I. From Figs. 14 and 15, we can see that this is cquivalenuo the statement that
thefrequency and phase response ofa first-order system are best when the time constant of
the system is small compared to the period offorr:ing, r « T, so that the system respollds
rapidly in comparison to the variations it is measuring.
In most mechanical systems, a moving mass exists and cannot be ignored. In such
cases, the system is of second order and has characteristics that are discussed in subseq uent
sections. However, for temperature-sensing systems, first-order response is usually a good
model, and we can use this model to examine the response characteristics of temperature
probes.
EXAMPLE 4
A temperature probe has a time constant of 10 s when used to measure a particular gas flow.
The gas temperature varies hannonically between 75°F and 300°F with a period of 20 s
(i.e., with a frequency of 0.05 Hz). What is the temperature readout in terms of the input
gas temperature? What time constant should the probe have to give 99% of the correct
temperature amplitude?
Solution
(3 ; ) (3 ; ) G� )
G� )
In this case the temperature input can be expressed as
Tps (I) =
00
75
+
= 1 87 + 1 1 2 cos
00
75
cos
i
1
From Eq. ( I 9a), we find that
Td
T,
= ,/
Td =
I + (211' x 10/20) 2
'
�
= 34°F
33
.
(211' ,:0 ) =
From Eq. ( 1 7), we find that the phase lag for a forcing period of T
ti>
=
tan- 1 211'(r / T ) = tan - 1
10
72.5
360
x
20 = 4 s
A graphical representation of the situation is shown in Fig. 16.
1 66
20 s is
tan - 1 71' = 72.5° (angle)
or
lime lag =
=
The Response of Measuring Systems
300 .0"
Gas
221 .s•
�
1 s1.s•
!
:I
i! 1 53.5°
!
{!.
,.
I
-+I
,.
�---
- - -
_ _
-
'
\
,,_
\
'
E
�-'
'
I
\
1- - - - -+I
\
- - -' '
�-Probe-
75.0"
0.0"
0
nme, t (s)
FIGURE 1 6: Response of temperature probe for conditions described in Example 4.
To obtain a
99% amplification ratio.
we requi re
I
Td
T, = ,/I + (2n /20)2
= 0.99
Solving, we find that the probe wou ld need a time constant of t' = 0.45 s. In that case, t'
would be very small compared to
16
T.
CHARACTERISTICS OF SECOND-ORDER SYSTEMS
Figure 1 7 illustrates the essen tials of a second-order system. Thi s arrangement approx­
imates many actual mechanical arrangeme n ts including simple weighing systems, such
as an el astic - type load cell supporting mass; D' Arsonval meter movements, including
the ordinary galvanometers; and many force-excited mechanical-vibration systems such as
a
accelerometers and vibrometers.
As w it h the firs t- order system, many excitation modes are po ssi ble , ranging from
the simple step and simple harmonic functions to complex periodic forms. These modes
approximate many actual situations, and , because all periodic i nputs can be reduced to
combinations of simple harmonic components the l atter can give us insight into system
performan ce when subject to most forms of dynamic input.
16. 1
The Step-Excited Second-Order System
Referring to Fig. 17, we let
and
F = 0, when t < 0
F
=
Fo , when I ::: 0
The Response of Measuring Systems
k
•
I
s :
m
t
F=
F(I)
FIGURE 1 7 : Schematic representation of a second-order system.
Application of Newton's second
law yields
(20)
where
s
"' displacement measured from the equilibrium position
we assume underdamping, that is, (kgc /ml >
general solution of Eq. {20) can be written as
If
(r;gc/2m) 2
(see Eq.
(2 l a)), the
(2 1 )
when:
A
and 8 are constants governed
Wnd
by initial conditions and
= damped
kgc
m
_
natural frequency
( r;gc ) 2
(2 1 a)
2m
Note th at the ex po nen t ia l multiplier may be written as e-• / r , where the time constant
T = 2m/r;gc. If we let s = 0 and ds/dt = 0 al t = 0 and evaluate A and 8, then by
rearrange ment and substitution of terms, we can write Eq. (2 1 ) as
An
te
!....
s,
=
�1 - e- t ""'' [-sin � w. r + cos ./ I
�
al rnative form is
!_
s,
=
f3 =
l
-
/ I -l
[ �]
I-
e-t.,,, r
tan - 1
v
1 68
�2
�-
COS{Wndl
-
{3),
- l r]
�
w.
(22)
(22a)
{22b)
The Response of Measuring Systems
where
w,,
l;c
=
,/kgc/ m = the undamped natural frequency i n radians per second,
=
2 ,/mk/8c = the critical damping coefficient,
� = I; I /;c = the critical damping ratio,
s, = Folk = the "static" amplitude, or the amplitude that is reached as t -+ oo
Here again we may introduce the general idea of an analogous process, P. or
p
P,
as
For the overdamped condition, � =
I;/ l;c
s,
>
1 , the solution of Eq. (20) can be written
(23)
Figure 18 shows the plots for Eqs. (22) and (23) for various damping ratios.2 When the
system has a nonzero damping, il approaches a static condition with P/ P,
. = 1 as the
· transient dies out.
,,. ....
2.0
(\
€=0
I
I
I
I
I
I
I
I
€ = 0.25 \
\
\
\
1 .5
PIP,
'
1 .0
I
/
'
\
0.5
2.0
6.0
4.0
8.0
"'•'· radians
FIGURE 1 8 : Response of a second-order system to a step input at t = 0.
2 Nocc lhat the cases of zero damping and critical damping require special treatment.
1 69
1 0.0
The Response of Measuring Systems
1 6.2
The Harmonically Exdted Second-Order System
Referring to Fig. 17, when
F(t) = Fo cos cut
we can write
For u nderd am ped systems
m d2s
-2
8c d I
ds
+ (" - + ks = Fo cos Ot
the solution becomes
s = e-((rd2mlt [ A COS <Und l + B sin Wnd l ) +
(
J[ i - �:7 e:r
='=-=�=)=
=''::("'
o/=k=)=COl
-;:=(=Fi=
= e -1 / < A cos ./ 1 - �2 "'• ' + B sin �
+
(24)
dt
s, cos(Ot - �)
./Cl - (r2/cu,, )l]2 + [� (r2/cun)J2
"'•' ]
+
(25)
(25a)
where A and B arc constants that depend on particular initial conditions, and
]
n = the frequency of excitation (rad/s),
� = tan - I
s,
= Folk
[
2� '2 /cun
I
- ( n /cun )
2
= the phase angle,
(25b)
We see that the first tenn on the right side of Eq. (25a) is transient and will disappear after
several time constants. The second tenn is the steady-state relationship, for which we may
write3
Sd
°i; =
pd =
P,
./[ I
-
= the amplification ratio
where
sd
1
(r2/cu0 ) 2 ) 2 +
[2� 0/w. J2
(25c)
= the amplitude of the periodic steady-state displacement
Figures 19 and 20 are plots of Eqs. (25c) and (25b), respectively, for various values
damping ratio. � - The ratio s,/sd is none other than the frequ•ency response, G, of
the system; we could write it in tenns of cyclic forcing frequency, J', by the su bstitution
n = 27r/.
of the
3Note lhat inasmuch as lhe complementary or homogeneous solution (transient) is nol involv.:d. this relationship
is valid for under·. over-. llld critically damped conditions.
1 70
The Response of Measuring Systems
2.0
1.5
1 .0
.; 0.7
j 0.6
II
o.5
j 0.4
0.3
0.2
I
- -'/ i.--. � I
// 0.3 1
- ....,,,. � �
0. 4
0.5 - - - ... v
=
.......
- ...... ' \,0.70 - - ......:..._
....
�
' I'\ "
2.0
'6
' ""'
' \ :\\
'
' -..
'
\. \ \\' 1.0
"'-.. "'-.. ., s.o "
\. v \
"\
"
"
"
"
t'\.
I
, EsO
�
0.1
0.1
rT
0
;..
0.20
"'
I'\.
"'
0.5
'\.
Frequency ratio, w....
'\
1 .0
�N
I\
I\ �
\.
2.0
��
3.0
FIGURE 19: Plot of Eq. (25c) illustrating the frequency response to harmonic excitation
of the system shown in Fig. 17.
180
150
:
!;. 120
...•
�
�
Iii
.
•
•
f
90
60
30
2
3
Frequency ratio, °'"'•
4
5
FIGURE 20: Plot of Eq. (2Sb) illustrating the phase response of the system shown in Fig.
17.
The Response of Measuring Systems
The ratio Sd /s, is a measure of the system response to the frequency input. Normally,
we hope that this relationship is constant with frequency; that is, we would like the system
to be insensitive to changes in the frequency of input F (t). Inspection of Fig. 19 shows
that the amplitude ratio is reasonably constant for only a limited frequency range and then
only for certain damping ratios. We see that for a given damping ratio, ideal response
(sd /s, = I ) may sometimes occur at only one or two frequencies. If the system is to be
used for general dynamic measurement applications, rather definite damping must be used
and an upper frequency limit must be established. Practically, if a damping ratio in the
neighborhood of 65%-75% is used. then the amplitude ratio will approximate unity over
a range of frequency ratios of about 0%-40%. · Even for these conditions, inherent error
(sd/s, - exists, and a usable system can be had only through compromise.
I)
It should be made clear that the basic reason for optimizing the damping ratio is to
extend the usable range of exciting frequency Q. Certain devices, notably piezoelectric
sensors commonly possess such high undamped natural frequencies, "'•· that the range of
normal operating frequency ratios, Q/w. , may extend from zero to only 1 0% or even less.
In such cases, damping ratio magnitudes are of lesser interest.
Inspection of Fig. 20 indicates that damping ratios of the order on 65%-75% of critical
provide an approximately linear phase shifl for the frequency ratio range of 0%-40%. This
approach is desirable if a proper time relationship is to be maiqtained between the harmonic
components of a complex input.
EXAMPLE 5
A particular pressure transducer consists of a circular steel diaphragm mounted in a housing.
One side of the diaphragm is exposed to varying pressures, which cause the diaphragm to
deflect. The elastic deflection of the diaphragm is sensed by a piezoelectric quartz crystal
mounted within the housing. on the rear side of the diaphragm.
This transducer is effectively a second-order spring-mass system. The elastic stiffness
of the steel diaphragm provides the spring force. The effective mass of the vibrating
diaphragm contributes inertia. Damping in the system is slight (� = 0.025). The diaphragm
has a diameter of 1 2 mm and a thickness of I . 75 mm.
The transducer is being considered for measuring combustion-engine cylinder pres­
sures. Will this transducer have adequate frequency and phase response at typical engine
speeds of 3000 to 6000 rpm? Whal is the amplification ratio at the transducer's natural
frequency?
Solution The first step is to estimate the deHection constant of the diaphragm and its
effective mass while vibrating. Appropriate elasticity calculations show that k "" 1 3 1
MN/m and m "" 3. 7 g. Thus
1 72
The Response of Measuring Systems
or 29.9 kHz. The e ngine 's highest operaling frequency is
n
= (6000 rpm)(2n' rad/rev)/(00 s/min)
=
628 rad/s
or I 00 Hz. Thus
n
- = 0.0033
"'•
From Figs. 1 9 and 20, we see that the transducer will perform with negligi ble phase shift
and an amplification ralio or unity; subslilution of � and 0/wn i nlo Eqs. (25b) and (25c)
verifies this conclusion. The transducer's response characteristics are well suiled to the
applicalion.
Ir the lransdu cer were subjected to pressure variations near its natural frequency,
however, the ampl i fical ion ralio would be enormous; from Eq. (25c),
PJ
P..
=
I
Jo + c2<0.025>< 1 >12
= 20
This device is undoub1edly nol intended to operate at frequencies that high.
16.3
General Periodic Forcing
(d2s)
dt2 + (ds)
We now relum 10 the general c ase of periodic forcing as suggesled in
convenience we will restate Eqs. (8a), (9) and (9a)
111
g;:
where
F(t)
and
C.
m
=
j
dt
oo
A2
n;l
. + 82
.
and
(26a) i nlo Eq. ( 26), we oblain
2
+ ('
+ ks = F(t )
2° + L c. cos(nOt
A
(dls)
dt (ds)
dt
By su bs1 i1u 1 i ng Eq.
g"
=
('
-
Section 1 3. For
-
I/In )
Bn
tan !/>. = An
Ao
�
+ ks = - + L... C. cos (n Ot - !/>. )
2
n= t
(26)
(26a)
(26b)
(27)
Allhough 1his expression appears quite formidable, we can easily recognize that it
y ields a combination of the solulions given by Eqs. (2 1 ) and (25a). Using the reasoning
1 73
The Response of Measuring Systems
that the cosine terms on the right side of Eq. (27) will give results similar to those of a
harmonically forced system, we can write a solution for � < in the fonn
.
s = e-({ g</2m)r [A cos wndl +
1
00
B sin Wndl] + ro + L '• cos(nOI" - tl>n - '/10)
n =I
(28)
where
A
ro = 2ko .
rn =
[ 1 - ( 0 ) ] + [ n0 ] 2
c. /k
n
Wn
tan "'·
=
2 2
-;:=====================
� (�)
1 -
(no)2
2f Wn
Wn
-
It helps to recall that
Wn
= the undamped natural frequency = y{kg;
-;;;
As we discussed previously, the first tenn on the right side of Eq. (28) is transient and
dies out after several time constants. The remaining tenns, then, represent the steady-state
response.
EXAMPLE 6
(a) Write an expression for the steady-state response of the sing le-degree-of-freedom
system shown in Fig. 17 when subjected to the sawtooth forcing function.
(b) Let
m = I kg, or 2.2 lbm.
k = 1000 N/m, or 5 .7 1 lbf/in.,
' = 3 1 .6 N · s/m, or 0. 1 80 lbf · s/in.,
A = 10 N, or 2.248 lbf
Using these data (the SI values). obtain computer-plotted wavefonns for input frequencies
of 10. 30, and 50 rad/s.
1 74
The Response of Measuring Systems
Solution
(a) In general, the steady-slate response is given by
s
tan- 1
( !: ) ,
[ (�) ]
JA� + Bi,
c.
tan -1
�
1 - (:� r
In this case, the forc ing function, F(t), is
F(t) =
L -nI sin(11 Qt)
rr
2A 00
•= I
so that
both
Ao = 0 and
An
= 0. Thus
Tr
-1
tf>. = tan oo = z ·
Cn = Bn =
I
­
rr n
2A
Therefore,
s =
1
L
rr k • = I n
Z A co
[ (110) 2] 2 [2; % ]
(
cos n ot
- i - .P.)
- -=========
I -
"'•
+
n '2 2
The Response of Measuring Systems
and
"'•
{,·
=
=
� =
2
[fi;
-;;;
V
{
.
=
31.6
1000 "'63.3
=
v I 000 '°"
2v=
=
- -
{,·
=
rad/s,
3 1 .6
[2 l
63.3
N · s/m,
"' 0.50,
� = � = l cm '
IOOO
k
x o s (�)
. n �l.62
- ( 31.6 )
( i )
- I: n-1 -=========
[ I ( n O ) 2] 2 [ n1O ] 2
- 31.6
3 .6
( - � - 1/t. )
2
1/1. = tan - •
s =
2
:n:
=
I
oo
cos no1 -
n=I
•
- 1/I•
+
=========
; � ;; --==
[ · =_( ;.�6 r r + (;1�6 r
°"
cos 1101
I
Note that s is in centimeters.
The most practical approach to obtain i ng numerical resu lts for this part of the example
is through the use of a computer. Computer-plolled results are shown in Figs. 2 1 (a) through
2 l(c).
a
From the foregoing d isc uss ions it is apparent that if simple measurements are to be
made as rapidly as possible or. more importantly, if the i nput signal is continuous and
com p lex , rather definite limi t t i o ns are im posed by the measu ri ng system.
17
ELECTRICAL ELEMENTS
As we di scu ssed in Section 8, mos t measurement systems are c o mposed of a combination of
mechanic al and electrical e l eme n ts Ve ry often the basic detecting element of the sen sor is
mechanical and its output is immetliately transduced into an electrical signal by a secondary
element. The signal conditioning that follows is largely by e lec tric al means; however
termination sometimes requires convers io n lo someth i ng b icall y mech n ical such as
a con trol ler a galvanometer-type recorder or ploue r, and so on II is clear, the n , that
overall performance results from a co mb i n t i on of mechanical and electrical responses. In
previous sections we ha ve discussed the response of simple, pu rely mechanical systems. In
succeeding sect i ons we will look at the corresponding electrical elements.
.
.
a
1 76
as .
a
,
,
The Response of Measuring Systems
-1
411"
0
(a)
0/, Radians
- - - - Forclng/k
-- SY!ltam response
611"
10...
Oltd. • 0.9486
8
.;
[o
..
..
'
a:
-1
0
411"
211"
(b)
'
8
.;
[o
•
:
'
a:
-1
(c)
• '
I
'
0
FIGURE 2 1 :
,,
.
2w
"
'
•\
'
'
'
01, Radians
'
•
'
I
I
'
.
, ,
611"
4,,.
01.
'I
•
1 0...
0/010 = 1 .581 0
'
'
'
I
''
- - - - Forcing/k
-- SY!ltem response
611"
''
'
'
'
'
,
,
'
- - - - Forclnglk
-- System response
1 0...
Radians
Compulcr-plolled saWloolh wave response for the second-order system spec­
= IO rad/s; (b) for Q = 30 rad/s; (c) for Q = 50 rad/s.
ified in the text: (a) for Q
1 77
The Response of Measuring Systems
TABLE 1:
Some Basic Electrical Quantities and Relationships
Symbol
Definition
Unit
E
I
Q
Electric potential
Electric curre nt
Electric charge
volt
L
c
Electrical inductance
Electrical capacitance
R
Electrical resistance
(V)
ampere (A)
c ou lom b (C )
ohm (fl)
henry (H)
farad (F)
Some defining relationships:
For a capacitance: I = dQ/dt = C(d E/dt), E = Q/C
For a resistance: E =
(Ohm's law)
For an inductance: E =
I /dt) = L (d 2 Q/d1 2 )
IR
L(d
In preparation for the discussion that follows, Tab le I lists some fundamental electrical
quantities and defining relationships. Table 2 lists certain mechanical-electrical analogies.
For verification of these, the reader is referred to any basic electric circuits text (2) or physics
n
text (3 ]. I addition, at this point it is also useful to recall Kirchhoff's two laws for electrical
circuits, namely,
1. The al gebrai c sum of all currents en teri ng a ju nction point is 1.ero, and
2. The algebraic sum of all voltage drops taken in a given d irect i on around a closed
circuit is zero.
18
FIRST-ORDER ELECTRICAL SYSTEM
Consider the circuit shown in Fig. 22. Assume that the c apac i tor carries no in itial charge;
then let the SPOT (single-pole, double-throw) switc h be moved to contact A, thereby
insening the batte ry into the circuit. Now the capac i tor begins to c h arge ; what is the
response of the circuit in terms of the voltage across the c apac itor"/
I�
R
+
c
FIGURE 22: Series resistance-capacitance circuit.
1 78
r
TABLE 2: Dynamically Analogous Mechanical and Electrical System Elements
Synibol
m
le
lcr
f1
F
T
Symbol
Mechanical Quantity
Mass. kg (Ihm)
Momem of inenia, kg ·
m2 (Ihm . in.2 )
Deftection constant, Nim (lbf/in.)
Ton.ional deftection constant, N · m/rad (lbf · in./Jud)
Damping coefficient, N · s/m (lbf · s/in.)
Torsional damping coefficient, N · m · s/m (lbf · in. • s/in.)
Force, N (lbl)
Tmque, N · m (lbf · in.)
m (in.)
JC
Tnnslational displacemem,
Angular displacement, rad
dx/dt
d9/dl
Translational velocity, mis (in.ls)
Angular velocity, rad/s (rad/s)
8
ll
d2x/dr 2
d 28/d1 2
Electrical Quantity
L
lndUCWK:e,
l/C
Reciprocal of capacitance, F- 1
R
Resistance, g
E
Voltage, V
Q
Charge, C
H
dQ/dl
Current, A
Forcing frequency, rad/s (rad/s)
ll
Forcing frequency, rad/s
Translational acceleration, mts2 (inJs2)
2
Angular acx:eleralion, rad/s2 (rad/s )
d2 Q/dt 2
Rate of change of current, A/s
uppemise C his long been used as the symbol fot Clp8Cilll!Ce . It hlS also been assigned as the symbol far lhe SI unit for elec:lric
clwJI, the coulomb. Ullewise, n is the SI symbol far miSWICe, ohms. It Is also widely used IO iepn:sent an exciting fiequency in ndians per
second. In 1his text we will let the symbols ielain each moaning. Should the coniext not moire clear the nanings inteaded, we include clarifying
Nole: The
Sta1Cmcnts.
The Response of Measuring Systems
,
By employ i ng Kirchhoff's law of vo l tages we may write
(29)
bul
dQ
l = ­
dt
he nce
RC - �R = 0
dt + _g_
dQ
Solving, we have
Q
=
( -
cE I
We define the circuit time constant as
e-•/RC)
(29a)
t" = RC
The voltage drop across the capacitor is Ee = Q/C;
Ee = E
In
hence,
{I - e -• I•)
a similar manner, if the switch contact is
may write
(30)
moved from A to
B after the capacitor is
charged, we
IR + .f?.c = 0
(3 1)
for which
(3 1a)
(10),
It is appare n t, then, that Eqs. ( 14) and
which apply to a mechanical system,
hold equally well for the electrical circuit discussed here. Should the battery in Fig. 22 be
replaced with a s in uso id al voltage source analysis would show that equations such as (17)
and ( 1 9a,b,c) also apply to electrical systems. In each case, it is necessary to insert the
appropriate time constant and to properly interpret the response variable.
,
EXAMPLE 7
,
23(a) shows a s imple circuit c onsist i ng of a capac i tor a resistor and a 5-kHz voltage
source connec led in series. Delermine the amplitude and phase shift of the voltage appearing
across 1he capacitor. Compare this to lhe voltage across the resistor.
Figure
Solution
From Kirchhoff's law of voltages, we obtain
dQ
I
Rdt
+ Q = Eo cos !lt
C
1 80
The
Response of Measuring Systems
R = 1 200 0
C = 0.035 ,_.F
I
-
(a)
4> = 53°
4>!0. = 2.9 x 1 0-a s
1�
Eco
- = 0.6
Ee
(b)
FIGURE 23: Resist or and capacitor i n series: (a) excited by a 5000- H z voltage source; (b)
voltage across the capacitor for the ci rc ui t shown in (a). The angle </J is a phase lag.
By analogy to Section 1 5.2 [ Eqs. ( 1 5), ( 17), and ( 1 8)), the steadycstate solution of this
equation is
Q =
with
EoC
JI + (rQ) 2
cos(Ot
-
</J)
and
r = RC
for T the period of the excit i n g voltage. The voltage across the capacitor is
Therefore, the voltage amplitude across the capacitor i s
Ec0
£0
=
I
Jr + (rQ) 2
JI + (21'{ t/ T) 2
Observe t hat these resu lts are i de n tical to those ploucd in Figs. 14 and 15.
181
Ee
=
Q/C.
The Response of Measuring Systems
Numerically,
= 4.2
T = 1 200(0.035 x 10- 6 )
T = .!_
I
x 10- 5 s.
4
= 2 x 10- s
so
Eco
0.6
Eo =
and
"' = 53°
Figure 23(b) illustrates the resulting relationship.
The resistor behaves somewhat differently. In term s or Q, the voltage across the
resistor is
E,
=
EoRCQ
.
dQ
[- sm(Qt - </J)]
Rdi = J I + (TQ)2
or
E, =
Eo T rl
= cos( Q t + ir/2 - </J)
Ji + (trl )2
Thus, the resistor voltage has an amplitude
E,0
t" rl
2Tr T/ T
Ea = Ji + (tQ)2
Jl
+ (2Tr T/ T)2
and a phase lead ofTr /2-</J. Note thal lhe resistor voltage amplitude is small when rI T
«I
(at low excitation frequencies), whereas the capacitor voltage is small when T / T » I (al
high excitation frequencies). The contrasting behavior or the two elements is the basis ror
this circuit's application to high-pass and low-pass filters.
First-order ci rc u i ts may also be constructed using an inductor and a resistor. Similar
results can be developed, the primary difference being that time constant is instead r = L/ R.
19
SIMPLE SECOND-ORDER ELECTRICAL SYSTEM
Figure 24 illustrates a circuit consi sti ng or R, L, and C elements in series with a voltage
source. Referring to Table I for the voltage drop across each e le ment and then applying
Kirchhoff's law or voltages, we can write
( dt ) ( ddtQ )
2Q
L d-2
-
+
-
1 82
+
Q
-
c
= Eo cos nt
(32)
The Response of Measuring Systems
R
c
L
FIGURE 24 : An
RLC
circuit
We o i e this expression as having the same form as Eq. (24), and thus we can quickly
write a solution :
rec gn z
Q=
we
.
e-r/r [A cos w.JI + B sm
w0Jf] +
Eo cos(01 - fl)
(32a)
C,/[l/C - Lf22)2 + (Rf2)2
amp l tude across the capacitor,
If, for example, we consider the steady-state voltage
may write (see Tuble 1 )
Q
Ee = C
=
C ,/[ l / C
l
Using the ana ogies in Table 2 we may write
Wn =
and
i
Eo
- Lf22) 2 + (Rf2)2
(32b)
fu
(32c)
(32d )
where
a resonance f uency corresponding to the undamped
natural frequency of the mechanical system, and
Re = a critical resistance analogo s to critical damping
By algebraic manipulation, Eqs. (32a) and (32b) may now be wriuen in the dimensionless
forms, where Ee is the dynamic amplitude of the voltage across C; that is,
Wn =
req
u
Ee
=
Eo ,/[ I - (0/w.) 2) 2 + [2(R /Rc)(Q/w0 ) ) 2
I
and
(32e)
(32 0
Except
y l
follows,
electric
ha have i
for the s mbo s , we see that Eqs. (32e) and (320 are identical to Eqs. (25c)
and (25b), respectively. lt
then, that figs. 19 and 20 apply eq u l l well to the
circuit t t we
just nves tigated.
ay
1 83
The Response of Measuring Systems
:;;-
;:..
11
G-
G(O )
1-------...::
I
g
!
g-
u:
Dynamic calibration
Frequency, I
0
25: Comparison of s1atic and dy namic calibrations. In a stalic calibration, G(O)
is measured and used 10 approx i male G(/) at h ig her frequencies. In a dy namic calibration,
G ( /) ilSelf is measured.
RGURE
20
CALIBRATION OF SYSTEM RESPONSE
The final and positive proo f of a measuring system's performance is direct measurement of
the actual syslem's response lo a completely defined and known input. Physical modeling
of system response (as in Sections 8-1 9) provides importan l understanding of the device's
essential characteristics, such as its time co ns 1 an t or natural frequency. However, if lhe
device is too complex to model well or if the modeling assump1ions are lenu ous , greaier
accuracy can be o b1ai ned 1hrough an experimental test of lhe sysle m ' s response. Such
tesli ng is always required i n hi gh -accu racy ex peri men tati on .
Testing a meas urin g sys1em 's response is really a process of calibration. Whal output
is obtained for an i n put of given arnplilude and freq uency ? Can we experimentally evaluale
the behav i or of the system when ii is con fronted with an input that is rapidly changing wilh
lime? To do this effectively, we must also have a time-varying input, or calibration source,
thal is precisely know n .
Calibration sources h avi ng sinusoidal lime varia1ion are undoubtedly the easiest lo
produce and lhe mosl used (recall Sec1ion 7). With electrical inpul sign als , commercial
vollage sources are easily a pp l ied to th i s lask. Various classical complex waveforms, such
as square waves or sawlooth waves, may also be employed. f'or exam p le , the response of
many electrical componen1s can be judged through the use of square-wave inputs. A s ki l led
technician can frequently pinpoint reasons for distortion by observing 1he lested apparatus's
treatment of square-wave i npu 1 . 4 More importantly, such tests can identi fy limits beyond
which system performance is q ue stion able .
For some measurands, even a sine-wave test is relatively hard to implement: How do
you create a well-defined sinusoidal variation in tem perature or H u id velocily? For such
variabl�. slep inputs or pulse inputs arc often easier to produce, and they can also yield
useful i n format ion about lhe syslem's rise lime, time constant , or resonant frequencies.
In cases where any time variation of th e i npu l is too difficull or loo expensive to
4Square-wave testing is so useful in adjusting lhe frequency response of h()(-wirc anemometer bridges lhal
many bridges � sold with built-in sqwue-wave generators.
1 84
The Response of Measuring Systems
produce, constant i nputs are occasionally substituted. For example, we may measure the
output of a temperature sensor at several different constant temperatures to obtain Tmeuuml
as a function of Tac1ual . This tells us how the sensor responds to i nputs of different ampl itude.
We would then assume that the measured amplitude relationship appl ied for all frequencies
of interest.
In this sense, we can distinguish two types of calibrations: static calibration and
dynamic calibration. In Section 7, we noted that the frequency response of a system was
the ratio of output to input amplitude at a given frequency: G(/) = V0/ Vi. In a static
calibration, we measure V0 and Vi at zero frequency to obtain G(O). We then assume that
G(/) ::::; G (O) over the frequency range of interest (Fig. 25). In a dynamic calibration,
we measure V0 and V; over a range of frequencies to obtain G(/) itself. Static calibration
approximates dynamic calibration at frequencies low enough that the dev ice 's frequency
response is flat. Static calibrations must always be used with caution and an awareness of
their inherent frequency limitations.
21
SUMMARY
Response is a vital feature of a measuring system's ability to acc urately resolve time-varying
inputs. I n this chapter, we have examined various important types of response, procedures
for testing response, and simplified physical models of the response of instruments.
1. Amplitude response, frequency response, phase response, and rise time are each
imponant aspects of measuring-system performance. Amplitude response is gener­
ally defined by the overload condition of an instrument. Frequency and phase respon se
are considerations in determining the range of frequencies over which an instrument
is accurate (Sections 2-7).
2. Each clement in a measuring system must have adequate frequency and phase response
for the measurand at hand. Some system elements, like filters, are selected specifically
for the limitations of their frequency response (Section 7).
3. Often, a physical model of a measuring device can identify the imponant character­
istics of its behavior. Mechanical models rely on the identification of masses, spring
forces, and damping (Sections 8- 1 1).
4. Selection of the correct amount of damping is vital to obtai ni ng optimal frequency
response (Sections 1 1 , 15, 1 6, 1 8).
S. The spring-mass·damper system is a fundamental model of the dynam ic response of
mechanical systems (Sections 1 2-16). Some systems behave as i f massless or first
order, especially temperature sensors (Section 1 5). Other systems have mass and are
second order; examples include accelerometers, elastic diaphragm transducers, and
various moving mechanisms (Section 16).
6. Electrical-system response is similar to mechanical-system response. Models for
electrical systems yield results analogous to those for mechanical systems (Sec­
tions 1 7- 1 9).
7. For a first-order system, the system's time constant, t , is of critical imponance to
ils response. The time constant should normally be small compared to the period
of forcing, T, i n order to obtain good response. First-order response to a step i n put
1 85
The Response of Measuring Systems
is described by Eq. ( 14) and Fig. 1 3 ; response to hannonit: forcing is described by
Eqs. ( 17) and ( 1 9a,b,c) and Figs. 14 and 15 (Sections 15, 1 8).
8. For a second-order system, both the natural frequency, "'n• and the critical damping
ratio, t, m ust be considered. For most systems, good rei;ponse is obtained when
the natural frequency is large compared to the forcing frequency,
and when the
damping ratio is 65%-75%. Second-order response to a sitep input is described by
Eqs . (22) and (23) and Fig. 1 8; response to harmonic input is described by Bqs. (25a,
b, c) and Figs. 19 and 20 (Sections 16, 1 9).
0,
9. Experimental determination of system res ponse , or calibral ion, is often required . A
sine-wave test is sometimes useful (Sec tion 7). Other kinds of test signals, such as
square-wave or step input, can also be applied. Static calill>rations (for f = 0) are
occasionally used when dynamic calibrations (for f > 0) are too difficult (Section 20).
SUGGESTED READINGS
Dorf. R. C . • and R. H.
Hall, 2004.
B i shop.
Modem Control Systems. 1 0th ed. Upper Saddle River,
Floyd. T. L. Principles of E�ctric Circuits. 7th ed. Upper Saddle River,
Meriam, J.
N.J.: Prentice
N.J.: Prentice Hall, 2003 .
L, L G. Kraige , and W. J. Palm. EngiMering Mechanics: �L 2-Dynamics. Sth ed.
New York: John Wiley, 2002.
Raven, F. Automatic Conrrol Engineering. 5th
ed. New York: McGraw-Hill, 1 994.
Kreyszig, E. Advanced Engineering Marhemarics.
8 th ed. New York: John Wiley, 1 998.
PROBLEMS
1.
2.
3.
A temperature measuring system (assumed to be a Hist-order sy:stem) is e xci ted by a 0.25Hz harmonically varying i n put . lhhe ti me constant of the system is 4.0 s and the indicated
amplitude is IO"F, what is the true temperature"!
A mercury- i n- g lass thermometer i ni ti al ly at 25°C is suddenly im mersed into a liquid that is
maintained at l 00°C. After a time i nterval of 2.0 s, the thermome:ter reads 76°C. Assuming
a first-order system, estimate the time constant of the thermom1!ter.
A 5-kg mass is statically suspended from a load (force) cell arid the load cell deflection
caused by this mass is 0.0 1 mm. Est i m ate the natu ra l frequenci1 of the load cell.
The Response of Measuring Systems
1000 lbl
-1000 lb!
>--
I+--
0.002 s
--+!
FIGURE 26: Dynamic load for Problem 6.
4.
If the load cell of Proble m 3 is assumed to be a second·order system with negligible
damping, detennine the practical freq uency range over which it can measure dynamic
loads with an inherent error of less than 5%.
5.
has a damping ratio of 0. 707, determine the practical
frequency range over which it can measure dynamic loads w ith an inherent error of less
than 5%.
6.
A manufacturer lists the specifications of a dynamic tension<ompression load cell as
lf the load cell of Problem 3 actually
follows:
Undamped natural frequency = I 000 Hz
Damping ratio = 0.707
Sensitivity = 10 mV/lbf
Stiffness = 1 00,000 lbf/in.
If a dynamic force as indicated in Fi g . 26 is applied to the load cell. determine the
steady·state output voltage as a function of time .
7.
8.
9.
Plot the voltage output of Problem 6 as a function of time using al leasl lhe first
five terms.
An RC c ircuit as shown in Fig. 22 is required lo have a time constant of I ms. Determine
three dilfereru combinations of R and C lo accomplish this .
A tank containing an initial volume of water Vo. discharges waler from an opening at the
bottom of the tank. lf the discharge rate is directly proportional lo the volume of water in
the tank, determine the time constant for this system.
10.
For the tank of Problem 9, determine the time constant if the initial discharge rate is Q
ll.
A temperature sensor is expected to measure an input having freque ncy components as
high as 50 Hz with an error no greater than 5%. What is the maximum time constant for
the temperature sensor that will permit this measurement?
12.
(liters/min).
A thermocouple with a time constant of 0.05 s is considered to behave as a fir.i-order
system. Over what frequency range can the thermocouple measure dynamic temperature
lluctuations (assumed to be harmonic) with an error less than 5%?
1 87
f{I)
The Response of Measuring Systems
1 00 1bl
0.10 S
O.OS s
0.1 5 s
FIGURE 27: Fon:ing for Problem 18.
13.
A I 00-µF capacilor is charged to a vol1age or 100 V. Al li me t = O. ii is discharged lluoogh
a 1 .0-MO resis1or. Detennine the lime for the vollage across the capacilor to reach 10 V.
14.
A pressure lransducer behaves as a second-order system. Ir the undamped natural f're­
quency is 4000 Hz and the damping is 75% or critical. detennine the frequency range(s)
over which the measurement error is not greater than 5%.
15.
What will be the frequency range(s) for Problem 14 if the damping ralio is changed to
0.5?
16.
Con sider the pressure lransducer or Problem 14 to be damaged such that its viscous
damping ralio is unknown. When the lransducer is subjecled lo a harmonic input or
2400 Hz, the phase angle between the oulput and inpul is measured as 45°. With lhis in
mi nd , detennine the error when the transducer is used to measure a hannonic pressure
signal or 1 800 Hz. What is the phase angle between the input and output at lhis frequency?
17.
A force transducer behaves as a second-order system. If the undamped natural frequency
or lhe transducer is 1 800 Hz and its damping is 30% or critical, detennine the error in
lhe measured force for a hannonic input or 950 Hz. What is the magnitude or the phase
angle'!
18.
Consider a second-order system wilh a damping ratio or 0. 70 and a undamped nawral
frequency of 50 Hz. Ir the value of is 100 lbf/in., detennine the steady-slate OUlpUI if
the forcing is as shown in Fi g . 27.
k
19.
Refer to Fig. 28.
( a ) What is the lime conslant if X = 200 µF and R= 10 kO?
( b ) Whal is the vollage across lhe resislor 0.5 s after the swilch is closed?
20.
Refer to Fig. 28.
( a ) What is the time constant if X
= 500 mH and R = l 0 O?
( b) What is the voltage across lhe resistor 0.02 s after lhe switch is closed?
( c ) What is the vollage across lhe resistor 0.05 s after lhe swilch is closed?
1 88
The Response of Measuring Systems
E -=-
FIGURE 28: Circuit for Problems · 1 9 and
20.
REFERENCES
(11 Dorf, R. C., and R. H. Bishop. Modem Control Systems.
NJ.: Prentice Hall, 2004.
[2) Floyd, T. L.
Hall, 2003.
Principles of Electric Cin:uits. 7th ed. Upper Saddle River, NJ.:
[3) Halliday, D., R. Resnick, and J. Walker.
New York: John Wi ley, 2004.
ANSWERS TO SELECTED PROBLEMS
l
3
l .57°F
f "' 1 60 Hz
4
35 Hz
9
T
= Vo/m0
14
f ::::: 1 , 1 24 Hz
17
Error
19
=
10th ed. UpPc:r Saddle River,
27 .8%: <f> ::::: 24°
(a) 2.0 s (b) e = 0.78£ volts
Prentice
Fundamentals of Physics. 11h ed., exlcnded.
Sensors
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
INTRODUCTION
LOADING OF THE SIGNAL SOURCE
THE SECON DARY TRANSDUCER
CLASSIFICATION OF FIRST-STAGE DEVICES
VARIABLE-RESISTANCE TRANSDUCER ELEMENTS
SLIDING-CONTACT DEVICES
THE RESISTANCE STRAIN GAGE
THERMISTORS
THE THERMOCOUPLE
VARIABLE-INDUCTANCE TRANSDUCERS
THE DIFFERENTIAL TRANSFORMER
VARIABLE-RELUCTANCE TRANSDUCERS
CAPACITIVE TRANSDUCERS
PIEZOELECTRIC SENSORS
SEMICONDUCTOR SENSORS
LIGHT-DETECTING TRANSDUCERS
HALL-EFFECT SENSORS
SOM E DESIGN-RELATED PROBLEMS
SUM MARY
INTRODUCTION
The !irst conlacl th al a measuring s ysie m has wilh lhe measurand is lhrough the i nput sample·
accepted by lhc detecting element of lhe first slage. This acl is u su ally accompanied by lhe
immediate lransduction of lhe i nput inlo an analogous fonn.
The medium handled is information. The detector senses the information input, fm.
and then lransduces or conve rt s it to a more convenient form, foul · The relationship may be
expressed as
foul = /Uin l
( l a)
furiher,
Transfer e fficie ncy
1 92
=
lout
/in
-
( l b)
Sensors
This cannot be more than unity, because the pi ckup cannot generate information but can
only receive and process it. Obviously, as high a transfer efficiency as possible is desirable.
Sensitivity may be expressed as
dloui
11 = -­
dlu.
( le)
Very often sensitivity approximates a constant; that is, the output is the linear function of
the inpul
2
LOADING OF THE SIGNAL SOURCE
Energy will always be taken from the signal source by the measuring system, which means
that the information source will always be changed by the act of measurement. This is an
axiom of measurement. This effect is referred to as loading. The smaller the load placed
on the signal source by the measuring system, the better.
Of course, the problem of loading occurs not only in the first stage, but throughout the
entire chain of elements. While the first-stage detector-transducer loads the input source,
the second stage loads the first stage, and finally the third stage loads the second stage. In
fact, the loading problem may be carried right down to the basic elements themselves.
In measuri ng systems made up primarily of electrical elements, the loading of the
signal source is almost exclusively a func tion of the detector. Intermediate modifying
devices and output indicators or recorders receive most of the energy necessary for their
functioning from sources other than the signal source. A measure of the quality of the first
stage. therefore, is its ability to provide a usable output without draining an undue amount
of energy from the signal.
3
THE SECONDARY TRANSDUCER
As an example of a system of mechanical elements only, consider the Bourdon-tube pressure
gage, shown in Fig. I. The primary detecting-transducing element consists of a circular
tube of approximately e11iptical cross section. When pressure is introduced, the section of
the ftattened tube tends toward a more circular form. This in tum causes the free end A to
move outward and the resulting motion is transmitted by link B to sector gear C and hence
lo pinion D, thereby causing the indicator hand to move over the scale.
In this exam ple, the tube serves as the primary detector-transducer, changing pressure
into near linear displacement. The linkage-gear arrangement acts as a secondary transducer
(linear to rotary motion) and as an amplifier, yielding a magnified output.
A modification of this basic arrangement is to replace the linkage-gear arrangement
with either a differential transformer (Section 1 1 ) or a voltage-dividing potentiometer (Sec­
tion 6). In either case the electrical device serves as a secondary transducer, transforming
displacement to voltage.
As another example, let us analyze a simplified compression-type force-measuring
load cell consisting of a short column or strut, with electrical resistance-type strain gages
(see Section 7) attached (Fig. 2). When an applied force dell.eels or strains the block, the
force effect is transduccd to deflection (we are interested in the unit deflection in this case).
The load is transduced to strain. In tum, the strain is transformed into an electrical resistance
change. with the strain gages serving as secondary transducers.
1 93
Sensors
t
Pressure
FIGURE I : Essentials of a Bourdon-tube pressure gage.
Bonded wire
slrain-gage
elements
FIGURE 2: Schematic representation of a strain-gage load cell. The block forms the primary
detector-transducer and the gages are secondary transducers.
Sensors
Primary
detector-
Intermediate
analogous I
I
I
I
signal
,___...___-!
I
I
I
I
i- - - - - - -1
I
I
Input signal
...._
..
....__
...
�
I transducer
I
L_ ____J
FIGURE
4
3:
i- - - - - - -1
Secondary
transducer
Analogous
driving
I
I
signal
t--1---''---<�
I
I
L _____J
Block diagram or a first-stage device with primary and secondary transducers.
CLASSIFICATION OF FIRST-STAGE DEVICES
It appears, therefore, that the stage-one ins1rumenta1ion may be or varying basic complexity.
depending on the number or operations perfonned. This leads 10 a classification or firsl­
stage devices as follows:
Class L First-stage element used as detector only
Class II. F'Jl'St-stage elements used as detector and single transducer
Class III. First-stage elements used as detector with two transducer stages
A generalized first stage may therefore be shown schematically, as in Fig. 3.
Stage-one instrumentation may be very simple, consisting or no more than a mechan­
ical spindle or contacting member used to transmit the quantity lo be measured to a sec­
ondary transducer. Or ii may consist of a much more complex assembly of elements. In
any event the primary detector-transducer is an in1egral assembly whose function is ( 1 ) 10
sense selectively the quantity of interest. and (2) lo process the sensed information into
a form acceptable 10 stage-two operations. It does not present an output in immediately
usable fonn.
More often than not the initial operation perfonned by the first-stage device is 10
transduce the input quantity into an analogous displacemcnl. Without auempting 10 for­
mulate a completely comprehensive list, lei us consider Table I as representing the general
area of lhe primary detector-transducer in mechanical measurements.
We make no auempt now lo discuss all the many combinations of elements listed in
Table 1. In most cases we have referred in the table to sections where thorough discussions
can be found. The general nature of many of the elements is self-evident. A few are of
minimal importance, included merely 10 round out the list. However. we can make several
pertinent observations at this point.
Close scrutiny ofTable I reveals Iha!, whereas many of the mechanical sensors trans·
duce the input 10 displacement, many of the electrical sensors change displacement 10 an
electrical output. This is quite fortunate, for it yields practical combinations in which the
mechanical sensor serves as the primary transducer and the electrical sensor as the sec ­
ondary. The two most commonly used electrical means are variable resistance and variable
inductance, although others, such as photoelectric and piezoelectric effects, are also of
considerable importance.
Sensors
TABLE
1: Some Primary Detector-Transducer Elements and Operations They Perform
Element
Operation
I. Mechanical
A. Contacting spindle, pin, or finger
Displacement to displacement
B. Elastic member
I . Load cells
a. Tension/compression
b. Bending
c. Torsion
2. Proving ring
3. Bourdon tube
4. Bellows
S. Diaphragm
6. Helical spring
7. Torsional spring
Fort:e to linear displacement
Fort:e to linear displacement
Torque to angular displacement
Force to linear displacement
Pressure to displacement
Pressure to displacement
Pressure to displacement
Fort:e to linear displacement
Torque to angular displacement
C. Mass
I. Seismic mass
2. Pendulum
3. Pendulum
4. Liquid column .
Fort:ing runction 10 relative displacement
Gravi1a1ional acceleration 10 frequency or period
Force to displacement
Pressure to displaccmenl
D. Thermal
I . Thermocouple
2. Bimaterial (includes meocury in glass)
3. Thermistor/RID
4. Chemical phase
S. Pressure thennometer
Temperature
Temperature
Temperature
Temperature
Temperalure
E. Hydropneumatic
Static
a. Float
b. Hydrometer
2. Dynamic
a. Orifice
b. Venturi
c. Pitot tube
d. Vanes
e. Turbines
to electtic potential
to displacement
to resistance change
to phase change
10 pressure
I.
Fluid level 10 displacement
Specific gravi1y to relative displacement
Fluid veloci1y to pressure change
Fluid vcloci1y to pressure change
Fluid velocily to pressure change
Velocity 10 force
Linear to angular vclocily
1 96
Sensors
TABLE 1: (continued)
Element
II. Electrical
A. Resistive (Sections 5�)
Operation
I. Contacting
Displacement to resistance change
2. Variable-length conductor
Displacement to resistance change
4. Variable dimensions of conductor
Displacement to resistance change
lo resistance change
Temperature to resistance change
3. Variable-area conductor
5. Variable resistivity of conductor
B. Inducti ve (Sectio ns 10-12)
I. Variable coil dimensions
2. Variable air gap
3. Changing core material
4. Changing core positions
5. Changing coil positions
6. Moving coil
7. Moving permanent magnet
8. Moving core
Strain
Displacement to change in inductance
in inductance
to change in inductance
Displacement to change in inductance
Displacement to change in inductance
Velocity lo change in induced voltage
Velocity to change in induced voltage
Velocity to change in induced voltage
Displacement to change
Displacement
C. Capacitive (Section 1 3)
I. Changing air gap
2. Changing plate areas
3. Changing dielectric constant
D.
Piezoelectric (Section 14)
Displacement
Displacement
to change
lo change
in capacitance
in capacitance
Displacement 10 change in capacitance
Displacement
to voltage and/or
voltage lo displacement
E. Semiconductor junction (Section 1 5 )
I . Junction threshold voltage
2. Photodiode current
Temperature 10 voltage change
Light i ntensity 10 current
F. Photoelectric (Section 1 6)
I.
Photovoltaic
2. Photoconductive
3. Photoemissive
G. Hall Effect (Section 17)
• Also sensitive lo wavelength of light.
Light intensity to voltage•
Light intensity
Light intensity
Displacement
lo resistance change'
lo current'
lo vollage
Sensors
In addition to the inherent compatibility of the mechano-electric transducer combination, electrical elements have several important relative advantages:
I. Amplification or attenuation can be easily obtained.
2. Mass-inertia effects are minimized.
3. The effects of friction are minimized.
4.
An output power of almost any magnitude can be provided.
S. Remote indication or recording is feasible.
6. The transducers can often be miniaturized.
Most of the remainder of this chapter is devoted to a discussion of electrical trans­
ducers.
5
VARIABLE-RESISTANCE TRANSDUCER ELEMENTS
Resistance of an electrical conductor varies according to the following relation:
pL
R=­
A
(2)
where
R = resistance(Q) ,
l = the length of the conductor (cm),
2
A = cross-sectional area of the conductor (cm ) ,
p
= the resistivity o f material (Q · cm)
Many sensors are based on changes in the factors determining resistance. Some exam­
ples include sliding-contact devices and potentiometers, in which l changes (Section 6);
resistance strain-gages, in which L. A, and p change (Section 7); and thermistors, photocon­
ductive light detectors, piezoresistive strain gages, and resistance temperature detectors, in
which p changes (see, respectively, Sections 8, 15 and 1 6).
Probably the simplest mechanical-to-electrical transducer is the ordinary
in
which resistance is either zero or infinity. It is a yes-no, conducting-nonconducting device
that can be used to operate an indicator. Here a lamp is fully as useful for readout as a
meter, since only two values of quantitative information can be obtained. In its simplest
fonn, the switch may be used as a limiting device operated by direct mechanical contact (as
for limiting the travel of machine tool carriages) or it may be used as a position indicator.
When actuated by a diaphragm or bellows, it becomes a pressure-limit indicator, or if
controlled by a bi me tal strip, it is a temperature limit indicator. It may also be combined
with a proving ring to serve as either an overload warning device or a device actually limiting
load carrying, such as a safety device for a crane.
switch
6
SLIDING-CONTACT DEVICES
Sliding-contact resistive transducers convert a mechanical displacement input into an elec­
trical output, either voltage or current. This is accomplished by changing the effective
1 98
Sensors
Slider or brush
Resistance wire
FIGURE 4: Variable resistance consisting of a w i re and movable contactor or brush . This
is often referred to as a slide wire.
length L of the conductor in Eq. (2). Some form of electrical resistance element is used,
with which a contactor or brush mainlains electrical contact as it moves. In its ·simplest
form, the device may consist of a s tretched resistance wire and s lider, as in Fig. 4. The
effective resistance existing between either end of the wire and the brush thereby becomes
a measure of the mechanical displacement. Devices of this type have been used for sensing
relatively large displacements I I [.
More commonly, the resistance element is formed by wrapping a resistance wire
around a form, or card. The turns are spaced to prevent shorting, and the brush slides across
the turns from one tum to the next. In actual practice, either the arrangement may be wound
for a rectilinear moveme nt or the resistance clement may be formed in to an arc and angular
movement used. as shown in Fig. 5(a).
Sliding-contact devices are also made using conductive films as the variable-resistance
elements, rather than wire.� [Fig. 5(b)]. Common examples include carbon-composition
films, in which graph i te or carbon particles are suspended i n an epoxy or polyester binder,
and ceramic-metal com pos i t ion films (or cermet), in which ceramic and precious metal
powders are com bi ned . In each case. the th i n film is supported by a ceramic or p lastic
(a)
(b)
FIGURE 5: Angular-mOlion variable resistance, or potentiometer: (a) wire wound, (b)
carbon composition.
Sensors
backing. Conductive-film devices are less expensive than wire-wound devices. The carbon­
film devices, in particular, have outstanding wear characteristics and long life [2), although
they are more susceptible to temperature drift and humidity effects.
These dev ices are commonly called resistance potentiometers, ' or simply pots. Vari­
ations of the basic angular or rotary form are the multitum, the low-torque. and various
nonlinear types (4). Multitum potentiometers are available with various numbers of revo­
lutions, sometimes as many as 40.
6.1
Potentiometer Resolution
The resistance variation available from a sliding-contact moving over a wire-wound resis­
tance element is not a continuous function of contact movement. The smallest increment
into which the w hole may be divided detennines the resolution. In the case of a wire-wound
resistor, the limiting resolution equals the reciprocal of the number of turns. I f 1 200 turns
of wire are used and the winding is linear, the resolution w i l l be 111200, or 0.09083%. The
meaning of this quantity is apparent: No matter how refined the remainder of the system
may be, it w i l l be impossible to divide, or resolve, the input into parts smaller than l/ 1 200
of the total po ten tiometer range.
For conductive-film potentiometers, resolution i s negligibly small and variation of
the slider's contact resistance is a more significant l i mitati on.
6.2
Potentiometer Linearity
When used as a measurement transducer, a linear potentiometer is normally required. Use
of the term linear assumes that the resistance measured between one of the ends of the
element and the contactor is a direct linear function of the contactor position in relation lo
that end. Linearity is never completely achieved, however, and deviation limits are usually
supplied by the manufacturer.
7
THE RESISTANCE STRAIN GAGE
Experiment has shown that the application of a strain to a resistance clement changes its
resistance. This is the basis for the resistance strain gage. A resistance element, such as a
fine wire, is cemented to the surface of a member to be strained. Assuming a tensile strain,
the element lengthens and its cross-sectional area decreases. From Eq. (2), we see that both
changes increase the wire's resistance. In addition, the wire's resistivity changes when the
wire is strained. Thus, every factor in Eq. (2) varies simultaneously.
This device is of sufficient importance in mechanical measurements to warra nt a more
complete discussion than can be given at this point.
8
THERMISTORS
Thermistors are .1!!5!:!a
!. ll y sensitive variable resistors made of ceramic-like semiconduct­
ing materials. Ox ides of manganese, nickel, and cobalt are used in formulations having
resi stances of 100 to 450,000 Q · cm.
1 U ro
n nunalely. another entirely different device is also called a po1cntiomc1er. h is the \'oltage-measuring
instrument wherein a slalldanl reference vollage is adjusled 10 counlerbalance !he unkoown volragc (see [31). The
1wo devices arc different and must not be confused.
200
Sensors
These devices have two basic applications: ( I ) as temperature-detecting elements
used for the purpose of measurement or control, and (2) as electric-power-sensing devices
wherein the thermistor temperature-and hence resistance-are a function of the power
being dissipated by the device. The second application is particularly useful for measuring
radio frequency power.
9
THE THERMOCOUPLE
two dissimilar metal s are in contact, an electromotive force ex i sts whose magnitude
is a function of several factors, including temperature. Junctions of this sort, when used to
measure temperature, are cal led thermocouples. Often t he j u nction is formed by twisting
and welding together two wires.
While
Because of its small size, its reliability, and its relatively large range of usefulness,
the thermocouple is a very important primary sensing element.
10
VARIABLE-IN DUCTANCE TRANSDUCERS
Inductive transducers are based on the voltage output of an inductor (or coil) whose induc­
tance changes in response to changes in the measurand. The coil is often driven by an ac
excitation, although in dynamic measurements the motion of the coil relative to a permanent
magnet may create sufficient voltage. A classification of inductive transducers, based on
the fundamental principle used, is as follows.
I. Variable self-inductance
(a) Single coil (simple variable
reluctance)
(b) Two coil (or single coil with center tap ) connected for inductance rat i o
2. Variable mutual i nd uctan ce
(a) Simple two coil
(b) Three coil (using series oppos it ion)
3. Variable reluctance with
(a) Movi ng
pcrmanem magnet
iron
(b) Moving coil
(c) Moving magnet
The inductance of a coil, L. is i n fl u e nced by a number of factors, including the number
of turns in the coil, the coil size, and especially the permeabil ity of the magnetic flux path
t hat passes through the center of the coi l. The mag ne tic Hux path forms a closed loop that
ex tends outside the coi l. Ofte n , a magnetic material, such as iron, will be used in the Hux
path, commonly in conjunction with one or more air gaps . Because the air gaps have a
much lower magnetic permeab i l i ty than the iron, they control the inductance of the coi l .
1l1us, the variation in the th ic kness of an air g a p is often the primary mea.�urand sensed by
a variable-inductance device.
201
Sensors
Some coils are wound with only air as the core material. They are generally USed
only with relatively high frequency excitation; however, t hey will occasionally be found
in transducer circuitry. An express ion that may be used to estimate the inductance of a
straight, cylindrical air-i:ore coil is as follows:
(3a)
where
L = inductance (JLH)
d = coil diameter (in.)
n
I = coil length (in.)
= number of lums
This expression appli es for I > 0.4d (5].
When the flux path includes both a magnetic material and an air gap, the inductance
may be estimated as follows:
where
L
hm
ha
L
=
=
=
µ.m =
µ..
=
n2
-h-,n--�h-.­
-µ._m_A_m + -µ..-A-a
(3b)
inductance (H)
length of flux path in magnetic material (m)
length of flux path in air gap (m)
permeability or magnetic material (Him)
=
permeability or air
=
4rr
"" µ.o , permeabi l i ty or free space
Am
Aa
=
=
n ==
x
10- 1 (Him)
cross-sectional area of Hux path in iron (m2 )
cross-sectional area of flux path in air (m2 )
number of turns
The quantities (Ir /JLA) in the denominator are called reluctances and have the same relation­
ship to magnetic Hux that resistance has to electric cunent. If additional gaps or magnetic
materials lie in the Hux path, the associated values of
would be added to the sum
i n the denominator (6).
In many instances, the permeability of the magnetic material is sufficiently high that
only the air gaps need to be considered. In such cases, Eq. (3b) reduces to
(h /µ.A)
(3c)
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Sensors
When an ac exciiation is used, the inductive reactance and de resistance of the coil
detennine the electrical response of the measuring circuit. The inductive reaciance is
(4a)
w here
==
inductive reac tance (Q)
f
==
frequency of applied vol!age (Hz)
L
==
induc!ancc (H)
XL
The total impedance of the coil is
(4b)
in which R i s the de resisiance. The highec the inductance of a coil relative to its resistance,
the higher is said to be its quality, which is designated by the symbol Q XL/ R. In most
cases, high Q is desired.
Inductive transducers may be based on variation of any of the variables indicated in
the foregoing equations, and most have been tried at one time or another. The fol lowi ng
are representative.
==
10.1
Simple Self-Inductance Arrangements
When a s i mp le s i ng le coil is used as a transducer element, the mechanical input usually
changes the reluciance of the flux path generated by the coil, thereby changing its ind uct ance.
The change in induciance is then measured by suiiable c irc ui try, indicating the value of the
input. The Hux path may be changed by a change in air gap; however, a change in either
the amount or type of core material may also be used [7 ,8).
The arrangement shown in Fig . 6 i ncl udes two air gaps whose thickness will vary in
respon se 10 the movement of an armature. The flux path runs through the upper horseshoe
of iron, across one air gap, through the iron armature, and back across the other air gap.
Each of the four parts of the path contributes a re luctance, but those from the air gaps will
likely dominate the total reluctance, owing to the low permeability of air relative 10 iron.
Figure 7 illustrates a form of rwo-coil self-inductance. (This may also be thought of
as a single coil with a center tap.) Movement of the core or armature alters the re lat i ve
inductance of the two coils. Devices of this type are usual ly incorporated in some form
of inductive bridge circuit in which variation in the induc!ance ratio between the two coils
provides the output.
10.2
Two-Coil
Mutual-Inductance Arrangements
Mutual-inductance arrangements using two coils are shown in Figs. 8 and 9. Figure 8
il lustrates the manner in which these devices function. The magnetic flux from a power coil
is cou pled to a pickup coil, which su pplies the output. Input information, in the form of
armature displacement, changes the coupling between the c oi l s . In the arrangement shown,
the air gaps between the core and the armature govern the degree of coupling. In other
Sensors
FIGURE 6: A simple self-inductance arrangement wherein a change in the air gap changes
the pickup output.
arrangements the coupling may be varied by changing the rela1ive positions of the coils and
armature, either linearly or angularly.
Figure 9 shows the detector portion of an electronic micrometer (9) . Inductive cou­
pling between the coils, which depends on the reluctance of the magnetic-flux path, is
Nonmagnetic
material
Core or armalure of
magnetic material
FIGURE 7: TWo-coil (or center-lapped single coil) inductance-ra1io transducer.
204
Sensors
To second-stage
circtJitry
8:
FIGURE A mutual-inductance transducer. Coil A is the energizing coil and B is the
pickup coil. As the armature is moved, thereby altering the air gap, the output from coil B
is changed, and this change may be used as a measure of armature movement.
changed by the relative proximity2 of a permeable material. A variation of this has been
used in a transducer for measuring small inside diameters [ 10). In that case the coupl ing is
varied by relative movement between the two coils.
-Excilallon
v
-
Measured
distance
_l_
,
)
(
�
�
lb stage·two cirwHry
�
FIGURE 9: Two-coil inductive pickup for "an elec1ronic microme1cr."
2Sensors that detect the prcselK."C or position or a nearby objc.;t are often called proximity 1�ruors.
205
Sensors
Output voltage, e0
Displacement
(al
Input \IOllag &, e1
Input
displacement
(b)
FIGURE I O: The differential transfonner: (a) schematic arrangement (b) section through
typical transformer.
11
THE DIFFERENTIAL TRANSFORMER
Undoubtedly the most broadly used of the variable-inductance transducers is the differential
transfonner (Fig. IO), which provides an ac voltage output proportional to the displacement
of a core passing through the windings. It is a mutual-inductance device making use of
three coils generally arranged as shown.
The center coil is energized from an ac power source, and the two end coils, connected
in phase opposition, are used as pickup coils. This device is often called a linear variable
differential transformer or LVDT.
12
VARIABLE-RELUCTANCE TRANSDUCERS
In transducer practice, the tenn variable reluctance implies some form of inductance device
incorporating a permanent magnet. In most cases these devices are limited to dynamic
application, either periodic or transient, where the Hux lines supplied by the magnet are
cut by the turns of the coil. Some means of providing relative motion is incorporated into
the device.
In its simplest fonn, the variable-reluctance device consists of a coil wound on a
permanent magnet core (Fig. 1 1 ) . Any variation of the reluctance of the magnetic Hux
path causes a change in the Hux. As the Hux field varies, a voltage is induced in the coil,
according to Faraday's law:
V
=
206
d
-n - CI>
dt
(5 )
Sensors
Permanent
magnet
/
N
To lrequency merer
or oscllloscope
-0
s
FIGURE 1 1 : A simple variable-reluctance pickup.
where
V = induced voltage (V)
n = nu m ber of turns i n coil
<I> = magne ti c Hux lhrough coi l (Wb)
Since rhe rare of change of lhe flux depends direclly on lhc speed at w hich the teelh move
past the magnet in Fig. 1 1 , lhe variable-reluclance transducer is sensitive to velocity, rather
than displ ace me n t.
Whereas the preceding arrangement depends upon changes of the reluctance of rhe
mag n etic Hux palh, other devices separale the mag net from the coil and depend upon relarive
movemenl berween lhe coil and the Hux field.
13
CAPACmVE TltANSDUCERS
When a capacilor is fanned from a pair of parallel Hat plales, ils capacitance is given by 1he
following equarion:
C=
EoKA
d
(6a)
Sensors
where
C
= capacitance
(pf),
Eo = permittivity of free space, 8.8542 pF/m.
K
= dielectric constant of med iu m between p lates (
d
= separation of plate
A = area of one side of one plate (m2),
I for air) ,
surfaces (m)
be obtained by using several capacitors in paralle l . This may
n equally s paced p l ates in w h ic h alternate plates are
connected to one another. For example, if five plates were slacked. plates I . 3, and 5 would
be connected to one voltage, while plates 2 and 4 would be co nnected lo another. The
Greater sensitivity can
be
accomplished with a stack of
capacitance of such a stack is
C=
All the terms represented i n Eqs.
EoKA(n
(6),
- I)
(6b)
d
except possibly the number of plates, have been
used in transducer applications [7,8). The following are examp les of each.
Changing Dielectric Constant
Figure 12 shows a device developed for the measurement of level in a container of
liquid hydrogen [ 1 1 ) . The capacitance betwee n the cemrnl rod and the surrounding tube
varies with changing dielectric constant brough t about by changing liquid
readily detects liquid level even thoug h the difference
liquid and vapor states may
level. The device
in dielectric constant between the
be as low as 0.05.
I
_....., ...,.__
.
_,, Liquid 18¥81
1 0-_
--_
� ��
;
-
FIGURE
� -=-
-=-- ---=-
12: Capacitance pickup for determining level of liquid hydrogen.
208
Sensors
F\>s�ion of teeth for minimum
value of capacitance
FIGURE 1 3:
Section
showing relative arrangement of teeth in capacitance-type
torque meter.
Changing Area
Capacitance change depending on changing effective area has been used for the
secondary transducing element of a torque meter ( 1 2). The device uses a sleeve with teeth
or serrations cut ax i al l y, and a matching internal member or shaft with similar axially cut
teeth. Figure 13 illustrates the arrangeme nt . A clearance is provided between the tips
of the teeth, as shown. Torque carried by an elastic member causes a shift in the relative
positions of the teeth , thereby changi ng the effective area. The resulti ng capacitance change
is calibrated in terms of torque.
Changing Distance
Varying the distance between the places of a capacitor is undoubtedly the most common
method for using capacitance in a pickup.
Figure 1 4 illustrates a ca paci t i ve- ty pe pressure transducer, wherein the capacitance
between the diaphragm to which the pressure is appl ied and the electrode foot is used as
a measure of the diaphragm's relative position [ 1 3- 1 5 ) . Flexing of the diaphragm under
pressure allers the distance between it and the ekctrodc.
209
Sensors
lJbe for clrculadng
cooling water
FIGURE 14: Section through capacillllcl e-1ype pressure pickup.
14
PIEZOELECTRIC SENSORS
Cenain materi als can genera te an electrical charge when subje<:ted to mechanical suain or.
converse ly, can cha nge dimensions when subjected to voltage [Fig. 1 5(a)). This is known
as the piezoelectric3 effecc. Pierre and Jacques Curie are credited with its discovery in 1880.
Notable among these materials are quartz, Rochelle salt (Potass ium sodium wtarate), prop­
erly polarized barium titanate, am mon i um di h ydrogen phosphalte, cenain organic polymers.
and even ordinary sugar.
Of a l l the materials that exhibit the effect, none possesses; all the desirable properties.
such as stability, h i gh output, insensitivity to temperature exll'emes and humid ity, and the
abi li ty to be formed into any desired shape. Rochelle salt provides a very high output, but ii
requires protection from moisture in the air and cannot be used above about 45°C (I 1s•F).
Quanz is u ndo ubtedly the most stable, but its output is low. B(:cause of its stability, quartz
is commonly used for regulating electronic osc ill ators. Often the quartz is shaped into 1
thin disk with each face silvered for the attachment of elecll'odes. The th ickness of lhe plate
is ground to the dimension that provides a mech an ica l resonance frequency
to the desired electrical frequency. This crystal may then be i ncorporated i n an appropnate
correspo�ing
electron i c circuit whose frequency it controls.
3Thc
prefix pi�r.o is derived from the Greek pieuin. meaning "lo press" or .-lo squcci.e ."
210
Sensors
~
Tlickness shear
ltansverse
chenge
(a)
c,
(b)
FIGURE IS: (a) Basic deformation modes for piezoelectric plates; electrodes (+ and -)
shown; (b) equivalent circuit for a piezoelectric element.
are
Rather than existing as a single crystal, as
many piezoelectric materials, barium
titanate is polycrystalline; thus it may be formed into a variety of sizes and shapes. The
piezoelectric effect is not present until the element is subjected to polarizing treatment.
Although exact polarizing procedure varies with the manufacturer, the following procedure
has been used [ 16). The element is heated to a temperature above the Curie point of 1 20°C,
and a high de potential is applied across the faces of the element. The magnitude of this
voltage depends on the thickness of the element and is on the order of I 0,000 V/cm. The
element is then cooled with the voltage applied, which results in an element that exhibits
the piezoelectric effect.
Piezoelectric polymers, such as polyvinylidene Huoride [ 1 7 ) , provide low-<:ost piezo­
electric transducers with relatively high voltage outputs. These semicrystalline polymers
are formed into thin filnis, perhaps 30 �m thick, with silvered electrodes on either side
211
Sensors
and are coated onto a somewhat-thicker Mylar backing. The resulting transducer is light,
flexible, and easily manipul ated .
Figure I 5(b) shows an equivalent circuit for a piezoelement, consisti ng of a cluuge
generator and a shunting capacitance, C, . When mechanically strained, the piezoelement
generates a charge Q (t ) , which is temporari ly stored in the element's inherent capacitance.
As with all capacitors, however, the charge dissipates with time owing to leakage, a fact
which makes piezodevices more valuable for dynamic measurements than for static mea­
surements.
The voltage across the piezoelement at any time is simply
v. <t> =
Q(t )
c,
(7a)
Measurement of this voltage, however, requ ires a very high impedance circuit to prevent
charge loss. A charge amplifier is nonnally used for this purpose. The voltage can alterna­
tively be ex pressed in tenns of the stress on the piezoelement,
V0 = Ghu
(7b)
h is the thickness of the element between the electrodes and u is the stress. G is a
constant equal to 0.055 Vm/N for quartz in compressive stress (thickness change)
and 0.22 Vm/N for polyvinylidene ftuoride in axial stress (transverse change ) .
Piezoelectric transducers are used to measure surface roughness, force and torque,
pressure, motion, and sound. In addition, the piezofilm transducers are used to sense !henna!
radiation. Since the film expands in response 10 temperature change, a c harge is developed
when infrared radiation is absorbed (the pyroelectric e ffec t). Pyroelectric transducers are
common elements in household motion detectors.
Ultrasonic transd ucers may use barium titanate. Such elements are found in industrial
cleaning apparatus and in underwater detection systems known as sonar.
where
material
15
SEMICONDUCTOR SENSORS
The semiconductor revolution has profo u nd l y inftuenced measurement technology. In
addition to digital voltmeters. comp uter data-acquisition systems, and other readout and
data-processing systems , semiconductor technology has produced compact and inexpen­
sive sensors. A principal strength of semiconductor sensors is that they take advantage of
microelectronic fabrication techn iq ues . Thus, the sensors can be quite small, mechanical
structures (such as diaphragms and beams) can be etc hed into the device, and other elec­
tronic components (resistors, transistors, etc.) can be directly i mplanted with the sensor to
form a transducer having onboard s ignal cond i t ioning.
1 5. 1
Electrlc:al Behavior of Semiconductors
Semiconducting materials include elements, such as silicon and gennanium, and com­
pounds, such as gallium arsenide and cadmium sulphide. Semiconductors differ from
metals in that relatively few free e lectron s are available to carry current. Instead, when a
bound electron i s separated from a particular atom in the material, a positively charged hole
21 2
Sensors
is formed and will move in the direction opposite the electron. Both negatively charged
eleclrons and positively charged holes contribute to the How of current in a semiconductor.
The number of charge carriers (electrons or holes) in a semiconductor, nc, is a strong
function of temperature, T. Typically,
nc
=
number per unit volumt:
oc
·
T31,
· exp
(
)
-constant
--T--
(8)
Since the n:sistivity of a material is proportional lo I i 11.-. a semiconductor's resistance
decreases rapidly with increasing temperalurt:. For silicon near room temperature, the
resistivity decreases by about 8%/°C [ 1 8] .
Greater control over a semiconductor's electrical behavior i s obtained b y doping i t
with impurity atoms. Thest: atoms may be either elec1ro11 donors or electron acceptors.
Electron donor atoms (such as phosphorus or arsenic) raise the number of free electrons in
the material. Electron acceptor atoms (such as gallium and aluminum) hold electrons, thus
raising the number of holes. Since the number of dopi ng atoms is usually large relative to the
number offree electrons in the undoped material. the dopant sets the majority current carrier
of the material. Specifically, doping with donor atoms c�ales a predominance of negative
charge carriers (electrons), giving an n-type semiconductor. Doping with acceptor atoms
creates a predominance of positive charge carriers (holes), giving a p-type semiconductor.
Semiconductors, either doped or undoped, are useful as temperature sensors. For
undoped semiconductors, the number of carriers increases rapidly with temperature [Eq.
(8)), so that the resistance is a strongly decreasing function of temperature. Thermistors
(Section 8) are based on this effect. Because such sensors have a negative temperature
coefficient of n:sistance, they are some ti mes called NTC sensors. When semiconductors
are heavily doped, the mobility of the carriers decreases with increasing temperature, so that
the resistance increases with temperature; these positive-temperature-coefficient devices
are called PTC sensors.
Semiconductors also respond to strain. For example. a p-type region diffused into an
n-type base functions as a resistor whose resistance increases strongly when it is strained
(this behavior is called piewresistivity). Such resistors arc the basis of semiconductor strain
gages, semiconductor diaphragm pressure sensors (Fig. 1 6). and semiconductor accelerom­
eters [ 1 9,20).
15.2
pn-Junctlons
Most semiconductor devices involve a junctiu11, at which 11-lype and p-type doping meet
(Fig. 1 7 ). Current flows easily from the p-typc 10 1he 11 -1ypc material. since holes (+charge)
easily enter the n·typc material a nd electrons ( -charge) easily enter the p-type material.
Current How in the opposite direction meets much greater resistance. Thus. this junction
behaves like a diode.
When a voltage is applied to the junction, the current through it varies as shown
i n Fig. 1 8. When V is positive, we say that the junction is fonvard biased. The current
becomes very large once the voltage reaches a threshold level. If the voltage is instead
negative, the junction is rel"erse biased, and only a very small current Hows. As lhe reverse­
bias voltage is raised, the current quickly reaches a value - lo. which is nearly independent
of V. The current lo is called t he reverse sa111ratio11 currelll. Typical values of lo are on the
order of nanoamperes for silicon and microampcrcs for germanium. The reverse saturation
21 3
Sensors
Diaphragm (0.001 in. typ.)
Diffused
strain gage
MetallizatiOl1
Etched
cavity
Silicon constraint
waler
.....
Sealed reference vacuum
FIGURE 1 6: Cross section of a semiconductor-diaphragm absolute-pressure sensor
(Motorola, not to scale). External pressure change causes dia.phragm to deflect, sttain­
ing the gage.
n-type
p-type
t
I
Metal
' elecbOde
Junction
v
+
II;
FIGURE 1 7:
11
(a)
/pn-junction diode
J
. �: J
v
(b)
(a) pn-junction with applied voltage (b) circuit �epresentation as a diode.
214
Sensors
.,.,_:-:::
_:=
_- ---
____
- lo
v,
v
Forward bias
Reverse bias
FIGURE 18: Voltage-current curve for a pn-junction. lo is the reverse saturation curre nt
and V, is the forward threshold voltage.
current. is more nearly voltage-independent for germanium than for silicon, as a result of
various secondary effects [ 2 1 ) .
Th e voltage current curve i n Fig. 1 8 i s described b y the equation
(9 )
which shows a strong dependence of current on temperature (q is the charge of an electron
and k is Boltzmann's constant). The saturation current is also a function of temperature,
roughly doubling with every l o•c increase in temperature for silicon and germanium diodes
near room te mperatu re [ 1 8).
When forward biased, the voltage drop (or threshold voltage), V, . across a silicon
pn-junction l imits to about 0. 7 V at room temperature. This voltage drop decreases by
about 2 m V /°C as temperature increases, and the changill'g voltage forms the basis of
some semiconductor-junction temperature sensors. Through integrated-circuit techniques,
junction-temperature dependencies have bee n applied to make linear-response, temperature­
sensing chips.
One disadvantage of many semiconductor-junction sensors is that the operating tem­
perature must remain below about I 50°C to prevent degradation of the junction's electrical
characteristics.
15.3
Photodiodes
Semiconductorjunctions are sensitive to light as well as heat. If a junction is formed near the
surface of a semiconductor, photons reaching the junction can create new pairs of electrons
and holes, which then separate and ftow in opposite directions. Thus, the irradiating light
21 5
Sensors
Dark charactenstic {H = OJ
v
H,
Increasing H
Reverse bias: i - -Ii. " -H
aracteristic for various i ncident l ight intensities, ff.
FIGURE 1 9 : Photodiode i· V ch
produces an additional current, !,., at the j unction :
i =
Io [exp ( k� v) - 1 ] - 1;.
( 10)
The photocurrent, /;. , is d i rec t ly proportional to lhe intensity, H, of the i ncomi ng light (in
W/m2):
h = constant x ff
e
expos�-d
to light . The voltage­
where the constant is proportional to th area of the diode
current characteristics of a photodi ode are shown in Fig. 19 [22 1.
The photoc urre nt is typ ical ly on the order of m il l ia m peres and thus is much larger
than the reverse saturation current ( h » Io). By operating a photodiode with a reverse­
bias vol tage, the output current is made directly proportional to the inc i dent light intensity
(i "" - h oc - H J .
Semiconductor junctions are most respons i ve to infra�-d wavelengths, but sensitivi ty
can e tend lo vi s i ble wavelengths and near-ultraviolet wave le ng th s as well. Collllo
ll n
photod i odes are usually made from i nex pen sive si l ico n j u nction s, although several other
semiconductors, suc h as gerrnanium, are also in use.
The sens iti v i ty of a photod i ode is limited by its "dark urre nt ," which is the usual
junction urren t with no incident l ight [i.e., Eq. (9)]. Since this current decreases with
x
c
c
21 6
Sensors
temperature, sensitivity can be improved by cooling the diode lo very low temperatures.
For example, in high-perfonnance infrared sensing, photodiodes may be operated al liquid­
nittogen temperatures or even liquid-helium temperatures (- l 98°C or -269°C, respec­
tively) [23).
Some bulk semiconductors, without a pn-junction,
also
respond to light. Photons
create additional electron-hole pairs in the material, thereby reducing its resistance. Com­
mon examples of such
photoconductive
materials include cadmium sulphide (CdS) and
cadmium selenide (CdSe). For h i gh - performance infrared sensing, low-temperature doped
gennan ium may
be
used.
Photodetectors are discussed further in the next section.
16
UGHT-DmCTING TRANSDUCERS
Light-sensitive transducers, or photosensors,
are used to detect light of all types: lhennal
radiation from wann objects, laser light, light emitted by diodes, or even sunshine. These
transducers may be categorized as either thermal detectors or photon detectors.
The lhenna l
detectors use a temperature-sensitive element which is heated by incident light. The photon
detectors respond directly to absorbed photons, either by emitting an electron from a surface
(the photoelectric effect) or by creating additional electron-hole pairs in a semiconductor
(as discussed in Section 1 5 .3).
Among the issues to be considered in selecting a photodetector
are the wavelength
the speed of response needed, and the sensitivity required. In general, ther­
mal detec tors are much slower than photon detectors but respond lo a broader range of
wavelengths. For any detector, the speed of response will also depend upon the supporting
circuitry. For visible and near-infrared light, semiconductor detectors are commonly used.
Photoemissive detectors can be sensitive well into the ultraviolet range. To detect long-wave
to be sensed,
infrared light, which is the heat emitted by objects near room temperature, either thermal
detectors or cryogenically cooled semiconductors may be used.
16.1
Thermal Detectors
Thermal detectors create
the detector.
a
tempemture change in the detecting clement when light heats
Some other property of
the detecting element changes in response to the
temperature c hange , and that property is measured 10 determine the light intensity [24].
Some examples follow.
Thermopile detec t ing elements use several thennocouples in series lo produce a volt­
age output proportional to the detector temperature. Pneumotic detecting elements use a
When the gas is heated, its pressure rises, and the pressure is
Bolometers use temperature-dependent electrical
resistance as a sen sor. Pyroelectric detectors respond to temperature changes by generat­
ing an electric charge; they are widely used a� infrared motion sensors for automotic light
switches and intruder alarms [ 24.25 ].
chamber containing a ga� .
measured to dctennine the light intensity.
Th e response times
of typical thermal detectors
range from a fe w milliseconds to
several seconds, depending main l y upon the size and configuration of the detecting element.
Specialized pyroelectric sensors can achieve response times below
21 7
1
ns.
Sensors
16.2
Photon Detectors
Photoemissive detectors (Type A, Tab le 2) consist of a cathode anode combi nation in an
evacuated e nvelope made of glass or sy nthetic quartz. In the proper circuit (commonly
requiring a de source of several hundred volts), l ig ht i mpi ngemen t on the cathode causes
electrons to be emitted The electrons travel lo t he anode, thereby providing a small current.
By addi ng several successi vely higher- voltage electrodes (or dyru;•des) to the envelope,
substantial current amplification is obtained, producing a photomultiplier tube, or PMT.
PMTs can be extraord i nari l y sensitive, to the point of detecting single ph otons, and they
have rise times as short as 1 ns. Since the invention of smaller, cheaper semiconductor
photoscnsors, these devices are used on ly in rather specialized applications that require
extremely high sens itivi ty (26).
Semiconductor sensors are of several types. In general, they perform best at near­
infrared wavelengths. The major classes of semiconductor photodetc:ctors are as follow.
Photoconductive sensors (l'Ype B, Table 2) consist of a lay·er of semiconducti ng
material between two electrodes. When the layer is exposed to l igh t, abs orbed photons create
addi tional conductors in the semiconductor (electrons and holes), lowering its resistance. In
conjunction with resistance-sensitive circuitry an output may be obtai n ed that is a function
of the i ntensi ty of the l ight source. l'Ypical photoconducting detectors: use cadmium sulfide,
cadmium selenide, lead sulfide, indium antimonide, or mercury cadmium telluride. Among
these, inexpensi ve CdS and CdSe are probably most common ly used, although they are
limited to visible and near-infrared waveleng ths and respond rather slllw l y (with typical rise
times of 50 ms). Devices using In S h or HgCdTe can operate at far--infrared wave lengths
and have rise t imes on the order of 100 ns. For best performance. these devices require
coo l in g to li qu id nitrogen temperatures (27).
Photodiodes (l'Ypes Cl, C2, and CJ. Table 2) u ti l i ze a pn-ju nclion and are similar to
photoconduc tive detectors (see Sec tion 1 5.3). When operated w i th a reverse- bias voltage
(Fig. 1 9), a p hotodi ode be haves as a l ight - sens i tive current source. Rise times may be as low
as I ns. The pin · photod iodc differs from the common variety in that a layer of undoped (or
intrinsic) semiconductor is sandwiched between the p and n laye rs . This increases both the
speed of response and the sen s iti vi ty. Some pin - p hotod iodcs have ri.se ti mes below 0. 1 ns
(28).
Avalanche photodiodes are obtai ned by operating photod i odes with a reverse bias
vol tage that approaches the diode's breakdow n vol tage (typically st:veral hundred volts).
The result is that the current generated al a given l ig h t level is 10 lo 100 t i mes greater than
that produced by a pn or pin - pho tod iode. APDs have the advan tages of hig h g ai n and fast
response (rise ti mes down to 0. 1 ns), with the di sadvan tages of requ iring a stable voltage
supply and a stable operating temperature.
Phototransistors and photodarlingtons (Ty pes DI and 02, Tab le 2) are basically
photodiodes fol lowed one or two stages of ampl ification incorporated iinto the same package
to enhance the se n sitivity. Pholotransistors have gains similar to AP'Ds , but much slower
rise times (tens of µs). Photodarl ingtons have higher gain , but with r i se times of hundreds
of µs [29).
16.3
Applications
Applications of photodetectors in mechanical measuremen ts are wide ranging. They include
simple coun t i ng , where the interrupti on of a beam of light is used :strain measureme nts
,
21 8
TABLE 2: Photon Detectors
Type
A. Photoemissive or
pho«>multiplier
B. Photoconductive
(or photoresistive)
Symbol and
Typical Circuit
ct:Jload
""u�
Form
of Output
Relative
Frequency
Reaponae
Cathode-uode in evacualed glass or uartz
envelope. PMT gain can be U>3 to 1 • Bulky;
requires high voltage. Highest sensitivity.
J
Current
Very fast
Resistance
Extremely slow
Light-sensitive resistor. Increased light i ntensi ty
causes reduced resistance. Expensive types can
be fast.
Very fast
Primary disadvantage is low output current.
chsnge
de
C l . Phntodiode
(p11-junction)
C2. pi11- photodiode
c[JLoad
C3. Avalanche
photodiode
DI.
Phototransistor
02. Photodarlington
co load
�load
Current
"Dllk c:um:nt" very low (nanoampere range), but
Current
Extremely fast
Current
Extremely
fast
not zero.
has "intrinsic" layer between p
layers, which creates faster response.
pin diode
APO is operated near breakdown
voltage.
and n
Has
much higher internal gain and is much more
Current
Slow
I
I
Commenta
Current
Very slow
sensitive than other pho todiodes.
Produces much higher current for given input
than photodiode does because of its amplifying
ability. Base lead, i£ accessible, is seldom used.
Much more sensitive than phototransistor.
Sensors
LED source
can
FIGURE 20: A photointerrupter consisting of an LED light source (often in frared) and a
photodetector. Mechanical interruption of the l ight path
be used for various purposes,
such as counting, triggering, and synchronization.
temperature measurements dewpoint controls, and a wide range of process monitors. In
conjunction with lasers, they form the basis of fluid velocity measurement systems and
vibration measurement systems.
Photodetectors are available in one- and two-dimensional arrays, as well. These
configurations allow the contruction of digital cameras, which reduce a visual image to a
discrete set of photodctector voltages. Of particular importance are charge-coupled diode
array s (CCD arra ys), which use silicon photodiodes to capture digital images. Arrays of
photodetectors enable whole-field temperature detection, machine vision systems, ftuid
flow visualization, and many other measurements.
Other special packages include optointerrupters and optoisolators, in which a pho­
todiode and a light-emitting di ooe (LED) are packaged so that the light from the LED
impinges on the photodiode (see Figs. 20 and 2 1 ). The interrupter is configured so that
some form of mechanical mask may be used to break the light beam between the LED and
the detector, thereby providing on off switching for counting or a variety of other purposes.
�ocr=
lotranslstor :
, - - - - - - - - - - - - - -,
I
:
I
.......
I
LED
I
:
I Dip pad<age
'- - - - - - - - - - - - - - '
FIGURE 2 1 : Th e essentials of a photoisolator, used fo r connecting low-impedance cur·
rent circuits to high-impedance voltage cin:uils. The isolator is also useful for providing
complete electrical isolation between cirucits, sometimes imperative in health-related elec­
tronics.
220
Sensors
The optoisolator is used to match low-impedance current circuits to high-impedance voltage
circuits, or vice versa. It also provides a high-impedance isolation between circuits, which
is an important feature in some forms of health-related electronics (29).
17
HALL-EFFECT SENSORS
The Hall effect is the appearance of a transverse voltage difference on a conductor carry ing
a current perpendicular to a magnetic field (30). This voltage is directly proportional to the
magnetic field strength. If the magnetic field is made to vary with the position of a nearby
object, the Hal l effect can be the basis of a proximity sensor.
In Fig. 22(a), a conductor carries current in the x-dircction, so that electrons How in
the x-direction with a velocity VJ . The magnetic field runs in the y-dire<:tion. Because the
electrons carry a charge -q, they experience a magnetic force Fs in the z-direction:
Fs
=
-qv4 x B
(a)
IT
FB
t
B
•
i
FE
+ ... + +
(b)
FIGURE 22: The Hall effect: (a) A conductor carries current in a perpendicular magnetic
field; (b) electrons arc driven upward by magnetic force, creating an opposing electric field.
221
Sensors
Permanent magnet
Hall oonduclOr
Housing
Lead wires
(power and output)
\
FIGURE
N
s
(t
\
23: Hall-effect gear-tooth sensor [30).
This force deflects electrons upward and so creates a negative charge along the top of the
conductor and a positive charge along the bottom [Fig. 22(b)). This charge distribution
in tum creates an electric field, E, whose force in steady state is equal and opposite the
magnetic force on the electrons:
FE =
-
q E = - FB
From these relationships, the magnitude of the electric field is E = Vd B. Since the
electric field is the gradient of voltage, the voltage difference across a conductor of height
I is
(11)
This i s the Hall-effect voltage.
The Hall beffect is present in any conductor carrying current in a magnetic field.
but it is much more pronounced in semiconductors than in metals. Thus most Hall-effect
transducers use a semiconducting material as the conductor, often in conjunction with an
integrated-circuit signal conditioner. A permanent magnet may be built into the transducer
to provide the needed magnetic field (Fig. 23). If a passing object, such as another magnet
or a ferrous metal, alters the magnetic field, the change in the Hall-effect voltage is seen at
the transducer's output terminals. Hall-effect transducers are used as position sensors, as
solid-state keyboards actuators, and as current sensors ( 3 1 ) . Low-cost, ruggedly packaged
versions are used as automotive crankshaft-timing sensors (32) .
1.8
SOM E DESIGN-RELATED PROBLEMS
Accuracy, sensitivity, dynamic response, repeatability, and the ability to reject unwanted
inputs are all qualities highly desired in each component of a measuring system. Many of the
parameters that combine to provide these qualities present conflicting problems and must be
222
Sensors
compromised in the final design. At this point we will discuss some additional design
problems.
1 8. 1
Manufacturing Tolerances
The conception or a component or system on paper is a necessary and imponant beginning;
to be useful, however, the apparatus must be produced, and no manufacturing process can
reproduce
exact
length or angular dimensions. Dimensions must always be assigned with
some specified or implied tolerances. How can one predict the effect or such variations on
perfonnance? The following example describes one approach to the problem.
EXAMPLE 1
Suppose a spring scale such as that shown in Fig. 24 is to be designed. We will assume
a force capacity of 50 N and a maximum deflection or 10 cm. This gives us a deOection
·
constant of 5 N/cm.
Solulion
Using conventional coil-spring design relations [33), we find that a spring made
with a mean coil diameter Dm of 2 cm and a steel wire diameter of 2 mm will meet stress
requirements prov ided we ensure against overload by including appropriate deflection limits
or stops. The deflection equation commonly used
K
F
y
= - =
for coil springs is
E, D�
3--
80,.n
FIGURE 24: Common spring-type "fish" scale.
223
Sensors
where
K = the deHection constant (Nim),
F = lhe design load (N),
y = l he corresponding deHeclion ( m),
E,
= lhe torsional elaslic modulus (about 80 x 1 09 Pa for steel),
n = lhe number of coils
Using this relalion and the preceeding des ign values. we find that 40 coils are needed 10
provide the required deHeclion constant. If we apply reasonable tolerances, our specifica­
tions become
D,. = 2 ± 0.0 1 mm,
D,. =
2 ± 0.05 cm,
n = 40 ±
E,
�
coils, and
= 80 x 106 ± 3.5 x 106 kPa
Let us now consider how various uncertainties (manufacturing tolerances) may affect
the deHec tion cons1ant, K . (We direcl allention to re ference (341 at this point.) We assume
th at 99% of the coils will not exceed any of these tolerances; in other words, we take the
tolerances to represent 99% uncen ai nty limils for eac h variable. Then, we have
UK
[( =
=
( .o
4 x0
2
0.0895
""
l )2
+
9%
or
K
Note that mm, cm, and m have
(3
( 0.33 ) 2
x 0.05 ) 2
2
+
40
(99% confidence)
= 500 ± 45 Nim
been used i n
( 3.5 ) 2
+ so
(99%)
1he preceding example: hence care must be .
used in placing decimal points.
We see then thal if we lay out 1he gradua1ions corresponding lo nominal values, a force
of 50 N may actually be indicated as anything in lhe range from 45.5 to 54.5 N, depending on
how the manufac1uring tolerances may fall. Should lhis resull not be salisfactory, our only
recourse is ( I ) 10 provide belier control or the manufacturing tolerances or (2) to provide
some means for adjusting calibration.
Various melhods of calibration can be used, depen d ing on lhe inlended "quality" of
the device. In this instance, two or three faceplates can be provided. each wi1h gradua1ions
scaled lo cover a ponion of lhe calibra1ion r.mge. Al the lime of assembly, a simple
calibration would delcrmine lhe most appropriate p late to use.
224
Sensors
Al lhis point, it is appropriate to make an additional observation. Weight is basically a
force; hence we should express the calibration in newtons rather than in kilograms. Should
we wish a scale calibrated in kilograms, !hen, 10 be complelely correct we should include
lhe assumed value of gravitational acceleration on the faceplate. 'The standard acceleration
due lo gravity is 9.80665 m/s2 , and lhe kilogram range corresponding to our SO N range
becomes 0 lo 50/9.80665 "" 5 . 1 kg. (Use of the non-SI symbol kgf is discouraged.)
It should be clear that the procedures used in the preceding example are applicable to
mosl elastic transducer configurations, as well as to many olher lolerance problems.
18.2
Some Temperature-Related Problems
An ideal measuring system will reacl 10 lhe design signal only and ignore all else. Of course,
this ideal is never complelely fulfilled. One of the more insidious adverse slimuli affecting
instrumenl operation is temperalure. It is insidious in that it is almost impossible lo maintain
a constant-temperature environment for a general-purpose measuring system. The usual
solution is lo accepl the lemperature varialion and to devise methods to compensate for it.
Temperature variations cause dimensional changes and changes in physical properties,
both elastic and eleclrical, resulting in deviations (bias error) referred to as zero shift and
scale error. [35). Zero shift, as the name implies, resul!s in a change in the no-input reading.
Factors other than temperature may cause zero shift; however, temperature is probably the
most common cause. In most applications the zero indication on the outpul scale would be
made lo correspond to the no-input condilion. For example, the indicalor or the spring scales
referred lo earlier should be set at zero when !here is no weight in the pan. If the temperature
changes after the scale has been set lo zero, !here may be a differential dimensional change
between spring and scale. altering lhe no-load reading. This change would be referred to
as zero shift. Zero shift is primarily a funclion of linear dimensional change caused by
expansion or contraction with changing lemperature.
Dimensional changes are expressed in lcrms of the coefficient of expansion by the
following familiar relalions:
a = --
1 t:.L
t:.T Lo
(12)
and
L r = Lo(!
+ a t:.T)
( 1 3)
where
a = the coefficient of linear expansion (ppm/deg temp. x 1 o-6),
L/ Lo = the uni! change in length,
t:. T = lhe change in tempcralure, Tr - To,
Lo = lhe lcnglh dimension at the reference temperalure To,
L 1 = lhe length dimension at any other temperature Tr
In addition to causing zero shift. temperature changes usually affect scale calibration
when resilient load-carrying members arc involwd. The coil and wire diamelers of our
spring would be altered with temperature change, and so too would the modulus of elasticity
225
Sensors
of the spring material.
These variations would cause a changed spri n g constant, and hence
referred 10 as scale error.
changed load-deflection calibration, resulting in what is
The thermoelastic coefficient is defined by the relations
I 6£
Eo
-
c=
( 14)
6T
and
E1 = Eo( I
+c
6 T)
( 1 5)
where
c
= the temperature coefficient for the tensile modullus of elasticity
6
(ppm/deg temp. x 1 0- ) ,
t. E/Eo = the unit change in the tensile modulus o f elasticity,
Eo = the tensile modulus of elasticity at temperature To.
E 1 = the tensile modulus of elasticity at temperature T1
Similarly, the coefficient for torsional modulus may be written
I t. E,
6T E,,,
m = - --
( 1 6)
and
E,, = £..., ( !
+ m 6T)
( 1 7)
where
m
= the temperature coefficient for the torsional modulus o f elas1ici1y
1 0- 6 ) ,
(ppm/deg temp. x
t. E, / E,,, = the u n i t change in the torsional modulus o f elas11ici1y,
Eso = the torsional modulus of elasticity at temperatm·c To,
E,1 = the torsional modulus of elasticity at tem perature T1
Representative val u es of these quantit ies are given in Table 3.
The manner in which temperature changes in elastic properties affect instrument
performance can
element in
for which
be demonstrated by the fol low ing example. Assume that a restoring
an instrume n t is essenti al ly a single-leaf cantilever spri n ;g of rectangular section,
the deflection equation al reference temperature To is
F
Ko = - =
y
3Eolo
--
226
Li
Eo wotJ
= --
4l�
(18)
TABLE 3: Temperature Characteristics for Some Materials
Coefficient of Unear
Expansion, u
Coefficient ofTenslle
Modulus of Elasticity, c•
ppmf°C
ppm/°F
ppmf°C
ppm/°F
7.93
( 1 1 .5)
1 1 .6
(6.5)
-220
( - 1 22)
20.7
(30)
7.93
( 1 1 .5)
12.2
(6.8)
-260
( - 145)
Stainless steel, 1ype 302
19.3
(28)
6.9
( 1 0)
1 6.7
(9.3)
-439
(-244)
Spring brass
1 0.3
( 1 5)
3.8
20.2
(5.5)
-39 1
(-2 17)
1 0 .3
( 15)
4.3
(6.3)
( 1 1 .2)
(9.9)
17.8
-380
( -2 1 1 )
14.8
(2 1 .4)
5.6
(8. 1 )
1.1
(0.6)
+48. 1
(+27)
1 8.0
( 26)
6.3
(9.2)
7.2
(4)
( -20 to +7.3)
2.6
(3.8)
23
( 1 3)
Tensile Modulus
of Elasticity, E
Pa x 10- 10
(psi x 10-6 )
Torsional Modulus
of Elasticity, E
Pa x 10- 10
(psi x 10- 6 )
Chrome-vanadium steel
Material
High-carbon spring steel
Phosphor bronze
lnvart
Isoelaslic t
Aluminum
20.7
(30)
6.9
( I O)
• c may be used for torsional modulus also.
tTrade names.
-36 10 + 1 3
-270 10 -400
( - 1 50 IO -220)
Sensors
where
= the deflection c onstant
Ko
Io
=
the moment of inertia,
,
wo = the width of the secti.on at reference temperature,
to
beam
pe
= the thickness of the section at reference temperature,
Lo = the length of the
A second equation may be wri
K1
_
-
[
Eo(
Thus we have
ucn
at the reference tem rature
for any other temperature, Ti , as follows:
l + c 6.T)J[UJo( I + a
4(lo( I + a
fl
Percent error in de ection scale =
6.T)J[to ( I
6.T)J3
( Ko;0 K t )
= (1 -
+ a 6.T)] 3
( 19)
x 1 00
( I + c 6. T)( l + a 6. T)) x 100
which we may simplify, by expanding and neglecting the second-order tenn, to read
ri
Percent scale error
If our sp ng is made of spring brass,
=
-
(c + a) 6. T x 100
(20)
Percent scale error/°F = - ( - 2 1 7 + 1 1 .2) x 10-6 x 100 = 0.02 1%
c g
Hence a temperature han e of +50°F would result in a scale error of about + 1%. (This
means that the reading is too high; our spring is too flexible, and a given load deflects the
spring more than it should.)
It is n te t ing to note that for our example the scale error is a function of material or
materials. It should be clear that we are s a n of the load-deflection re l ation for resilient
memhers in this connection and that this would not i c l
members whose duty it is simply
to transmit motion, such as the linkage in a Bourdon-tube pressure age
A ltho ugh not a mechanical quantity, another item affected by t mpe ture change is
electrical resistance. The basic resistance equation may be written in the fonn
i res
e
pe ki g n ude
g .
e ra
(2 1 )
w her
R
p
t
= he electrical resistance (Q),
= the resistivity (Q · cm) ,
e
cro -sec
L = the l ngth of the conductor (cm),
A = the
ss
tional area of the conductor (cm 2 )
228
Sensors
As temperalurc changes, a change in the resistance of an eleclrical conduclor will be
noled. This will be caused by 1wo different faclors: dimensional changes due to expansion
or contraction and changes in lhe currenl-opposing properties of the malerial ilSelf. For an
unconstrained conductor, the taller is much more significant than the former, causing more
than 99% of the toial change for copper (36). Therefore, in most cases it is not very important
whether the dimensional effect is accounted for or not. If dimensional changes caused by
1emperature are ignored, the change in resistivity wilh temperature may be expressed as
(22)
or
Pl ;
Po(l + b 6 T)
(23)
where
b ; the 1empera1ure coefficienl of resistivity [(Q · cm)/(O · cm · deg)) ,
6T
·
;
the 1empera1ure change(deg).
6p /Po ; the unit change in resistivity,
Po ; the resistivity at the reference 1empera1ure To(O · cm) .
pi
;
the resis1ivi1y al any temperalure T1 (Q · cm)
I f we account for temperature-dimensional changes,
Pl ;
Ro A o
--
Lo
( I + b 6 T)( l
; Po ( I + b 6 T)( I
the equation reads
+ a 6T)
+ a 6 T)
(24)
Table 4 lists values of the cocfficienlS of resistivity for selected malerials.
18.3
Methods for Limiting Temperature Errors
Three approaches to a solulion of the temperalure problem in ins1ntmen1ation are as follows:
( I ) minimization lhrough careful seleclion of materials and operating 1emperature ranges,
(2) compe11satio11 through balancing of inversely reacling elements or effeclS, and (3) elim·
ination through 1emperature control. Although each si1ua1ion is a problem unto i1self,
thereby making specific recommendalions difficult, a few general remarks wi1h regard 10
these possibilities may be made.
Minimization As we poinled out earlier, lemperalure errors may be caused by
lhermal expansion in the case of simple mo1ion-1ransmiuing elemenls, by thermal expansion
and modulus change in the case of calibra1ed resi lient transducer elements, and by thermal
expansion and resistivity change in the case of electrical resislance transducers. All these
effects may be minimized by selecting materials with low-temperature coefficients in each
of the respective categories. Of course, minimum temperature coefficients are not always
combined with other desirable features such as high strength, low cosl, corrosion resistance,
and so on.
229
Sensors
TABLE 4:
Resistivity and Temperature Coefficients of Resistivity fo1r Selected Materials
Resistivity at
Material
Aluminum
Constantan •
Copper (annealed)
Iron
Isoelastic•
Manganin•
Monet•
Nichrome•
Nickel
Silver
Composition (for alloys)
20°c {68°F)
S2 · cm
x
2.8
44
60% Cu, 40% Ni
99.9% pure
36% Ni, 8% Cr, 4% Mn, Si, and
Mo, remainder Fe
9-18% Mn, I -4% Ni;
remainder Cu
33% Cu, 67% Ni
75% Ni, 12% Fe, 1 1 % Cr, 2%
Mn
!
106
·Coefficient of
Resistivity, b
�!/ 9. . deg x 106
Per"C
Per"F
3900
11
1 .72
3900
10
48
5000
470
2 1 70
.6
2 1 80
2800
260
44
II
6
2000
1 100
220
42
1 00
400
7
6400
1 .6
4000
3550
2250
Note: Values should be considered as quite approximate. Actual values d1:pend on exact
composition and, in certain cases, degree of cold work.
•Trade names.
Compensation Compensation may take a number of d:ifferent forms, depending
the basic characteristics of the system. If a mechanical system is being used, a form of
compensation making use of a composite cons truc ti o n may be em pl oyed . If the system is
electrical, compensation is generally possible in the e l ec trica l circuitry.
An example of composite construction is the balance wheel in a watch or clock. As the
temperature rises, the modulus of the spring material reduces and, in addition, the moment
of inertia of the wheel (if of simple form) increases because of thermal expansion, both of
which cause the watch to slow down. If we i ncorporate a bimei:al element of appropriate
characteristics in the rim of the wheel, the moment of inertia dt:creases with temperature
enough to compensate for both expansion of the wheel spokes and change in spring modulus.
(See also Section 6 for a discussion of tem peratu re effects on linear measuring devices.)
Electrical c ircu itry may use various means of compensating for temperature effects .
The therm istor, discussed in some detail in Section 8, is quilt: useful for this purpose.
Most circuit elements possess the characteristic of i nc reasing de resistance with rising
temperature. The thermistor has an opposite temperature-resistance property, along with
reasonably good stabili ty, both of wh i ch make it ideal for simple temperature-resistance
compensation.
Resistance strain gages are particularly susceptible to temperature variations. The
actual situation is quite complex, involving thermal-expansion 1:haracteristics of both the
base material and all the gage materials (support, cement, and grid ) and temperature·
resi stivi ty properties of the grid material , combined with the f.act that heat is dissi pated
by the gri d since it is a resistance device. Temperature compen1sation is very nicely hanon
230
Sensors
died, however,
by pitt ing the temperature effect output from like gages against one another
subjecting them differentially to strain. This outcome is accom pl is hed by use of a
resistance bridge circuit arrangement, which is used e x tens i vel y in strai n -gage work. In
addition through careful selection of grid materials, so-called self-compensating gages have
while
been developed.
Elimination The third method-el im inating the temperature prob lem by temper­
ature control-really requires no discussion. Many methods are possible, extending from
the careful conirol of large environments to the maintenance of constant te mperatu re in
small insirument enclosures. An example of the latter is the "crystal oven," often used to
stabilize a frequency-<letermining quartz crystal.
19
SUMMARY
We have in no sense exhausted the list of possible devices or princ ipl es suitable for sensing
mechanical i n puts. In certain instances, we discuss others elsewhere in
Table
this book, and in
I have attempted to reference some of these. For further information on basic sensing
devices, we refer you to the Suggested Readings.
1. Sensors often include both a first, detecting stage and a second, transducing stage.
Each stage may convert the sensed information into a different form, often resulting
a fi nal electrical signal. Specific sensor design and selection is normally guided by
the requ i rements of sufficient sens itiv i ty and minimal source loadi ng ( S ec t ions 1-4).
2. A wide range of transducers are based on changes in the resistance of a sensing
e lemen t (Sections 5-8). Vari atio n s in inductance (Sect i o ns I 0- 1 2) and capacitance
(Section 13) are also used frequently.
3. Some sensing e le ments are sel f-poweri ng . These i nclude thermocoup les (Section 9),
which generate an electromotive force depe ndent upon temperature. and pi ezoelectric
sensors (Section 14), which generate a charge w hen loaded.
4. Semiconductors devices are i nc reas i n g ly common among sensing elements (Sec­
tion 15). These sensors can take advantage of mi croe lec tronic fabricat i on tec h nology.
S. L i g ht-detec ti n g transducers may be divided into thermal and photon device.•. A variety
of photon dev i ce s are de scri bed in Sec ti on 16.
Hall-effect sensors are often used in pos i t io n sensing and related applications (Sec­
17).
7. Manufacturing tolerance and temperature errors must be cons idered when designing a
sensing system . Often, the m agni tude of these problems can be est i ma ted in advance,
using the methods of uncenainly a nal y sis (Section 1 8). Such estimates can provide
gu idel i nes for improving iransduccr performance.
6.
tion
Sensors
SUGGESTED READINGS
/SA Directo ry of lnstrwnentation. Research Triangle
yearly editions.
Park, N.C. : Instrument Society of
America,
Gautschi. G. H. Piewel�crric St!nsorics. Berlin: Springer-Verlag, 2002 .
Juds, S. M. Photoekctric Sensors and Controls. New York: Marcel Dekker, 1988.
Khazan. A. D. Transducers and Their Ekments. Englewood Cliffs, NJ.: Prentice Hall, 1994.
Kovacs. G. T. A. Micromachined Transducers Sorm:ebook. New York: McGraw-HiU,
1 998.
Norton. H. N. The Handbook of Transducers. Englewood Clifl's. NJ.: Prentice HaU, 19119.
Nunley, W.,
and J. S. Bechtel. Infrared Oproell!ctronics: Devices and App/ictJlians. New Yort:
Marcel Dekker, 1987.
Pallls·Areny,
R .. and J. G. Webster. Sensors und Signal
Wiley. 200 1 .
Sydenham, P.
Conditioning. 2nd ed. New Yort: Jabn
H. (ed.). Handbook of Measurement Science. New York: John Wiley, 1 992.
Tudd, C. D. Tlie Potentiometer Handbook. New York: McGraw-Hill, 1 975.
Trietley, H. L. Transducers in Mechanical and Ekctronic Design. New York: Man:el Dekker, 1986.
Wilson, J., and J. F. B. Hawkes. Optoelectronics: An Introduction. Jnl ed. Harlow, UK: �
Hall Europe, 1 998.
PROBLEMS
I.
Consider an i nduc ti ve displacement probe having a diameter of 0.25 in. If the probe is
2.
It is desired to construct a dynamic compression force ce l l capable of measuring forces
J.
to a shaft, determine the probe sensitivity
(mV/0.001 in. d i sp lacement ) when the probe is used as shown in the circuit shown in
Figure 25. Assu me Eq. (3c) is valid here w i th n = 100 and that the excilation mquency
is IOOO Hz.
set al a "sland-off" d islance of 0.050 in. relative
in the ra ng e of ± 1 000 N. If a quartz disk 1 .0 mm thick and 1 0 mm in diameter is used as
t h e sensing element. determine the force cell sensitivity (mV/N).
For a capacitive displacement transducer whose behavior can be represented by Eq. (6a).
determi ne an ex press ion for the sensitivity
transducer is used as shown in Fi g. 26.
t
I
de0 /d (d) for an exci1ation frequency I iflhe
1 000 0
Displacement
probe
e1 = 1 0 V
FIGURE 25: Circuit
232
for Problem I .
Sensors
R
Capacltiw
transducer
FIGURE 26: Circuit for Problems 3, 4, and 6.
4.
Consider the capacitive displacement transducer in Fig. 26 to be governed by the following
relationship:
C
=
where
C = capacitance (pF),
0.225A
d
A = cross-sectional area of IJ'anSducer tip (in. 2).
d = air-gap diSlallce (in.)
Determine the change in e0 when the air gap changes from 0.010 in. to O.O I S in.
S.
A capacitive displacement transducer as shown in Fig. 27 is constructed of two plates with
area 2.0 in.2 separated by a distance of0.006 in. If air is the separating medium, determine
the sensi ti vity of the transducer in picofarads per 0.001 in. change in x .
,_,
L
FIGURE 27: Transducer arrangement for Problem 5 .
Sensors
light �
Incident
Photodiode
H
v,,,.
0------0 +
Rioad
FIGURE 28: Circuit for Problem 8.
6.
7.
If the transducer of Problem 5 is insened into the cin:uit of Fig. 26, determine the change
in output voltage when x changes from 0.0 10 in. to 0.0 15 in. ls the sensitivity conslanl in
this range?
( a ) A commen:ial fon:e sensor uses a piezoelectric quanzcrystal as the sensing eleme111.
The quanz element is about 0.2 in. thick and has a cross S1:ction or about 0.3 in. by
0.3 in. The sensing element is compressed in the thicknesi; direction when a load is
appl ied over its cross section. The output voltage is measured across the thickness.
W hat is the output of the sensor in volts per newton?
( b) A polyvinylidene . fluorid e film is used as a piezoe lectri c load sensor. The film is
25 µm thick, 1 cm wide, and 2 cm in the axial direction. J!t i s stretched in the axial
direction by the load. The output voltage is measured acrross the thickness. What
is the output in volts per newton?
8.
The cin:uit of Figure 28 may be used to operate a photod iode. Tiie voltage V, is a reverse­
bias vol tage large enough lo make diode current, i , �roponional 10· the incident light
intensity, H. Under this cond it on , i / H = l µA/(W/m ).
i
( a ) Show that the output voltage, V""" varies linearly with H .
2
= 1000 W/m , V, = 5 V, and an output voltage o r I V is desired, determine
an appropriate va lue of R1oad ·
(b) IfH
Sensors
Assuming that the speci fications for Examp le 1 in Section 18. 1 are for a nominal temper­
ature of 20"C, calculate the nominal value for the deHection constant K for temperatures
or (a) 40°C and (b) -20°C. (Use values for h igh-c arbon spring steel.)
9.
REFERENCES
[1) Knecn, W. A. A revi ew of the electric d i splaceme nt gages used in railroad car test i ng.
/SA Proc. , 6:74, 1 95 1 .
[2] Michael, P. C., N . Saka, and E . Rabinowicz. Burn i sh ing and adhesive wear o f an
electrical ly conductive po lyester-c arbon film. Wear. 1 32 : 265-285, 1 989.
[3) Beckwith, T. G., and R. D. Marangoni. Mechanical Measurements. 4th ed. Read i ng,
Mass : Addison-Wesley, 1990, Section 8.
[4) Todd , C. D. The Potentiometer Handbook. New York: McGraw-Hill, 1975.
[5] Hutchinson , C. (ed). The ARRL Handbook for Radio Amateurs. 78th ed. Newi ngton ,
Conn.: American Radio Relay League, 2000, p. 6.22. Rev ised annually.
[6] Carl son, A. B., and D. G. Gisser. Electrical Engineeri11g. Reading, Mass.: Addison­
Wesley, 1 98 1 , Chapter 17.
[7] Khazan, A. D. Transducers and Their Elements. En g lewood Cliffs, NJ.:
1 994, Chapter 3.
Prentice Hall,
[8] Pallls-Areny, R., and J. G. Webster. Sensor a1ul SigMI Conditioning. 2 n d e d . New
York: John Wi ley, 200 1 , Chapter .
[9] Electronic micrometer uses d u al coils. Prod. Engr. , 19: 1 34, January 1 948.
[10) Brenner, A., and E. Kellogg. An e lectric gage for measuring the inside diameter of
tubes. NBS J. Res. , 42:46 1 , May 1 949.
[11) Low- temperature liquid-level i nd ic ator for conde n sed gases. NBS Tech. News Bull. ,
38: 1 , January 1 954.
[ 12) Heteny, M. Handbook of Experimental Stress Analysis. New York: John Wi ley 1 950,
.
p. 287.
,
(13) Sihvonen, Y. T., G. M. Rassweiler, A. F. Welc h and J. W. Bergstrom. Recent i m prove
ments in a capacitor-type pressure transd ucer /SA J. , 2, November 1 955.
,
­
.
J. W., G. M. Rass weiler, A. F. We l c h , and Y. T. Sihvonen. Eng i ne pressure
indicators, application of a capacitor type. /SA J. , 2, August 1955.
[ 14) Leggat,
(IS] Welch, Weller, Hanysz, and Berg strom . Au xil iary eq ui pment for the capacitor-type
transducer.
J. , 2, December 1955.
/SA
235
Sensors
(16) Flemi ng , L. T. A ceramic accelerometer of wide frequency range. /SA Proc.,
5 :62, 1 950.
( 17) Kynar Piezo Film Technical Manual. Valley Forge, Pa.: Penwalt Corporation, 1987 .
1 1
( 18) Carlson, A. B . , and D. G. Gisser. Electrical Engineering. Readi ng, Mass.: AddisonWesley, 98 , pp. 287-294.
[19) Pressure Sensors, Catalog BR 1 2 1/D. Phoeni x , Ariz.: Motoro la, Inc., 1 991 .
(20) Solid-State Sensor Handbook. Sunnyvale, Calif.: SenSym, Inc., 1 989.
(21) Alley, C. L., and K. W. Atwood. Microelectronics. Englewood Cliffs, N.J.: Prentice
Hall, 1986, pp. 56-57.
(22) Wilson, 1., and J. F. B. Hawkes. Optoelectronics, an Introduction. 2nd
Hempstead, UK: Prentice Hall International, 1989, pp. 280-286.
ed: Hemel
(23) Handbook of Infrared Radiation Measurements. Stamford, Conn . : Barnes Engineer­
ing Company, 1983, pp. 5 1 -56.
(24) Dennis, P. N. J. Photodetectors: An Introduction to Current Technology. New York:
Plenum Press, 1986.
(25) Pallas-Areny, R . • and J. G. Webster. Sensors and Signal Conditioning, 2nd ed. New
York: John Wiley. 2001 , Chapter .
(26) Photomultiplier Ha11dbook. Lancaster, Pa.: Burle Technologies, Inc., 1980.
(27) Wil son, J., and J. F. B. Hawkes. Optoelectronics: An Introduction. 3rd ed. Harlow,
UK: Prentice Hall Europe, 1 998, Section 7.3.5.
(28) Kasap, S. 0. Optoelectronics and Photonics: Principles and Practices. Upper Saddle
River, N.J.: Prentice Hall, 2001 .
[29) Horowitz, P. . and W. Hill. The A n of Electronics. 2nd ed . New York: Cambridge
University Press, 1989.
(30) Tipler, P. A. Physics. New York: Worth Publishers, Inc., 1976, pp. 848-850.
(31) Hall Effect Transducers: How to Apply Them as Sensors. Freeport, Ill.: Micro Switch,
1
A Honeywell Division, 982.
[32) Shutler, J., and A. Lee . Personal Communication 10 J. H. Lienhard, Chrysler Motors
Corporation, Februacy 1992.
[33) Shigley, J. E., and C. R. Mischke. Mechanical Engineering Design. 5th ed. New York:
McGraw- Hill, 1 989, Ch. 1 0.
(34) Haugen, E. B. Probabilistic Approaches to Design. New York: John Wiley,
1968.
[35) Gitlin, R. How temperature affects instrument accuracy. Control Eng. , 2, May 1 955.
(36) Laws, F. A. Electrical Measuremenls. 2nd ed. New York: McGraw-Hill, 1 938, P·
236
2 17.
Sensors
ANSWERS TO SELECTED PROBLEMS
4
UK / K ':::: ±3%
11
UK / K ,,,. ±5.7%
13
K = 498 Nim
Signal Cond iti o n i n g
1
2
3
4
5
6
7
8
18
19
20
21
22
INTRODUCTION
ADVANTAGES OF ELECTRICAL SIGNAL CONDITIONING
MODULATED ANO UNMODULATED SIGNALS
INPUT CIRCUITRY
THE SIMPLE CURRENT-SENSITIVE CIRCUIT
THE BALLAST CIRCUIT
VOLTAGE-DIVIDING CIRCUITS
SMALL CHANGES IN TRANSDUCER RESISTANCE
RESISTANCE BRIDGES
REACTANCE OR IMPEDANCE BRIDGES
RESONANT CIRCUITS
ELECTRONIC AMPLIFICATION OR GAIN
ELECTRONIC AMPLIFIERS
OPERATIONAL AMPURERS
SPECIAL AMPLIFIER CIRCUITS
FILTERS
SOME FILTER THEORY
ACTIVE FILTERS
DIFFERENTIATORS AND INTEGRATORS
SHIELDING AND GROUNDING
COMPONENT COUPLING M ETHODS
SUMMARY
1
INTRODUCTION
9
10
11
12
13
14
15
16
17
Once a mechanical quantity has been detected and possibly transduced, it is usually neces­
sary to modify the stage-one output further before it is in satisfactory form for driving an
indicator or becoming the input to an electronic control or display. We will now consider
some of the methods used in this intermediate, signal-conditioning step.
Measurement of dynamic mechanical quantities places special requirements on the
elements in the signal-conditioning stage. Large amplifications, as well a.• good transient
response, are often desired, both of which arc difficult to obtain by mechanical, hydrau lic,
or pneumatic methods. As a result, electrical or electronic elements are usually required.
A n input signal is often converted by the detector-transducer to a mechanical
displacement. It is then commonly fed to a secondary transducer, which converts it into
a form, usually electrical, that is more easily processed by the intermediate stage.
In
some cases, however, such a displacement is fed 10 mechanical intermediate elements,
such as l inkages, gearing, or cams; these mechanical elements present design problems of
From Mecha11ica/ Measu,.ments,
Si.ih Edition. Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V.
Inc. Published hy Prentice Hall. All right< reserved.
Copyright «:I 2007 by Pearson Education.
239
Signal Conditioning
considerable magnitude, particularly if dynamic inputs are to be handled.
In the field of dy namic measurements, strictly mechanical systems are much more
uncommon than they were in years past, largely because of several inherent disadvantages,
which we will discuss only briefly. 1
Mechanical amplification by these elements is q uite limited. When ampli fication is
req u i red frictional forces are also amp li fied, resulting in considerable undesirable signal
loading. These effects, coupled with backlash and elastic deformations, result in poor
response . lnenial loading results in reduced frequency response and in certain cases,
depending on the panicular configuration of th e system, phase response is also a prob­
lem.
2
ADVANTAGES OF ELECTRICAL SIGNAL CONDITIONING
As we have already seen, many detector-transducer combinations provide an output in elec­
In these cases, of course, it is conve nie nt to perform funher signal condi tioning
ndi tioning may typically include converting resistance changes to volt­
age changes, subtracting offset voltages, increasing signal vol tages, or removing unwanted
frequency components. In addition, in order to minimize friction, inertia, and structural
flexibility requirements. we also prefer electrical methods for their ease of power amplifi­
cation. Additional power may be fed into the system to provide a greater output power than
input by the use of power amplifiers, which have no important mechanical counterpan in
most instrumentation. 2 Electronic signal c o n di t ioni ng is, obviously, always needed when
the output is to be recorded or processed by a computer, electronic control, or digital display.
trical
form.
electrically. Such co
3
MODULATED AND UNMODULATED SIGNALS
Measurands may be "pure" in the sense that the analog electrical signal contains nothi ng
the real-time variation of the mcasurand information itself. On the other hand,
the signal may be .. mixed" with a carrier, wh ich consists of a voltage oscillation at some
freq ue ncy h ig he r than that of the signal. A common rule of thumb is that the frequency ratio
should be at least I 0 10 I . The signal is said to modulate the carrier. The meas u rand affects
the carrier by vary i ng either i ts amplitude or its frequency. In the former case the Clllrier
freq uency is held constant and its ampli tude is varied by the measurand. This process is
k now n as amplitude modulation, or AM [Fig. l (a)). In the latter case the carrier amplitude
is held constant and its freque ncy is varied by the measurand. This is known as frequency
modulation, or FM [Fig. l (b ) ] . The most familiar use of AM and FM transfer of s i g nal s is
in AM and FM radio broadcasting.
When modulation is u sed in instrumentation, ampl itude modulation is the more com­
m on form . Nearly any mechanical signal from a passive pickup can be transduced into an
analogous AM fonn. Sensors based on either inductance or capacitance require an ac exci­
tation. The differential transformer is an example of the former, whereas. The ca11acitive
other than
11'he first and second editions or lb.is book
oondilioning methods and problems.
contain a more 1 horough discussion of stric1ly mechani cal sipal·
2 h is uuc th.al hydraulic and pneumatic systems may be set up to increase signal power; however. their use is
10 relaiively slow-acting control appljcations. primarily in the fields or chemical processi ng and electric
power generation. As in lhe cme of mcclumi cal systems. friction and inertia severely limit transient response of
1hc type required for measurement or dynamic inputs.
limited
240
Signal Conditioning
Modulaled signal
RecUHed signal
Oemodulaled signal
(a)
continued
FIGURE I : (a) Ampli1ude modulation, whereby the envelope of the carrier contains the
signal information; (b) frequency modulation, whereby the s ignal infonnation is contained
in the fmiuency variation of the carrier.
pickup for liquid level is an example of the lauer. In addition, some resistive- type sensors
use ac excitation.
Extracting the signal infonnation from the modulated carrie r is required. When AM
is used, this operation may take several fonns. The simplest is merely to display the entire
signal using an oscilloscope or osc i l lograph , and then to "read" the result from the en velope
of the carrier. More common! y, the mixed signal and carri er arc demodulaled by rectification
and filtering, as show n in Fig. l (a). FM demodulati on is a more complex operation and
may be accomplished through the use of frequency discrimination, ratio detection, or IC
phase-locked loops. Furt her discussion is beyond the scope of this text.
4
INPUT CIRCUITRY
Electrical detector-transd ucers are of two general types: ( I ) passive, those requ i ri ng an
auxiliary source of energy in order to produce a sig na l ; and (2) active, those that are
self-powering. The simple bonded strain gage is an example of the fonner, whereas the
piezoelectric accelerometer is an example of the lauer.
241
Signal Conditioning
Unmodulated carrier
�--------�- - - Modulated carrier
-------�-
Modulating signal
(b)
FIGURE I : Conti1111ed
Whereas it may be possible to use an active, or self-powering, detector-{ransducer
directly with a minimum of circuitry, the passive type, in general, requires special arrange­
ments to introduce the auxiliary energy. The part i cu lar arrangement required will depend
on the operating principle involved. For example. resisti ve- ty pe pickups may be powered
by either an ac or a de source, whereas capac i ti ve and inductive types, with an exception or
two, requ i re an ac source.
Although not al l inclusive, the following list c l assi fie s the most common forms ofinput
circuits used in transducer work: ( I ) sim ple current-sensitive circuits, (2) ballast circuits,
(3) vollage-dividing circ uits, (4) b ridge circuits, (5) resonant c i rcu i ts , and (6) ampli fier
input circuits. Often. the input circuits will be followed by some type of filter circuit. These
circuits are discussed i n the following sections.
5
THE SIMPLE CURRENT-SENSITIVE CIRCUIT
Figure 2(a) illustrates a simple current-sensitive circuit in which the transducer may use
any one or the vari o us forms of variable-resistance elements. We will let the transducer
resistance be k Rr , where R, represents the maximum value or transducer resistance and k
represen ts a percentage factor that may vary between 0.0 and 1 .0 (0% and I 00%) , depending
on the magnitude of the input signal . Should the transducer element be i n the form of a
sliding contact resistor, the value of k cou ld vary through the complete range of 0% to
242
Signal Conditioning
Rm
.--.i'Vl.N'--1
Current Indicator
or recorder
sensing output
current, i0
Resistance-type
transducer
kR,
(a)
o.&O i-;.-....+----'...,._
..
+---+'"'""::-+:-- -----i
�-
0.40 1---"<-+----'"'d--jl---""-..::::l--I
0.4
0.2
k
(b)
0.6
0.8
1 .0
AGURE 2: (a) Simple current-sensitive circuit; (b) plot of Eq. (2), showing variation of
current in terms of input signal k for a simple current-sensitive circuit.
100%. On the other hand, if R, represents, say, a thermistor, then k would fall within some
limiting range not i ncludi ng 0.0%. We will let R,. represent the remaining circuit resistance,
inc ludi ng both the meter resistance and the internal resistance of the voltage source.
If i0 is the current flowing through the circuit and hence the current indicated by the
readout device, we have, using Ohm's law,
.
10 =
kR,
e;
(I)
+ Rm
Note that maximum current ftows when k = 0 , at which point the current i s imax
Equation ( I ) may thus be rewritten as
=
e; / R,,, .
(2)
243
Signal Conditioning
Figure 2(b) shows plots of Eq. (2) for various values of resistance ratio. The abscissa
is a measure of signal input and the ordinate a measure of output. First of all . it is observed
that the input-output relation is nonlinear, which of course would generally be undesirable.
In addition, the higher the relative value of transducer resistance R1 to Rm . the greater will
be the output variation or sensitivity. It will al so he noted that the output is a function
of imax. which i n !urn i s dependen t on e; . Thus careful control of the drivi ng vol tage is
necessary if calibration is to be maintained.
6
THE BALLAST CIRCUIT
Now let us look at a variation of the current-sensitive circuit, often referred to as the ballast
circuit, shown in Fig. 3. Instead of a current-sensitive indicator or recorder through which
the total current ftows, we shall use a voltage-sensitive device (some fonn of voltmeter)
placed across the transducer. The ballast resistor Rb is i nserted in much the same manner
as Rm was used in the previous circu it . It will be observed that i n this case, were it not for
Rb, the i ndicator would show no change with variation in R1 ; i t would always indicate full
source voltage. So some value of res is tance Rb is necessary for the proper functioning of
the circuit.
Two different situations may exisl. depending on the relative impedance of the meter.
Firs!, the meter may be of high impedance. as wou ld be the case if some form of electronic
voltmeter were used; in this case any current How through the meter may be neglected.
Second, the meter may be of low impedance, so that consideration of such current ftow
is required .
Assu ming a high-impedance meter. we have, by Ohm's law,
e'·
.
• = ---
(3)
Rb + k R,
Then, if e0 = the vo l !age across k R, (which is indicated or recorded by the readout device),
e,. =
This equation may be wriuen as
�
e;
.
t (k R, )
=
= --­
e;k R1
(4)
Rb + k R,
k R, / Rb
(5 )
I + (k Rr / Rb)
For a given circuit, e0/e; is a measure of the output, and k R1 / Rb is a measure of the input.
Voltage Indicator
or recorder
sensing output
voltage
FIGURE
3: Schematic diagram of a ballast circuit.
244
Signa l Conditioning
Defining
we have
'1
as the sens itivi ty, or the ratio of change i n output lo change in input,
'I
=
de0
e; RbR1
dk = (Rb + kR,)2
(6)
We may change Rb by i nsert ing different values of resistance. In that case the sensitivity
would be altered, which would mean that there may be some optimum value of Rb so far
as sensitivity is concerned. By di fferen ti ation with respect to Rb, we should be able 10
determine this value:
.!!.!!.._ = e1 R, (k R, - Rb)
dRb
(7)
(Rb + kR1)3
The derivative will be zero under two conditions: ( I ) for Rb = oo, which results in
minimum sensitivity, and (2) for Rb = kR, , for which maximum sensi tivity is obtained.
The second relation indicates that for full-range usefulness, the value Rb must be
based on compromise because Rb. a constant, cannot always have the value of k R,, a
variable. However, Rb may be selected to give maximum sensitivity for a certain point in
the range by sett i ng its value to correspond to that value of k R, .
This circuit is occasionally used for dynamic appl ications of resistance-type strain
gages ( I , 2]. I n this case the change in res istance i s quite small compared with the total gage
resistance, and the relations above indicate that a ballast resistance equal to gage resistance
is optimal.
F igure 4 shows the relation between input and output for a circuit of this type as given
by Eq. (5).
0
FIGURE 4:
0.2
0.4
0.6
0.8
1 .0
k
Curves showing relation between input and output for a ballast circuit.
245
Signa l Conditioning
FIGURE 5:
It will
be
The vo ltage-d ivid er c i rc u i t .
noted that the same disadvantages apply lo Ibis circuit as IO
lhe curren1-
sensitive circuit discussed previously-namely, ( I ) a percentage variation in lhe supply
voltage, e1 , results in a greater change in output lhan does a si mil ar percentage change in
k, so that very careful voltage regulation must be used; and (2) the relation between output
and input is not linear.
7
VOLTAGE-DIVIDING CIRCUITS
voltage divider (Fig. 5) i s a ubiquitous element of instrumentation circuits. Very
e; , into a smaller
output voltage, ea. If a negligible currem is d rawn from the output terminals. lhe current
through the resistors follows from Ohm's law:
The
simply, this circuit uses a pair of resistors lo divide an input voltage,
The output voltage meas u red across Ri
is then
ea = i R2 = R 1
R1
+
i e;
R
(8)
chapter. The ballast ci rcuit of the precedi ng
[see Eq. (4)] i n which the fraction of input voltage al
the output depends on the transducer resistance; bridge ci rcu its (Section 9) are essentially
pairs of voltage dividers; and the noninvening a mpl i fier (Example 5) also incorporates a
Voltage dividers appear throughout this
section is essentially a voltage divider
voltage divider.
7.1
The Voltage-Dividing Potentiometer
Figure 6 is
a very
useful voltage-divider arrangement for sliding conlacl resistance trans­
voltage-dividing potentiometer circuit.. Note that the c ircuit is
ducers. It is known as the
connected not to the slider.
resistance elemenL
The
as ii
would
be
in the ballast circuit. but
1ermina1ing, or readout. device is c on nected
R p determ i ned by k ..
with this arrangement, dlepend i ng on the relatiw:
impedance of the resistance element and the indicator-recorder. If the terminating instrU­
ment is of sufficiently high relative impedance, no apprec i able current will flow through ii.
d rop across the
ponion of res i st ance element
across the complete
lo sense the voltage
Two different si1ua1ions may
occur
246
Signal Conditioning
T
FIGURE 6: Simple voltage-dividing potentiometer circuit.
The circuit then becomes a true voltage divider, and the indicated output voltage e0 may be
determined from Eq.
(8),
k Rp
e0 = -e; = k e;
Rp
or
(8a)
k=�
e;
On the other hand, if the readout device draws appreciable current, a loading error will
result.
7.1.
Loading Error
The loading error may be analyzed as follows. Referring to Fig. 6, we find that the total
resistance seen by the source of e; will be
and
;=
+
e; (k Rp + RL)
!!.
=
R k R � ( I - k ) Rp RL
The output voltage will then be
or
ea
;; =
I+
k
(Rp/ RL ) k - ( Rp/ R L)k2
(9)
If we assume the simpler relation given by Eq. (Sa) to hold, an error in ea will be introduced
according to the following relation:
[
[
+] I ]
k
____
k ( I - k)(Rp/RL)
k2 ( 1 - k)
e
= ; k ( I - k ) + (R1. / Rp)
Error =
e·' k
-
____
247
(10)
Signal Conditioning
k
ng potentiometer cir­
FIGURE 7: Curves showing error caused by loading a vol tage-divid i
cuit.
[ k2(1 -k)
J
By comparing to the full-scale output, e; , this relation may be written as
Perce nt error
=
k(I
-
k) + ( RL/ Rp)
x 1 00
( I I)
Except for the en dpoin ts (k = 0.0 or 1 .0), where the error is zero, the error will
always be on the negative side; that is, the measured value of vol tage will be lower than
wou ld be the case if the system perfonned as a li near voltage divider. Figure 7 shows a
plot of the variation i n error w ith slider position for various ratios of load to potentiometer
resistance. Obviously, the higher the value of load resi stance compared with potentiometer
resistance, the lower will be the error; thus high-input resistance is a desirable feature in
voltage-reading devices.
7.3
8
Use of End Resistors
II will be observed th at the nonlinearity in the relation between the potentiometer output
and the i nput d isplace me nt k may be reduced if only a portion of the available potentiometer
range is used. For example, a 1 000- 0 potentiometer may be selected, but the input could
be limited to onl y a 500-'2 ponion of the total range. This limitation would reduce the
potentiometer resolution and would be ge nerall y impractical; however, it would result in a
reduction in the dev iation from li nearity. A similar result may be obtilined through use of
what are known as end resistors (Fig. 8). When either an upper- or lower-end resistor, or
both, is used. it is often possible to compensate for the reduced potentiometer output caused
by the increased resistance by inc reas ing the voltage inpu t e; by a proportional amou n t.
SMALL CHANGES IN TRANSDUCER RESISTANCE
Some resistance transducers sh ow only very small changes in their resistance. For example,
the resistance of a foil strain gage may vary by only about 0.000 1% during use! The
248
Signal Conditioning
e,
-+
FIGURE 8 : Method for i m provi ng linearity of potentiometer circuilS when low-impedance
indicating devices are used. Res istors R, are termed end resistors.
smallness of
the re s is tance change has important ramifications for the choice of signal­
conditioning circuit.
Suppose that a voltage-divider (or ballast) circuit is formed from a transducer of
resistance Ri and a second resistor Rt [Fig. 9(a)). 1be resistances are initially made equal,
Rt = Ri = Ro. so that the initial output voltage is
e0
=
Ri
---
R1
+
R2
e;
=
Ro
Ro + Ro
---
e;
e;
= -
2
(a)
+
(b)
FIGURE 9: The use of voltage dividers in measuring small resistance changes.
249
Signal Conditioning
If the resistance of the transducer then increases from Rz
output changes to
eo
+
A eo =
=
=
(Ro + AR)
Ro + (Ro + A R )
Ro + !!i.R
e
2Ro + l!i. R ;
e;
=2
e;
(
(l
Ro
2Ro
e;
1 + l!i.R/Ro
1 + AR/2Ro
+
e;
( )(
AR/2Ro
2Ro
1
)
e
1 + A R/2Ro
l!i. R
=2+2
)
=
Ro to Rz
;
1
+ A R/2Ro
=
(Ro + AR), the
)
Assuming that the resistance change is small, so that fl. R /2Ro « I , we can approximate
the last factor on the right-hand side as unity; hence,
eo + l!i.eo �
� (� )
+
2
2
= eo +
( )
AR
( 12)
2Ro
e;
AR
4Ro
( I la)
Thus, for small resistance changes, the output voltage shows straight-line variation with
A R .3 S uc h variation is advantageous because it simplifies data reduction. Unfortunately,
the disadvantages of this circuit become quite apparent when we look at the numbers.
Taking the strain gage transducer as an example, a typical resistance change might be
l!i. R = 240 µQ in a gage of initial resistance Ro = 1 20 Q. Hence,
l!i.e0
e.
=
(l!i. R/4Ro)e;
e; /2
=
AR
2Ro
= 10
-6
Since we measure the su m e0 + t.e0, we will need a meter with a resolution of better
than one part in a m i l l ion in order to see any change in e0 at all. This excludes common
voltmeters. which may resolve to only 0.0 1%, although it is within the reach of the very
best commercial meters.
An even more i m porta nt limitation is the stability of the input voltage, e; . If e; drifts
slightly between the initial and final readings of the output (to e; + l!i.e;), then Eq. (12)
shows that the output becomes
eo + l!i.eo �
e; + l!i.e;
--2--
+(
e; + l!i.e;
--2--
� eo + - + -e;
lie;
2
l!i. R
4 Ro
)( )
l!i. R
2Ro
3 tn much the same way, end resiston in the voltage-dividing potentiomeu:r (Section 7.3) serve 10 make !he
transduo:erresistano:echangc small relative 1olheolher resistances, creating an appro•imately slrllighl-line varialioa
or the output
Signal Conditioning
Thus if e; drifts by even 0. 1 % (t.e; = 0.00 l ei ) , the change in 6e0 caused by voltage
drift will be 0.00 l (ei /2)-a thousand times larger than the strain-induced voltage change
(6 R/4Ro)ei = 1 0- 6 (ei /2) !
The difficulty, of course, is that we are ttying to resolve a voltage change which is
a tiny fraction of the total output voltage, and it illustrates an important principle in mea­
surement: Avoid measurements based on a small difference between large numbers. Such
measurements are limited by the accuracy with which the large numbers can be measured.
In this case, the solution is to design a circuit having output voltage proportional
to 6e0 itself, without the large offset voltage, e0 • We can do this by introducing another
voltage divider with lixed resistors Ro [Fig. 9(b)), which has a midpoint voltage of e0• We
now measure the difference between the midpoint voltages of the two dividers as the output
voltage of the circuit:
( 1 3)
The problems caused by the offset voltage, e0, are thus eliminated.
This arrangement of two voltage dividers is, in fact, identical to the Wheatstone bridge
circuit discussed in the next section; howeveT, the Wheatstone bridge is not always restricted
to small resistance changes.
9
RESISTANCE BRIDGES
Bridge circuits are the most common method of connecting passive transducers to measuring
systems. Of all the possible configurations, the Wheatstone resistance bridge devised by
S. H. Christie in 1833 [3, 4) is undoubtedly used to the greatest extent. Figure 10 shows a
de Wheatstone bridge circuit consisting of four resistor anns with a voltage source (battery)
and a detector (meter). In applications, one or more of the arms is a resistance transducer
whose resistance is to be detennined. Typical resistance transducers used with a circuit of
A
+
B
D
c
.... - ....
\
'+
1 e• ..,,_____.
.___..,
, _ _ ...
FIGURE t o: Simple Wheatstone bridge circuit.
Signa l Conditioning
this kind include resistance strain gages, resistance thennometers, or thermistors.
Bridge circuits enable high -accuracy resistance measurements. These measurements
are accomplished either by balancing the bridge-making known adju stments in one or
more of the bridge anns until the voltage across the meter is zero-or by determining the
magnitude of unbalance from the meter readi ng . If the circuit appears complicated to you,
it may help to recognize that, when negligible current flows through the meter. the bridge is
simply a pai r o f vol tage-divider circuits (ABC and ADC) with the output taken between the
midpoi nts of the two dividers ( B to D). The great advantage of the bridge circuit is that the
offset vol tages of the two dividers cancel, so that the bridge output voltage can be di rectly
related to ch anges in transducer resistance (see Sec t ion 8).
U s ing Fig. 10. we may analyze the requirements for balance. At balance, the voltage
across the meter is zero and no cu rrent flows through it; hence, i1 = 0. In t hat case, we also
know th at i 1 = i2 and il = i4. S i nce the potential across the meter is zero, i 1 R1 = i3R3
and iz R2 = i4 R4 . By elimi nating i 1 and i) from these relations, we obtain the condition
for balance, namely,
( 14)
or
( 14a)
From these two equations we may formulate a statement to assist us in remembering
the necessary balance relation. In order for the Wheatstone resistance bridge to balance.
the ratio of resistcmces ofany two adjacent arms must equal the ratio of resistances of the
remaining two arms, taken in tire same sense. (Note: "Taken in the same sense" means
that if t he first resistance ratio is formed from two adjacent resistances reading from left to
right, the balancing ratio must also be formed by reading from left to right, etc.)
Basic bridge types are summarized in Table I . When a null bridge is used, the
res istance of one u n known ann i s determined by finding values of the other ann s for which
the bridge is balanced. Thus, some provision must be made for adjusting the resistance
of one or two arms so as to reach balance . Some balancing arrangements are shown in
Fig. 1 1 . An important factor in determining t he type lo use is bridge sensitiv ity. If large
resistance changes are lo be accommodated, large resistance adjustments must be provided;
thus one of the series arrangements would be most useful and could well be the type to use
for sliding-contact variable-resistance transducers or thennistors. When small resistance
changes are to take place, as in the case of resistance strain gages. then the s hunt balance
would be u sed . In order to provide for a range of re si stances, a bridge with both series and
shunt balances might be uti lized.
When the deflection bridge is used, bridge unbalance, as indicated by the meter
readi ng, is the measure of i nput . Usually, the deftection bridge is balanced initiall y and
later change.� in transducer resi stance cause the unbalance. Manufac t uring variations in
real res istors make it virtually i mpossible to obtain three resistors that will match the i nitial
transducer resistance well enough to satisfy Eq. (I 4a) precisely; hence, provi s ion is gen-
252
Signal Conditioning
TABLE
Bridge Type
Null balam:e bridge
1:
Types of Electrical Bridge Circuits
Bridge Features
Adjustment is required to maintain balance. This becomes
source of readout (e.g., a manually adjusled strain indicator).
vs.
Defl ect i on bridge
Readout is deviation of bridge output from initial balance
(e .g. , as required by a computer's analog-to-digital converter
[AID)).
Voltage-sensitive bridge
vs.
Curre n t-se ns iti ve bridge
ac bridge
Readout instrument does not "load" bridge; that is, it requires
no current (e.g., electronic voltmeter or analog-to-digital con­
verter).
Readout requires current (e.g., a low-impedance indicator
such as a simple galvanometer is used).
Alternating-current voltage excitation is used.
vs.
de bridge
Direct-current voltage excitation is used.
Constant voltage
Voltage input 10 bridge remains constant (e.g., battery or
voltage-regulated power supply is used).
vs.
Constant current
Resistance br idge
vs.
Impedance bridge
Current input lo bridge remains constant regardless of bridge
unbala nce (e.g., current-regulated power supply is used).
Bridge arms made up of "pure" resistance elemen ts .
Bridge arms may include reactance elements.
erally made for initial balanci ng by adjusting one or more arms, again using an arrangement
from Fig. 1 1 . Fo r static inputs, an ordinary voltmeter may be used to disp l ay the output; for
dy nam ic si gn als , however. the output may be displayed by an oscilloscope or the output may
be fed 10 an analog - to -d i gi tal converter and a computer for display, recording, or immediate
app l ic at ion .
The output from a deflection bridge may be connecled to either a high- or a low­
impedance device. If the bridge is con nected to a simple D' A rsonval meter or most gal­
vanometers, the output circuit will be of low impedance, and an appreciable current (ig ) is
drawn from the bridge. In most cases in which amplification or digital processing is neces­
sary, the bridge output will be connected 10 a high-impedance device and the bridge would
supply esse n t i ally no current. Such is the case when either an an electronic voltmeter or an
analog-to-Oigital convener is used. In the former instance the bridge is current sensitive;
in the latter it is wltage se11si1ive.
253
Signal Conditioning
(a) Series balance
(b) Differential seneu
balance
(c) Shunt balance
(d) Differential shunt
balance
FIGURE 1 1 : Arrangements used to balance de resistance !bridges.
9.1
The Voltage-Sensitive Wheatstone Bridge
Let us consider the si mple st case first, in which the bridge output i s c'onnected directly to
a high-impedance device, say an oscilloscope. Referring to Fig. I O, we see that the output
voltage is the difference between the voltages at 8 and D
and. making use of the voltage-divider relati o n [Eq. (8)]. we may write
( 1 5)
( I Sa)
We will n o w assume that the resistance
254
R1 changes by an amou nt
t:;. R1,
or
Signal Conditioning
The relation may be si mplified by assuming all resistances to be initially equal to
which case e0 = 0). Then
R (in
( 1 7)
Figure 1 2(a), plotted from Eq. ( 17), shows the relation for the output of a voltagc­
sensitive de flection bridge whose resistance anns are ini tial ly equal. Inspection of the curve
indicates that this type of resi stance bridge is inherently nonlinear. In many cases, however,
Plot of Eq. ( 1 7)
0.50
0.40
�-
.,.o
0.30
0.20
0.10
0
""I --0.10
--0.20
--0.30
--0.40
--0.50
-
--
./
--
--
--0.6 --0.4 --0.2
0
(a)
0.30
0.20
....._..
""I
\
I\
0. 10 r-...
0
--0.1 0
0.2
.d R,r'R
�'
I'.
�� l!!i..
0.4
0.6
0.8
1 .0
Plot of Eq.(20)
... ��
�
--
--0.20
--- -
� ......::::
R/R = 2
-
1
f::::: ::::::::- -;;;-�
- 1 .0 --0.6 -0.2 0 0.2
0.6
1 .0
1 .5
2.0
2.5
3.0
(b)
FIGURE 12: (a) Output from a voltage-sensitive deflection bridge whose resistance anns
are initially equal; (b) output from a current-sensitive deflection bridge whose resistance
arms are initially equal, plotted for different relative gal vanometer resi s tances.
255
Signal Conditioning
the actual resistance change is so small that the arrangement may be assumed linear. This
assumption applies Lo most resistance strain-gage circuits. In those cases, 6R2/2 R « I
and lhe linearized output is
( 1 8)
which is identical to Eq. ( 1 3).
9.2 The Current-Sensitive Wheatstone Bridge
When Lhc deOection-bridge output is connected to a low-impedance device such as a gal­
vanometer. appreciable current flows and the galvanometer resistance must be considered in
lhe bridge equation. Galvanometer current may be expressed by the following relation [5]:
( 19)
where
i1 = the galvanometer current,
i1 = the input current,
R8 = the galvanometer resistance
The remaining symbols are as defi ned in Fig. 10.
If we assume that an initial bridge balance is upset by an incremental change in
resistance tl.R1 in arm R1 and all arms are of equal initial resistance R, we may write
l:> ig
i;
=
- t.. R i / R
4( I
+ ( R8 / R )) +
[2 + ( R8 / R)J( t.. R 1 / R)
(20)
Figure 12(b) shows Eq. (20) ploued for various values of Rg/ R.
93
The Constant-Current Bridge
To this point oor discussion of bridge circuits has assumed a constant-voltage energ izing
source(a battery, for example). As the bridge resistance is changed, the total current through
1hc bridge will, therefore, also change. In certain instances, use of a constant-current
bridge� may be desirable [6, 7 ] . Such a circuit is usually obtained through the application
of a commercially available current-regulated de power supply, S whereby the total current
How i; 1hrough the bridge (Fig. I 0) is maintained at a constant value. It should be noted
lhal such a bridge may still be either voltage sensitive or current sensitive, depending on
1he rela1ive impedance of the readout device.
"Tiie lem'I Whtatslonie-. as appl ied co bridge �in:uils. is common I)' lin\iced 10 the cons1ant-t10l1aae resistana
btidg<. We shall abide by this convention and l'oid n:fening to the constant-current bridge as a Wheatstone bridge.
Sc111111a111 cumnt is obtained by us ing the voltage drop across a series resistor in the supply-output line to
pnwide a regulating feedback voltage. It may also be approximated by placing a large ballast resistor bcl1he bridg.le and the vollage source� lhe resistor is made large enough 1ha1 variations in the bridge resistors have a
nqlipble elfCCI on i; .
256
Signal Conditioning
Relationships for the voltage-sensitive constant-cu�nt bridge may be developed as
follows. Referring to Fig. 10, we may write
(2 1 )
or
Substituting in Eq. ( 1 5a), we have
(22)
which is the basic equation for the voltage-sensitive constant-current bridge, provided that
i; is maintained at a constant value. Ir the resistance of one arm, say Rz, is changed by an
amount ll. R , then
and
For equal initial resistances (R1
= Rz
tieo - i I·
9.4
[
]
= RJ = R4 = R),
6R
4 + (6 R/R)
(24)
The constant-current bridge has better linearity than the constant-voltage bridge, as is appar­
ent upon comparing Eqs. ( 17) and (24 ) .
The AC Resistance Bridge
Resistance bridges powered by ac sources may also be used. An additional problem, how­
ever, is the necessity for providing reactance balance. In spite of the fact that the Wheatstone
bridge, strictly speaking, is a resistance bridge, it is impossible to completely eliminate stray
capacitances and inductances resulting from such factors as closely placed lead wires in
cables to and from the transducer, and wiring and component placement in associated equip­
ment. In any system of reasonable sensitivity, such unimentional reactive components must
be accounted for before satisfactory bridge balance can be achieved.
Reactive balance can usually be accomplished by introducing an additional balance
adjustment in the circuit. Figure 1 3 shows how this may be provided. Balance is accom­
plished by alternately adjusting the resistance and reactance balance controls, each time
reducing bridge output, until proper balance is finally achieved.
257
Signal Conditioning
Indicator
FIGURE 1 3: Circuit arrangement for balanc i ng an ac bridge.
Compensation for Leads
9.5
Frequently a sensor and a bridge-type instrument must be separated by an appreciable
Wires, or leads, are used to connect the two as illustrated i n Fig. l4(a), which
shows the sensor as some type of resistance element such as a resistance thennometer
or strain gage. ln addition lo the extra resistance introduced by lhe leads, temperalWe
grad ie nts along lhe wires may, in cerlain cases, cause error. We can compensalc for lhis
distance.
Resistance·
type sensor
(a)
Resistance­
type sensor
Compensating lead
(b)
FIGURE 14: (a) Simple bridge wilh re motel y localed sensor; (b) circuil similar to thal shown in (a).
a compensating wire.
but with
Signal Conditioning
fl,
FIGURE 15: Melhod for adj u sti n g bridge sensitivity th rou gh use of variable series
1ance,
resis-
R,.
usin g a lhree- wire circui1 as ill ustrated in Fi g . 1 4(b). lnspccl ion shows thal
the additional lead serves lo balance the total lead-wire lengths in the two adj acent anns,
thereby eli minating any u nbalance from this source.
1ype of error by
9.6
Adjusting Bridge Sensitivity
Adjustable bridge sensitivity may be desired for several reasons: ( I ) Such adjustment may
be used to attenuate inputs that are larger than desired; (2) i i may be used to provide
a convenient relation between system calibration and the scale of the readout i n s1ru men1 ;
(3) ii may be used to provide adjustmenl for adapting individual transducer characleristics to
precalibrated syslems (this method is used 10 insert the gage faclor for resistance strain gages
in some commercial circuils); (4) ii provides a means for controlling certain extraneous
temperature effects.
A very simple method of adjusting bridge output is lo i n se n a variable series resi stor
in one or both of lhe input leads, as shown in Fig. 1 5 . If we assume equal initial resistance
R in all bridge arms, the resislance see n by the voltage source will also be R. If a series
resistance is inserted as shown, lhen, trunking in terms of a voltage-dividing circuil, we see
that lhe inpul lo the bridge will be reduced by the faclor
inputs such as
n
We call n lhe
which
10
bridge factor.
=
R
-R + R,
=
I
(25)
I + ( R, / R)
The bridge oulput will be reduced by a proport i on al amount,
makes this melhod very useful for controlling bridge sensitivity.
REACTANCE OR IMPEDANCE BRIDGES
Reactance or impedance bridge configurations an: of the same general form as the Wheal­
lhat reaclive elements (capac i lors and induc1ors) are involved in one
slone bridge, except
�r
more of the arms. Because such elements are inherently frequency sensitive, impedance
bridges are ac excited. Obviously the mullitude
of varia1ions 1ha1 are possible preclude
more lhan a general discussion in a work of this nature; 1hus the reader is referred 10 more
specialized works for detailed coverage (8, 9 ).
Figure 16 shows several of the more common ac bridges, alo ng wit h t he lype of
element usually measured and the balance rcquiremcnls.
259
Measures l or C
Belance equations:
R, = R,
'tI
If inductive, L, • L,
..
II capacitive, C,=
�
1
R
C•R;,
Measures l (best lorwl,IR, < 1 0)
Belance equations:
L, = flo�C,
N
0)
0
I
R,v.
.
R =
•
flo
�
R,
�
�:;.
A
Resonant circuit
Measures l (best lor wl/R, > 10)
Betance equations:
l =
x
R
'
•
flo R3 C,
1 +w2Cl R,2
w2C�R, flo R3
1 +w2Cl R12
R.
A R.
��
Wein, or RC, frequency bridge
Hay circuit
FIGURE 1 6:
c• . c; !!i
R, = flo
Ro
�
Scherlng circuit
�· ��il.
Maxwel circuit
Measures C
Balance equations:
Impedance bridge arrangements.
Measures l or C (/known),
f (l end C known)
Balance equations:
XL . Xe; or L
f
-
.c �
�
1
- 2ric
Measures f
Balance equations:
f-
1
- 2:rrJR,R4C;iC4
�-�
+�
R• C3
flo
Signal Conditioning
l�I
R
L
c
Frequency
(a)
(b)
RGURE 17: Parallel LC c i rc ui t with curve showi ng ou tpu t versu s freque ncy charac teristics.
11
RESONANT CIRCUITS
Capac i tance - i nductance combinations present varying impedance, depending on their rel­
ative values and the frequency of the appl i ed voltage. When connected in parallel, as in
Fig. I 7(a), the inductance offers small opposi ti on to current flow at low frequencies, whereas
the capac itive reactance is low at high frequencies. At some intermediate freq ue ncy, the
opposi ti on to current flow, or im peda nce , of the combination is a maximum [Fig. I 7(b) ]. A
similar but opposi te variation in impedance is o btained in the series-connected combination.
The frequency correspond i ng to maximum effect, k now n as the resonance frequency,
may be de termi ned by the relati on
f
I
= 2rr ./LC
(26)
where
f
=
the frequency (Hz),
= the inductance (H),
C = the capacitance (F)
L
It is evident that should, say, a capacitive transducer e lem en t be used , it could be
in combination with an i nducti ve element to form a resonant combination. Variation in
capacitance caused by variation in an input signal (e.g., mechanical pressure) wou ld then
alter the resona nce frequency, which could then be used as a measure of input.
11.1
Undesirable Resonance Conditions
On occ as i on , resonance conditions that occur may introduce spu riou s outputs. Most cir­
cuits arc susceptible because they use some combi nation of inductance and capacitance
and most are called on to handle dynamic signal in pu ts . In certain cases the capacirance
261
Signa l Conditioning
and inductance may be not more than the stray values eitisting between the circuit com­
ponents, including the wiring. Hence there may be resonance conditions that can result in
nonlinearities at certain input or exciting frequencies.
Normally such situations are avoided in the design of commercial equip ment insofar
as possible. However, the instrument designer is not always in a position to predict lhe
exact manner in which general-purpose components may be asse mbled or the exact nature
of the input signal fed to the equipment. As a result, it is possible to unintentionally set
up arrangements of circuit elements combined with frequency conditions that result in
undesirable resonance conditions.
12
ELECTRONIC AMPLIFICATION O R GAIN
The ratio of output to input for an electronic signal-conditioning device is referred to vari­
ously as gain, amplification ratio (if greater than unity), or attenuation (if less than unity).
It may be defined in terms of voltages, currents, or powers, that is,
Voltage gain = voltage output/voltage input,
Curren t gain = current output/current input,
Power gain = power output/power input
Another way of expressing power gain is through use of t he decibel. A decibel (dB)
is one-tenth of a be/ and is based on a ratio of powers:
Decibel (d B ) =
1 0 log 1 0 ( Po / P; l
(27)
where P0 = the output power and P; = the input power. both eitpressed in the same units.
The average human ear can just detec t a lo udness change from an audio amplifier
when a power ratio change of one decibel is made. It has also bee n observed that this is
nearly true regardless of the power level.
Solving Eq. (27) for the ratio ( P0/ P; ) corresponding to one decibel yields a ratio
of 1 .26. In other words, for the average human ear to j u s t detect an increase in sound
output from an amplifier (feeding some form of earphone or loudspeaker), an increase of
approximately 26% in power is required. Some other useful power ratios, as eitpressed in
decibels, are
P0 = 2
x
P; :
Po = 1 0 x P; :
Po = 100
x
P; :
3 dB .
2°
I
Pu =
l O dB .
I
Pu =
P0 =
20 d B ,
I
W
OO
I
x
P; :
-3 d B .
x
P; :
- l O dB,
x
P; :
-20 d B
The half-power point, - 3 dB, is often used in characterizing the frequency response of
amplifiers and, especially, of filters.
For a pure resistance, electric power may be expressed as
Power = ei
262
=
e2 / R = i 2 R
Signal Conditioning
where e = the voltage, i = the current, and R is a pure resistance. Substituting either of
the last two fonns into Eq. (27) yields
dB = 20 1og10
or
R0,
dB = 20 1og 10
(z)
(io)
i;
-
IO logto
+ 10 1og 1o
(�;)
(28)
( Ro);
R
zero.
R;
then the last tenn in each case reduces to
Should R; =
decibel lo power and voltage ratios is illustrated in Fig. 1 8 for
(28a)
The relationship of the
= R0 •
1 00.0
80.0
60.0
50.0
40.0
30.0
20.0
15.0
....
�
"!.
�
.2
�
1 0.0
8.0
6.0
5.0
4.0
3.0
2.0
1 .5
1 .0
0.8
0.6
0.5
0.4
0.3
0.2
0.15
0.1
-20
-10
0
10
20
30
Decibels (dB)
FIGURE 1 8: Rela1ionship of power and vohage to lite decibel.
40
Signal Conditioning
One should remember that the decibel is fundamentally a power ratio and that "for.
gelling" the R's in the preceding equations is strictly legitimate only if the two loads, with
and without amplification, are equal. Nevertheless, outpulfinput ratios are often described
using the decibel even when no load is directly involved, and one frequently sees voltage
ratios eitpressed in decibels as
dB = 20 1og 10
( �)
(28b)
Another common use of the decibel is in constructing a Bode plot of frequency response.
In such a graph, the gain in decibels is plotted against a logarithmic frequency axis, rather
than showing e0/e; versus f on linear coordinates (compare Figs. 26 and 27).
Amplification calculations based on the decibel offer two important advantages:
( I ) reasonably small numbers are involved, and (2) combining the effects of various stages
of a system may be accomplished by simple addition.
Voltmeters often carry a decibel scale. When using such a scale one must always be
cognizant of three important factors: ( 1 ) in reality the measurement is not in decibels, but
in voltage; (2) because the decibel is a ratio, the scale must be based on some reference
voltage; and (3) reference to Eq. (28) shows that the scale must assume a reference load.
Most voltmeter scales are based on a reference of l mW across 600 n, or
el
P = ­
R
hence.
e
=
(PR) tfl = (0.001 x 600) l /2 = 0.7746 V
which means that zero on the decibel scale has been arbitrarily set to correspond to 0.7746 V.
In some instances the references are indicated directly on the meter face. Often the abbre­
viation d B m is used lo indicate the aforementioned conventions. Why the 600-0 load
rather than something else"! The answer is that this is a long-established industrial standard,
preda1ing the field of electronics and originated by telegraph and telephone practices.
Suppose we use a voltmeter to indicate decibels. Suppose also that the signal source
imped ance is R, rather than R, , where the laner is the reference. What correction should
be applied'? The following provides the proper result:
dB (coirectcdl = dB(indicatcd)
+
I 0 log
( :: )
(28c)
EXAMPLE 1
Suppose a reading of 50 dBm is obtained across a 16-0 load, using a voltmeter with scale
referenced to 600 n. What is the true dB value?
Solution
dB (correctcd) = 5 0 + I O log
264
(�)
= 65.7 dB
Signa l Conditioning
Corrections must be made to obtain true dB values when load and refere nce conditions
differ. Very conveniently, however, ir we require only differences or changes in decibels,
!hen we may not need corrections in individual readings. This situation exists if the loads
remain unchanged during the actual m easuremen ts.
13
ELECTRONIC AMPUAERS
Some form o f amplification is almost always used in circuitry intended for mecha n ical mea­
surement. ll is not the purpose of !his section to be concerned with eleclronics or electronic
theory beyond the barest minimum required to make intelligent use of such equipment for
the purposes of mechanical measurement. The following discussion, therefore, is brief and
is directed primarily to applications rather than to specific lheory of operation.
Electronic amplification originated wilh the invention of lhe triode vacuu m tube.
Thomas Alva Edison discovered that elec!rons could How from a heated cathode to an anode
in an evacuated space; hence the term Edison effect. Lee de Forest is credited with show ing
that lhe How could be controlled by i nserti n g a third el ement , the grid, between lhe cathode
and lhe anode. This resulted in the triode electron tube and, in various configurations, many
with additional elements, provided the basis for electronic amplification.
or course, vacuum
tubes are little used in instrumentation today, having been almost
and i ncreasin gly sophisticated semiconductor
devices. Historically, lhe term electronic, as opposed to the word electrical, meant that in
some part of the circuit electrons are caused to How through space in the absence of a
physical conductor, thus i m pl ying the use of vacuum tubes. Today, lhe word electronics
has taken on a broader meani ng, encompassing the solid-state devices most often used in
instrumentation: diodes, transistors, integrated circuits, and the like.
entirely superseded by less fragile, less costly,
Electronic amplifiers are used in mechanical measurements to provide one or a com­
bination of the following basic services: (a) voltage gain ; (b) current gain, or power, gai n:
and (c) impedance lransformations. In most cases in wh i c h mechanical or eleclrical trans­
duction is used, vol tage is the electrical output that is the analogous signal. Often the
voltage level available from the transducer is very low; thus a voltage amplifier is used to
increase the level for subsequent processing. Occasionally, the input si g nal must finally be
used to drive another device, such as a control mechanical or mechanical indicator. In this
case, voltag e gain may not be sufficient in itself because power must be increased; hence a
current or power amplifier is needed.
In cenain instances a transducer produces sufficient signal level but is accompanied
"
by an unacceptably high output impedance level. This is true of most piezoelectric-type
lransd ucers, for example. High - i mpedance lines have the disadvantage of susceptibility
to noise. If the s ig nal is to be lransmitted any appreciable distance (even a few inches i n
some cases), the noise pickup from the environment may be unacceptable. Low-im pedance
lines arc much less prone to this problem . Hence it may well be desirable to insen an
impedance transformation in the form of an amplifier that will accept a high-impedance
input but produce a low-impedance output. This type of ampl i fier is often called a buffer.
There are several ge neral i ties that
lronic am plifier:
I.
can
be listed
for
the ideal {but nonexistent) elec­
In fi n i te input impedance: no input current, hence no load on lhe previous stage
or device
265
Signa l Conditioning
(+) Supply
voltage, V..,
lnvertln
input
Clutput
�-----o
Noninverttn
Input
(-) Supply
voltage, v..
_L�
Offset
null adjust
FIGURE 19: D i agram show i ng typical operat ional
ampli fier C•Jnnec ti ons .
2. Infinite gain (lower gain can be obtained by adding attenuation c ircuits)
3. Zero output impedance ( low noise)
4. Instant response (wide frequency bandwidth)
S. Zero output for zero in put
6. A bi li ty to i gnore or reject ex tra neous inputs
A l th ough none of these aims can be completely realized,
them, and their assumption simplifies circuit analysis.
it
is often possible
to approach
Today's amplifiers are most often constructed as integrated cin=uits. As the name
implies, integrated circuits are groups of c irc u i t elements combined i1nto a s i ngle device.
For the most part the elements consist of transistors, diodes, resistors, anid, to a lesser extent,
capaci tors, all connected a nd packaged in co nve n i ent plu g- i n or surface:-mount u n i ts . They
form the bui lding blocks used 10 construct more complex circuits: amp l i fiers , mixers (for
combining signals), timers, filters, audio preamps, audio power ampli.fiers, vol tage refer­
ences, regulators and comparators, and many of the digital devices. Of particular importance
to mec ha n ical measurements is the operati onal amplifier, or op amp. In the following para­
graphs we will discuss th is device in more detail.
14
OPERATIONAL AMPLIFIERS
The op amp is an integrated circuit that functions as a de differential voltage amplifier. By
de we mean that it wi ll process input signals over a frequency range extend i ng down to
and including a de vol tage. As a differential amp l i fier it accepts two inputs a nd responds
to the difference in the vo l tages appl ied to the input terminals. One of these inputs, called
noninverting, is conventionally identified with the (+) symbol (Fig. 19). The other, called
the inverting i nput, carries the ( - ) symbol. The vo ltage at the output terminal, e0, is the
product of the am pli fier gai n , G , and the voltage di fference :
(29)
266
Signal Conditioning
••
(e.
Saturated
- e_ )
SalUrated
FIGURE 20: Op-amp output response.
The output voltage is roughly limited to the power supply voltages, Vee and V,,, as the
voltage difference increases; if the voltage difference becomes too large, the output saturates
near one of these values and remains constant if the voltage differential increases further.
Op-amp response is illustrated in Fig. 20.
The op amp's differential characteristic has great importance in instrumentation
because it eliminates offset voltages and noise signals common to both input tenninals.
For eumple, nearby power lines may induce 60-cycle noise in the exteri or circuitry leading
to the amplifier. Such line noise is often present in identical form at both input tenninals,
and ii is thus canceled by the differential amplification. This be h av ior is known as common­
mode rejection. I f, instead. an op amp receives the output of a voltage-sensitive Wheatstone
bridge, the common offset voltages of the two voltage dividers are canceled, and only the
desired difference voltage is amplified.
Figure 19 shows the configuration of the exterior circuitry of the op amp. Two power
sources of equal magnitude but opposite polarity are general ly required ( - V,, = V.,).
These voltages usually fall somewhere in the range of 5 to 30 V de. Quite often, common
9-V de batteries may be used. Op amps are usually packaged in dual-in-line package (DIP)
form, one of the standard ''TO" cans, or in surface-mount packages [Figs. 21 (a), (b), and (c)).
The op amp very nearly satis fies the ideal voltage amplifier characteristics of Sec­
tion 13 for the following rea�ons:
1. It has very high input impedance (megaohms lo teraohms).
2. It is capable of very high gain (I 05 - I 06 or I 00 dB- 120 dB).
3. It has very low output impedance (down to a fraction of an ohm with feedback).
4. It has very fast response or high slew rate (output can change several volts per µs.)
5. It is quite effective in rejecting common-mode inputs.
One nonideal characteristic of most op amps is that they do not completely satisfy
the differential amplifying property: With both inputs grounded, a residual output voltage
Signal Conditioning
(b)
(a)
(c)
AGURE 2 1 : (a) 'fypical DIP (dual in-line package) integrated circuit; (b) typical TO
integrated circuit package; (c) typical surface-mountable SOP (small outline package).
remains. The multitude of transistors, resistors, and other elements within the op amp are
never perfectly matched, so the amp output actually reaches zero at some small nonzero input
voltage. To accommodate this input offset voltage, the commo n op amp is provided with
pins marked "offset null" or "bal ance," which provide a means for adjusting the unwanted
offset voltage toward zero (see Example 7).
A second limitation is that the actual common-mode rejection is finite. If the two
input signals each include a common-mode vo l tage , "cm • the op amp's actual output will be
The finite common-mode rejection is characterized by the common-mode rejection
(CMRR) in decibels:
CMRR
=
20 log 10
(.E.. )
G em
ralio
(30)
'fypical op amps have a CMRR of 60 to 1 20 dB; thus, the common-mode gain is typically
lol to 106 times smaller than the differential gain. Obviously, a high CMRR is desirable.
In addition, thermal drift can limit op-amp performance. Bo th internal and external
circuit elements may be tempera tu re sensitive, and the design of each circuit usually includes
compensating features. A wide variety of op amps are available, and their differences
largely represent attempts to improve thermal stability, CMRR, offset voltages, or frequency
response. Understandably, such refinements are reflected in cost.
1 4. t
Typical Op-Amp Specifications
Op amps are often designed to optimize those aspects of performance needed for a spec ific
application. One common general-purpose amplifier is the LF4 l l . In comparison to more
soph ist i cated op amps, the LF4 l I is quite simple; yet in a package the size of a finger·
1
nail, it incorporates 23 transistors, 1 1 resistors, 3 diodes, and 1 capacitor. Typical LF4 1
specifications are as follows:
268
Signal Conditioning
Open- loop gai n
Input i mpedance
Input offset voltage
Input offset voltage drift
Input offset curre nt
Input bias current
CMRR
Maximum output curren t
Slew rate
Maximum power supply voltages
Power supply current
Maximum input voltage range
Maximum differential input
Short-circuit output time
14.2
2 x I OS (depends on frequency)
10 12 n
0.8 mV
7 µVJ°C
25 pA
50 pA
IOO dB
25 mA
IS V /µ.s
±18 V
1 .8 mA
±IS V
±30 V
Indefinite
Applications of the Op Amp
Operational amplifiers may be used as the basic components of linear voltage amp l i fiers,
differential amplifiers, i n tegrators and differentiators, voltage comparators, function gen­
erators, fil ters, impedance transformers, and many other devices. They are not power
ampl ifiers, nor do they have exceptionally wide bandwidth cap ilit ies . Undistorted fre­
quency response is typically limited lo about I MHz when the circuit gain is low, and it
decreases as the gai n is raised. In general , an op amp's maximum voltage output is limited
ab
the supply voltage.
Since the number of applications of the op amp to mechan ical measurements is almost
limitless, we can describe here on l y a few. Yet this will give the reader some idea of the
tremendous versa t ility of the device and will suggest additional uses (see also lhe Suggested
Read ings at the end of the chapter).
One feature common to most op-amp circuits is a negative feedback loop. Because
op-amp gain is so high, even a s l igh t i npu t- voltage difference will drive the amplifier to
saturatio n . To prevent this, a connection is made between the output terminal and lhe
i nverti ng (-) terminal. With this connection in place, an increase in e0 w i l l be fed back
to e_ , red uc i ng the in pu t voltage-dilTerence. The net effect is to produce a circuit that
holds e_ "" e + , preventing saturation. Ex ample 2 describes a circuit with no feedbac k , and
several subseque nt exampl es treat circuits h aving feedback loops.
by
EXAMPLE 2
The open-loop configuration6 has the following characteristic.�:
I. No feedback loop. RL is the load resistance powered by e0• The circuit may be free
O oat i ng or grounded.
2. Amplifier is run wide open: Any input other than zero will drive t he amp l ifier to
saturation (i.e., a very small i n pu t will drive the output to the limit pcnnittcd
power supply).
by
the
6 1t is
conventional in op-amp circuil diagrams to show only 1hose terminals lhat are used in the particular
configuration. Power supply inputs are always requin:d. whether shown or not Null adjustment is often 101
shown. allhough ii may be required for oplimal performance (see Example 7).
269
Signal Conditioning
3. It is seldom used: however, it may be employed as a voltage comparator With
.
different voltages app l ied to ( +) a nd ( - ), open-loop output polarity (positi 1
ve
negative saturation) will be controlled by the larger input. For sinusoidal input,
square-wave output would result.
�
6;
r e·
=� Rl
·)
______
EXAMPLE 3
The voltage follower, or impedance transformer. has a feedback loop connecting the full
output to the inverting i nput
;_
-
9>�·
•
,_
+6;
1
l
The feedback loop prevents saturalion by h old ing e_ "" e+. Since e; = e+ and
e + . the output voltage is equal lo (follows) the in pu t voltage: e0 = e; . The
circuit gain is G = I .
This circuit capitalizes on the h i gh input impedancu o f the op amp: Since the inpul
impedance is so large. the input current , is in nanoamps (nA) or even picoamps (pA).
Source loading is minimized and can often be entirely nes;lected (i+ "" 0). In contraSt, the
output terminal can deliver up to the maximum current o f the op amp. This circuit
an impedance transfonner in that the input impedance is i n gigaohms, whereas the output
impedance is a fraction of an ohm.
e0 = e_ ""
acts as
i
This example demonstrates lwo important rules of thumb
op-amp circuit having negative feedback:
that can
1. The input currents, i+ and L, are essentially zero: i +• L "" 0.
2. The input voltages,
e+
be applied any
and e_, are h eld equal by the: negative feedback:
270
e+ "" t- ·
Signal Conditioning
EXAMPLE 4
inverting amplifier is one of the most used op-amp circuits. Feedback is provided
throllgh resistor R2 .
The
+
;, _.,
+
e;
lBo
l
-
=
Since i+ "" 0, Ohm's law shows e+ = 0. Because negative feedback is present,
e- = e+ = 0. The inverting input also draws no current, so that i 1 = i2 . Thus we can
apply Ohm's law to resistors R 1 and Ri t o find th e relation between e 1 and e0:
e1 - 0
.
e1
1 1 = -- = R1
.
0 - e0
• 2 = -- =
R2
or
R1
eo
- -
Ri
eo = - R; e;
R2
The output is opposile in sign from the input (inverted, or 1 80° oul of phase), and the gain
of lhe circuit is G = - R2 / R 1 .
The resistor R3 is commonly made approximalely equal 10 the parallel value of R 1
and Ri, i.e., RJ "" R1 R2/( R 1 + R2 ). This choice provides nearly equal inpul impedances
at lhe ( - ) and ( +) lerminals.
EXAMPLE 5
The noninverring amplifier is as shown at lhe top of lhe next page.
The inpul voltage is applied 10 the ( +) tenninal (e1 = e+); becaus� nega tive feedback
is present, e- = e+ = e1 . The output voltage is related to lhe voltage at the inverting
lerminal by the voltage-divider relation:
e_ =
Rearra nging,
(�
)
+
271
R1
R2
e0
Signal Conditioning
R,
Thus, lhe oulput and the i np u t are in phase and the circuit gain is G = (R1 + R2 )/ R i .
Res i stor R1 serves the same purpose as in the inverting amplifier.
EXAMPLE &
In lhe differential, or difference, amplifier:
I. If R 1 = R1 and RJ = R4, then e0 = - (R3/ R 1 ) (e;1 - e;2 ) (see Problem 23).
2. The need for offset null adjustmen t (see Ex ample 7) i s minimized by making input
resistances al ( - ) and ( +) equal.
3. Precise resistor matching is necessary to achieve h i gh CMRR.
EXAMPLE
7
An amplifier with o ffse l null adjustment is exemplified by the accom pany i n g diagram.
I . The ci rc u i t
allows trimming to zero ou1pu1 wit h zero input.
2. Specific example shown i llus trates pin numbering.
272
Signal Conditioning
v.. H
EXAMPLE I
The voltage comparator has the following features:
l. A small voltage difference between e1 and e,..r swings output to
limit permitted by
power supplies; erer is set to desired reference voltage. No feedback is used.
2. When e; > e.-.r. output is positively saturated; when e; < erer . output is negatively
saturated. This provides output indication for the size of e; re lative to e,.r . For
example, should e1 be gradually rising, when its value reaches e..,r the output polarity
would reverse. This could be used to trigger external action.
3. Diodes serve to limit differential input.
e,
e.,,
EXAMPLE 9
The summi11g amplifier shown has the fol lowing characteristics:
l. ea = -[ei ( R4 / R 1 ) + e2 ( R4 / R2) + eJ( R4/R1)) (see Problem 24).
2. If R 1 = Ri = RJ = R, then e0 = - ( R4/ R)(e1 + ei + eJ).
273
Signal Conditioning
3. This circuit has application to digital-to-analog co nverters Also note the similari
ty
to the inverting am pl i fier (Example 4).
.
15
15.1
SPECIAL AMPLIRER CIRCUITS
Instrumentation Amplifiers
In practice, transducer signal s are often small voltage differences that must be accuralcly
amplified in the presence of large common-mode signals Simultan1eously, the current drawn
.
from the transducer must remain small to avoid loading the tran sducer and degrad ing its
signal. Standard op-amp circuits, such as the differential amplifie r (Example 6), may not
provide adequate input impedance or CMRR when high-accuracy nneasurements are neeckd.
The instmmentation amplifier uses three op amps to remedy these problems (Fig. 22).
The instrumentation amp is essentially a d i fferential amplifier with a voltage follower placed
at each in put (this is easi ly seen if R 1 is temporarily removed) . The voltage followers
increase the ( +) and ( - ) input impedances to the op-amp i mpeda.nces. The addi tion of Rt
between the two followers has the effect of raising CMRR. Resistor match i ng is less critical
for this circuit than for a differential op-amp circuit alone.
Instrumentation ampli fiers may be bui l t from d i screte comiponents, or they may be
purchased as single integrated circuits. The typical instrumentation amp may have CMRR
reac hi ng 1 30 dB, input impedance of 1 09 0 or more, and circuit ;gain of up to 1000.
1 5.2
The Charge Amplifier
The charge amplifier is used with piezoelectric transd ucers . Th·ese transducers are com­
posed of a h igh - impeda nce material that generates e lectric c har.ge Q(r) in response to a
varying load. The charge amp produces an output proportional to l:he charge while avoiding
the potential noise difficulties of a high-impedance source. The c:omplete circuit is shown
in Fig. 23.
The transducer, cable, and feedback capaci tances are C, , Cc. and CI • respectively.
the large feedback resistor RI is ignored, the output of the circui1t can be expressed as
eo
- Q (r)
= -----"-----C1 + (C1 + Cc + C1) / G
If
Signal Conditioning
Differential 811l>lifle r
Input stage
(voltage lolloweis
when R1 - 00)
G=
(1 + �R, x�)
R•
R,, = R3
=
R4
� R5
= R1
FIGURE 22: An instrumentation amplifier circuil
7
l- - - - - - - -R - - - - - -I
1- - - - - - - - - - - -,
I
I
I
I
I
I Q(�
I
I
I
I1
I
_ _ _ _ _ _ _ _ _ _ _ _1
Transducer
Gable
I
I
I
I
I
>--+-+--<> ••
I
I
!_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I
Amplifier
AG URE 2 3 : A charge amplifier circuit.
Signal Conditioning
I
"'*" Center
/ rotloll frequency
Cutoff or
I
,, - '2
@ �----....__
�
�
frequency
I
•
Bandwidth at
.:idB down
.dG/.dl,
measured in dB/octave
(where one octave
corresponds to a doubling,
or a halving, of frequency)
Frequency f2
Frequency
(a)
f,
(b)
FIGURE 24: (a) Some tenninology as applied to a low-pass filter; (b) band-pass filter characteristics.
where G is the open-loop gain of the op amp. Because op-amp gains are enormous, the
second term in the denominator is usually negligible, and the effective output is just
Q(t)
ea = - -Ct
Note that the charge amp's output is independent of cable and transducer capacitance ( I 01.
The resistor RI limits the response of the charge amp at frequencies below f =
l /27r RtC I · Such parallel resistance is often introduced to eliminate low-frequency con­
tributions to output; however, some parallel resistance is always present, owing to the finite
resistances of real capacitors.
Although the piezoelectric effect was known in the nineteenth century, it did not
become technologically important until very-high-input-impedance amplifiers were devel­
oped in the 1950s and 1 960s. The charge amp itself was patented by W. P. Kistler in
1 950 and gained wide use following the development of MOSFET circuit< and high-grade
electrical insulators such as Teflon and Kap ton [ 1 1 ) .
16
FILTERS
As we have seen, time-varying measurands commonly consist of a combination of many
frequency components or harmonics. In addition, unwanted inputs (noise) are often picked
up, thereby resulting in distortion and maslting of the true signal. It is usually possible to
use appropriate circuitry to selectively filter out some or all of the unwanted noise.
Filtering is the process of attenuating unwanted components of a mcasurand while
pennitting the desired components to pass. Filters are of two basic classes, actfre and
passive. An active filter uses powered components, commonly configurations of op amps.
whereas a passive filter is made up of some form of RLC arrangement. In addition. filters
may be classified by the descriptive terms high-pass, low-pass. band-pass, and 1101ch or
band-reject. In each case, reference is to the signal frequency; for example, the high-pass
276
Signal Conditioning
filter permits components above a certain cutoff frequency to pass through. The notch filter
auenuates a selected band of frequency components, whereas the band-pass filter permits
only a range of components about its center frequency to pass. Figures 24(a) and (b)
illuslrate certain terms applied in filter design and use. Similar terms are applicable to the
high-pass and notch filters, respectively.
17
SOME FILTER TH EORY
The simplest low-pass and high-pass filters are made from a single resistor and capacitor.
Electrically, these passive RC filters are first-order systems. The RC low-pass and high-pass
filters are shown in Figs. 25(a) and (b), respectively.
Consider first the RC low-pass filter. Since a capacitor tends to block low- frequency
currents and pass high-frequency currents, the basic effect of the capacitor in this filter is to
short-circuit the high-frequency components of the input signal. To determine the frequency
characteristics, we must find the filter output, e0, for a harmonic input voltage, e; :
e;
=
V; sin(21f/t)
If negligible current is drawn at the output, the currents through the resistor and the
capacitor are equal, so that
or
(3 1 )
o ----NI'./'
+ oR
e,
i'
0
(a)
c
I
I
o+
e.
1
fc = 2,;fjC
yg
_ ____
v, -
"' =
fc =
-
yg_
,
+ (f/f,,)2
,/1
tan" 1 (flfcl
1
21TRC
(flf,j
v, - \11 + (flfl
if> = 90°
- tan-1(//fc)
(b)
FIGURE 25: First-order RC filters: (a) low pa�s. (b) high pass.
277
Signal Conditioni ng
Solution of this equation gives.
e0
V'·
V0 sin(2ir/t + </I) =
=
J i + (2ir RC/)2
sin(211Jt + I/>)
(32)
where the phase lag If>, is
,
(32a)
Filter performance is normally characterized by defining a c14tofffrequency; k
fc
5
I
2ir RC
(33)
In terms of fc . the frequency response (or gain), from Eq. (32), is
V,,
v;
=
I
Ji + ( f/fc )2
(33a)
and the phase response, from Eq. (32a). is
(33b)
Al the cutoff frequency,
V0
V;
=
I
v'2°
a d
n
P0
I
=
2
P;
(33c)
either of which indicates a -3 dB change in the signal strength (st:e Section 12).
The frequency response is shown on linear coordinates in Fig. 26. Graphed this
way, the filter response seems to change only slightly with frequency. However, the graph
�
v,
1
-12
0.5
--2
0
3
FIGURE 26: Frequency response of the RC low-pass filter ( linear coordinates).
278
Signal Conditioning
0
-10
!'R_ (dB)
v,
-20
-30
-40
O.o 1
0. 1
1 .0
10
1 00
"·
FIGURE 27: Frequency response of RC low-pass and high-pass filters (Bode plot).
shows only a factor-of-three increase in frequency, while, in practice, filters are used to
separate freq uencies that may differ by orders of magnitude. A logarithmic graph, such as
a Bode plot (Section 1 2), is needed to illustrate such variation.
A Bode plot of the low-pass filter's response is gi ven in Fig. 27, illustrating the -3 dB
reduction in signal at the cutoff frequency. The frequency range ploued spans four orders
of magnitude, and the amplitude attenuation runs from 0 to -40 dB. For frequencies well
below fc . the filler's response is flat and shows no signal reduction. The transition from the
passband to rejection band occurs gradually with increasing frequency. In the rejection band
iiself, al frequencies well above fc. the amplitude rolloff is -6 dB/octave (an octave being
a factor-of-two change in frequency) or - 20 dB/decade (a decade being a factor of ten ).
In addition to reducing amplitude, this filter also produces an increasing phase shirt
as signal frequency rises (Fig. 28). At the -3 dB point, the output lags the input by 45°.
;i
.t:.
�
..
..
m
�
�
45°
00
O.Q1
-45"
FIGURE 28: Phase response of RC high-pass and low-pass filters.
Signa l Conditioning
The RC high-pass filter is obtained by interchanging the resistor and capacjl()[
[Fig. 25(b)]. Now the capacitor blocks low frequencies while pas si n g high frequencies.
The results are quite similar:
e0
=
.
V0 sm(2ir/r + <f>)
,
where the phase sh i ft
.
,/I + (211" RCJ)l
t/J, is now a lead (q, > 0) rather than a lag (<f>
q, = 90°
-
fc
+ <f>)
< 0)
tan- I (211" RC/)
The high-pass filter's cutoff frequency is identical to the
In
sm (2ir/t
V; ( 2ir RCJ)
=
low-pass filter's:
(34)
I
=
2'r RC
terms of the cutoff frequency, the frequency response and phase lead are
v.
V;
The -3 dB poi nt is again
-20 dB/decade.
=
(fife)
Jl + (f/fc>2 '
t/J = 90°
(34a)
- 1 (f)
tan -
(34b)
fc. and the rolloff in the rejection band is again -6 dB/octave or
The high-pass frequency and phase response are shown in Figs.
28. One common use of this filter is lo remove de (/ =
0) offsets.
27 and
(
First-order RC filters have a fairly slow rolloff above the cutoff frequency not many
d ec i bel s
per octave), but their simplicity still gains
can
them wide use
in situations where
-
the
desired and undesired frequencies are widely separated. Similar high-pass and low-pass
filters
be made usi ng a single resistor-inductor pair. However, fi rst order Rl filters are
se ldom used.
EXAMPLE 1 0
e
A transduc r
output is
responding t o a
5000- Hz signal
{5 sin(2.7r
·
also picks u p 60 Hz noise. Th e resulting
60 · t) + 25 c os( 2ir - 5000 · t ) ) mV
To remove the 60-cycle noise, a high-pass fi l ter with cutoff of
i s the filtered output?
Solution
1000 Hz is introduced. Whal
The amplitude and phase shift are computed separately for each component:
( ) -- ,/1 (60/ 1000) 2 / 1 000)
_
( ) SOCIO ,/t (5000
-0
(5000/ 1000)2
- ( ,:0)
- (�:)
v.
Vi
v0
Vi
60
(60/ 1 000)
+
=
t/J60 = 90°
<Psooo = 90°
- 0 060
.
+
tan- 1
tan- t
280
=
.
•
98
•
86.6° = 1
= 1 1 .3°
=
.5
1 rad,
0. 1 97 rad
Signa l Conditioning
Then
eo = (0 . 3
sin(2ir 60 t + I .S I ) + 24.5 cos(2ir 5000 t + 0. 197 ) } mV
·
·
·
·
The noise amplitude is reduced from 20% of the signal amplitude to o n l y 1 .2%. Note
that the signal itself undergoes a slight amplitude reduction as well as a small phase shift.
Such changes in the signal are undesirable, and they often motivate the use of more complex
filters.
Three desirable elements of filter
performance are as follows:
I. Nearly flat response over the pass and rejection bands;
2. High values of rolloff for low-and high-pass filters, as measured in decibels per octave;
3. Steep skirts for band-pass and band-rejection filters.
Significant improvements in perfonnance are obtained by using combinations of sev­
eral capacitors, inductors, or resistors to produce second-order {or higher-order) electrical
response. Such filters can have steeper rolloff and sharper transition from pass to rejection
bands. In addition, such compound RLC arrangements can produce band-pass and notch
filters. For example, Figs. 29(a) and (b) show an RC band-pass filter and its response:
V0
V;
= Jll
1
+ R1 / R2 + C2/Ci ) 2 + [2ir R1 C2 / - ( l /211" R2 C1 /) ] 2
�
R
1
(35)
C1
o----+-----0
(a)
?,
(dB)
Frequency �og scale)
(b)
FIGURE 29: (a) A circuit for a simple band-pass filter; (b) performance characteristics of
band-pass filter shown in (a).
281
Signal Conditioning
Inductors and capacitors used toge!her allow resonant behav.ior, which can P!'OCh
steeper filter skirts !ban are possible with first-order RC circuits. In fact, the resonaot�
of Section 1 1 is someti mes used to build very narrow band-pass filtem known as tunedfiJte
Some additional LC designs are shown in Fig. 30.
Two practical issues influence !he design and use of passive filters. Fust, the fill
considered here are all designed as if negligible current is drawn fro:m the oulput termiru
If several filters are placed in series, to steepen rolloff, then the curr"Cnt drawn by one Iii
Low pass
High pass
Band pass
L-section
L-sectlon
L-section
r-section
r-section
1T·section
1T·SecilOl1
1]»�
r-section
,,..section
Frequency
Frequency
Frequency
performance
performance
performance
FIGURE 30: Examples of LC filter arrangements and their output characteristics.
282
Signal Conditioning
Passlw
filter
network
FIGURE 3 1 : Basic active filtt:r circuit.
can alter the perfonnance of the filter that precedes it. To avoid lhis output loading, a voltage
follower (Examp le 3) should be
in trod uced as a buffer between each successive filter.
The inductors lhemselves are the second problem. At lhe frequencies encountered
in mechanical measurements, which rarely exceed
100 kHz, lhe required inductors may be
quite large and bulky. In add ition, the optimal inductor values are not al ways easily obtained,
happens in
much less satisfactory in practice lhan lhey seem on paper.
The usual way of avoiding these p ro blems is to employ an active fil ter, as described next.
and lhe
inductors may have substantial internal resistances as well. As often
engineering, inductors can be
ACTIVE ALTERS
can be used to construct filter circuils without i nductors and without lhe problems
These ac ti ve filters can also have very steep rolloff, arbi trarily Hat
passbands, and even adjustable cutoff frequencies. Ac t i ve filters are a rich s ubject , and
entire textbooks have been devoted 10 lheir design.
The bas ic active filter is shown in Fig. 3 1 . Passive filler networks are linked to an op
Op amps
of output load i ng .
amp, which provides power and improves impedance characteristics. The passive network
is buil t from resistors and capacitors o nl y : Inductive characteristics are simply simulated by
the circuit. Since lhe ou tput impedance is general l y low, these fillers can deliver an output
current wi thout reduced performance. Some 1ypical active filters
are shown
in Fig. 32.
of 80 dB/octave and more lhan 60 dB a11en­
ua1ion in the rejection band. H igh- oi:der ac ti ve fil ters are even sold as integrated circuits
contained in a single DIP package . For furlher study, see the Suggested Readings al lhe
end of lhis chapter.
Active fillers are available w ith rolloffs
DIFFERENTIATORS AND INTEGRATORS
A final op-amp application is in circuits that respond to lhe rate of change or the time history
of an input signal, called
·
differentiators and integrators,
respectively [Figs. 33(a) and (b)).
In the differentiator. lhe curn:nls through the resistor and cap aci tor are equal, and
e_ = e+
=
0. Thus
d
C-e1 =
dt
ea
R
--
Signal Conditioning
�
(a)
,c = -121T A;.
c;
fl;
"•
-
(b)
�
�
R,
c,
1<
e,
. 1
=
21TR1 C1
"•
-
R2
(c)
I
1_
_ _
""" 21T R,, G.!
�
e,
R,
= -1q.._, 21TR1 C1
f
"•
c,
FIGURE 32: First-order active filters: (a) low pass; (b) high pass, (c) band pass.
or
e0 =
d
- RC di e;
(36)
In the integrator, the capacitor charges in proportion to the time summation of e; . Again,
the resistor and capacitor currents are equal :
or
eo =
-
I
RC
J e;
dt + constant
(36a)
To prevent drift in the capacitor's charge over long time intervals, a l arge resistor, Rf.
may be placed in parallel with it. In that case, the integrator circuit is restricted 10 s ig nal
frequencies high enough that f » 2ir R ,c.
284
Signal Conditioning
R
- f,
(a)
c
(b)
FIGURE 33: (a) Op-amp differentiator; (b) op-amp integrator.
20
20.1
SHIELDING AND GROUNDING
Shielding
S hieldi ng applied to electrical or electronic circuitry is used for either or both of two related
but different purposes:
I. To isolate or retain electrical energy within an apparatus
2. To isolate or protect the apparatus from outside sources of energy
An example of the former is the shielding required by 1he Federal Communications Commis­
sion lo minimize radio frequency radiation from computers. In the second case, shielding
may be required to protect low-level circuitry from the entry of unwanted outside signals.
A very common source of ouiside energy is the ubiquitous 60- Hz power l i ne .
Sh ield ing is of two basic types: (a) elecuos1a1ic and (b) electromagnetic. Jn each case
the shielding normally consists of some form of meta l lic enclosure; for example, metallic
braid may be used lo shield signal-carry ing wiring, or circuitry may be partially or entirely
enclosed in melal boxes.
Only nonmagnetic metals may be used for electromagnetic shielding, whereas almosl
any conduciing metal, such as steel, aluminum, or copper, may be used for electrostalic
shielding. Circuits within a device oflen mus! be shielded from each other; however,
connections must still be made between the subcircuits through use of special amplifiers or
transformers. For example, power transformers are often provided with copper shielding
between primary and secondary windings. The copper provides elecuostatic shielding
285
Signal Conditioning
without hindering the transfer of electromagnetic power. .Some rules for shi
eldin
8
follows [ l 2, 1 3 l :
Rule I.
Rule 2.
Rule 3.
20.2
- as
An ele trostatic shield enclo ure, to be ffec:tive, should be connected to the
�
�
�
zero-signal reference potential of any c1rcu •try contained within the shield.
The shield conductor should be connected to· the zero-signal reference po1en1ial at the signal-to-earth connection.
The number of separate shields required in a system is equal to the number
of independent signals being processed, plus one for each power entrance_
Grounding
When low-level circuitry is employed, some form of grounding is i nevi tably
required.
Grounding is needed for one, or both, of two reasons: ( I ) to provide an electrical refmnce
for the various sections of a device, or (2) to provide a drainage path for unwanted currems.
A grou11d reference may be either of two types, ( I ) earth ground, or (2) chassis
ground. In the latter c a<e, chassis commonly refers to tht: basic mou nting structure (e.g..
the ground plane of a circ uit board) or the enclosure w i th i n which the circuitry is mounted.
Conventional schematic symbols for the two are as shown in Fig. 34 .
In a text such as this, only superficial coverage of this complicated topic is possible.
However, cenain "rules.. and observations may be listed as follows:
l.
A n entire system can be "grounded" and need not involve earth at all. For example.
2.
The
c i rcuitry in aircraft and spacecraft are referenced to some common datum.
word circuit need not imply wires or componemts. Each of the various elemenu
in a device may, unless effectively shielded, possess capacitive paths to one or lllOR
of the others.
3. Shielding can be at any potential and still provide shielding.
4.
The assumption that two nearby poi nts are at the sarne potential is often invalid-not
only earth points, but also points in any ground plane.
5. Potential characteristics of an clement are not the saJ111e at "radio" or high frequencies
as they are at "power" or low frequencies. For example, a capacitance ex i sts betwcel
a bonded strain gage element and the structure on1 which it is mounted . A l ck or
low frequency, such capacitance may be unimponant, but at radio frequencies lhe
capacitance may provide a ready electrical path.
6. A ground bus is protection against effects of equipment faulting. but it is
source
7. All
of zero potential
for the solution of instrume:ntation processes.
metal enclosures and housings should be earthed and bonded together.
current should be permitted to How in these connections.
Earth
FIGURE
34:
Chassis
Conventional symbols
286
for g;round references.
not
lhe
but no
Signal Conditioning
a. Good practice suggests that it is wise to i nsu l ate an equipment rack from the obvious
ties. such as bui lding earths and conduit connections, so Iha< the rack can be ohmically
connected to a potential most favorable to the instrumentation processes .
9. Rules that are applicable at one frequency range may be inadequate at another.
10. Safely practices demanded by various civil codes can seem to be in direct conflict
with good instrumentation practice.
11. E.lecuostatic shields are s imply metallic enclosures that surround s i gnal processes .
To be effective, these shields shou ld be tied lo a zero signal poten tial where the signal
makes its external, or ground connection.
To reiterate, shielding and grounding are very complex subjects, and often some
degree of uial and error, coupled with experience , is required to find a solution.
COMPONENT COUPLING METHODS
When elecuical circuit elements
are connected. special atte n tion must often be given to the
coupling methods used. In certain cases, transducer-amplifier, amplifier-recorder, or other
component combinations are inherently incompatible, malting direct coupling impossible
at best, causing nonoptimal operation. Coupling problems include obtaining proper
as damping. These prob­
lems are usually caused by the desire for maximum energy transfer and optimum fidelity
of response.
The importance of impedance matching, however, varies considerably from applica­
tion lo application. For example, the input impedances of most cathode-ray oscilloscopes
and elec tron ic voltmeters are relatively high, but satisfactory operation may be obtained
from directly connected low-impedance transducers. In th is case, vol tage is the measured
quantity and power transfer is i nc i dental . In most cases, driving a high-impedance circuit
component with a low-impedance source presents fewer problems than does the reverse.
As a simple example of transfer, consider Fig. 35. Shown is a source of energy Es and
a sink or load having impedance ZL . Zs is the so urce impedance. To simplify our example
further, let the i mpedances be s i m ple electrical resistances, Rs and R L , respectively. Then
the voltage across R1. wi II be
or,
impedance matching and maintaining circuit requirements such
AG URE 35 : S imple circuit
for demonstrat ing tranSfer concepts.
' I--i l
1---H--' ..--Signa l Conditioning
Zs
.___.
(a)
�
(b)
FIGURE 36: Impedance matchi ng (a) by means of a coupling transformer and (b) by means
of a resistance pad.
and the power delivered to RL is
(37)
To determine the values of Rl for maximum power transfer, set d P /d RL to zero and
solve. We find that maximum power is transferred i f RL = Rs. Although this is not ieally
a proof, in general terms, maximum power is transferred when ZL = Zs.
In addition to, or instead of, opti m i zi ng power transfer. proper coupling may be
important in providi ng adequate dynamic response. Three spec i al methods of coupling,
depe nding on the circuit elements, are common; they utilize ( I ) matc h ing transfonners
(discussed shortly), (2) i m pedance transfonning (see Example 3 ), and (3) coupling networks
(discussed shortly).
The problem is similar to that of connecting a speech amplifier to a loudspeaker. In
both cases the output impedance of the driving power source in the amplifier is often higher
than the load impedance to wh ich it must be connected. Transfonner coupling is generally
used as shown in Fig. 36(a). Matching requirements may be ex pressed by the rel ation
(38)
where
Zs = the source impedance,
ZL = the load impedance,
Ns/ NL = the turns rntio of the transformer
General - purpose devices such as simple voltage ampli fiers and oscillators often incor­
porate a final amplifier stage, called a buffer, to s up pl y a low-impedance output. By reducing
288
Signal Conditioning
the impedance source, one can minimize losses in the connecti ng lines and the possibility
of extraneous signal pickup.
Proper coupling may also be accomplished through use of malching resistance pads.
Figure 36(b) illustrales one s imp le form.
If we assume that the driver outpul and load impedances
resistive, lhen the match­
ing requirements may be put in simple form as follows: The driving device, which may be
a voltage amplifier, looks into the resistance network and sees the resistance R, in series
with the paralleled combination of R L and Rp . Hence, for proper matching,
are
;.1; •
(39)
where
fl<I = the output impedance of the driver (Q),
Ji!; : !
i!·-•.
:J.' .
Rl = the load resistance (Q),
Rp = t he paralleling resistance (Q),
.e.:;.
· .� '
R,
The
connected
= the series resistance (Q)
e,
driven device sees two parallel resistances,
resistances R, and fl<I. Henc for matching,
Rl =
Solving fo r R , an d
Rp , w e have
and
Rp
=
[
made
up of Rp and
Rp(R, + fl<I)
R p + ( R, + fl<I )
Rl
(
Rd R L
Rd - R L
the series­
(39a)
)]1 /2
(40)
(40a)
Now if Rd and Rl are known, values of R, and Rp may be dc1ermined to salisfy the
matching requirements by use of Eqs. (40) and (40a). In using resistive clements, a loss
in signal energy is unavoidable. Such losses are often referred to as insertion losses. In
genera l , however, by providing proper match, the network w i l l provide oplimum gain.
22
SUMMARY
Electrical signal conditioning can serve many purposes: to convert sensor outputs to more
easil y used forms (e.g., resistance change into voltage); to separate small signals from l arge
offsets or noise; lo increase signal vohage: to remove unwanted frequency components
from the signal : or to enable the signal to drive output devices.
I.
-e
For resistance-type transducers, several simple signal-conditioning ci rcu i ts arc useful
if the resistance change is re latively large. These include cu rrc nt s nsi1 i ve circuits,
ballast circuits, and voltage-dividing circuits (Sections 5, 6. 7).
289
Signal Conditioning
resata.:!
2. For small resistance changes, bridge circuits provide a more sensiti"Ve lllelbod
detection , in which offsets are eliminated. Oulpul can vary linearl y with
change, but when the resistance changes are too large, bridge circ uits beco111e
non.
linear (Sections 8, 9).
3. AC exc itation of sign al-condi tio ni ng circuits is necessary with inductive and Clplci.
tive sensors. AC-excited circuits include reactance bridgc>S (Section 10) and resonan
t
circuits (Section I I ).
4. Decibels (dB) provide a co nven ie n t method of quanti fying gain or attenuation. The
decibel is a logarithmic measure of signal power (Section 1 2).
5. Electronic ampl i fiers are ubiqu itous elements of signal -conditioning circuits. Ampli­
fiers serve to increase voltage, to increase power or curre,nt, or to change impedance
(Section 13).
6. Operational ampl i fiers are among the most common elements in instrumentation cir­
cuits. Many different op-amp configurations are possible, with the choice depending
on the characteristics of the specific sensor involved and the type of output �
desired (Sections 14, 15, 18, 1 9).
7. Some signal -condition ing techniques app ly pri marily to• ti me-varyi ng measurands.
These include carrier modulation (Section 3), filtering (Sections 16-18), and differ­
entiating and i ntegrating (19).
8. Filters allow unwanted frequency components to be removed from a signal . Filters1R
characteri zed by a cutoff frequency (or -3 dB point) separating the pass and rejection
bands and by the steepness of ro lloff from the passband! to the rejection band. RC
filters are the simplest type of passi ve filter. Common appl ications of filters include
noise removal, de-offset removal, and carrier demodulation (Sections 1 6- 1 8).
9. Shielding and grounding are essential considerations in building and using measure­
ment circuits. Shieldi ng for noise prevention is especially important when precise
measurements are to be made ( Section 20).
10.
Component coupling is often designed to maximize power transfer or dynamic response.
If vol tage detection is of greater importance than power l:ransfer, however, it is ofien
sufficient that a device's i nput impedance be large compared to the output impedance
of the device driving ii (Section 2 1 ).
SUGGESTED READINGS
Carr. J. J. Otsigner's Handbook of lnsmunentmion and Control Cio'Cuits.
Press. 199 1 .
C lay ton. G . B . • and S . Wi nde r. Operalional Amplifiers.
San Diego: Academic
41h ed. Boston: Newnes, 2000.
Coughlin, R. F., and F. F. Driscoll. Operational Amplifiers and Une1lr Integrated Cin:uits.
Upper Saddle River, N.J.: Prentice H all , 1998.
Slh ed.
Deliyannis. T., Y. Sun. and J .K Fidler. Continuous-lime Active Fihrer Design. Boca Raton,
CRC Press, 1999.
AL:
Royd. T. L. Electronic Devices: Electron Flow W.rsion. 5lh ed. Upper Saddle River, NJ.: Prentice
Hall. 2005 .
Royd . T. L. Principles of Electric Cirr:uits. 7th ed. Upper Saddle Ri>"er, NJ.: Prentice Hall, 2003.
Hague. B., and T. R
Foord . Alternating Curnnt Bridge Methods. 6th ed. London: Pitman, 1 97 1 .
290
Signa l Conditioning
Horowitz. P. . and W. Hill. � Art of Electronics. 2nd ed. New York: Cambridge University
Press. 1 989.
Kibble. B. P. . and G. II Rayner. Coaxial AC Bridges. B ristol ,
UK: Adam Hi lger,
1984.
Morrison, R. Groundi11g and Shielding Techniques. 4th ed New York: Wiley-lnterscicnce, 1998.
Newby. 8. W. G. Electronic Signal Corulitiomng.
Oxfonl:
Butterworth-Heinemann, 1 994.
Ott. H . W. Noise Reduction Techniques in Electronic Systems. 2nd ed. New York: John Wiley, 1 988.
Pallls·Areny. R . . and J . G. Webster. Sensors and Signal Coruliaoning. 2nd ed. New York: John
Wiley. 200 1 .
Schaumann. R., and M . E Van Valkenburg. Design ofAnalog Fitten. New York: Oxford
University
Press. 200 1 .
Stanley, W. D. Operational Amplifiers with Linear Integrated Cirruits. 4th ed.
I.
N.J.: Prentice Hall.
Zverev. A.
2002.
Upper Saddle River,
Handbook of Filter Synthesis. New York: John Wiley, 1 967.
PROBLEMS
·,.'I ;
L
2.
3.
4.
A force cell uses a resistance element as the sensing elcmenL It is connected in a simple
cunent-sensitive circuit in which the series resistance Rm is 100 S2, which is one-half the
nominal resistance of the force cell. Determine the cunent for force i nputs of (a) 25%,
(b) 50%, and (c) 7 5% of full range if the input voltage is JO V.
If the force cell of Problem
I
is placed into a ballast circui t
output vo lta ge for the conditions of Problem I .
For the ballast circuit of Problem
of full range.
Equations
6.
=
R., ), determine
the
I , determi ne the sensitivity, "' for the three percentages
(5) and (6) are deri ved on the basis of a high-impedance indicator. Analyze the
circuit assuming that the indicator
S.
( Rb
For £, = 10 V and R,
0 < Ri < 200 S2.
=
resistance R., is
7 5 S2,
use
Eq. (37)
comparable in magnitude to
to plot
P versus RL
R,.
over the range
The c i rcuit shown in Figure 37 is used to determine the value of the unknown resistance
Rz. If the voltmeter resistance, RL, is 10 MO and the voltmeter reads e0 = 4.65 V, what
is the value of Rz?
100 S2
::=: 12.0 v
FIGURE 37: Circuit for Problem 6.
Voltmeter
Signal Conditioning
FIGURE 38: Circuit
7.
for
Problem 7.
The voltage-dividing potentiometer shown in Fig. S is modified as shown in fig. 38.
Determine the relationship for e0/e1 as a function of le. Compare the R:$U(ts with F.q. (9).
What advantages or disadvantages does this circuit have over the general voltage-dividing
potentiometer?
8.
9.
10.
11.
Write a spreadsheet template to solve Eq. ( I Sa), penniuing each term to be varied by a
[Suggestion: Rewrite the equation, multiplying each tenn by (I + k), where
le is the delta plus/minus term-for example, R1 ( l + k1 ).)
delta amount.
Fig. 10 is used to determine accurately the value
of an unknown resistance R1 located in leg 1 . If upon initial null balance Rl is 127.5 Cl
and if, when Rz and R4 are interchan ged, null balance is achieved when R3 is 157.9 n,
what is the value of the unknown resistance R 1 ?
A si mple Wheatstone bridge as shown in
Consider the voltage-sensitive bridge shown in Fig. I 0. If a thermistor is placed in leg 1 or
the bridge while Rz = R3 = R4 = R0• determine lite bridge output when T = 400°C if
Ro = 1000 O at To = 27°C and {J = 3500. Plot the bridge output from T = 27°C to T =
soo•c and detennine the ma xi mu m deviation from linearity in this temperawre range.
Referring to Fig. 10, show lltat if initially R t = Ri = Ri = R4 = R and i f 6. Rt = - 6. Rz,
lite bridge output will be linear. (Note: This bridge configuration is very commonly used
when strain gages are applied to a beam in bending siwat i ons) .
12.
13.
14.
Referring to Fig. 1 0, initially let R1 = Rz = R3 = R4 = R. In addition, assume that
6. R4 IR = - A R 1 / R. Demonstrate lite nonlinearity of the brid ge output by plotting to/t;
over the range 0 < A R;/ R < 0. 1 . (Suggestion: Use a spreadsheet program.)
A resistive element of a force cell forms one leg of a Wheatstone bridge. If the no-load
resistance is SOO n and the sensitivity of the cell is 0.5 O/N, what will be the bridge
outputs for app lied loads or 1 00, 200. and 35 0 N i r the bridge excitation is I 0 V and each
arm of the b ridge is initially 500 O?
39 shows a differential shunt bridge con figuration . One or more or t he resistances.
R1 , may be resistance-type trans du cers (thermistor, resistance thermometer. strain gage.
etc.), with the remiining resistances fixed. Resistance R6 is a conventional vo���e­
dividing poten.tio meter. usually of lite multitum variety. It may be used either for 1ruual
Figure
292
Signal Conditioning
"•
I
FIGURE 39: Circuit for Problem 14.
nulling of the bridge output or as a readout means. The variable k is a propollional term
varying from 0 to (or 0% 10 100%); see Section 9. Rs is sometimes called a scaling
resistor. Its value largely determines lhe range of effectiveness of R6. R1 is used to adjust
bridge sensitivity. Devise a spreadsheet template lo be used for designing a bridge of
this type.
15.
Using lhe spreadsheet created in answer to the pn:ceding problem, determine the null­
balancc range of 6 R 1 that the bridge can accommodate if
R1
Rz
= 1 000 n
(nominal),
= RJ = R, =
Rs = 1 0. ooo n.
R6 = 1 2, 000 Sl,
R1 = 0 Sl
16.
17.
18.
1 000 n (fixed),
Using the spreadsheet from Problem 14 and the nominal res i stance values listed in Prob­
lem 1 5, investigate changes in measurement range of lhe bridge as affected by (a) changes
in Rs and (b) changes in R6. Investigate the linearity of the circuit when used in lhe
null-balance mode, using k as the calibrated readout.
Referring to Problem 14, assume that the tolerances for resistances Rz, R3, and R4 an:
±5%. What will now be the effective null-balance range, 6 R 1 , lhatcan be accommodated?
Figure 40 shows a shunt balance arrangement for nulling a Wheatstone bridge. Suppose
that Rt = R3 = 1 20 n, Rinm = 1 27 n, and Rpoc = 1 0 kSl. Whal is lhe maximum value
of Rz for which the bridge can be brought into balance by adjusting Rpoc'I Whal would
be the maximum value if R1 = 1 1 9 Q and R3 = 1 2 1 Sl'I
Signal Conditioning
+
FIGURE 40: Circuit for Problem 18 ..
19.
20.
21.
22.
Derive the relationships for the Wien bridge circuit shown ilB Fig. 16.
Derive the relationshi ps for the Maxwell bridge circuit shown in Fig. 16.
Show that an inc rease of I dB corresponds to a power increase of about 26%. Also show
that an increase of n dB corresponds to a power increase to approximately ( l.26)".
Equation (28c) may be wri uen as
K
where
=
dB (
og
c:om<U>dl = I + 1 0 t ( R,/ R,)
dB (indi<:al<dl
dB (indical<dl
K
dB(comc1ed)
dB(indicaled)
=
=
conection factor.
conected decibels,
= indicated decibels,
R, = reference impedance,
R,
23.
24.
25.
26.
27.
= source impedance
Plot K versus R,/ R, for dB(iodicatodl = 50, 100, I SO, and 200. Make separate families of
plots (a) for R, J R, ::: I and (b) for R, / R, � I .
S how that the input-output relationship for the differential amplifier described in Exam·
pie 6 is e0 = - ( R3/ R 1 ) (e;1 - e1, >.
Show that the input-output relation ship for the summing am1plifier described in Example 9
is "• = -(e1 ( R4 / R 1 ) + e2( R4/ R2) + e3( R4/ RJ)) .
The circuit shown in Fig. 41 is a voltage-to-current convcrtc:r. Determine the value of RJ
for which it in milliamps equals e; in volts and find e0•
The circuit shown in Fig. 42 accepts two input voltages, e1 and e2 . Detennine its oulpllt
voltage, e0•
1be circui l shown in Fig. 43 uses a potentiometer to achie•e variable gain. By adjusting
the potentiometer, the gain can be raised or lowered. The tol:al resistance of the pot is Rpo1
and the resistance of the pot between its movable wiper and ground is k Rpo1 for 0 ::: k ::: I .
Fi nd e+ in terms of e; and k, and then detennine e0 and the gai n.
294
Signal Conditioning
R,
,,
--:---­
FIGURE 4 1 : Circuit for Problem 25.
FIGURE 42: Circuil for Problem 26.
e,
R"°' > ·----1
FIGURE 43: Circuit for Problem
295
27.
Signal Conditioning
28.
29.
A noninverting amplifier (E><ample 5) is built using I -kn precision n:sisto111 R1, R2
and
Rl · which have a tolerance of ± I %. DelCnnine the circuit's gain and the tolccance � 1he
.
gain.
a semiconductor diode is related lo the voltage across ii by i ,.
I), where /0
I x 1 0-9A, A = 1 . 17 x Hf / T , and T is the absolute
1emperature in kelvin. For T = 300 K and e1 on the order of one volt, show that 1he
The cunenl tluough
/0(exp(Ae)
=
-
output of the circuit in Fig. 44 is proponional to the exponential of the input voltage.
R
FIGURE 44: Circuit for Problems 29 and 30.
JO.
31.
Show that, if the diode and the resistor in Fig. 44 are inlCrchanged, the circuit output is
proportional to
the logarithm of lhe input voltage.
&.,,o-o-A-c:
-��=-0-kil
ii is in position
B.
_
_
_
Bl
FIGURE 45:
32.
33.
Lt>J
_:
Determine the output of the circuit in Fig. 45 when the switch is in position A
and when
' ° '"___...__
_ e°8'
Circuit for Problem
31.
46, a photodiode is reverse biased with a voltage e6• In this condition,
i, which is proponional 10 the intensity, H, of the
light which irradiates it: i
k H. Determine the relation between the output voltage, "••
In the circuit of Fig.
the photodiode generates a current.
=
and ff.
For the summing amplifier of Example 9,
e1
ez
e1
and R1
=
R2
=
R4
if
=
8.0 sin(4001) V.
=
3.0 cos(4001 )
= - 2.0 sin(4001) V,
V
= 5 kil, and R1 = 2.5 kQ, what is lhe rms output voltage'/
296
Signal Conditioning
R,
+
.
t1o o-�-K1����....��
-
R,
FIGURE 46: Circuit for Problem 32.
FIGURE 47: Circuit for Problem 34.
34.
Consider the filter circuit shown in Fig. 47.
( a ) What type of filter is this? Calculate its cutoff frequency in Hz.
( b) The following input signal is applied to the circuit:
e;0
35.
=
(5 sin(211'200t) + 2.5 cos(211' 1 000t ) + L5 sin(2ir 100001 ) ) mV
Determine e°"' .
Consider the following data from an RC filter. From experimental testing . these data were
obtained for a LO V sine-wave input.
Frequency (Hz)
1 0.0
20.0
50.0
1 00.0
200.0
500.0
1 ,000.0
2.000 .0
5,000.0
1 0,000.0
If the capacitor C
=
tout
(volts)
LOO
LOO
0.97
0.92
0.7 1
0.37
0.2 1
O. I O
0.04
0.02
0.033 µ,F, determine the value of the resistor R.
297
Signal Conditioning
1 0 kQ
T:'':�?�/, .;" .
••
0.1 µF
�·
.....
-VVVV'-
• <f"
'"''
·'
k:
FIGURE 48: Circuit for Problem 37.
160 0
e1
+ o--..J\l\N"---<l>-----l
e,
0.1 µF
5 kQ
320 0
FIGURE 49: Circuit for Problem 38.
36.
A set of RC low-pa�s filters are buih from 1 00-kSl resistors and 0.0 1 -µF capaciiors. The
resistors' tolerance is ± 10% and the capaciiors' tolerance is ±20%. Determine the cutoff
frequency of these fillers and the tolerance of the cutoff frec1uency.
37.
Consider 1he amplifier circuit shown in Fig. 48. Determine the amplitude e0 if •1 =
5 cos(40irr) if e; is in millivolts and t is in seconds. What is the amplitude e0 if e; =
3.0 sin(4000ir t )'!
38.
Consider the circuit shown in Fig. 49.
( a ) Qualitatively speaking, what does this circuit do to am ac signal?
( b ) Make a Bode plot of the amplitude ratio 1•1 /e; I .
( c ) Add a Bode plot o f th e amplitude ratio 1•2/e; I to the graph o f p art (b).
( d ) Add a Bode plot of the amplitude ratio Je0/e; I to the graph of parts (b)
Referring 10 Fig. 35 and Eq. (37), show that the maximum power is transferred when Rt
R,.
39.
REFERENCES
[l)
and (c).
Geldmacher, R. C. Ballast circuit design. SESA Proc. 1 2( 1 ): 27, 1 954.
(2) Meier, J. H.
D i sc ussion of Ref. [I ), in same source, p. 33.
=
�:�\:�.'
/·;t-
�- 1
/'•t
�
·
c_c:
:��;���:;�
:
Signal Conditioning
[3) Wheatstone, C. An account of several new instru ments and processes fo r detennining
the constants of a vollaic dn:uit. Phil Trans. Roy. Soc. (London} 1 33:303, 1 843.
'· . [4)
!Stone's bridge. In En c vclopedia Britannica, vol. 23. Chicago: William Benton,
Publisher, 1 957. p. 566.
. ,. "
}."� �.-::-·
;�. :;
�
[SJ Laws, F. A. Electrical Meas11rements. 2nd ed. New York: McGraw-Hill, 1938, p. 2 1 7.
(6) Bowes, C. A. Variable resistance sensors work belier wi th constant current excitation.
Instrument Techno l .. March 1 967.
(7) Sion, N. B r idge networks in transducers. Instrument Control Systems, August 1 968.
[8) Hague. B .. and T. R. Foord. Alte rnating Current Bridge Methods. 6th ed. London:
Pitman, 197 l .
(9) Kibble, B. P., and G. H Rayner. Coaxial AC Bridges. Bristol, UK: Adam Hilger, 1984.
(10) Gauischi. G. H. Piezoelec1ric Sensorics. Berlin: Springer-Verlag, 2002, Chap. 1 1 .
(11) Kistler Instrument Corporation. Kistler Piezo-lnstrumentation General Catalog. 1st
ed. Arnhers1, N . Y. : Kistler Instrument Corp., 1 989.
(12) Morrison, R. Grounding and Shielding Techniques. 41h ·ed. New York:
lnlerscience. 1998.
Wiley­
(13) On, H. W. Noise Reduction Techniques in Electronic Syslems. 2nd ed. New York: John
Wiley, 1988.
ANSWERS TO SELECTED PROBLEMS
(k Rr / R s ) [k R r / RM + k Rr / R s + 1 1- 1
4
eo/e;
6
Rz = 63 Q
9
13
29
33
34
Rt
=
=
1 4 1 .9 Q
For 100 N, 6eo = eo = 0.24 V
R3 = I kn
enns
= 6.0 V
(a) fc
=
4.82 kHz
Dig ita l Tech n i q ues i n
M ech a n i ca l M eas u rements
2
3
4
5
6
7
B
9
10
11
12
13
14
INTRODUCTION
WHY USE DIGITAL M ETHODS?
DIGITIZING M ECHANICAL INPUTS
FUNDAMENTAL DIGITAL CIRCUIT ELEMENTS
NUMBER SYSTEMS
BINARY CODES
SOME SIMPLE DIGITAL ORCUITRY
THE COMPUTER AS A MEASUREM ENT SYSTEM
THE MICROPROCESSOR
THE MICROCOMPUTER
ANALOG-TO-DIGITAL AND DIGITAL-TO-ANALOG CONVERSION
DIGITAL IMAGES
GEmNG IT ALL TOGETHER
SUMMARY
INTRODUCTION
In this chapter we discuss some basic uses of digital logic and circuitry as they apply to
mechanical measurement�. but it must
be
understood at the outset that our purpose is not
to cover digital electronics in depth: The subject is too large
for such a treatment. Rather,
our i ntent is to survey the subject sufficiently so that those in fields of engineering other
than electrical will gain some appreciation for the advantages, disadvantages. and general
workings of digital circuitry in the context of measurements.
Most measurands originate in analog form. A n analog variable or signal is one that
varies smoothly in time, without discontinuity. In many cases the amplitude is the basic
variable; in others, the frequency or phase might be. A common example of a quantity
in analog form is the ordinary
120-V ac, 60-Hz power- l i ne voltage ( Fig. l (a )) . An analog
signal, however, need not be simple sinusoidal or periodic in fonn. The stress-time rela­
tionship accompanying a mechanical shock [Fig.
I (b JI is considered
an a log in
fonn. The
pressure variations associated with the transmission of the human voice through the air are
also analog. The readout from the pointer-over-scale D' Arsonval meter is also considered
analog because of the uninterrupted movement of the pointer over the scale. An analog
scale can be compared to the range of brightness between black and white, including all the
variations of gray in between. Digital information, on the other hand, would pennit only
the time variation between the two extreme levels of brightness-black or white.
302
Digital Techniques in M echa n ica l Measurements
,, r\JV
(\ '
(a)
1 byte
Previous byte
-l r-
f(I)
;:!
(b)
_ ,
Next byte, etc.
Bit interval
- --, I
I
I
I
I
I
I
--
I
I
I
I
I
I
I
__L__
- - - - - Logic 1
I
I
I
I
I
I
I
Logic 0
_
(c)
,
FIGURE I : Examples of voltage-time relationships for analog signals (a) and (b), and a
digital signal (c).
Digital information is transmitted and processed in the form of bits ( Fi g . l (c)), each
bit being defined by (a) one or the other of two predefi ned "logic levels" and (b) the time
interval assigned to it, called a bit interval. The most common basis for the two logi c states
is predetermined voltage levels, say 0 and 5 V de. Current or shifts in carri er frequency are
also u.'ICd. The time rate of the bits is closely controlled, commo nl y by a crystal-controlled
oscillator (or
clock). The information i s then carried by specific bit groupings, coded in
predeterm ined sequences; for example, alphanumeric data may be handled by sequences of
three, four, or more bits sent in the various possible combinations, with each combination
or group forming a word of information (see Section 6). The term byte is ap pl ied to an 8 -bit
word, whereas the term word may be applied to any unit of digital i n formation . A 1 6- bit
word is two bytes i n length. A 4-bit word is sometimes referred to as a nibble.
Figure l (c) shows one possible combination of bits grouped to form one byte. In
this case, the sequence of bit values i s 1010 01 10. From this we see that a bit need not
completed off/on/o ff sequence. Indeed a byte of
1 1 1 1 1 1 1 1 , in which case no bi t- lo- bit changes
occu r throughout the byte. One bit corresponds to either of two different logic states held
be a pulse in the sense that it must be a
information could well be 0000 0000 or
constant during the one-bit interval.
Because most measurement in pu ts originate in analog form, some type of analog­
to-digital (NO) converter (or ADC) is usually required (Section 1 1 ). In certain instances
it may also be desirable or necessary to use a digital-to-analog (D/A) converter (or DAC)
somewhere in the measurement c hai n . A sophisticated example of digital information
handling is pulse-code mqdulation (PCM) of the human voice, which is used in essen tial l y
303
Digital Techniques in Mechanical Measuremenu
all digital communication systems. For digital telephones, the spoken word is con\'Crted lo
digital fonn for trans mi sio n over telephone circ its .and lhen onverted back to its ori
ginat
analog fonn al the rece1 V 1 ng end. Some of the mouvauons for this appare ntl y circul ar PIOCess
�
�
�
are � i scus�ed in S�tion 2. . PCM is also the basis of digit_al audio on �omputers, music CDs,
.
d1g1tal radio, and, m the visual context, of DVDs and d1g1tal telev151on. It is interestin
g to
note that when PCM was proposed by Alec Reeves, in 1 937 , digital semiconductor circui ts
had not been developed . and his idea could not easily be implemented on the technologies of
lhe day
2
( I ] . I I was only in the late twe n t ieth century th at his invention came to full fruition.
WHY USE DIGITAL M ETHODS?
Over the past several decades, digital electronics have become u biquitous in both household
and technical setti ngs . Students today may be surprised to learn that when many of their
pro fessors were students, essent ial ly all instrumentation was analog and data were usu.
ally recorded by hand. Voltmeters used ana log pointer-over-scale (D' Arsonval) displays,
oscilloscopes used direct am p l i fication of the input signal to drive the cathode ray
tube,
storage of lhe oscilloscope signal was accomplished with a film-based camera, and com­
puter acquisition and processing of data would
be
found only in the best equipped research
laboratories-if indeed it could be found at all !
The analog instruments that have been replaced by digital instruments were often very
accurate, so we may reasonably ask why digital instrumentation has become predominant.
A number of factors have contributed. First, di g i tal electronics are usually easier to design
and
fo bricate than analog
g
electron ics . Inte ra ted circuit technologies make it possible at
low cost to mass produce relatively sophisticated ins tru ments . These devices can be quite
small, and they generally operate in the range of
5 to
12 V de. This may be compared to
the relative bulk of some analog devices ( for example, those requiring large, high-quality
capacitors) and to the hig h opera ting voltages of some a nalog instruments (perhaps ranging
to several hundred volts).
A second factor, of course, is the ease of data recording, storage, and display. A digital
voltmeter provides a direct numerical display of the voltage, often au tomat ical l y scaled
into the proper range. The analog voltmeter had to be v isually interpolated if the pointer
lay between adjacent g raduati ons of the scale.
A digital voltmeter may also be direc tl y
coupled to a computer for data recording and processing. Thus, good-quality graphs may
be
produced and printed w i t h ease-your professor may very well have
used graph paper
and a Fre nch curve to plot data when he was a student. Of equal importance is that many
steps necessary in data reduction can be handled automatically. For example, temperatures
and pressures acquired from a Huid-How process may be combined immediately and reduced
to provide an on-the-spot overall result, the mass How rate perhaps. Further, data stored on
a
com puter
can easily be rescaled and replotted so as to test various d i ffe re n t
theories for
its interpretation or to investigate th e cause of some anomaly.
Digital instrumentation has another tremendous ad van tage : Digital signals are inhcr·
The informational content of the digital signal i s not ampl i tude
dependent. Ralher, it is dependent on lhe particular seq ue nce of on/off p ulses thal apply.
Therefore, so long as the sequence is ide nti fi able , the true and complete Conn of the input
ently noise resistant.
remains unimpaired. Maintenance of accuracy, lack of distortion induced by signal pro·
to analog
cessi ng and noise pickup, and greater stability are all enhanced in comparison
methods. The nature of the digital signal and circuitry permits the signal to be regenerated
or reconstituted from point to point throughout the processing chain. The voltage ampli·
304
Digital Techniques in Mechanical Measurements
1ude of lhe i n fonnalio nal pulses is commonly
5 V de.
U n less noise pulses approach lhis
magnilude-a highly unusual condition-they are ignored. This is of particular value i n
lhe central conlrol of a l arge processing syslem, such as a refinery or power plant, where
low-frequency si g nals might
be relayed
over relatively great distances, perhaps a mile or
more. This advanlllge is even more obvious when radio links are used, as in lhe ground
recording of signals originating from a space vehicle. On the level of con sumer electron­
ics. one reason Iha! lhe digital compacl disc format for music d i spl aced lhe earlier analog
pho nograph record is lhat digillll sound reproduction is substainally less suscepli ble to the
effects of dust, vibralion, and wear lhan is lhe magnelic phonograp h needle; and CDs are
sma l ler and contain m ore music than did a vinyl LP.
3
DtGmZING MECHANICAL INPUTS
To be processed digitally, analog measurands mu s1 be
1. Converted 10 yes/no pulses;
2. Coded in a form mean i ng fu l 10 lhe remainder of the system; and
3. Synchronized so as 10 mesh properly wilh other inputs or control or command signals.
Meeting lhese three requ iremen ts is collectively referred to as
intetfacing.
When some fonn of computer is a part of lhe system, not only must lhe input
be
converted to digilal pulses, bul also lhe pulses musl be converted to lhe language used
by lhe compu te r, that is, binary words. In addilion, of course, lhe compuler is unable 10
give undivided auen tion 10 any one signal source: It will also be receiving i npu ts from
other sources, processing the i n putted dalll, and outputting data and conlrol commands.
Input from any one source must wait its turn for attentio n . In olher words, all lhe inputs
and outputs must be synchronized lhrough proper interfacing. Before attempting further
coverage of computer data acquisition, however, we will discuss some of the fundame ntals
and a few of the simpler types of digilal i nstrumentation .
S i ng le di gi tal - type instruments whose end purpose is simply to display the magni­
tude of an input in digilal fonn (as opposed to an inpul to be inlerfaced inlo a syslem) often
require o n l y lhal lhe i n put be transduced 10 a frequency. Convention al transducers may be
u sed to sense the m agnitude of the measurand and 10 convert it into an analogous vol tage .
The vo l lllge can then be amplified and, by a circuit cal led a voltage-controlled oscillator
(VCO) [2), transduced to a proportional frequency. A frequency - measuring circuil (Sec­
tion 7.3) might then be ca li brated lo display the magni tude of the i nput . Many transducers
in the field of mechanical measurements produce vol lllge ou1 puts (e . g . , s trai n - gage bridges,
thermistor bridges , differenlial transfonners, thennocouples, etc.). In addition, mechani­
cal motion, both rotational and translational, may often be qu ic kly, easily, and c ompl ete ly
converted to d i gi t a l vo l tage pulses by prox imity transducers, pho1ode1ec1ors, optical inter­
rupters, and so on.
4
4.1
FUNDAMENTAL DIGITAL CIRCUIT ELEMENTS
Basic Logic E,lements
An ordinary single-pole single-throw (SPST) switch [ Fig. 2(a)) is a digi ta l element in its
simplest form. Whe n ac tu ated , il is capable of producing and con trolling an on/off sequence.
It this sense, its output is binary. The ordinary electromechanical relay [ Fig . 2(b)] is a
305
Digita l Techniques in Mechanical Measurem•�nts
j
... ---......
I I I - _____
-----
__
(a)
I
I
_
I
(b)
+5 V dc
�---- --<> Output
+5 V dc
+S V de
�
o
-
o�
Input
(c)
FIGURE 2: Digital swi tching devices: (a) a simple mechanically operated switch, (b) an
electrically controlled relay, (c) a transistor-type switch. For the latter, when the input is
"high" the transistor conducts, thereby effectively shorting the: output to ground, and hence
providing near-zero or "low" output. When the input is low the transistor does not conducl,
thereby providing near +5 V de at the output. Additional arrangements are possible if signal
inversion is not desired.
Slightly more advanced digital device in which an electrical input may be used to change
the ou tpu t condition. A more sophisticated electronic switch is provided by the transistor
[Fig. 2(c)). When properly biased, a transistor can be made to conduct or not to conduct.
depending on the input signal. It is a near-ideal switching device. It can function al
relatively low control voltages. is capable of switching at ra.tes of hundreds of GHz. can
be made extremely small and rugged, and can be designed to consume relatively little
power. Originally, discrete transistors were hardwired into the various circuits, usually
as a replacement for vacu u m tubes. (Thus, one still hears the term transistor radio even
though vacuum tube radios have virtually vanished.) Today, transistors are almost always
fabricated together with other simple elements such as resistors, capacitors, and diodes
as a single special-purpose device on a single semiconductor· c h i p-a so-called integrated
circuit, or IC.
A multitude of special-purpose IC chips are available to I.he electronic design engineer.
Their variations and complexity have grown at an astonishing rate over the past decades.
306
Digita l Techniques in Mechanical Measurements
·�·
A
(a)
B
0 0 0
0 1 0
1 0 0
1 1 1
·y·
A
·:Jlfj :
(C)
A
B
:-f:.;
Em-y
A� lB
0
0
1
1
(d)
A
(e)
0 1
1 0
0 0
1 0
AUB
-y
-y
0 1
1 0
(g)
� �
B
0 0 0
0 1 1
1 0 1
1 1 0
B
NOR
OR
' · 1� !
·· :! '
·y·
NOR
0 0 0
0 1 1
1 0 1
1 1 1
�,.
B
0 0 1
0 1 1
1 0 1
1 1 0
(b)
OR
A
·�·
NANO
AND
(I)
0 0 1
0 1 0
1 0 0
1 1 1
FIGURE 3: Symbols for some common digital logic units. Also shown are the respective
truth tables for the units.
These range from very simple components such as op amps through a myriad of digital·
circuit building blocks, such as logic gates and ftip-ftops (which we discuss shortly). through
instrumenation circuits, such as voltmeters or frequency counters, up to highly sophisticated
microprocessors.
At the heart of most digiial devices are logic gates. These are simple circuits that
involve just a handful of transistors and provide an output that depends upon two or more
inputs. For convenience, these circuits are represented by special shorthand symbols. Fig­
ure 3 illustrates symbols for common logic elements used in various combinations to fonn
many of the IC chips. Elements 3(a) through 3(0 are gates. Figure J(g) represents a simple
inverter. Also shown are logic, or truth. tables, which list all the possible combinations of
inputs and their corresponding outputs. Recall that basic digital operations are based on
simple yes/no, on/off, 110, states. For example, the truth table for the AND gate shows that
the output is high only if the inputs to both A and B are high, hence, th e AND rule:
For the AND gate, any low input will cause a low output; that is, all i nputs
must be high to yield a high output.
307
Digital Techniques in Mechanical Measurements
+5 V dc
A
B
Out
FIGURE 4: A schematic diagram showing the internal structure of a NANO gate.
In like manner, we can also easily state rules for the other elements. Truth tables are of par.
ticular importance to the circuit designer, especiall y when combinations of circuits increase
their complexity.
As a mailer of i nterest , the NAND gate actually contains four transistors, three resis­
tors, and a diode, as shown in Fig. 4. Suppose a chain of pulses is applied to input B of lhe
AND gate. We can see that their passage may be controlled by input
If is high, lhe
chain will be permitted to pass; if low, the chain will be stopped. From this simple example
the origi n and significance of the term gate is clear.
The various gates may be expanded lo provide more than two inputs; see, for example,
Fig . 5. The IC shown is a three-input NANO gate, for which a zero input level at any one of
the input ports perm i ts the passage of pulses from any other port; in other words, all inputs
m ust be high in order for the output to be low. With this arrangement, a combi nation of
several control conditions must be met simultaneously to block passage of a signal.
Integrated circuits are often characterized by the effective number of gates they incor·
porate. The so-called MSI (medium-scale integration) chips incorporate 12 10 100 individ·
A. A
�
ABC
c 8 A
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
Out
1
1
1
1
1
1
1
0
FIGURE 5: NANO gate with three inputs and 23 = 8 possible input combinations. A
NANO gate with eight inputs has 28 = 256 input combinations of which only one causes
an output of logic 0.
308
Digital Techniques in Mechanical Measurements
·
q'
uaJ gates per chip. LSI (large-scale integration) chips contain 100 to 10,000, VLSI (very
large-scale integration) chips contain l if to IOS gales, and so on.
Combin ation of Logic Elements: The Fllp-Flop
Figure 6(a) shows two NANO gates connected lo fonn a very useful circuil The circuit
has two inputs. S and R, and two outputs, Q and Q, that are always meant to be opposite
one another (one at l ogic I and the other at logic 0). As a simpl e Hip-flop, element A is
called the S ET gate and element B the RESET gate. Consideration of the i nd iv idual bUth
tables, along with their panicular interconnections, shows that the fol low i ng are the only
workable conditions:
Condition I
Condition II
Condition Ill
Condition IV
s
I
I
0
R
I
I
I
0
Q
I
0
I
0
Q
0
I
0
I
Now suppose that both S and R are initially at logic 1 ; then either of Conditions I or
II may exist, depending on random or programmed preconditions. If either of two outputs
can correspond to a given input, the device is referred to as having bistable logic. In some
contexts, the circuit is called a latch.
I.et us momentarily ground input S-thal is, impose logic 0. Condition III, called
the SET condition, will result. This is true regard l ess of whether the circuit is initially in
Condition I or II. Return o f S to the high state will cause no change in Q or Q: It is latched.
Now, if R is momentarily grounded, Condition IV will be instituted, and this state
will continue even when R is returned to logic I . This is called RESET, and we see that the
outputs are caused to flip and flop between SET and RESET.
Grounding both inputs simultaneously-setting both S and R to logic 0-would
amount lo attempting SET and RESET simultaneously. This state is nonnally avoided.
Because the changes in state occur when S or R is at low logic, this circuit is referred to as
an active-low S-R latch.
This ciruit has various important uses. For example , as a latch it may be used to hold
(latch) a count in an electronic events counter, then await a RESET i npu t for initiating the
next count. The n i p- flop is used as a memory cell, capabl e of holding one bit of infonnation
for later use. It also provides the basis for the switch debouncer. When an ordinary electric
switch depending on mechanical contacts is closed, numerous contacts are actually made
and lost before solid contact is finalized. In a counting circuit, for example, this son of switch
hash cannot be tolerated. By placing a flip-Hop or latch in the switch circuit [Fig. 6(b)] we
cause the latch to respond lo that first momentary contact and then to ignore all that follows
until a RESET signal reinitializes the circuit. 111ese arc only several of the uses to which
the circuit may be applied; and the particular circuit discussed is only one of a number of
different circuits referred to as flip-flops (3).
43
IC
Families
Several families of integrated-circuit chips are available. each of which has special char­
acteristics but, in general. all of which perfom1 essentially the same basic tasks. The
309
Digital Techniques in Mechanical Measurenn ents
s
-a
R
(a)
+5 V dc
NC
dosed)
(normally
NO
(normally
open)
To circuit l>elng controlled
+5 V dc
(b)
FIGURE 6: (a) Two NANO gates configured to form a llii p - ll op, or latch; (b) a swi1ch­
debouncer circuit.
most commonly used groups are the newer comp/emenlary metal oxide semiconductor, or
CMOS, group, and the older transistor-transistor-logic, or 1m, group. In each family lhc
various logic units are com bined to perform special functions. For example, the TJ1. family
consists of more than 1 50 di fferent types of chips. Table 1 is a partial list. A l l have man:
or l ess the same outward appearance but each is designed to pcrfonn a different funclion.
Schematics of several of the 1m fa mily arc illustrated in Fig. 7. The circuits in Figs. 7(1),
(b), and (c) are simple enough that the functional perfo rman ce sy mbols may be shown. The
circuitry of Fig. 7 (d ), however, is so complex (because of the number of elements used) thal
we h ave made no anempt to indicate the internal architecture. Application of most of the
ch ips selected for listing in Table 1 is covered i n later sections of this chapter. All of these
basic devices are also available as CMOS IC chips.
Within each family of circuits, various subcategories are available, eac h wi th
_
what different current, voltage, and speed characteristics. The subtype is usually mdicat
by a set of letters that is placed between the 74 and the follow ing digits . For example.
ld be
an "advanced high speed" CMOS (AHC subtype) quad t wo- i nput AND gate wou
design ated 74AHC08. Here, the word quad indicates that four separate gates aie on 1
�
single chip .
Id-
��OS
CMOS c i rc u i ts are based on a different type of transi:>tor than the 111.. fam il y
effect transistors rather than b i pol ar junction transistors), and for most purposes
31 0
(8)
Quad 2·1nput AND gate.
(The wonl 'quad" lndlcatea
ttmt the chip Include& lour
Independent AND gates.)
(b)
Hex Inverter
(six Independent
Inverter elements).
Top view
Top view
(c)
Dual 4-input NANO gate.
(d)
BCD lo 7-segment
decoder-driver
Top view
Top view
FIGURE 7: Diagram showing four typical IC chips: (a) quad 2-input AND gate, (b) hex inverter (six independent inverter elements), (c) dual
4-inpul NANO gate, (d) BCD (binary-coded decimal) to 7-segment decoder-driver.
Digital Techniques in Mechanical Measurements
TABLE
Type Number
7400
7404
7408
74 1 4
74 1 6
7430
7432
744S
7447
7474
747S
7483
748S
7489
7490
7492
74 1 09
74 1 2 1
74 1 SO
74 1 S4
74 1 60
1:
A Partial Listing of Tll.. IC Chips
Description
Quad two·inpul NANO gale
Hex inverter
Quad 1wo-inpu1 AND gate
Hex Schmid! !rigger
Hex driver, invening
Eighl·inpul NANO gale
Quad lwo-inpul OR gale
BCD to decimal ( I -of- I 0) decoder driver
BCD to seven-segment decoder driver
Dual D-type Oip-ftop, positive-edge !rigger
Quad latch
Four-bit full adder
Four-bil magnilude comparator
64-bit ( 1 6 x 4) memory
Decade counier
Base- 1 2 counler
Dual J-K ftip-ftop, positive-edge trigger
Monos1able multivibrator
1 -of- 1 6 dala selector/mul1iplexer
l -of- 1 6 data distributor/demultiplexer
Synchronous BCD decade counter with asynchronous �•
circuits have subs1antially more desirable electrical charac1eristics than Tl1.. circuits. As •
result, lhe CMOS family is used in mosl new circuit designs, and the entire 111. family is
gradually becoming obsolele.
4.4
IC Oscillators and Clock Signals
Figure 8 illustrates an interesting IC oscillator-the SSS chip. This chip contains 23 11111·
sis1ors, IS resislors, and 2 diodes. The package is about half the size of a common posiagc
stamp, has eight terminals, and costs less than a cup of coffee and a doughnut. Whcft
configured with two or 1hree external resistors and a capacitor, ii can produce an osci111ling
outpul 1ha1 switches back and fonh between two slates. In this 1ypc of astable
lhe SSS is capable of yielding a square-wave oulpul covering a frequency range from 0. 1 Hz
to over I 00 kHz. With additional outboard circuil elements. triangular and linear ralllP
wavefonns can be obtained. In slill other configura1ions, 1his chip can produce a si ngle
outpul pulse in response to an inpul (so-called monostable operation), and many
have bee n devised (4 ). This IC oscillator is an example of the wide vcrsali lily f 1 smg
oscillation.
other. US:
o
.
special-purpose integrated circuil.
di�•·ia1
Chips such as lhe SSS are used 10 provide a time-base reference for a variety of
8
sys1ems, and so 1hey are sometimes referred 10 as timers. The oscillating signal
1
circuil can be used as a clock signal 10 synchronize the actions of several com ponents 1"
n
sys1em. In these synchronous sys1ems, the lime at which any su bcircuit's outpul ca chaRSC
of u�
312
Digital Techniques in Mechanical Measurements
+5 V dc
8
4
7
OUlpUI
S t---6
2
FIGURE 8: A SSS astable/monostable oscillator or multivibrator.
is controlled by the clock signal. For example, flip-flop can be designed so that when an
input is changed, the change in output is delayed until the clock signal changes; the D­
type and J-K flip-flops in Table I (7474 and 74 109, respectively) both have this feature.
Synchronous operation usually simplifies circuit design. Circuits that are not synchronous
are called asynchrono11s.
The output of a SSS-based clock circuit may be stable to about I % or so but it is
susceptible to drift. When higher accuracy is required, a quartz-crystal oscillator is used
instead. The heart of such a circuit is a small piece of quartz cut to precise dimensions,
which resonates at a highly stable frequency. Owing to the piezoelectric properties of quartz
the resonating crystal is easily coupled to an electrical circuit. Quartz-crystal oscillators are
available as off-the-shelf !Cs at low cost with frequencies from roughly I 0 kHz to I 00 MHz.
Frequency stability to a few parts per million is easily obtained. These crystals are used
in all microprocessors and microcomputer systems. Crystals resonating at 32.768 kHz are
al the heart of digital wristwatches: When additional digital circuits are used to divide this
frequency by 2 1 5 , the result is a I -Hz signal for counting seconds (5). Note that an error of
I 0 parts per million in the crystal frequency would correspond to about one second per day
of error in a watch.
4.5
Digital Displays
In some situations we may require only simple, single lights for an indication or readout,
perhaps to show that a device is powered or ready for use. In other cases we may use
combinations of discrete lights or indicators to produce an alphanumeric display.
Alphanumeric readout elements are usually of either the liquid-crystal diode (LCD)
type or the light-emilling diode (LED) type. The l iquid crystal has a decided advantage
in some cases, as, for example, in a digital watch requiring very low power consumption.
31 3
9: 'I)'pical solid-state numeric di splay. The +5 V de anode is common to all
Grounding pin a, for example. through a suitable vol tage-droppi ng resistor
lig h ts segment a .
FIGURE
segments.
LCDs are t he usual choice fo r bauery-powered i nstru me nts . A d isadvantage i s th e n eed for
illumination or a separate lamp for back.Ii ghting the display. The LED
prov ides a bright d ispl ay, usually in red or green, but it consumes much more power. It is
therefore more commonly used in i n stru men ta ti o n that c o nnec t:; to an external power source.
Figure 9 shows an arrangement of a ty pi c al LED seven-segment digital di splay. Each
segm ent consists of one or more LED e leme nts and with proper s witchi ng s uppl ied by an
appropriate IC driver, each of the decimal digits, 0 through 9, can be formed. Commonly.
the input sig n al and a 5-V de powe r source are all that is requ'ired for power.
The operation of a digital displ ay requi res a proce ss of conve rting information between
bi nary and deci mal forms. This process is known as e11coding when in format ion is converted
to a binary form, and it is called decoding when information is converted out of a binaiy
form. These operati ons clearly require some understanding of the binary and decimal
number systems. In the next two sections. we look at n um b er systems and the common
bi n ary codes.
either proper ex tern al
5
NUMBER SYSTEMS
Whether it i s
in digital o r anal og form, a measurement signal conveys a mag n itude . Mag­
in numbers and numbers im pl y some sort of numbering system or
structure. Digital devices with their high/low, y es/no. on/off s eq uenc ing suggest the use of
a base 2, or bi nary, system for cou n tin g . In this section we w il l present the essentials of
number systems other th an decimal that are pertinent to digitul o perations .
We know that the position of each of th e digits in a decimal number is important. For
ex am p le , consider the deci mal number 347.25. The 3 is the m ost s i gn i fican t digit and the 5
is the least significant digi t. We k now that the number can be expanded to read
n itudes are expressed
347 .25 = 3
x
ta2
+4
x
101 + 7
31 4
x
IOO + 2 x
1 0- I
+5
x
1 0-
2
Digital Techniques in Mechanical Measurements
It is clear that the various positions or the digits detennine the power to which the base, 10,
is raised.
For the binary number sysiem, only two different digits are required, a I and a 0. The
digit may correspond to a high condition, say +5 V de, and the digit 0. to a low condition,
say O V de. In the binary sysiem, as in the decimal system, position has meaning. Consider
the binary number 1 10 10.01 2 . (We add the subscript 2 to make it clear that we are using
the binary syslem.) The first digit, I, is the most significant digit and the last digit, l, the
least significant. Each digit is called a bit in the sense that it supplies an elemental "bit"
of information. Hence, the lerms most significant bit, or MSB. and least significant bit, or
I.SB, are used. What conesponds to the decimal point in the decimal system is called the
binary point in the binary system .
I
(l'i- , ,
·
. r· ·
As we have observed, in a decimal number each position corresponds to an integral
power of 1 0. B y the same token, in a binary number, each position corresponds to an
iniegral power of 2. Whereas the coefficients for the decimal number could be anythi11g
between 0 and 9, for the binary system we are limited to 0 and I .
Let's convert the binary number written above to the equivalent decimal number.
Equivalent, of course, means that the two numbers signify the same actual magnitude or
quantity (it may be convenient to refer to Table 2).
2
1 1 0 1 0.0h = I x 24 + 1 x 23 + 0 x 2 + I x 2 t + 0 x 2° + 0 x 2- 1 + I x 2- 2
I
= 16 + 8 + 0 + 2 + 0 + 0 +
4
= 26.25 10
TABLE 2: Decimal Values of Bases Raised to Various Powers'
n
-3
-2
-I
0
I
2
3
4
5
6
2"
4"
1/8
114
112
I
2
4
8
16
32
1/64
1/16
1/4
I
4
16
64
256
1 ,024
4 ,096
64
8"
1/5 1 2
1164
118
I
8
64
512
4,096
32,768
262, 144
16"
1/4096
11256
1116
I
16
256
4.096
65,536
1 ,048,576
1 6,777,2 1 6
• n i s both the power to which the base
is raised and the positio11al weight.
31 5
Digital Techniques in Mechanical Measurements
We can see that the positional signi c �ce �f the Os. a �d l s lies in the integral powers to
.
which the base (also called the radu) IS raised. llus is true for both the binary and the
decimal systems.
�
Two other systems are commonly used in digital manipulations: the
or base 8
system; and the hexadecimal, or base 1 6 , system. These two systems have the
advantage over the decimal system in the ease and convenience of their conversion, by
either machine or human, to binary.
octal,
6
marked
BINARY CODES
In addition to the binary number, which is, of course, limited in magnitude onl y by the
number of positional bits that may be arbitrarily permitted, there are various binary codes.
At this point we will consider several of them.
6.1
Binary-Coded Decimal
Suppose we limit the number of positions to four, i.e., we provide only four on/offswitches
or their equivalents. We are then limited to the decimal range of 0 to 15 inclusive. The
result is what is known as the four-bit word, also referred to as a/our-level code.
A modification of this code is known as binary-coded decimal, or BCD. Although
BCD is also a 4-bit word, it arbitrarily makes illegal all words greater than 10012 , or 910.
We see that the BCD code is useful because of its convenient relationship to the decimal
digits 0 through 9. Table 3 lists the equivalencies. When the BCD code is used, each digit
in a decimal number is processed separately as a 4-bit sequence. For example, the decimal
number 875 10 translates to 1 000 0 1 1 1 0 1 0 1 2 in BCD.
TABLE 3: Four-Bit Binary Numbers and Binary-Coded Decimal
Decimal Digit
Binary
0
I
0000
000 1
00 1 0
00 1 1
0100
0101
0 1 10
01 1 1
1000
1 00 1
1010
101 1
1 1 00
1 101
1 1 10
1111
2
3
4
5
6
7
8
9
10
II
12
13
14
15
31 6
BCD
()()()()
000 I
00 1 0
00 1 1
0100
0101
01 10
01 1 1
1000
1001
Illegal
filegal
lllegal
lllegal
lllegal
lllegal
Digital Techniques in Mechanical Measuremenu
6.2
Position Encoders and Gray Code
Figure lO shows a schematic diagram of one type of binary displacement encoder based
upon a 5-bit binary code. 1be "card" shown consists of five active tracks plus a reference
track, which may or may not be needed. Pickups (not shown) sense the relative disp l acement
of the card. One sensor is used per track. Optical sensors are most commonly used to detect
the on/off status of of each track at a given position. The cards may be transparent or opaque
for use with transmitted or reflected light, respectively. Printed-circuit methods may also be
used. The output from the sensors is easily processed by on board or remote digital circuitry.
A direct binary encoding such as shown in Figure 10 has an inherent disadvantage.
When several bits are different between adjacent levels, the assoc iated position sensors may
not all change at the exact same instant, leading to a drastic sensing error. For example, in
going from the eighth to the ninth position, the binary encoder shown would change from
a b c
d
e
8lnaty
00000
0000 1
000 1 0
000 1 1
00100
00101
001 1 0
001 1 1
01 000
01001
01010
0101 1
0 1 1 00
01 1 0 1
01 1 1 0
01 1 1 1
1 0000
1 0001
1 001 0
1 001 1
1 0100
1 0101
1 01 1 0
1 01 1 1
1 1 000
1 1 001
1 1 010
1 101 1
1 1 100
1 1 101
1 1110
11111
Declmsl
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
RGURE 10: Binary displacement encoder.
31 7
Digital Techniques in Mechanical Measurements
TABLE 4: Four- Bit Binary Numbers and Gray Code
Decimal Digit
Binary
0
I
2
3
4
5
6
7
8
9
to
II
12
13
14
15
000 1
00 1 0
001 1
0 1 00
OIOl
Ol lO
Ol l 1
1000
1 00 1
1010
101 1
1 100
l lOl
1 1 10
1111
0000
Gray Code
0000
000 1
00 1 1
00 10
0 1 10
0111
OlOI
0 100
1 100
1 101
1 1ll
1 1 10
1 0 10
101 1
ltJO l
It:)()(}
00 1 1 1 2 to O IOOOi (i.e., from 7 10 to 8 10). If just one bit wem to change Loo soon or loo lale,
lhe position could be sensed as 000002 or 0 1 1 1 1 2 , correspo0nding Lo either the first or the
sixteenlh level !
Gray code is a binary code that el imi nates 1his ly pe of error, and it is thus preferred for
use in position encoders. It is not a numeri ca l code, i n t he sense lhat there is no positional
weight to the digits. Instead, t he digits of the code are sequenced so that only one bit changes
in going from one level in the sequence to the next. If one 1:rack's sensor changes too early
or too late, lhe error is no more than one level-the drastic errors possible with direct binll}'
encoding are avoided. A 4-bit Gray code code is shown in Table 4. Figure 1 1 illustrates a
6.3
circular card for a rotatary shaft position encoder based upon four-bit Gray code.
Alphanumeric Codes
The preceding codes are for the transmission and processing of numeric data. Alphanumeric
information must also include provisions for the Letters of t he alphabet and
other symbols, such as punctuation marks.
perhaps cenain
pposed
One of lhe simplest binary codes for transmission of general inform ation (as o
to numeric data only) is the International Morse Code, which uses pulse-duration modu·
lation, or PDM (also sometimes called pulse-Length or pulse-widlh modulation). The two
different pulse widths, the "dot" and the "dash," are used in various combinations to inns·
l
mit the alphabet, the decimal digits, and certain olher special-purpose telegraphic sym s.
�
International Morse Code is an uneven-kngth code i n that various time intervals are requin:d
for lhe transmission of t he various characters. A related <:ode
teletypewriter, code, which is a live-unit,
even-length code.
31 8
All
is the Baudot, or common
characters are formed by 8
Digital Techniques in Mechanical Measurements
FIGURE 1 1 : Circular encoder using Gray code.
combination of five possible on/off states, each of uniform duration. These two codes are
not especially important in mechanical measurements.
The most welt-known code for alphanumeric data is the American St4ndard Codefor
lnfonna1ion Interchange or ASC11 (6). This code is of 7-bit binary form, and it thus has 27
or 128 characters. Both the uppercase and lowercase letters of the English alphabet, plus
the decimal digits zero through 9 and certain other control symbols, are included (Table 5).
ASCII originated as a 7-bit bit teleprinter code, essen tially as a more sophisticated successor
to the 5-bit Baudot code. Thus, the first 32 characters and the final character are control
codes that might govern such functions as carriage returns, line feeds, and form feeds-all
handy for driving a printer. These codes are not associated with displayed characters, and
so Ibey are omitted from the table.
When writing the binary equivalents of the ASCII code, one generally divides the
binary number into two groups of 3 and 4 bits each; for example, for lhe letter a, the binary
equivalent is written t IO 000 1 . The lefthand binary digit is the MSB (most significant
bil) and the righthand, the LSB. In processing, the ASCII number for a would appear as
319
Digital Techniques in Mechanical Measurements
Displayable Characters of the American Standard Code for InfOrtnali
on
Interchange (ASCil) with Decimal, Binary, and Hexadecimal Numbering
TABLE 5:
Dec
Binary
20
34
010 000 1
010 0010
010 00 1 1
21
22
36
37
010 0100
010 0101
010 01 10
010 01 1 1
40
41
010
010
010
010
1000
1001
1010
101 1
44
4S
46
47
010
010
010
010
1 100
1 101
1 1 10
llll
4B
49
50
51
01 1 0000
O t t 000 1
Oi l 0010
01 1 001 1
S3
Oi l 0100
0 1 1 0101
Oi l O l l O
01 1 01 1 1
56
S1
01 1
01 1
01 1
Oi l
42
43
52
54
SS
S9
58
60
61
62
63
Oi l
Oi l
Oi l
011
1000
1001
1010
IOl l
1 100
1 101
1 1 10
1111
100 0000
40
6S
100 000 1
100 0010
100 00 1 1
41
42
4J
@
A
D
E
46
(
)
.
+
100 0100
1000101
1 00 0 1 1 0
1 00 Ol l i
44
70
71
72
73
74
1S
100
100
100
100
1000
1001
1010
101 1
48
49
-
76
n
78
I
79
100
100
100
100
1 100
1 101
1 1 10
1111
I
80
Space
010 0000
39
Dec
Dee
3J
38
Char
Char
32
3S
Hex
64
Binary
Hex
2J
24
2S
26
27
28
29
2A
28
2C
2D
2E
2P
.
#
!
$
.,,
&:
30
0
34
4
3S
J6
s
6
32
31
33
37
38
39
38
3A
JC
JD
3E
JF
3
2
7
8
67
66
68
69
81
82
83
BS
84
86
87
88
89
:
;
90
91
=
9J
9
<
>
?
92
94
9S
4S
47
4A
96
1 10 0000
Hex
8
97
98
c
99
1 10 000 1
1 10 0010
1 10 001 1
62
•
b
•
100
IOI
102
103
1 10 0 100
1 10 0101
1 10 0 1 10
1 10 01 1 1
64
d
p
G
67
•
I
H
104
1 10 1000
1 10 1001
1 1 0 1010
1 10 101 1
68
h
J
I
48
K
40
M
IOI
IOI
soS I
p
IOI 0100
0101
101 0 1 1 0
IOI Ol l l
SS
101
101
101
101
SB
S9
SA
0000
000 1
101 0010
101 00 1 1
IOI
1000
1001
1010
101 1
101 1 100
101 1 101
IOI 1 1 10
101 1 1 1 1
4C
4E
4f
S2
S3
S4
S6
S1
SB
SC
SD
SE
SP
a..
Binary
106
IOS
107
60
61
63
6S
66
69
6A
68
N
0
109
1 10
111
1 10 1 100
1 10 1 101
1 10 l l lO
1 10 1 1 1 1
Q
1 12
1 13
1 14
l lS
Ill
Ill
111
111
T
u
v
w
l l6
1 17
l l8
l l9
I l l 0100 . 74
1 1 1 0 101
1S
76
1 1 1 0 1 10
n
1 1 1 01 1 1
L
108
R
s
x
y
z
[
\
I
-
"
0000
000 1
0010
001 1
120
121
122
12J
111
111
111
111
1000
1001
1010
101 1
124
125
126
1 1 1 1 100
1 1 1 1 101
1 1 1 1 1 10
6C
60
I
j
i
k
I
6F
m
n
•
72
73
p
q
r
s
6E
70
71
78
7A
78
19
7D
7E
7C
I
"
y
w
•
y
I
•
I
I
I LO 000 1 . As a function of lime, ii would appear as shown in Fig. 1 2.
ASCII does not include very many of the characters found for world's wriuen lan­
guages. By itself, ii is not even sufficient for the lang uages of Western Europe as it lacks
accented characters such as �. ii, and 6. This defect was partially addressed by adding one
more bi t to ASCII, creating an 8-bit extended ASCII hav i n g 256 characters. Unfortunately.
several versions of extended ASCil were promulgated by di fferenl manufacturers. and no
single standard exists under that name.
320
Digital Techniques in Mechanical Measurements
--
1
I
., - - r - ., - I
I
I
I
I
I
----·
High logic
I
Low logic
FIGURE 1 2: The ASCII logic sequence for lhe leller a.
Subseque!JI work did lead lo slandardized 8-bil characler selS, which are very com­
monly used in coding Web pages. For Weslern European languages, including English, 1he
mosl imponant of lhese is ISO 8859- 1 , lhe so-called Latin I character set (7). In lhe Lalin 1
encoding, the prinlable charac1ers below 12710 are lhe same as !hose in ASCII. Additional
standard encodings are have been defined for olher families of languages.
lnlernationalization of commerce and information exchange has grown dramatical! y
in recent decades, and so ii is desirable to have character encoding syslems lhal need nol
be changed whenever lhe language of application changes, or, more particularly, which
can support lhe use of several different languages wilhio a single document. Further, the
encoding system should ideally be lhe same for every piece of software. These needs have
driven the development of the Unicode standard (8), which aims to assign a unique numerical
code to every character in every language. When one considers that lens of thousands of
characters are found in some Asian languages, such as Han Chinese, ii becomes clear lhat
far more than 8 bilS are required. The early versions of Unicode ran to 32 bilS (facilitating
up to 65,535 characters), but thal barrier was removed in subsequenl versions. Unicode 4.0
assigns unique numbccs to more than 90,000 characters.
Serial and Parallel Transmission
The lnternalional Morse and 1he Baudot codes are necessarily of a serial type, i.e .. the
various on/off slates occur in sequence, one following another in "bucket-brigade" fashion.
If they were transferred by an electrical conductor, only a single lransmission circuil would
be required. In some cases, an alternative to serial lransmission, called parallel transnussion,
is used. This means that lhe bilS in a single word are transmiued simullaneously. In its
simplest form, this requires as many transmission circuits as there are code levels, but
with the obvious benefit of higher speed . Parallel circuitry is often used within a single
inslrument or device, whereas serial circuitry is used when long cables are involved, as, for
example, when connecting a computer to lhe internet through a telephone modem.
Parity and Error Detection
We have previously noted that digital signal transmission is less susceptible 10 trans­
mission errors lhan analog signal transmission. Nevertheless, errors still occur, albeit with
a low probability. Digital transmission systems are capable of sending many millions of
bits per second, so even with a low probablility of occurrence, errors w i l l arise often e1ough
that a means of detecting them is needed.
One approach to !his problem is the parity method of detection. In this scheme, an
extra bit is allached to each word of information. For example, a 7-bit ASCII code would
have an eighth pariry bit auached to it, so that the transmiued word would have 8 bits.
In an even parity method, the parity bit is selected so that the total number of ones in the
321
Digital Techniques in Mechanical Measurem1�nts
word is even. If I bit in the received word is incorrect, the number of ones would be
odd.
thus enabling the receiver to determine that a transmission error has occurred . In an odd
parity scheme, the total number of ones would be odd. The parity method clearly cannot
w:
detect errors in which 2 bits of one word are incorrect; howev1:r, in situations where ev
en
extremely improbable and is thus of li
single bit error is improbable, a 2-bit error will be
concern.
6.4
Bar Codes
Everyone is familiar with at least some of the applications of bar codes.
The familiar
and
black/white lines are arranged to codify data of various sorts-addresses, inventory,
virtually any other information that can be put into a binary representation. A low-intensity
laser may be reftected off of the bar pattern, with the signal received by an
appropriate
photodetector. The output is electronically processed and sent on to the cash register or
inventory tracking system.
A number of standards have been used to create
bar codes, the most common being
the Universal Product Code (UPC) and the European Articli• Numbering (EAN) system.
In a typical implementation, a blac k line corresponds to
I and a white line corresponds
Seven lines are used for each character; however, the fitrst and last lines arc always
opposite each other (black and white or white and black, respectively) in order that adjacent
to
0.
characters can be distinguished. The five remaining bits offor 32 possible combinations.
but not all
are used. Instead, only combinations yielding just t wo black bars and two white
bars are employed, where a bar's width may be one, two, th<ree, or four lines.
There are
20 such combinations. These combinations are assigned 1:0 the ten decimal digits from
0 to 9 in two different sequences, one "left handed" and the onher "right handed."
only
De pending upon the particular type of bar code, as many as 14 digits may be encoded.
In addition to the bars for the digits, the codes usually include guard bars at left, center,
and right. The numerical code itself is assigned to particular companies and products by a
standardization agency.
In addition to the basic one-dimensional bar code, various two-dimensional malrix
style codi ng systems are also in use. These systems can encodt: greater amounts of alphanu­
meric data. They are sometimes used on shipping labels from c ommercial delivery services.
7
7.1
SOME SIMPLE DIGITAL CIRCUITRY
Events Counter
Figure
13 shows the outward simplicity of one form of digital
events counter. 1\voconsec­
utive stages, or decades, are shown. Each decade consists of a seven-segment LED display,
IC chips, and a few current-limiting resistors. The 7490 chip is called a decade counter
14 and sends as output
parallel BCD pulses through pins d. c, b, and a, where d corresponds to the most signi ficant
bit and a lo the least significant. In addition, this chip provides an output pulse at the tnd
of the ninth input pulse. This pulse is available from pin 1 1 and can be used as the input
two
(see Table I ). It accepts input pulses in serial form through1 pin
�
to the next higher counting decade. We can see that additiomal decades may be
easi l y. As we will note later, this arrangement obviously also provides a "divide-by- IO
capability.
322
+S V dc
+S V dc
,
,
-
I I
/_a
I
I I
14
Til
I
7447 '
g
---
11
d
---
7-segment
display
d c
b
b
c
14
6
15
7e- - - -9
1 oc- - - 11
1 3b - - 12
1 8_ _ _ 13
c
d
6 12
b
t
qe
c
3
a
7
Count In
,
Parallel
r-binary
form
,.
I
d
8
4
BCD
to
7-segment
decoder
driver
10
Bd- - - -
TTL 7490
To next decade
GNO
19 1 1 2
b
a
TIL 7490
14
Decade
counter
5
+5 V dc
FIGURE 1 3 : Schematic circuit fo r a simple digital events counter.
Serial input
---
14
I 10
GND
Digital Techniques in Mechanica l Measurements
The BCD outputs from the 7490 chip are fed 10 a 7447 chip, cal led a BCD-to-seven.
segment decoder driver (Table 1). This chip accepts the parallel BCD input,
decodes the
� npul-i.e. , converts ii 10 a unit decim� outp�l-and ac1i:a1es th� appropriate
SC&me1Jls
m the seven-segment readout, thereby displaying the required decunal d igit. nus sim
le
circuit i s c apable of counting al rates of up 10 about 100 kHz. In addition 10 this
'
IC 7447 also prov ides control tenninals 3, 4, and 5, which may be used for
funcli::O
I.
reselling the readout lo zero;
2. blanking unused leading zeros-e .g., making prov ision so that a five-decade display
of the nu mber 25, for instance, would not show as 00025 but s imply as 25, with lhe
unnecessary zeros blanked;
3. making provisions for d ispl ay i ng appropriate dec i mal points .
Complete even t counter circ u i ts are avai l able as LSI ICs, u sual ly with addilional
features which we describe next.
7.2
Gating
Suppose the counter ci rc ui t shown in Fig. 13 is lo be used 10 count a sample of pulses
stemming from a continuing sequence of pulses. Can we not use a simple mechanical
switch to gate the input line? Probably not directly! When the contacts of the mechanical
switch (or re lay ) are closed, there exists a period of indecision. The contacts touch, break,
t ouch again, and so on, many times before a final and complete contact is establ ished . This
is the source of switch hash, w h ich sometimes shows on the screen of an osci lloscope. The
cou n ter is unable 10 distinguish between the "good guys" and the "bad guy s" and the speed
of the counter is su fficie nt to count them all ! Under such circumstances it is necessary
10 use debo unc i ng circuitry ( Sec ti on 4.2) in the i n put, which recognizes the first con1act,
latches, and then ignores the succeed i ng contac ts.
7 .3
Frequency Meter
Elec tro n i c sw i tch i ng or gati ng would probably be used to switch the counting circuit on
and off. Recall the AND gate [see Fi g . 3(a)] . The in pu t pulses may be conn ec ted to input
A and a con trol i npu t 10 8 . The output will be high on ly when both A and 8 are h igh . This
prov ides an essential part of a digital frequency meter.
Suppose we introduce an unknown freq uency al A and co ntro l 8 wi th a square-wave
osc i l lator (Fig. 14). W he n both inpu ts are high, the counter will count. Whe n 8 is low.
cou n t i ng will stop. Obviously the acc uracy of the count will be depe nden t on the accuracy
of the gated time interval. The oscil lator will probabl y be crystal co ntro l led , and crystals
osc i llate at relative ly high rates. For i ns tance , the fundamental oscillator might have a
freq uency of 5 MHz. The period of 0. 1 µ,s would certai nl y be 100 short for gating most
mechanic al inputs. Recall, however. that we already have discussed a divide-by- I 0 IC. the
IC 7490 used in the s im ple counter. By cascadi n g seven divide-by· IO ICs we can reduce
the fu ndamental period 10 2 s, I s of which will be high and I s low. By u s i ng an AND gate
as shown in Fig. 14, we can sample a second's worth of the i npu t. and the resulting count is
thus the frequency in hertz. Obv i ousl y, with the cascade of di v ide- by - IO ch ips we can use
panel-controlled switches for different gating times by simply lapping the cascade al other
points in the chain. Other features can be added as requi red .
324
Digita l Techniques in Mechanical Measurements
AND gate
Simple events
counter, e.g., as
shown in Fig. 1 3
5-MHz crystal·
controlled
oscilator
Cascade of seven
divide-by-1 0
I C chips
1'2 Hz
RGURE 14: Schematic diagram for a simple frequency meter circuit.
7.4
Wave Shaping
Digital logic circuits prefer instant toggling from low to high and back again. Suppose we
wish to feed the counter or frequency meter described previously with a sine wave or some
other varying waveform. If the signal level is sufficient. our counter may work; but. if the
peaks cannot easily be discriminated from the rest of the signal. the counter may miss or
double count them. We can convert the sine wave to a square wave signal, or an irregularly
varying wave into a sequence of high and low voltages. by using a Schmitt trigger (see
1TL 74 1 4 in Table l, for instance).
The Schmill trigger accepts a gradually rising input but stays at low-output voltage
until its "rising trigger level" is reached. It then switches to a high output voltage, and it
will remain there until the signal drops below its "falling trigger level."' These two trigger
levels arc made slightly different so that a signal that lingers near one of the trigger levels
will not cause multiple transitions. We are therefore able to reshape the in pu t into a series
of on/off pulses, much preferred by the common IC.
7,5
Integrated-Circuit Counter and Frequency Meter
The close relationship of event counters, frequency meters, and period or time-interval
meters allows all these functions to be incorporated on a single LSI chip. An example
popular in 1he 1990s was the lntersil ICM7226A, an eight-digit, multifunction, fri;quency
counter/timer chip. It incorporated a high-frequency oscillator. a decade time-base counter,
triggers, an eight-decade data counter, and seven-segment display decoder/drivers in a
single package. ll could function as any of the meters mentioned previously and could run
an eight-digit LED display. It measured periods from 0. 1 µs to l 0 s and frequencies from
de to 10 Mliz.
325
Digital Techniques in Mechanical Measurements
7.6
Multiplexing and Demultiplexing
Figure 15(a) illustrates an IC chip called a multiplexer, or a l -of- 1 6 data seleci« (
1TL 74 1 50, Table I ). In mechanical terms it may be considured a selectable co111111
quite simply, it is similar to a si ng l e - pole , 1 6-position switch [Fig. 1 5(b)) .
i n pu t (of up to 1 6) that is permitted to pass is determined by the binary number inserted 81
the d. c, b, and a ports, where d is the MSB and a is the LS B·.
�
The �
The demultiplexer, or l -of- 1 6 data distributor, is s imil air but with reversed action (see
1TL 74 1 54, Tabl e I ). It accepts a digital signal through its one i n pu t and then routes it lO
the particular output selected by the binary value at the control ports.
+5 V de
Ground
Output
-
Control ports ( t byte)
---
(8)
-o
� t � Output
�
�
---0
---0
--0
"'-·-·
'·-·'--, --Binary (yes/no)
control
---0
- - - -0
{b)
FIGURE 1 5 : (a) The 1TL 74 1 50 multipl exer or l -of- 1 6 sclc:ctor. Each of the 16 possible
logic combinations of the dcba control pons is employed to select a corresponding signal
input for connection to the output. (b) Illustration of a nearly •!quivalent mechanical switch­
ing arran geme nt . A difference , however. lies in the fact Ih a.I the mechanical device must
switch through t he sequence in order, whereas the multiplex•!r need not.
326
Digital Techniques in Mechanical Measurements
Ground
-15 v de
+S V dc
i,
:!?
;:!:
•
0
1
�
.5
...
...J
t.
�.. -
...J
'=- !!l
..
.. J5
� -
it!!
:; z
?U!
.0
"3
i!
E .,
=
111118rter
� ..
c
Ground
b
a
d
�
3l
J!l
::>
�
�
..__,.__,
Selector
bits
FIGURE 16: A multiplexer-<lemultiplexer circuit.
Why concern ourselves with multiplexers and demultiplexers? In many cases con­
tinuous monitoring or recording of a given data source is not necessary: Periodic sampling
will suffice. We can see that by sequencing the binary control through 0 to 15, we can
consecutively connect 16 different inputs to a given readout/recording/computing system.
This approach is economical in many cases.
Multiplexing may also be used, in certain instances, within the circuitry of a single
instrument. Recall the events counter we discussed in Section 7. 1 . Each decade required
a seven-segment decoder driver plus seven current-limiting resistors. More sophisticated
counter circuitry uses a multiplexer-<lemultiplexer combination arranged so that each read­
out element is sequentially connecled lo a single driver-resistor combination. By time
sharing in lhis manner we can reduce the number of circuit elements and realize quite
a saving in cosl. The sequencing rate is sufficiently high thal the readout appears 10 be
illuminated conlinuously.
Data processing may take place at quite some distance from the data source, as
in a large industrial complex such as a refinery or power planl. By using mulliplexer­
demulliplexercombinations, we may also use single, rather than separate, circuits toconnccl
the lwo positions (Fig. 1 6). We hasten lo add that this may not be precisely the case because
it would also be necessary lo synchronize our binary control circuits at the two locations.
A simple solution is to run four additional wires connecting ports d, c, b, and a. Thus we
would have 5 circuits instead of 16 (for the particular combination cited).
There is, however, another possibility lhrough use of more sophisticated !Cs. The
universal asynchronous receiver/trans mi Iler (UART) contains what amounts lo a multiplexer
Digital Techniques in Mechanical Measurements
and a demultiplexer-actually a set of two each-on a single chip. It is capable of con
verting
serial to parallel or parallel to serial data. The word asynchronous indicates that the openw
need not be sy nchronized Actual ly, this is not entirely true. What is meant is that
i nputs and outputs need not be perfectly synchronized: Approximate synchronization sa
±5%, is sufficient. This permits the use of separate control oscillators, or cloc
the transmitting and receiving ends, whose frequencies are very close but not Reeessarily
exactly the same. In this case, directly connected synchronizing circuits between the two
locations are not necessary.
.
"
8
:
b,M �
THE COMPUTER AS A MEASUREMENT SYSTEM
Computers have become u biqu i tous laboratory tools. They allow measurements to be
recorded, processed, displayed, and printed with great ease. In some si tuations, sensors
and transdu rs are connected directly to computers, which then take over the role of any
additional instruments that might otherwise be required-voltmeter, oscilloscope, spectrum
analyzer, and so on. Powerful soft ware packages are available to perform all such functions
o n the computer, and they can transfer measurements to spreadsheet packages or other codes
for data reduction and presentation. In the next several sections, we look at some of the
details that underlie this convenience.
ce
Computers are, of course, digital systems. The frequency meter mentioned in Sec­
tion 7 .5 is an example of how digital instrumentation can be built from integrated circuits.
However, the data received by that meter are esse n t ia ll y digital in character: All that is
req u i red is to determine whether the input voltages are high or low. More often, we may
need to deal with transducer s ig nals that are entirely analog, as when a thermocouple voltage
is used to measu re temperature. In this case, we must convert the voltage to digital fonn if
a digital device such as a computer is to process it. This trans formation is accomplished by
an analog-to-digital converter or ADC (Section 1 1 ). A computer system used for recording
analog s igna ls w i ll incorporate an ADC, either as an integ ral part of the computer or within
the transd ucer that is connected to the computer. In every case, the sample rate and voltage
range of an ADC must be chosen to avoid aliasing and other forms of signal distortion
(Section 1 1 .3).
Data from the ADC will be transferred to the computer's storage or central processing
units a long one of the computer's
w hi h are internal circuits for passing data and
con trol signals. For most computer users, only the external bus (Section I 0.4) will be of
concern. The external bus governs the operati on of the com pu ter s input and output pons,
which connnect printers, remotely operated instru m n ts or controls, external data storage
devices, modems, and other peri pherals Buses may use either serial or parallel transmission
of data. Common types of buses include the IEEE 1 394 (Fire Wire), the RS-232C, the IEEE
488 (GPIB), and the Universal Serial Bus or USB .
buses,
.
c
e
'
At the heart of most current computers is a V L S I in tegrated circuit called a micro­
processor (Section 9). The mic roprocessor is a collection of digital circuits that fonn the
central processing unit, or CPU, of the com puter. The m icroprocesso r executes the instruc­
tions ofcomputer programs and controls the transfer of information between the input ports,
memory, and the output ports. Computers using mi croprocessor CPUs are someti mes called
microcomputers, even though today's m icrocomputers arc so powerful that the tenn micro
may be a bit misleading! We describe the internal architecture of t he microcomputer in
Section 10.
328
Digital Techniques in Mechanical Measurements
THE MICROPROCESSOR
A microcomputer is built is around a microprocessor chip that acts as its central processing
unit (CPU). The CPU serves as the control center for directing the flow of digitized infor­
mation. It is more than a traffic controller because it not only provides the organizational
plan for the flow of elemental bits of information but also assigns the pathways, tempo­
rary "parking" spaces, and stop/go gating and can perform a limited manipulation of the
traffic. It accepts inputs in digital form, either data or command instructions. and rou tes
them to predetermined (programmed ) destinations over buses (path ways) to displays, mem ­
ories, controllable devices, and so on. Sources of the data or commands may he e x terna l
memories, keyboards, transducers, or other devices.
A wide range of CPUs are available, with many levels of complexity--4-bit. 832-bit, 64- bit, and so on. The CPU selected for a given purpose will de pend
on the application. For some simple low-end dedicated uses a 4-bit or 8-hit CPU may
suffice, whereas 32-bit or 64-bit CPUs may be selected for general-purpose or h igh -end
microcomputers. For purposes of illustration, the 8- bit Motorola 6800 microprocessor
serves very well. This chip is obsolete for most purposes, but the general
principles of its
·
design are characteristic of most CPUs.
bit. 16-bit,
Figure 17 is a highly simplified schematic diagram of the external connections and
some of the internal features of the Motorola 6800. The diagram shows the primary buses
into and out of the processor plus some essential internal devices. To be functionally useful,
the system requires additional s u pporting circuitry external to the CPU, including interfacing
devices sometimes referred to as buffers, input/output (UO) facilities, synchronizing clocks,
memory, and so on.
Status
AO
Zl
!
,,
�
Index register
�
U)
Program counter
Stack
pointer
A12
A15
.5
..
ACC-A
_,.
A8
DO
ACC-B
A4
�
+5 V dc
Ground
Arithmetic logic un�
(ALU)
04
!I
"'
0
07
Clock 1
Clock 2
Reset
NM interrupt
Halt
Interrupt requesl
3-stale conlrOI
Dara bus enable
Bus available
Valid memory address
Read/write
Ground
AG URE 1 7 :. Simplified schemalic diagram of the Motorola 6800 microprocessor.
329
Digital Techniques in Mechanical Measurements
The Motorola 6800 microprocessor is a single integratc:d circuit chi housed .
p
dual-in- ine package (DIP). The dimensions, exclusive o the pins that provide e
connections, are about 6 x 20 x 50 mm . The power requ1Temf:nt for this CPU by i
lSclr is
0.6 to 1 .2 W at 5 V de-less than I % of the wattage of a large CPU today.
�
�
�
The figure shows the following:
l. An address bus consisting of 16 parallel lines for accesi;ing (connecting to) 2 •6 =
FFFF16 = 65,53610 different memory locations. The actual number available is, of
course, dependent on what may be provided by supportin1g hardware.
2. A dara bus consisting of eight bidirectional lines for sim ultaneously handling 8 bils
( I byte) of data. This bus is a two-way street, and the direction of How is controlled
by gating (Section 7.2). Eight bits provide for 28 = FF16 = 25610 combinations.
3. Various control and decode lines. Bus synchronization, cl osely akin to multipaiag­
demulliplexing (Section 7 .6), is controlled through use of two external clock signals.
Additional lines are shown, and in many cases their titles provide a c.lue to their uses.
Figure 17 also shows some of the internal structure of the 6800 chip. Various regis­
ters are used . These may be considered as temporary storage bins or momentary ''parking"
locations for data, instructions, or addresses as they are being shunted from one location to
another. Essentially, all data and instructions must pass through at least one of the accu­
mulators, A or
8, each having a I -byte (8-bit) handling capabil ity. Primarily for providing
index register (I R) , program counter (PC),
and stack pointer (SP) are added.
manipulation of 2-byte ( 1 6-bit) addresses, the
As illustrated in Fig. 1 7 , the stack consists ofthe registers as shown. Their contenls
are stored in contiguous addresses, and the purpose of the sta1ck pointer is lo keep lrack
(RAM).
co11nter controls the sequencing of any program :1teps, including the starting
point, and the index register, in addition to other functions, provides a channel through
which 2-byte addresses may be handled in a program. The contents of the registm are
of where the stack information is stored in the external random access memory
The prog ram
always available on command.
The arithmetic logic unit (ALU) has a relatively limited capability of manipulating
numbers. It can add one byte to another or determine their diifference, or it can perform
several logic functions, such as AND, OR, and EOR (Section 4. 1 ). To handle dala in
magnitudes requiring several bytes, the add and subtract functions of the ALU may involve
carryovers (for addition) or borrows (for subtraction). It is the responsibility of the status
register (SR) (also called the condition code register) to monitor such requirements for
possible further program use. Flags (primary data bits) in the status register are either set
(made equal to I ) or not set (made equal to 0), depending on predetennined conditions. Ir an
addition results in a carry over from one byte lo the next byte, a. bit momentarily indicating
that fact, stored in the status register, will
be added to the byte of the next higher order. Or,
if the operation results in a zero, a zero flag in the SR may be used to trigger the decision to
branch (or not to branch) to some other point in a program. These are only two of a number
of powerful functions of the status register.
For a more detailed discussion of microprocessors and for information on current
designs, refer to the Suggested Readings at the end of lhe chap ter.
330
Digital Techniques in Mechanical Measurements
COMPUTER
CPU
1--�����.-���.--��������.--... Contra bus
1----.,...�
...+
��+-��.-+-���-+-���.-+-... Add�ss bus
1-..-<1-+�-.--+-+-�..,...+-+-�....
-+
.-+-��....-+-+-... Dam bus
.
1.----'
Bidirectional
l/O polls
Monitor
Mouse
Printer
External
buses
Networ1<
Disk
Drives
FIGURE 18: Typical microcomputer system.
THE MICROCOMPUTER
r.l ,.
As noted previously, a microprocessor must be surrounded by a number of servants before
it can claim to be a microcomputer. Figure 18 is a schematic diagram of a microcomputer
system showing some of the essential components. The various peripheral devices com­
municate with the CPU through the three microprocessor buses mentioned in the previous
section.
10.1
Read-Only Memory (ROM)
· . .l'!>. ··:
. �·,
, ri :
ROM is used lo store programs that direct the computer's operation at start-up and that guide
basic video and keyboard communications. Collectively, these instructions are usually
referred to as the system BIOS. or basic input-output system. When lhe computer is turned
on, the BIOS will direct the computer to read and load the operating system software from
the c omputer's hard disk drive. Data stored in ROM are usually semipennanenl, in the
sense that they are preserved when the compuler is powered off but can be updated by the
user from time to li me .
10.2
Random Access Memory (RAM)
RAM is the "bank" that is used for te mporary deposits and withdrawals of data or infor­
mation required for the operation of programs. RAM may be both read and written, as
program instructions are loaded, sent to the CPU, or replaced by other program instruc­
tions. All addresses in RAM may be accessed in any order-thus the term random. Random
access memory is usually volatile-if the power is turned off or if the system is rebooted,
all stored data are lost.
Data can usually be accessed much faster when they are in RAM than when they are
on a nonvolatile mass storage device such as a hard disk drive or CD-ROM. Thus, computers
Digital Techniques in Mechanical Measurements
will hold frequently accessed programs or data i n RAM so as to avoid havi ng 10 rec
them from a mass storage device each time they are required. Hence, having a large
.:::1:
of RAM on a computer usually improves its performance.
1 0.3
Input and Output Ports (110 Ports)
Input aild output ports are the portals through which the microprocessor exchanges data
other devices. An
output pon
input port is essentially an address from which data are read, whereas 30
with
is an address to which data are sent. Examples of devices that connect to inp
ut
ports include keyboards and mice. Examples of devices that connect to output ports include
video monitors and printers. Some ports are bidirectional-<:apable of both sendi ng
receiving data-including those associated with hard disk drives, network interfaces
,
computer-controlled instruments.
1 0.4
and
and
External Buses
The computer's ports may connect to a wide variety ofdevices, and these connections in tum
have a wide variety of possible fonnats and speeds for data transfer. The spec i fic interface
arrangement is determined by the type of external bus or communications
which
the port reaches a particular device.
Some common types
Either serial or parallel data transfer (Section
6.3)
are
interface through
listed in Table 6.
may be used to connect measuring
instruments or sensors to a computer. The speed of a serial connection is measured in bits of
data transferred per second. The speed of a parallel connection is measured in bytes trans­
ferred per second; parallel connections
are
usually 8 or
1 6 bits wide;
that i s , I or 2 bytes.
Parallel cables generally have a large number of w ires , ra ngi n g from 24 to 68 i n most cases.
Network and wireless connections
are also
used by some measuring devices
(data transfer
on these buses is serial). An instrument with a network interface has a network address that
computers and devices on the network may use to obtain readings or send instructions.
Like most other areas of digital technology, communications arrangements are con­
stantly evolving and the available speed of data transfer is constantly increasing. Recentl y,
the IEEE 1 394 and USB 2 buses have become the most common serial standards for con·
necting devices to computers. Older serial buses, based on RS-232C or USB I, are being
phased out quickly. Parallel connections were standard
but they
are
also being replaced by USB 2. In
the
for
printers until the late 1 990s,
past, m uc h effort might
be required to
establish proper connections of devices to computers, possibly involving a great deal of time
soldering wires to RS-232 connectors and trial-and-error adjustment of software settings.
com m un i cat io n over bid irection
are better standardized and better inte­
A particular problem was the synchronization of two-way
lines, a process called
handshaking.
Today's buses
grated with computer operating systems, so that establishing two-way communication is
usually as simple as plugging in a connector.
11
ANALOG-TO-DIGITAL AND DIGITAL-TO-ANALOG CONVERSION
As we discussed in Section 3, some measurands originate i n digital form. Most mechan­
ical inputs, however, exist in analog form. Hence, before d i g i tal data processing can
be
accomplished, an analog-to-digital conversion is necessary. In a like manner, if a com­
puter's digital output is used to drive an analog device, a digital - to - an alog conversion must
be performed.
332
Standard
Maximum
Speed •
Serial Interlaces
RS-232C
RS-422A
US B I
- I O kbps
I O Mbps
12 Mbps
IEEE 1 394
400 Mbps
US B 2
480 Mbps
Parallel Interlaces
Ce ntronics
IEEE 488
IEEE 1 284
SCSI
1 50 kBps
I
M B ps
2 M Bps
320 MBps
TABLE 6: Exiemal Buses.:: '
::
-. ·'.'"· �··" .;.:-
' .
•( '·;�·
.\r� ;.r�
Comments
Cables limited to 15 m. 9- or 25- pi n connectors. Was once vecy widely used.
Up to 1 200 m cables at lower speeds.
Universal Serial Bus. Also has 1 .5 Mbps speed. Up to 1 27 devices may be daisy-chained
on a single port. Four-pin connector. Supcrceded by USB 2.
Trade-named FireWire. Commonly used for digital video. Up to 63 devices may be
daisy-chained on a single port. S ix-pin connector.
Fast enough for digital video.
Orignal PC parallel port connector. Output only. S peed often much sl ower than maxi­
mum. Up to I 0 m cable. 8 bit 25- and 36-pin coMectors. Obsolete.
Developed by Hewlen Packard Corp. in 1960s as the HP-IB bus. Up to 14 instruments
controlled b y one dev ice . Up to 20 m cable length.
8 bit. 24-pin coMectors. Also cal led
General Purpose Interface Bus, or GPIB.
Bidirectional successor to Centronics bus. Released in 1 994. Up to 10 m cable . 8 bit .
25- and 36-pin connectors.
Small Computer System Interface. Many different vers ions exist, including 8 and 16 bit.
50- and 68-pin connectors. Up to 12 m cable. Up to 1 6 devices. Commonly used as hard
disk drive bus.
Network and Wireless Interlaces
Dial-up modem
ADSL madem
Bluetooth
IEEE 802-1 l b
IEEE 802 . 1 l g
IEEE 802 - 1 l a
IEEE 802-3
56 kbps
varies
3 Mbps
I I Mbps
54 Mbps
S4 Mbps
IO Gbps
Early modems were just 300 bps!
Maximum speeds of several Mbps; actual speeds often a few hundred kbps.
Wireless connection in 2.4-GHz band. 10 m range .
Wireless etbemet in the 2.4-GHz band.
Wireless ethemet in the 2.4-GHz band.
Wireless ethemet in the 5 -GHz band.
Ethernet. 10-Mbps and 100-Mbps networks are common. Higher-speed networks some-
•l.�:�be=::�����--�
�-u�se�op�u_·c�
�������������������u-·mes
•bps = bits per second; Bps = by tes per second.
Digital Techniques in Mechanical MeasuremenU
Analog-to-digital (AID) and digital-to-analog (DIA) conversions can be el!Cellled
using a variety of different circuits. AID and DIA converters aire often. manufactwtc1 as
integrated-circuit chips for incorporation into other devices, but they may also be obtailled
as cards that plug into computers or as stand-alone data recording systems. The basic
measures of perfonnance are the number of bits used and the spc::ed of processing. A wide
range of designs exist for both AID and DIA converters. Here we describe only typical
examples of each, leaving further development to the Suggested Readings.
11.1
A Digital-to-Analog Converter
A simple DIA converter is based on the summing amplifier. Referring to Fig. 19, we see
that the currents summed are controlled by a set of digital switches. Four basic elements
are involved:
= I V de).
A ladder arrangement of summing resistors. For this 8-bit converter, eight ladder
resistors are used. The resistance values increase in a sc:quence of powers of 2
from 2° R to 27 R.
3. A series of switches. These are not mechanical switches; ra.ther, they are solid-state
gates [e.g., simple AND !Cs (Section 4. 1 )). The eight switches can be' activaled by
digital inpu ts, so their operation may be controlled by the respective bits contained
in a single byte of data.
4. Op-amp output circuitry. The op-amp output voltage is equal to - Ra times the sum
of the currents from each ladder branch. The gain-controlling resistor, Ra, is selected
to scale the output voltage, perhaps to a maximum magnitude of 10 V de.
1. A stable reference voltage (for instance, Erer
2.
When a switch is closed, a current is delivered to the op amp in proportion to the
power of 2 for that circuit's resistor: Switch 0 contributes io = Erer/ 1 28 R = r1 Erer/ R,
switch I contributes ;1 = E,.r/64R = 2- 6 E,.r/ R, and so on. Hence, switch 0 corresponds
Reference
voltage,
E,.,
Ra
7
1R
6
2R
2
32R
l
Solid-state
switches
(gates)
64R
AGURE 1 9: A simple 8-bit DAC (digital-to-analog converter).
334
Output
Digital Techniques in Mechanical Measurements
to the least significant bit, bo ; switch 7 corresponds 10 the most significant bit, b-,; and
so on. By closing selected switches, the output voltage can be made proportional to any
particular 8-bit number. For example, if switches 0, 5 , and 6 are closed, the output voltage is
proportional 10 0 1 10 000 1 2 (= 6 1 16 = 97 10). Since Ra sets the constant of proportionality,
the output gain can be scaled lo provide an appropriate range of analog values. Additional
output circuitry could be used to invert the signal or to add offset voltages.
An Analog-to-Digital Converter
One typical AID converter is the parallel encoder (Fig. 20). This circuit uses a set of voltage
comparators and a series of resistors lo compare simultaneously an analog input signal to a
£..,
R
Yollage
comparators
,---A---..
Binary
encoder
,---A---..
7
R
6
R
5
R
4
R
3
R
b,
�
�
R
2
R
FIGURE 20: A 3-bil parallel AID converter.
3-bil
binary
output
Digital Techniques in Mecnanical Measuremenu
Output blls
/
111
110
/
/
I
I
I
I
+-
Sab.nated
1 01
1 00
01 1
010
001
/
,
ooo �'�...___..__���������'--�.L-���
�E,,,
0
i Em
� Em
Analog Input voltage, E.n.1og
FIGURE 2 1 : B i n ary output versus analog input voltage for the 3-bil parallel ND convener.
c
set of refe ren e vollages. The basic elements are as follows.
l. A stable reference voltage (such as Erer = I V de).
2. A series of equal resistors. These resi stors fonn a voltage-dividing sequence between
Erer and ground. For this l-bit convener, 23 = 8 resistors are used. The voltages
at the nodes between resistors increase in i ncreme nts of (R/8R) · E,.r, specifically,
( j ) E,.r, ( j) E,.r, . . . , <l>E..r.
J. A se t of voltage comparators. The analog voltage is simultaneously compared lo each
node's voltage. A comparator's output voltage is high (on) when
given reference voltage and low (off) when it is below. Seven (23
Eanaros is above a
-
I ) comparators
are needed for the 3-bit conmter.
4. An encoder circuit. The encoder reads the comparator outputs (each a high or low
vol tage) and produces a 3-bil binary output corres pon di ng 10 one of the eight poss ible
on/off conditions of inputs I through 7. (Note that one condition is to have all seven
comparators off.)
For exam ple, if the input signal is tween <i>Erer and (a) E,.1, comparators 1-3
read high and 4-7 read low. The input stale is 3, corresponding 10 a binary output of
all comparators read high and t he output is 1 1 1 2 ; and
01 1 2( =3 10). If Eana108 >
all are low and the output is OOQi . The d i gita l output is shown as a
if Eona1og < (
be
<l>E..i,
t) E,.r,
fu nc ti on of Eana1og in Fig. 2 1 .
336
Digital Techniques in Mechanical Measurements
Parallel AID conversion is particularly fast, since all bits are set simultaneously, for
which reason it is sometimes called flash encoding. Other AID conveners may use a
successive approximation technique that sets one bit al a time. Yet another type of AID
converter uses a voltage-controlled oscillator (VCO) to conven voltage to frequency and a
digital frequency meter to digitize the resulting frequency.
If the output of an AID converter is sent to a binary-to-BCD encoder and from there to
a BCD-to-seven-segment decoder/driver, and then to an LED display, the result is a digital
voltmeter. Integrated-circuit technology allows all these functions, and others, lo be placed
on a single (and inexpensive) chip, which might form the heart of a handheld instrument.
11.3
Analog-to-Digital Conversion Considerations
High-quality analog-to-digital conveners are readily available to the experimentalist, so it
may be tempting treat them as black boxes. They cannot, howevet, be used carelessly if
accurate results are to be obtained. In the following, we outline some of the most important
considerations for the end user of an ADC.
Saturation Error
The most obvious limitation of an AID converter is that it has definite upper and lower
limits of voltage response. 'JYpical full-scale input ranges are 0 to I 0 V and - I 0 to +I 0 V.
If the input signal exceeds the upper or lower limits of response, the converter saturate.�
and the recorded signal does not vary with the input. This situation can be prevented by
appropriate signal conditioning, such as amplitude attenuation or de offset removal.
Resolution and Quantization Error
As we have seen, an AID convenet responds to discrete changes in the input voltage.
For example, the 3-bit parallel encoder's output steps correspond to changes in £analog of
£,.c/2l (Fig. 2 1 ). Thus there is a smallest increment of voltage change that can be resolved
by an AID convener. In general, the voltage resolution per bit, E v , depends on the full-scale
voltage range and the number of bits of the convener:
rs
Ev = 2n
6V
( I)
where
6Vc5 = the full-scale voltage range, and
n
=
the number of bits of the AID convener
Typical AID conveners have 8, 1 2, or 16 bits, corresponding to division of 6Vrs into a total
of 28 = 25 6, 2 12 = 4096, and 2 1 6 = 65,536 increments. A 1 6-bit convener with a - 1 0 lo
+ 1 0 V range has a voltage resolution of 0.3 mV. The voltage resolution is a known value
for a given ND convetter, and i t i s needed to process the digitized data.
The finite resolution of the AID converter introduces error in the recorded values,
since the actual analog voltage usually lies between the available bit levels. This is
called quantization error (since the bit levels arc "quanti1.ed"), and it is en1irely analo­
gous to the reading error o f a digital display. An estimate for the quantization uncertainty
337
Digital Techniques in Mechanical Measurements
= &v /2 (95% ). Quantization errors may be reduced by usi1ng an AID convener
is
with more bits.
uq
Conversion Errors
AID converters may also suffer from slight nonlinearity, zem-offset errors, scale
errors, or hysteresis. Such errors are a direct by-product of the particular method of input
quantization. For example, the conversion illustrated in Fig. 2 1 is, on the average, low by
one-half of the least significant bit (the amount by which the solid c1urve lies beneath the
dashed line). Normally, the manufacturer will provide specifications for the potential size
of such conversion errors.
Sample Rate
The rate at which an AID converter records successive values of a time-varying input
is called the sample rate. Each AID converter has a maximum posi;ible sample rate, of
which typical values range from about 1000 Hz to more than 100 MHz. Software often
allows the user to specify any sample rate up to this maximum value. Miasing andfrequency
resolution are of particular concern.
Signal Conditioning for AID Conversion
To make the best use of an AID converter, conditioning of the analog signal is often
required. The most important considerations are the prevention of aliasing, the minimization
of quantization errors, and the prevention of saturation errors. Aliasi ng can be prevented
by using a low-pass, or a11tialiasing, filter to remove frequencies of /.,1I11pl e/2 or more from
the analog signal. Quantization error can be minimized by amplifying the signal to span as
much of the full-scale range as possible. However, this approach som•:times conflicts with
the need to avoid saturation errors.
As an example, a hot-wire anemometer signal may include both a 3-V de component
and important ac components ofonly 5 mV rms. Suppose that a ± I 0-V, 1 6-bitA/D converter
is to be used for sampling. The ac components are near the AID converter's resolution
(20 V/2t6 = 0.3 mV), but large amplification of the sum of the ac and de components
will saturate the convener. To resolve the ac components accurately, a:n offset-and-gain (or
"buck-and-gain") amplifier may be used. This amplifier subtracts (or "bucks") a precisely
specified de voltage from the analog signal and amplifies the remaining ac signal by a gain
of several hundred. The de offset voltage is recorded with one AID converter channel and
the amplified ac component is recorded with another.
In this way, both components can be
'
digiti1.ed accurately.
1 1 .4
Dlgital Signal Processing
Digital techniques make it possible to filter or enhance analog signals such as those carry ing
sound or images in telecommunications systems, audio systems, and video systems. A
digital signal processor, or DSP, is an integrated circuit that perfom1s functions such as
filtering using digital algorithms. Upon combining an analog-to-digital converter, an digital
signal processor, and a digital-to-analog convener, one obtains a circuit that can , say, filter
noise out of a telephone signal while the users are speaking.
338
Digital Techniques in Mechanical Measurements
Analog input
signal
t
Analog
�
Anti-aliasing
filter
Digital
signals
+
Recoostruction
filter
Analog
signals
�
Analog OUlpUt
signal
FIGURE 22: A digilal signal processing system.
How does this work'! Digitized data can be processed numerically, for example, by
using the Fast Fourier Transform (l'FT). The FFT allows identification of the frequencies
present in a signal. and with that information it becomes possible to filter the sign al digi­
tally: Numerical algorithms can
used to eliminate unwanted frequenc ies or to enhance
freque ncies of interest (9) . With sufficient computational power, these di gita l filteri ng tech ­
niques can prOduce sharp frequency cutoffs that would be very difficult t o obtain from the
analog filters. Funhennore, digital filters are not subject to drift when the temperature
shifts, as a nalog filters are, and they may be programmed to achieve different effects as
required without changing any components in the circuit.
be
Figure 22 shows a schematic arrangement for a digital sign al processing system,
all of which might reside on a single IC chip. An an alog signal is passed through an
antialiasing filter to an AID convener. The DSP then processes the signal and sends the
output to a D/A converter. The reconstructed analog signal is smoothed by a final low pass
(or reconstruction) filter as i t leaves the chip. The DSP itself may be controlled by through
an external 1/0 interface. !'or an audio DSP, this might enable varying levels of artificial
reverberation to be added to the original signal, so as to simulate a concert hall sound for
part of a music track.
A DSP is essentially a specialized microprocessor, designed for high-speed numerical
calculations. Like other microprocessors, as DSPs have become cheaper, smaller, and
more powerful, their applications have multiplied. They are now found in cell phones,
video recorders, CD players, computer sound cards, and digital television sets. DSP is
used in medical image processing, as for magnetic resonance imagi n g (MRI) and computed
tomography (CT) scanning. DSP is essential to modem telecommunications sytems, where
it provides echo reduction, signal compression, and sig nal multiplexing. DSP has found
applic ation in speec h recognition systems, radar, sonar, sei smology, h igh-speed modems,
MP3 players, and many other areas.
Digital Techniques in Mechanical Measuremenu
A particularly �levant aspec t of digital signal p�essing is that it has come
10 do111.
.
_
mate many areas o f signal processing lhat were trad11lonally done with analog tec
h .
especially filtering. This trend is likely to continue, and the results should continu 1
remarkable.
: ��
12
DIGITAL IMAGES
i�ital i �ages have become u quitous �wing to digital cam �ras,
��anal computers, and
_
digital video recorders. In add111on to their consum.er apphcauon, d 1 g1 tal im ages and digi
lal
image processing are also powerful tools for data collection and analysis. In this section. we
will briefly describe some basic aspects of lhis vast and important subj ect. More detailed
information may be found in the Suggested Readings for this chapter.
�
��
An image, be it a camera photograph or an x-ray lilm, is digitized by dividing it into
a rectangular grid of cells, called pixels. I n general, the pixel sizes must be small relative
to the features in lhe image if the digital representation is to be accurate. When it is noc,
the image takes on a jagged appearance that is sometimes referred to as pixehition. The
number of pixels required to represent an image thus depends on both lhe physical size or
the image and the amount of fine detail to be recorded. For an image digitized to a grid or
640 pixels by 480 pixels, a total of 640 x 480 = 307,200 pixels are required. Ir 1 600 by
1 200 pixels are used, the image has 19.2 million pixels !
Each pixel carries information about the intensity and color or the image at that poinL
For a purely black-and-white image, only one bit is needed per pixel: 0 = black and I =
white, say. For a grayscale image, 8 bits (or I byte) might be used to carry the in tensity,
allowing 2 8 = 25 6 possible shades of gray to be applied to a single pixel. For color images.
a common approach is to use three colors per pixel : red, green, and blue, or RGB. Each
color has a different intensity in each pixel. If 8 bits are assigned to each or the three colors
(for a total of 24 bits per pixel), then each pixel has 2 8 x 28 x 2 8 = 1 6.8 million possible
mixtures of color.
The total number of bits needed per image can thus be quite large. A black-and-white
image at 640 by 480 pixels needs only 307,000 bits (or 38,400 bytes). The same image
in a 24-bit color format would be 24 x 640 x 480 = 7.4 million bits (or 92 1 ,600 byteS).
and a 24-bit color image at 1 600 by 1 200 pixels would require a total of 46 million bi1s (or
about 5.8 million bytes). For comparison, the full text James Joyce's novel U/ysus can be
stored in ASCII format in about 12 million bits (or 1 .5 million bytes) [ IO). Perhaps this
gives added meaning to the saying "A picture is worth 1 000 words."
A digital video is simply a succession of digital images, or frames, shown al high
speed . A typical framing rale for normal viewing is 30 fram es/second . For purposes
of measurement, much higher framing rates may be employed, ranging up to 1 00.000
frameslsecond or more, as when it is desired to record processes happe ning on m illisecond
el
or microsecond timescales. At 30 frameslsecond and a video image size of 640 by 480 pix
22
in 24-bit color, the rate of data transfer for recording or display is 30 x 640 x 480 x 24 =
�
1
million bitslsecond. Comparison to Table 6 shows that connection of a v ideo camera 10
. CD
. ·
We may compare video data rates to those for high-quality audio, as on a music
s
fn:que
contain
The CD must reproduce the full spectrum of human hearing, which
up to 20 kHz, so it is sampled at 44. 1 kHz (giving a Nyquist frequency of 22 kHz). haS
1
re
sample records the volume in a 1 6-bit format. Two-channel audio (stereo) merefo
computer will require either a USB 2 or IEEE 1 394 bus.
340
=
.
Digital Techniques in Mechanical Measurements
data rate of 1 .4 million bits/second. Interfacing audio signal to computers is clearly much
lesS demanding of speed than is video. Similarly, audio files usually require less storage
space than video files.
Once an image has been digitized, computer software may be used lo manipulate it in
various ways, perhaps to enhance dim features or to smooth out pixelation. One of the most
common digital processes applied to images is inuige compression. Because much of data
i n an image will be similar in adjacent pixels, it is possible to use numerical algorithms to
reduce the amount of information that must be stored. A very common compression method
for single images is JPEG compression; for video streams, MPEG compression is a typical
format. ' Many other compression algorithms are in use. With appropriate compression,
digital image or video files can be reduced in size by an order of magnitude with little
apparent loss of clarity.
Digital techniques may also be used to identify features and to extract measurements
from images. The possibilities are almost limitless. To name just two, digital video may
be used to record the motion of particles suspended in a liquid, and software may be used
to locate the particles in the image and to compute the velocity field from the sequence of
particle positions. A second example is infrared temperature measurement. An appropriate
detector array can be used to photograph the infrared radiation from a surface, and soft­
ware can then be used to calculate the temperature at the location of each pixel giving the
temperature distribution on the surface.
Digital images can be produced by nonphotographic methods, such as computer
graphics programs, but for photographic recording the essential detecting element is often
a CCD array. The CCD array, or charge-coupled diode array. is a semiconductor device. It
consists of an array of photodetecting cells that generate charge in response to photons they
absorb [ 1 1 ]. A CCD array is situated behind the focusi ng lenses of a camera (in the place
where one would have once found film). After the camera shutter is. opened and closed,
the amount of charge accumulated is proportional to the intensity of l ight received. The
charge is converter to voltage using a charge amplifier, and the voltages for each pixel in
the array are digitized by an ND converter and recorded. The digital number associated
with each pixel (or numbers, for color photography) define the intensity for th at cell of the
digital image.
13
GEITING IT ALL TOGETHER
The possibilities for pulling together a sophisticated d igi tal measurements system for a given
project are entirely open ended. The degree of completeness is usually limited by resources.
In many situations the de si gn and assembly or a complex system may be more costly in time
and money than the project warrants. On the other hand. when great masses of data are to be
collcctcd, requiring extensive computational lime lo digest, or when con1inuous monitoring
and control is needed, funds and lime expended in putting together an "automatic" system
may very well be cost-conservative.
Figure 23 shows som.: of the tremendous possibilities for advanced data gathering
and processing. It should be noted that the transducers and ND subsystem may be partially
or entirely incorporated into some type or digital measuring instrument which connects
1 JPEG is an acronym for Joint Pbo1ographic F.xpcn:s Group. MPEG is an acronym for Moving Piclurc Experts
Gtoup. These groups work in coordination with the International Organization for Standardization (ISO) lo
develop inremational standards in lheir respective areas.
341
Digital Techniques in Mechanical Measurements
Aeal·time process
Contact douures
Keyboard
Display
Mouse
Printout
FIGURE 23:
Hard disk
drive
To other
computer systems
CAT
Printer,
plolter
A block diagram i l lustrati ng the potenti al o.f an integrated measure­
ment/control system.
lo the computer
Chapters.
14
via
;an
appropriate external bus. We
consider such
instruments further
SUMMARY
In bringing this chapter to a close, we reiterate that it is presumpnuous to attempt to summa­
rize digital techniques in so few pages. We hope, however, that the material presented w il l
serve as an introduction to further study. There is no doubt as tc• the value of the topic as it
relates to mechanical measurements, and further developments are c ontinually increasing
this importance.
I. A digital signal has only two valu�n or off, 0 or I, yes or no, black or white, low
voltage or hig h . Digital information may be composed of sets of bits , each of which
takes one of the two possible values (Section I ).
2. Digital instrumentation has the ad vantage of direc t computer interfacing, i nherent
noise resistance, low system voltages, and d irect numerical display of readings (Sec­
tion 2). Analog signals must be convened to an appropriate digital form to be pro­
cessed by a digital measu ring system (Section 3).
342
Digital Techniques in Mechanical Measurements
J. The basic elements of digital circuits are switches (transistors) and l ogic gates. These
elements are the building blocks for more complex devices, such as fl ip-fl ops , innu­
merable integrated circuits, clock or timer circuits, and digital displays (Section 4).
4. Digi tal information, when expressed as a string of bi ts, may be i nterpre ted using the
binary number system by assigning the values 0 and I to the bit levels (Section 5).
Bi nary numbers, in turn, allow information to be expressed in di gi tal codes, such
as bi nary -coded decimal (BCD), the American Standard Code for Information Inter­
change (ASCII), and the Latin I character set (ISO 8859- 1 ) . Digital patterns (strips
of black and wh ite) may be used to encode positional information or product labels
(Section 6).
S. Sim p le di gital circuits include events counters, frequency meters, waveform - shapi ng
devices, and multichannel s w i tches or multiplexers (Section 7).
6. The digital computer is a natural component of measuring systems, both for record­
ing and processing of data. The microcomputer is composed of a mic roproce ssor,
permanent and temporary memory (ROM and RAM), input and output ports, and
external buses or communications interfaces (Sections 8- 1 0).
7. Conversion of signals between analog and digital form is essential when analog trans­
ducers or devices are coupled to a digital computer or microprocessor. These oper­
ations are achieved using either analog-to-digital (AID) or digital-to-analog (D/A)
converters. B y combi ni ng an AID and a D/A converter with a digital s ig n al processor
(DSP), analog signals can be efficiently filtered, enhanced, and processed (Section 1 1 ).
8. Images may be digitized by breaking them into a two-dimensional grid of pixels.
Each pixel is assigned a digital number or numbers that describe the i nten s ity and/or
color at that point in the grid. Digi tal video is a succession of digi tal images played al
h igh speed . Digital image and video files may be mani pu lated by com pu ter 10 extract
measurements of various sorts. The associated files and data transfer rates can be
relatively large (Section 12).
SUGGESTED READINGS
Balch. M. Complete Digilal Design. New York: McGraw-Hill. 2003.
Brey, 8. B. The Intel Microprocessors: 8086/8088, 8018618()188, 80286, 80386. 80486. Pentium,
Pentium Pro Processor, Pentium II, Pentium Ill, and Pentium 4: Architecture. Programming. and
Interfacing. Upper Saddle Ri ve r, N.J.: Prentice Hall, 2003 .
Floyd, T.L. Digital Fundamentals. 8th ed. Upper Saddle R iver, N.J.: Prentice Hall, 2003.
Gonzalez, R. C., and R. E. Woods. Digital ltnllge Processing. 2nd ed. Upper Saddle River. N.J.:
Prentice Hall, 2002.
Horowitz. P., and W. Hill. The An of Electronics. 2nd ed. Cambridge, U.K.: Cambridge University
Press. 1 989.
Komcev, V. V. . and A. Kiselev. Modem Microprocessors. 3rd ed. Hingham, Mass.: Charles River
Media, 2004.
Kularatna. N. Digital
neers, 2003.
and Analogue lnstTUmJ!ntation.
London: The Institution or Eleclrical Engi­
Rathore, T. S. Digital Measurement Techniques. 2nd ed. Pangbome. U.K.: Alpha Science Interna­
tional. 2003.
Digital Techniques in Mechanical Measurements
Smith, S. W. Digital Signal Procwing: A Practical Guide for Engineers
Newes, 2002.
and Scielllilts. Boetoa:
Tooci, R. J., N. S . :Widmer, and R. L. Moss. Digital Systems: Principles and ApplicaJioru.
.
Upper Saddle River, N.J.: Prentice
Hall , 2004.
9tb cd.
PROBLEMS
1.
2.
Wriie the following binary numbers as
Wrile the rollowing decimal numbers as
(a)
(b)
(c )
(d)
3.
base 1 0 numbers.
( a ) 10101 1 1
( b ) 1010
(c) I l ll
( d ) 101 1 1 0 1 1
binary numbers.
16
87
419
40 1 77
Use a 1ruth table 10 show that the output from the cin:uit in Fig. 24 is high except when
the two inpu�� to either one or both or the AND gales are simultaneously high.
FIGURE 24: AND/OR c irc u i t for Problem 3.
4.
Use a truth table 10 show that the output from the circuit in Fig.
both inputs of either of the OR gales
are simultaneously low.
25 will be high only when
Q
Output
FIGURE 25: OR/NANO circuit for Prob l em 4.
344
Digital Techniques in Mechanical Measurements
Find the percent resol ution of each.
( a ) An 8- bit ND convener
S.
( b) A 1 2-bit ND convener
vary from 2.500 mV to 3.500 mV.
If the signal is fed to a 1 2-bit ND converter having a ±5.0 V range, estimate the voltage
increment represented by LSB. By what gain should the signal be amplified?
6.
The output from a temperature sensor is expected 10
Each step of an 8-bil 0/A convener represents 0. 10 V. What will the output or the DIA
7.
convener be for the following digital inputs?
( a ) 01 000 1 1 1
( b ) 1 0 10 1 1 0 1
U se a truth table to detennine under what se t o f conditions the output from the three- i nput
NANO gate shown in Fig. 26 will be low.
8.
A
8
c
D
FIGURE 26: Circuit for Problem 8.
9.
Compact disc d i gita l audio tracks are usually recorded using 1 6 bits to digi1i1.e the vo lu me
of each sample on each of the two SICreo tracks. Samples are taken al a frequency of
44. l kHz. Some recordi ngs are made using 24-bit samples and sample rates of 96 kHz.
( a ) What Ille si1.e Cin bi ts) docs each fonnat require to record a three minute song?
( b ) What is the Nyquist frequency for each fonnat and how does it relate IO dte range
of human hearing, wh ic h ends al about 20 kHz?
( c ) If the dynamic range compares the highest voltage amplitude recordable IO the
lowest nonzero a mpli tude recordable, what is the dynamic range of each format in
dB ? Assume that the DA convener driving the speakers i s noise free . How docs
this compare to human hearing, which spans roughly 1 20 dB?
REFERENCES
( 1 ) Robenson, D. Alec Reeves 1 902-1971. Sacramento, Calif.: Privatelinc.com, 2002.
(2) Horowitz, P. • and W. Hill. The Art ofElectronics. 2 nd ed. Cambridge, U .K.: Cambridge
University Press, 1 989, p. 240.
(3) Floyd, T. L. Digital F11ndame11tals. 8 t h ed. Upper Saddle R ive r, NJ.: Prentice Hall,
2003. Chap. 8.
345
Digital Techniques in Mechanical MeasuremEmts
[4) Berlin, H. M. The 555 Timer Application Source Book with Experimen1s. Derby, Conn .
··
E & L Instruments, Inc., 1 97 6 .
[SJ Horowitz, P., and W. Hill. The Art of Electronics. 2nd ed. Cmnbridge, U.K.: Cambridge
University Press , 1 989, Section 5. 19.
(6) American National Standards Institute. Coded Character Se•t-7-BitAmerican Nazional
Standard Codefor Information Interchange (7-Bit ASC/l), ANSI X3.4- 1986. New York:
ANSI, 1 986.
(7) ISO 8859- 1 : 1 998. Information technolog�-Bit Single-8.yte Coded Graphic Charac­
ter Sets-Part /: Latin Alphabet No. I. Geneva, Switzerla1nd: International Otganiza­
tion for Standardization, 1 998.
(8) Aliprand, J., et al . (eds.) The Unicode Standard. Version 4.0. Boston: Addison­
Wesley, 2003.
(9) Oppenheim, A. V., and R. W. Schafer Digital Signal Proceesing. Englewood Qiffs,
NJ.: Prentice Hall, 1975.
(10) Joyce. J. Ulysses. Oxford, Miss.: Project Gutenberg Literary Foundation. 2002.
(Online text available at http:l/www.gutenberg.org/)
(11) Wilson, J., and J. F. B. Hawkes. Optoelectronics: An Introduction. 3rd ed. Harlow,
U.K.: Prentice Hall Europe, 1 998, Section 7.3.7.
ANSWERS TO SELECTED PROBLEMS
1
(b) 1 0
5
(a) e v / tJ. Vr,
=
�
·;,I.,
'.' . ·:·
... ,
,.,·::-
-f�J
Ii
,�{
� �Strain and
Stress:
�
��� Me a surement and Ana lysis
}i
2
3
4
5
. 6
' '"·• · ·
'�
.
18
9
10
11
12
13
14
15
16
17
18
INTRODUCTION
STRAIN MEASUREMENT
THE ELECTRICAL RESISTANCE STRAIN GAGE
THE METAWC RESISTANCE STRAIN GAGE
SELECTION AND INSTALLATION FACTORS FOR BONDED METALLIC STRAIN GAGES
CRCUITRY FOR THE METALLIC STRAIN GAGE
THE STRAIN-GAGE BALLAST CIRCUIT
THE STRAIN-GAGE BRIDGE CIRCUIT
THE SIMPLE CONSTAtn:-CURRENT STRAIN-GAGE CIRCUIT
TEMPERATURE COMPENSATION
CALIBRATION
COMMERCALLY AVAILABLE STRAIN-MEASURING SYSTEMS
STRAIN-GAGE SWITCHING
USE OF STRAIN GAGES ON ROTATING SHAFTS
STRESS-STRAIN RELATIONSHIPS
GAGE ORIENTATION AND INTERPRETATION OF RESULTS
SPECIAL PROBLEMS
FINAL REMARKS
INTRODUCTION
All machine or structural members deform to some extent when subjected to external loads
or forces. The deformations result in relative displacements that may be normalized as
percentage displacement, or strain. For simple axial loading (Fig. I ).
dL
Li - L 1
6. L
L
Li
Li
Ea = - � --- = -
(I)
where
Eu =
axial strain,
L 1 = initial linear dimension o r gage length,
Li
·
= final strained linear di mension
More correctly. the term 11ni1 strain should be used for the precedi ng quantity and is generally
intended when the word strain is used alone. Throughout the following discussion, when
From Meclianical Measurtments. Sixth Edition, Thomas G. Heckwith, Roy D. Marangoni . John H.
Copyri ght © 2007 by Pearson Education, Inc. Published by Prentice Hall. All ri ght s reserved.
Lienhanl V.
Strain and Stress: Measurement and Analysis
:�-....
/
�-,....-
\
--i.------..-----.--�
1 ' - - - "" I
I
I
�I o,, I�
I
I
I
I
I
I
L, - L,
L,
o,
L, - L,
Ea = ,L-
.J
0,, - D,
El = -o;-
FIGURE I : Defin i ng rela1ions for axial and lateral strai n.
lhe word strain is used, we mean lhe quanlity defined by Eq. ( I ). If the net change in a
dime n s io n is required. the term total strain will be used.
Because the quantity s1rain. as applied to mosl engineering materials, is a very small
number, it is commonly multiplied by one million; the resu l ti ng number is then called
microstrain, 1 or parts per million (ppm).
The stress strain relation for a uniaxial condi1ion, such as exists in a simple tension
test specimen or at the outer fiber of a beam in bending. is expressed by
(2)
1 Considerable use or the lenn micn1l·trai1� will be made throughout
viarion ,,.strain will often be used.
348
this chapter. For convenience.
the :abbn:·
Strain and Stress: Measurement and Analysis
where
E
<Ta
=
=
Young 's modulus,
uniaxial stress,
Ea = lhe strain in
the direction of lhe stress
This relalion is linear: that i s , E is a conslanl for most malerials so long as lhe stress i s kept
below the proponional limit
When a member is subjected lo simple uniaxial stress in the elastic range (Fig. I ),
lateral strain resul ls in accordance with the fol lowing relation:
-El
v = --
(2a)
Ea
where
v
= Poisson's ratio,
El = lateral strain
A more general condition commonly exists on lhefret! surface of a stressed member. Let us
consider an element subjecl to onhogonal stresses Ux and a1 as shown in Fig. 2. Suppose
,,.,
,- - - - - - - - - 1
I
1---+---
u,
�------.---� - - -L... .J
FIGURE 2: An element laken from a biaxi a lly slrcssed condition with normal slresses
known.
349
Strain and Stress: Measurement and Analysis
that th� stresses _ ax �d O"y are applied one at a time . If O"x is applied lint, there will be 1
.
_ m the x-d1rect1on equal
steam
to O"x f E. At the same Ume, because of Poisson's ralio' there
will be a strain in the y-direction equal to -VO"x/ E.
Now suppose that the stress in the y-direction, a1, is applied. This stress will l"eSlll
in a y-strain of a1/ E and an x-strain equal to - va1/ E . The net strains are then
by the relations
O'y - VO"x
O"x --VO"y
and
Ex = --E
( 3)
ey = --E
--
exixesscc!
If these rel atio ns are solved simultaneously for O"x and a1 , we obtain the equations
(Ix =
E (6x + VEy)
I - v2
and
(4)
When a stress az exists , acting in the third orthogonal diJ1ection, the more general lluec­
dimensional relations are
Ex = E (O"x - v (ay + a,)),
I
6y
= I [ay - v (a, + O"x >I.
E
I
(S)
Ez = E (a, - v(O"x + a1) ]
2
STRAIN MEASUREMENT
Strain may be measured either directly o r indirectly. Modern strain gages an: i nherently
sensitive to strain; that is, the unit output is directly proportional to the unit dimensional
change (strain). However, until abou t 1 930 the common experimental procedure consisted
of measuring the displacement tJ. L over some initial gag': length L and then calculating
the resulting average strain usi ng Eq. ( ! ). An apparatus called a n v:tensometer was used.
Th is device generally incorporated either a mechanical or <Jptical lever system and sensed
displacements over gage lengths ranging from about 50 mm to as great as 25 cm (about
10 in.). The Huggenberger and the Tuckerman e xte nsometers are representative of the
more advanced mechanical and optical types, respectively. Reference ( I ) prov ides a good
summary of extensometer pract ices.
Electrical-type strain gages are devices that use simple resistive, capacitive (2 , 31.
inductive (4 ], or photoelectric principles. The resistive types are by far the most common
and are discussed in considerable detail in the following s•::ctions. They have advantages.
primarily of size and mass , over the other types of electr:ical gages. On 1he other hand.
strain-sensitive gaging elements used in calibrated devices for measuring other mechanical
quantities are often of the inductive type, whereas the ca.paci t i vc kind is used more for
special-purpose applications. Inductive and capacitive gage:> are generally more rugged than
resistive ones and better able to maintain calibration over a J ong period of lime. Inductive
gages are sometimes used for permanent installations, suc:h as on rolling-mill frames for
monitoring roll loads. Torque meters often use strain gages i n one form or another, includi ng
inductive (5] and capacitive [2 ] .
Strain and Stress: Measurement and Ana l ysis
i·�i
r.�T
Other strain measuring techn.iques include optical methods [6), such asphotoelasticity,
the Moire techn.ique and holographic interferometry.
,
HE ELECTRICAL RESISTANCE STRAIN GAGE
Io 1 856 Lord Kelvin demonstrated that the resistances of copper wire and iron wire change
when the wires are subject to mechanical strain. He used a Wheatstone bridge circuit with
a galvanometer as the indicator [7]. Probably the first wire resistance strain gage was that
made by Carlson in 193 1 [8). It was of the unbounded type: Pillan; were mounted, separated
by the gage length, with wires stretched between them. What was probably the first bonded
strain gage was used by Bloach (9]. It consisted of a carbon film resistance element applied
directly
to the surface of the strained member.
lo 1938 Edward Simmons made use of a bonded wire gage in a study of stress-strain
relations under tension impact [ 10]. His basic idea i s covered in U.S. Patent No. 2,292,549.
At about the same time. Ruge of the Massachusetts Institute of Technology (MIT) conceived
the idea of making a preassembly by mounting wire between thin pieces of paper. Figure 3
shows the general construction.
During the 1950s advances in materials and fabricating methods produced the foil­
type·gage, which soon replaced the wire gage. The common form consists of a metal foil
element on a thin epoxy support and is manufactured using printed-circuit techniques. An
important advantage of this type is that almost unlimited plane configurations are possible;
a few examples are shown in Fig. 4.
Assembled gage
FIGURE 3: Construction of bonded-wire-type strain gage.
Strain and Stress: Measurement and Analysis
I
(a)
(c)
(b)
(d)
FIGURE 4: Typ ic a l foil gages illustrating the followi n g types: (a) si ng le element, (b) two-elemenl
(c) t h ree e le ment rosette, (d) one example of many different special-purpose gages. The
l ane r is for use on pressurized diaph mg m s .
rosette.
4
-
TH E METALLIC RESISTANCE STRAIN GAGE
The theory of o pe rali on of the me la l lic resislance strain gage is rel at ivel y simple.
a l c n gl h of wire (or foil) is mechanically stretched, a
longer
When
length of smaller sectioned
c o nductor resuhs; hence lhe e l ec l r ical resislance changes. lf lhe lenglh of resistanceelemenl
is imimalcly an ac hed lo a
slrai ncd member in such
a
way that lhe ele ment will a lso be
strained. then the measu red c h ange in resistance can be calibrated i n terms of suain.
A general relation between the electrical and mechanical properties may be derived
conductor length L, having a cross-sectional area CD2. ( In
ge ne ra l the section need not be circular; hence D will be a sec t io nal dimension and C will
be a proportionalily constanl. If the section is square, C
I; if it is circular, C = "/4.
e lc. ) If lhe co nd ucto r is strained axially in 1ension, thereby caus ing an increase in length.
the lalcral dimension should reduce as a fu nct io n of Poisson's ratio.
We w i l l start wi lh l h e rel al i on eq u at i on
as follows: Assume an i n i t i a l
=
pl
R=A=
(6)
pl
CD2
ept for
If 1he conduclor is s irai n cd we m a y assume that each o f 1 h e q u anl il ies in Eq. ( 6 ) exc
C may change. Differentiating, we have
dR
C D 2 ( L dp + p dL ) - 2Cp L D dD
(CD 2 ) 2
I
dD
= coi ( L dp + p dL) - 2pL
=
(
D
352
)
( 7)
Strain and Stress: Measurement and Analysis
Dividing Eq. (7) by Eq. (6) yields
dR = dL _ 2 dD + dp
R L D p
(8)
which may be wriuen
( 8a )
Now
and
dL =
T
dD
0=
.
.
Ea
= axial stram,
Ei
.
= lateral stram
v = Poisson's ratio =
_ dD/D
dL/L
Making lhese substitutions gives us the basic relation for what is known as the gage factor,
for which we shall use the symbol F:
dR/R = l + 2v + dp/p
F = dR/R
dL/L =
dL/L
Ea
(9)
This relation is basic for the resistance-type strain gage.
Assuming for the moment that resistivity should remain constant with strain, !hen
according 10 Eq . (9) the gage factor should be a function of Poisson's ratio alone, and in the
elastic range should not vary much from I +
= 1 .6. Table I lists typical values for
various materials. Obviously, more than Poisson's ratio must be involved, and if resistivity
is lhe only olher variable. apparently its effect is not consistent for all materials. Note the
value of lhe gage factor for nickel. The negative value indicates that a stretched element wilh
increased length and decreased diameter (assuming elastic conditions) actually exhibits a
reduced resistance.
In spite of our incomplete knowledge of the physical mechanism involved, lhe factor
F for metallic gages is essentially a constant in the usual range of required strains, and its
value, determined experimentally, is reasonably consistent for a given material.
By rewriting Eq. (9) and replacing the differential by an incremental resistance change,
we obtain the following equation:
(2)(0.3)
E
=
R
I
6R
- -
F R
(10)
In practical application, values of F and are supplied by the gage manufacturer, and the
user determines A R corresponding lo the input situation being measured. This procedure
is lhe fundamental one for using resistance strain gages.
TABLE 1 : Representative Properties of Various Grid Materials
Grid Material
Composition
A ppprox.
Gage
Factor, F
Approximate Resistivity
Ohms per
Micromil . foot
ohm - cm
A pproximate
Temperature
Coefficient of
Resistance,
ppml"C
Maximum
Operating
Temp., °C
(approx.)
Nichrome V*
80% Ni; 20% Cr
2.0
108
650
400
1 100
Constantan*, Copel*,
45% Ni; 55% Cu
2.0
49
290
11
480
lsoelastic*
36% Ni; 8% Cr;
0.5% Mo; Fe remainder
3.5
1 12
680
470
Karma•
74% Ni; 20% Cr;
3% Al; 3% Fe
2.4
1 30
800
18
Manganin*
4% Ni; 1 2% Mn;
84% Cu
0.47
48
260
11
5. !
24
42
1 37
240
7.8
45
6000
60
3000
Advance•
Platinum-lri<!iYm
Monel*
95% P!; 5% !r
67% Ni; 33% Cu
Nickel
1 .9
- 1 2t
Platinum
4.8
10
1 '\ C i\
l .t. JV
2000
815
1 t '"'
l l VV
Strain and Stress: Measurement and Analysis
SELECTION AND INSTALLATION FACTORS FOR BONDED METALLIC STRAIN GAGES
Perfonnance of bonded metallic strain gages is governed by five gage parameters: (a) grid
material and configuration; (b) backing material; (c) bonding material and method; (d) gage
protection; and (e) associated electrical circuitry.
Desirable properties of grid material include (a) high gage factor,
F; (b) high resistiv­
ity, p; (c) low temperature sensitivity; (d) high electrical stability; (e) high yield strength;
(0 high endurance limit; (g) good workability; (h) good solderability or weldability; (i) low
hysteresis; G) low thennal emf when joined to other materials; and (k) good corrosion
resistance.
Temperature sensitivity is one of the most worrisome factors in the use of resistance
strain gages. In many applications, compensation is provided in the electrical circuitry;
however, this technique does not always eliminate the problem. Two factors are involved:
( I ) the differential expansion existing between the grid support and the grid proper, resulting
in a strain that the gage is unable to distinguish from load strain; and ( 2) the change in
resistivity p with temperature change.
Thermal emf superimposed on gage output obviously must be avoided i f de circuitry
is used . For ac circuitry this factor would be of little importance. Corrosion at a junction
between grid and lead could conceivably result in a miniature rectifier, which would
be
more serious in an ac than in a de circuit.
Table 1 lists several possi ble grid materials and some of the properties influencing
their use for strain gages. Commercial gages are usually of constantan or isoelastic. The
former provides a relatively low temperature coefficient along with reasonable gage factors.
Isoelastic gages are some 40 times more sensitive to temperature than are Constantan gages.
However, they have appreciably higher output, along with generally good characteristics
otherwise. They are therefore made available primarily for dynamic applications where the
short time of strain variation minimizes the temperature problem.
The gage factor listed for nickel is of particular interest, not only because of its
relatively high value, but also because of its negative sign. It should be noted, however,
that the value of
F
for nickel varies over a relatively wide range, depending on how it is
processed. Cold working has a rather marked effect on the strain- and temperature-related
characteristics of nickel and its alloys, and this feature is used advantageously to produce
special temperature self-compensating gages (sec Section
1 0.2).
Common backing materials include phenolic-impregnated paper. epoxy-type plastic
for a
Table 2
films, and epoxy-impregnated fiberglass. Most foil gages i ntended
moderate range
of temperatures (- 75°C to 100°C) use an epoxy fi l m backing.
l ists commonly
recommended temperature ranges.
No particular difficulty should be experienced in mounting s trai n gages if the man­
ufacturer's recommended techniques are followed carefu lly. However, we may make one
observation that is universally applicable: C/ea11/iness is an absolute requirement if con­
sislently satisfactory results are to be expected. The mounting area must be cleaned of all
corrosion, paint, and so on, and bare base ma1erial musl be exposed. All !races of greasy
film must be removed. Several of the gage suppliers offer kits of c l e aning materials along
with instructions for their use. These materials are very sa1isfac1ory.
MQSt gage installations arc not complete until provision is made to protect the gage
from ambient conditions. The latter may include mechanical abuse, moisture, oil, dust
Strain and Stress: Measurement and Analysis
FIGURE 5: Ballast circuit for use with strain gages.
If our indicator is 10 provide an indication for strain of, say, l µ.-strain, il must sensea4variation in 4 V, or 0.000 l 0%. This severe requirement practically eliminates lhe ballast
circuit for static strain work. We may use il, however, in cenain cases for dynamic straia
measurement when any static strain component may be ignored. [f a capacitor is insened
into an output lead, the de exciting voltage is blocked and only the variable component is
allowed to pass (Fig. 5). Temperature compensation is not prov ided; however, when only
transient strains are of interest, this type of compensation is often of no importance.
µ. V
8
THE STRAIN-GAGE BRIDGE CIRCUIT
A resistance-bridge arrangement is particularly convenient for use with strain gages because
it may be easily adjusted to a null for zero strain, and it provides means for elfecli11ely
reducing or eliminating the temperature effects previously discussed (Section 5). Figure 6
shows a minimum bridge arrangement, where arm I consists of the strain-sensitive gage
mounted on the te.�I item. Arm 2 is formed by a similar gage mounted on a piece of unstrained
material as nearly like the test material as possible and placed near the test location so that
the temperature will be the same. Arms 3 and 4 may simply be fixed resistors selected for
good stability, plus ponions of slide-wire resistance, D, required for balancing the bridge.
If we assume a voltage-sensitive deftection bridge with all initial resistances nominally
equal, we have
In addition,
e = .!._
l>R i
F R
Then
For e;
=
8 V and
F
= 2,
l>eo =
358
or ll R
=
(2)t
4+....,
(2""")"""
8 x 2 x e
FRt
Strain and Stress: Measurement and Analysis
l
!
FIGURE 6:
Simple resistance-bridge arrangement for strain measurement.
If we neglect the second term in the denominator, which is normally negligible, then
t:.eo = eo = 4E v
or for e = I µ.-strain, e0 = 4 µ.V.
We see that under similar conditions the output increment for the bridge and ballast
arrangements is the same. The tremendous advantage that the bridge possesses, however, is
that the incremental output is not superimposed on a large fixed-voltage component. Another
imponant advantage, which is discussed in Section I 0, is that temperature compensation is
easily auained through the use of a bridge circuit incorporating a "dummy," or compensating,
gage.
8.1
Bridges with 1\No and Four Arms Sensitive to Strain
In many cases bridge configuration permits the use of more than one arm for measurement.
This is panicularly true if a known relation exists between two Slrains, notably the case of
bending. For a beam section symmetrical about the neutral axis, we know that the tensile
and compressive strains arc equal except for sign. In this case, both gages I and 2 may be
used for strain measurement. This is done by mounting gage I on the tensile side of the
beam and mounting gage 2 on the compressive side, a� shown in Fig. 7 (see also case F in
Table 4 ). The resistance changes will be alike but of opposite sign, and a doubled bridge
output will be realized.
This may be carri ed funher, and all four arms of the bridge made strain sensitive,
thereby quadrupling the output that would be obtained if only a single gage were used. In
359
Strain and Stress: Measurement and Anal)isis
AGURE 7: Bridge arrangement with two gages sensitive to strain.
this case gages 1 and 4 would be mounted to record like strain1 (say tension) and 2 and 3 to
record the opposite type (case G in Table 4).
Bridge circuits of these kinds may be used either as null ..balance bridges or as deftec­
tion bridges. In the former the slide-wire movement becomes the indicated measwe of
strain. This bridge is most valuable for strain-indicating devi:ces used for static measure­
ment. Most dynamic strain-measuring systems, however, use a, voltage- or current-sensitive
deftection bridge. After initial balance is accomplished, the output, amplified as necessaiy.
is used to deftect an indicator, such as an oscilloscope beam or input to a digital recorder.
In addition, the constant-current bridge may offer certain adv11ntages.
8.2
The Bridge Constant
At this point we introduce the term bridge constant, which we shall define by the following
equation:
k=�
8
where
k
A
8
=
=
=
the bridge constant,
the aciual bridge output,
the output from the bridge if only a single gage,
sensing maximum strain, were effectiv·e
( 13}
Strain and Stress: Measurement and Analysis
In the example illustrated i n Fig. 7, the bridge constant would be 2. This is true because
the bridge provides an output double of that which would be obtained if only gage I were
strain sensitive. If all four gages were used, quadrupling the output, the bridge constant
would be 4. In certain other cases (Section 1 6), gages may be mounted sensitive to lateral
strains that arc functions of Poisson's ratio. In such cases bridge constants of 1 .3 and 2.6
(for Poisson's ratio = 0.3) are common.
8.3
g
Lead-Wire Error
When it is necessary lo use u nusually long leads between a strain gage and other instru­
mentation, lead-wire error may be introduced.
THE SIMPLE CONSTANT-CURRENT STRAIN-GAGE CIRCUIT
Measurement of dynamic strains may be accomplished by the simple circuit shown in Fig. 8.
It is assumed that the power source is a true constant-current supply and that the indicator
(an oscilloscope is shown) possesses near-infinite input impedance compared to the gage
resistance. As the gage resistance changes as a result of strain, the voltage across the gage,
hence the input to the oscilloscope, will be
e;
= i; R
( 14)
and
t:.e; = i; t:. R
Dividing Eq. ( 1 4a) by Eq. ( 14), w e have
t:.e;
-;:
Constanl·
current
power supply
=
t:. R
ff
Gage. R
FIGURE 8: Single-gage constant-current circuit.
( 14a)
Strain and Stress: Measurement and Analysis
I nserting A R/ R in Eq. ( 1 0) gives us
I A e;
e = -­
F e;
( 14b)
The oscilloscope should be set in the ac mode to cancel the direct de component, and. of
course, the oscilloscope amplification capability must be sufficient to prov ide an adequa11:
readout.
10
TEMPERATURE COMPENSATION
As already implied, resistive-type strain gages are normally quite sensitive 10 tcmper.uUR
Both the differential expansion between the grid and the tested material and the te mper.u�
coefficient of the resistivity of the grid material contribute to the problem. It has been shown
(Table I) that the temperature effect may be large enough to require careful consideralion.
Temperature effects may be handled by ( I ) cancellation or compensation or (2) eval uation
as a part of the data reduction problem.
Compensation may be provided ( 1) through use of adjacent-arm balancing or com­
pensati ng gage or gages or (2) by means of self-compensation.
10.1
The Adjacent-Arm Compensating Gage
Consider bridge configurations such as those shown in Figs. 6 and 7. Initial electrical
balance is obtained when
R4
R1
If the gages in arms I and 2 are alike and mounted on similar materials and if both gages
experience the same resistance shift, A R, , caused by temperature change, then from Eq. ( 19)
in Section 16 with 11 1 = e + er and e2 = er
de0
F
F
- = - (£ 1 - E 2 ) = -E
4
4
e;
We see that the output is unaffected by the change in temperature. When the compensatins
gage is used merely to complete the bridge and to balance out the temperatwe component.
it is often referred to as the "dummy" gage.
1 0.2
Self-Temperature Compensation
In certain cases it may be difficult or impossible 10 obtain temperature compensation b!
means of an adjacent-arm compensating or dummy gage. For exam ple. tem perature lfldi·
11
ents in the test part may be sufficiently great 10 make it impossible to hold any !WO gages
similar temperatures. Or, in certain instances, it may be desirable to use the ballast ra�
uon.
than the bridge circuit, thereby eliminating the possibility of adjacent-arm compensa
Situations of this sort make self-compensation h igh l y desirable.
I
The two general types of self-compensated gages avai l ab le are the selected-me t gaF
and the dual-element gage. The former is based on the disc overy that
manipulation of alloy and processing, particularly through cold worki ng, so�e con
the temperature sensitivity of the grid material may be exercised. Throu gh this approac
throUg�
h�
362
Strain and Stress: Measurement and Analysis
t
400 1--�-+��-+��-t-���
200 1--�-+��-+��--±=
f l---�_._.....
..
i -200
I-@
0
0
50
1 00
1 50
200
Temperature,
°F
300
400
FIGURE 9: Approxi mate range of apparent strain versus temperature for a ty pical selected- melt
gage mounled on the appropriate material (e.g., steel at about 1 1 ppml"C).
materials may be prepared that show very low apparent strain versus temperature c ha nge
over certain temperature ranges when the gage is mounted on a particu lar test material.
Figure 9 shows typical characteristics of selected-melt gages compensated for use with a
material having a coefficient of expansion of 6 ppml"F, which corresponds to the coefficient
of expansion of most carbon steels. In this case, practical compensation is accom p l ished
over a temperature range of approximately 50"F-250°F. Other gages may be compensated
for different thermal e xpansion s and temperature ranges. These curves give some idea of
the degree of control that may be obtained through manipulation of the grid material.
The second approach to se l f-compensati on makes use of two grid elements connected
in series in one gage assembly. 1be two e le me nts have different temperature characteristics
and are selected so that the net temperature ind uced strain is minimized when the gage is
mounted on the specified test material. In general, the performance of this type of gage is
similar to that of the selected-melt gage shown in Fig. 9.
Neither the selected-melt nor the dual-element gage has a distinctive outward appear­
ance . One company uses color-coded backings to ass i st in identifying gages of different
specifications
-
.
11
CALIBRATION
Ideally, calibration
of any
meas u ring
system consists of in troducing
an accurately known
sample ofthe variable that is to be measured and the n observing the system's response.
This
be realized i n bonded resistance strain-gage work because of the nature
of the transducer. Normally, the gage is bonded to a test item for the simple reason that
the strains (or stresses) are unknown. Once bonded, the gage can hardly be tra ns ferred to a
known strai n situation for cali bration . Of course, this is not necessaril y the case if the gage
or gages are used as secondary transducers app lied to an appropriate elastic member for
the purpose of measuring force, pressure, torque, and so on. In cases of this sort, i t may be
ideal cannot often
perfectly feasible to introduce known inputs and carry out satisfactory calibrations. When
the gage is used for the purpose of experi mental l y determining strains, however, some other
approach to the cal ibrat ion prob le m is required.
363
Strain and Stress: Measurement and Analy:sis
Resistance strain gages are manufactured under carefully controlled CoodiliOni,
the gage factor for each lot of ga�es is provided by the manu fac�urer wilhi.n an
tolerance of about ±0.2%. Knowmg the gage factor and gage resistance makes �
'"1,k
simple method for calibrating any resistance strain-gage syst1:m. The method coasisls
determining the system's � ponse to t�e introduc�ion of a known small resistance
at the gage and of calculaung an eqmvalent stram therefrom. The resistance chanae
introduced by shunting a relatively high-value precision res is tance across the gage, 15
shown in Fig. I 0. When switch S is closed, the resistance of bridge arm I is changed byaa
a
small amount, as determined by the following calculations.
�
��
•
Let
R11 = the gage resistance,
R, = the shunt resistance
Then the resistance of ann I before the switch is closed equals .R8, and the resistance of ann
1 after the switch is closed equals ( R8 R,) I ( R8 + R,), as detem11i ned for parallel resistances.
Therefore, the change in resistance is
t!> R =
R1R,
(R8 + R, ) 2
-
R1 =
R:
R1 + R,
- -- -
Now to determine the equivalent strain, we may use the relation given by Eq. (1 1):
E=
+..!.F.
(
l!> Rg
R1
By substituting !!. R for 6 R8 , the equivalent strain is found to be
Ee =
-
l
F
R8
Rg
+ R,
)
FIGURE 1 0: Bridge employing a shunt resistance for calibration.
364
(15)
Strain and Stress: Measurement and Analysis
EXAMPLE 1
Su ppose that
Rg = 1 20 Q ,
F = 2. 1 ,
R., = 1 00 k Q (i.e., 100,000 Q)
What equ ivalent strai n
gage?
Solution
��.f- ����� ·
·�·--i
·I;
; . f�f:�
·
i ndicated when the shunt resistance is connected across the
Ee = [
From Eq. ( 1 5 ),
=
·;:�9.iie
12
will be
I
1 20
- 2.1 100,0000 + 1 20
-570 jL-strain
]
=
- 0.00057
Dynamic calibration is sometimes provided by replacing the manual calibration
switch with an electrically driven switch, often referred to as a chopper, which makes
and breaks the contact 60 or 100 times per second. When displayed on .a oscilloscope
screen or recorded, the trace obtained is found to be a square wave. The step in the trace
represents the equivalent strain calculated from Eq. ( 1 5).
There arc other methods of electrical calibration. One system replaces the strain­
gage bridge with a substitute load, initially adjusted to equal the bridge l oad ( 1 2]. A series
resistance is then used for calibration. Another method injects an accurately known voltage
into the bridge network.
COMMERCIALLY AVAILABLE STRAIN-MEASURING SYSTEMS
Commerci all y available systems intended for use with metallic-type gages fall within three
general categories:
The basic strain indicator, useful for static, single-channel readings
The single-channel system either external to or an integral part of a oscilloscope or a
computer data acquisition system.
J. Data acquisition systems whereby the strain data may be
1.
2.
(a) displayed (digitally and/or by a video terminal)
(b) recorded (on hard disk or hard-copy printout)
(c) fed back into the system for control purposes
The wide range of availability and divergence of such systems makes it impractical to
attempt any but a superficial coverage in this tellt. Better sources of state-of-the-art details
arc the brochures and technical "aids" provided by many of commercial suppliers.
Strain and Stress: Measurement and Analysis
12.1
The Basic Indicator
Typically, the basic indicator consists of a manually or self-balancing Wheatsto
ne
with meter-type or digital readout, an ampli fier, and adjustments to accommodate
bridge
active-=
�s
a
of gage factors. Provision is also common for handling bridges with a single
two-gage, and four-gage con figurations. For fewer than four gages, the bridge
loop .
completed within the instrument The measurement process consists of zeroing the
n:
under initial conditions, then, after applying test conditions, rebalancing the bridge.
difference between initial and final readings provides the strain increment. Such instrume
nts
are generally precalibrated to provide direct strain readout, often in digital Conn. Most
strain
indicators now have analog output tenninals 10 display dynamic strain data to oscilloscopes
or computer data acquisition systems.
13
STRAIN-GAGE SWITCHING
Mechanical development problems often require the use of many gages mounted tluough­
or
necessary to pro.
oul the test item, and simultaneous or nearly simultaneous readings are often necessary.
course, if the data must be recorded at precisely the same instant, it will be
vi de separate channels for each gage involved. However, frequently steady-state conditions
may be maintained or the test cycle repeated, and readings may be made in succession until
all the data have bee n recorded. In other cases, the budget may prohibit duplication of the
required instrumentation for simultaneous multiple readings or recordings, and it becomes
desirable 10 switch from gage to gage, taking data in sequence.
For high-speed multiple strain measurement.�. the digital techniques described in
are recommended, possibly including multiplexing methods or multichannel
Chapter
14
AID converters.
USE OF STRAIN GAGES ON ROTATING SHAFTS
al least three different
(3) by usc of slip rings.
When a shaft rotates slowly enough and when only a sampling of data is required, direct
connections may be made between the gages and the remainder of the measuring system.
Strain-gage information may be conducted
ways: ( I ) by direct connection,
(2)
from ro tat ing
shafts in
by wireless telemetery, and
Sufficient lead length is provided, and the cable is permiued 10 wrap itself onto the shaft
In fact, the available time may be doubled with a given length of cable if it is first wrapped
on the shaft so that the shaft rotation causes it to unwrap and then 10 wrap
up
again in
of lhe
or automatic disconnecting arrangement may be provided. This
the opposite direction. If the machine cannot be stopped quickly enough as the end
cable is approached, a fast
actually need be no more than soldered connections that can be quickly peel ed off. Shielded
cable should be used to minimize reactive effects resulting from the coil of cable on the
sh"aft. This technique is somewhat limited, of course, but should not be overlooked,
because
it is quite workable at slow speeds and avoids many of the problems inherent in the other
methods.
A second method is that of actually transmiuing the strain-gage infonnatio n through
radio frequency transmitter mounted on the shaft and with the signal picked up by 3
receiver placed nearby. This method has been used successfully for a long ti me [ 1 3), and a
a
366
Strain and Stress: Measurement and Analysis
Slip rings
Input
Complete
strain-gage bridge
mounted on rotating
meni:Jer
Output
FIGURE I I : S l i p rings external to the bridge.
diverse array of wireless ins tru mentation is commercially available. For strain appl i cati ons
in panicular, digital te lemetry systems are avai labl e with onboard multiplexing that allows
many sensors to be rou ted through a si ng le transmitler. Such a system is practical when the
added cost can be j ustified .
Undoubtedl y the most common m ethod for obtaining s trai n- gage information from
rotating shafts is through the use of sl ip rings. Slip-ring problems are similar to swi tc h i ng
problems , as discussed in the prec edi ng section, except t ha t additional variables make
the problem more difficuh. Factors such as ri ng and brush wear and changing contact
temperatures make it i mperati ve that the full bridge be used at the test poi nt and that the
sl i p ri ngs be i ntrod uced externally to the bridge as shown in Fig. 1 1 .
Commerc i al s l i p- ri ng asse mbl ies are avai lable whose performances are quite satisfac­
tory. Their use , however, presents a problem that is often difficult to solve. The assembly
is nonnally self-contained, consisting of brush supports and a shaft with rings mounted
between two beari ngs . The construction requires that the rings be used at a free end of a
shaft, which more often than not is separated from the test poi nt by some form of beari n g.
This arrangement presen ts the problem of gell i ng the leads from the gage located o n one side
of a beari ng to the s l i p ri ngs located on the oppos i te side. It is necessary to feed t he leads
through the shaft in some manner, which is not alway s convenient. Where this presents no
panicular problem , the commercially available sl i p - ri ng assemblies are practica l and also
probably the most i nexpe n sive solution to the problem.
367
Strain and Stress: Measurement a nd Analysis
15
STRESS-STRAIN RELATIONSHIPS
As previously stated . strain gages are generally used for one: of two reasons : to
determine
s ess cond i ti ons through strain measurements or to act as secondary transducers calibrated
tr
� n te�s of such qua� tities as force, pressure, di splacement, and _the like.
In either
case,
use of stram gages demands a good grasp of stresiHtrain relationships. Know(.
edge of the plane. rat her than of the general three-dimensional case, is usually su lliciau
for strain-gage work bt.cause it is o n l y in the very unusual situation that a strai n gage
is
mounted anywhere except o n the unloaded suiface of a stressed member.
m te lhgen t
-
1 5. 1
The Simple Uniaxial Stress Situation
e
c
in a t ns io n or om press io n member, the u1nloaded outer fiber is subject
condition resul in a tria:Xial strai n condi on,
we know t hat there will be lateral strain in addition to the strain in the direction of Sln:ss.
Because of the si m p l i ity of the ordinary tensile (orcompressi·ve) situation and its prevalence
(see Fi g . I ). the fundamental stress-strau1 relationship is based on il Young's modulus is
defined by the relation expressed by Bq. (2), and Poisson's: ratio is defined by Eq. (2a).
It is imponant to realize that th these defi ni tions are ma e on the basis of the simple
o ne-direc tion stress system.
For s it uations of th is sort, calculation of s
s from strain measuremenlS is simple.
The stress is detennined merely by m u lti pl yi ng the strain,
i n the axial direction
in microstrains, by lhc modulus of e lastic i ty for the test mate:rial.
In hendi ng.
or
ts
to a u niax i a l stress. However, this
ti
because
c
bo
d
tres
m1easured
EXAMPLE 2
S uppose the tens ile member in Fig. 6 is of a l.umi n um having a modulus of elasticity equal
to 6.9 x 10 10 Pa ( 10 x 1 1>6 lb ffin . 2 and the strain measured by the gage is 326 µ-slrain.
What axial stress exists at the gage?
)
Solution
a0 = Eea
=
(6.9 x 1 0 10 )
= 22.4
x
x
(325
x
10-6 )
1 06 Pa (3250 lbffin.2 )
EXAMPLE 3
beam
beam
Strain gages are mounted on a
as s hown i n Fig. 7. 1l1e
is of steel having an
est im ated modulus of e l as t i i ty of 20 . 3 x 1 0 10 Pa (29.5 x
lbffi n . 2 ). If the total readout
from the two gages is 390 -st rai n , what stress exists at the lo1ngitudinal center of the gage?
Note that the bridge on s tan t is 2.
Solution
c
c
µ
Clb =
Eeb
106
=
(20.3
x
=
3958
x
1 0 10 ) x (390 x 1 0- 6 /2)
104 Pa (5700 l bffi n . 2 )
Strain and Stress: Measurement and Analysis
y
-4-.--
x
1
FIGURE 12: Eleme nt located on the shel l of a cy lindrical pressure vessel.
15.2
The Biaxial Stress Situation
Often gages are used at locations su bject to stresses in more than one direction. If the test
poi nt is on a free surface, as is usually the case, the c ondition is termed biaxial. A good
example of this condition exists on the outer surface, or shell, of a cylindrical pressure
vessel. In this case, we know that there are hoop stresses, acting circumferentially, tending
to open up a longitudinal seam. There are also longitudi nal stresses tending to blow the
heads off. The situation may be represented as shown i n Fig. 12.
The stress-Slrain condition on the o uter s urface corres ponds to that shown in Fig. 2.
The two stresses uL and uH are principal stresses (no shear in the l ongitudi nal and hoop
directions) , and the corresponding stresses may be calculated using Eq. (4), i f we know (or
can esti ma te ) You ng 's modulus and Poisson's ratio.
EXAMPLE 4
Suppose we wish to detennine, by strain measurement, the stress in the circumferential or
hoop direction on the outer surface of a cyl i ndrical pressure vessel. The modulus of el astic i ty
of the material is 1 0.3 x t o 10 Pa, and Poisson's ratio is 0.28. By strain measurement the
hoop and longitudinal strains (in microstrain) are detenni ned to be
Solution
l;ff
v2
= 425
Using Eq. (3), we have
<1ff
=
"L = 1 15
10.3 x 10 10 (425 + 0.28 )(" 1 1 5) x 10- 6
1 - (0.28)2
E(EH + VEL)
I -
and
=
7
5. 1 1 x 1 0 Pa (7.42 x iol lbf/in.2)
Although we may not be directly interested, we have the necessary information to determi ne
the longi tudinal stress al so. as follows:
"L =
E(EL +
veH>
I - v2
10.3 x 1 0 10 c 1 1 s + 0.28 x 425) x 10-6
I - (0.28)2
7
= 2.6 1 x 10 Pa (3.8 x
tol lbf/in.2 )
Strain and Stress: Measurement and Analysis
It may be noted that the 2-to- 1 stress ratio traditionally expec ted for the
lhio-walJ
cylindrical pressure vessel does not yield a like ratio of strains. 11le strai n ratio is more
nearly 4 to I .
Use o f Eq . (4 ) pemtlts u s to detemtlne the stresses i n two orthogonal directio
However, this information gives the complete stress-1>train picture only when the two
angled directions coincide with the principal directions. I f we do not know the principal
directions, our readings would only by chan(.-e yield the maximum stress. In gcneraI,
if 1
plane stress condition is completely unknown, at least three strain measure ments must be
made, and it becomes necessary to use some form of three-element rosene (see Fig. 4).
From the strain data secured in the three directions, we obtain the complete stress-strain
picture. Stress-strain relations for roseue gages are given in Table 3.
Although only three strain measurements are necessary to define a stress situation
completely, the T-delta rosette, which includes a fourth gage element, is sometimes used
to advantage for the following reasons:
ri;:
1. The fourth gage may be used as a check on the results obtained from the other three
elements.
2. I f the principal directions are approximately known, gage d may be aligned wilh
the estimated direction. Then, if the readings from gages b and c are o f about lhe
same magnitude, it is known that the estimate is reasonably correct, and the principal
stresses may be calculated directly from Eqs. (4 ), greatly simplifying the arithmetic.
If the estimate of direction turns out to be incorrect. complete data are still available
for use in the equations from Table 3.
3. If the four readings are used in the T -delta equations in Table 3, an averaging effect
results in belier accuracy than if only three readings are used.
In spite of the advantages of the T-delta roselle, the rectangular one is probably the
most popular, with the equiangular (delta) kind receiving second greatest use.
EXAMPLE 5
Figure 13 illustrates a rectangular rosette used to determine the stress situation near a
pressure vessel nozzle. For thin-walled vessels, the assumption that principal directions
correspond to the hoop and longitudinal directions is valid for the shell areas removed from
discontinuities. Near an opening, however, the stress condition is completely unknown,
and a roseue with at least three elements must be used.
Let us assume that the rosette provides the following data (in microstrain):
Ea = 72,
Eb =
1 20,
Ee =
248
In addition, we shall say that
E=
II = 0.3,
20.7 x 1 0
10
Pa
Solution A study of the equation forms in Table 3 shows that for each case. the principal
strain, the principal stress, and maximum shear relations involve similar radical terms.
Therefore, in evaluating roseue data, it is convenient 10 calculate the value of the radical �
1
the first step. _ It will· also be noted that the second term in the principal s tress relations 5
370
TABLE 3: S1ress-Strain Relations for Rosette Gages•
c
b
Principal strains,
Principal s1resses,
a 1 . a2
./2(ea - Eb)Z + 2 (eb - Ec)Z
2eb - Ea - e"
Eu -
1an W
(I,
1 4,
! [ £• + ec ± _l _ x
l+v
2
1-v
,/2(ea - eb)2 + 2(eb - e,-)2)
E
Tmax
Ee
I Si-
Eb > �
2
�
a
--- x
2(1 + •,..>_______
Maximum shear.
0 < 8 < +90°
Rectangular
Hea + sc
__
2-+-2'""(± ./r
eb_
e_
eb-)'"'
l2]
c .,.
2-(e-.--
e 1 , e2
• Rererenca
r
8
-
Type of Rosette
t
�
�
�
�
� ''\ 8
� _L
,,
a
Equiangular (Delta)
j [sa + e11 + ec
±
[
,/2 (ea - Eb)2 + 2(eb - ec)2 + 2 (£c - £al2]
E ea + e11 + £c
-
3
1 -v
I
l+v
,/2(ea - £1>)2 + 2(eb - ec)2 + 2(£c - ea>t]
± -- x
E
+ •)
2-+-""
.;"'i-(e-.--e11--e""'
>2
+"""'2'"'<2 <-ee11-)...
Zc --s-."'
c >"'
--- x
3( 1
./3(ec - Eb)
(2£11 - ta - Ee)
Ee >
lb
t Note: 8 = the ansle of n:fereoce, """'5un:d posi1ive in the coun1erdockwise din:ction from the a-ll<ls of the ro.ae
lO 1he axis or the al&d>niically lalger stn:ss .
8
e £
H·+ 4
T·Delta
± /<ea - £ )2 + � (£11 - Sc)2
4
[
! e• + e4
2
1-v
/<ea E
'- x
± -
l+v
£4 ) 2 +
� (e11 - e..>2
]
]
>)--- --2( 1 + 1'"'..,..r
J<ea - £d)2 + � (£b - ec)2
--- x
2 (£,- - Eb)
.,/3 (ta - Ed)
Ee >
Eb
Strain and Stress: Measurement and Analysis
Flange
Weld
/
•'
/
B
�:---· - --·
/
/
A
Rectangular rosette
FIGURE
1 3:
Rosette installation near a pressure vessel nozzle.
equal to the shear stress; thus arithmetical manipulations may be kept to a minimum ir lhc
shear stress is calculated before the principal stresses are d•elenni ned. Hence,
J2(e0 - Eb) + 2(6b - ec)'l
and
e1
=
£2 =
� (72
1
+
248 + 193)
=
J2(72 - 120):t + 2( 120 - 248)2
= 193 µ-strain
=
256 µ-strain,
.
2 112 + 248 - 193) = 63 µ-stram,
20 7 x 1010
( 193) x 10- 6 = 1537 x !Cl4 Pa (2230 lbf/inh.
fmax =
2(1 + 0.3)
20.7 x 1010 72 + 248
o-1 =
x --- x 10 6 + 15'1
' 7 x 1..
u ·.4
0.7
2
= (473 1 + 1537) x 10" = 6268 x 10" Pa (909 1 lbflinh.
0-2 = (473 1 - 1537) x 10" = 3 1 94 x 10" Pa (4632 lbf/in.2 )
To
·
detennine the principal planes, we have
tan 28 =
_(2_s_b_-_s_a_-_sc_)
(Sa - Ee)
(2 x 120) - 72 - 250
=
(72 - 1 50)
205°,
or
28 = 25°
or
9
=
12.5°
or
==
102.5 °
0.46.
Strain and Stress: Measurement and Analysis
q1
•
62680 kPa
:_
u, :
__
,-
31940 k� - - - -
--
__
,_
12.3•
�i�n�
__
direction
FIGURE 14: Stress conditions determined from data oblained by the rosette shown in Fig. 1 3 .
measwcd counterclockwise from the axis o f element A . We must test fo r the proper quadrant
as follows (see the last line in Table 3):
e
Ea + Ee
2
=
72 + 248
= l 60
2
es
which is gr ater than Eb. Therefore, the axis of maximum principal stress does
between 00 and 90° . Hence, 8 = 102° . Figure 14 i l l ustrat this condition.
16
GAGE ORIENTATION AND INTERPRETATION OF RESULTS
not
fall
In a given situation it is often possible to place gages in several different arrangements
to obtain the desired data. Often there is a best way, however, and in certain instances
unwanted strain components may be canceled by proper gage orientation. For example,
it is often desirable to eliminate unintentional bending when only direct axial loading is
of primary interest ; or perhaps only th bending component in a shaft is desired, to the
exclusion of tors io n al strai ns.
e
Strain and Stress: Measurement and Analysis
The following discussion should be helpful in determining the proper posilioain r
gages and interpretation of the results. We will assume a standard bridge
0
shown in Table 4, and the gages will be numbered � n the following examples according
_
standard. When fewer than four gages are used, 11 1s assumed that the bridge configllrali
on
is completed with fixed resistors insensitive to strain.
This equation evaluates bridge output e0 for a given input e; . namely,
arrangeme!
lo�
( 16)
If we assume the resislance of each bridge
deo
=
arm
lo be variable, then
aeo
aeo
aeo
aeo
-dR1 + -dR2 + -dR1 + -dR4
a�
a�
a�
a�
·
(17)
By using Eq. ( 1 6) we can evaluate the various partial derivatives and write
deo
-;;
=
R4dR3
R2dR 1
( R 1 + R1)2
( R3 + R4)2
+
RJ �
( R3 + R4)2
(17a)
where dR 1 , dR2, dR3 , and dR4 are the various resistance changes in each of the bridge arms.
Ordinarily lhe gages used to make up a bridge will be from lhe same lot and
Rt
=
R2
=
RJ
=
R4 = R
Each gage may experience a different resistance change: hence we must retain the subscriplS
on the d R's; however, we can drop them from the R's. Doing so yields
de0
dR 1 - dR2 - dR3 + dR4
e;
4R
( 17b)
From Eq. ( I O), we have
( 18)
where
F = lhe gage factor.
En
= the main sensed by gage n
Combining Eq. ( 1 8) and ( 1 7b) gives us
( 1 9)
where the e's are 1he strains sensed by 1he respective gages.
Equation ( 1 9) aids in the proper interpretation of the strain results obtained from the
standard four-arm bridge in addition to assisting 1he stress analysl in lhe proper placemen!
and oricnlalion of gages for experimental measuremenls.
374
Strain and Stress: Measurement and Analysis
TABLE 4: Slrlli n-Gage Orientation
':'�rd Bridge Configuration
.
Requuemem for null:
le
z:b
/
•'"'1-----"'
===----./.]
v-
k = I
B
=
Bridge constant =
Compensates
RJ
R1
i = 4
R
R
Output of bridge
Output of primary gage
for temperature if "dummy" gage is used
in arm 2 or arm 3.
Does not compensate for bending.
Compensates for bending.
k=2
1\vo-arm bridge does nol provide temperature
compensation.
Four-arm bridge ("dummy" gages in arms 2 and
c
·&l---$9
/
..:
28
D
E
k
=
2(1 +
k= I
J,,'
.,!_
18
Two-arm bridge compensates for temperature and
bending.
--"
v)
2
3) provides temperature compensation.
1
��-b
Four-arm bridge compensates for temperature and
bending.
Temperature compensation accomplished when
"dummy" gage is used in arm 2 or arm 3.
Bridge is also sensitive to axial and torsional
components of loading.
375
Strain and Stress: Measurement and Analysis
TABLE 4:
Standard Bridge Configuration
Conlinued
Requirement for null:
k = Bridge constant =
F
�-- - - - -�
Rt
R2
=
R3
R4
Output of bridge
Output of primmy gage
Temperature effects and a>:ial and torsional eomponen1s
are compensated.
2
G
k =4
./
H
k
=
Four-arm bridge.
Temperature effects and a:tial and torsional
componenis are compensa.ted.
'\
2
a +b
a
�$
:
Tempera.lure effecis and v ial and torsional components
are compensated.
2
k= l+
G)
·
�··:v
Temperature effects are compensated.
Axial and torsional load componenis an: not
compensated.
376
Strain and Stress: Measurement and Analysis
TABLE 4:
_,.
Standard Bridge Configuration
Continued
.
Requirement for nuU:
k = Bridge conslant =
�
I�
R1
R
= R.t3
R2
Output of bridge
Output of primary gage
Torsion
J
k=2
$=·ffi·
fii}·
"J'Wo.arm bridge.
fimperature and axial load components arc
compensated.
Bending componenlS are
K
k=2
accentualed.
Two-arm bridge.
Temperature effcclS and axial load components
are compensated.
Relatively insensitive to bending.
L
k=4
Four-arm bridge.
-
_A
�
'
Sensitive to torsion only.
'- Ii:
(Gages I and 3 a re on opposite sides o f the shan
from gages 2 and 4.)
Strain and Stress: Measurement and Analysis
For example, when only one active gage is used (gage I, say),
FE1
=
-;;-- 4
E.q. (19) reduces to
deo
In further discussions we will assume that the tenn bridge corutanl abides by the defi nition
given in Section 8.2.
EXAM PLE 6
The simplest application uses a single measuring gage with an external compensating gage
as shown in Fig. 6 (also Case A, Table 4). This arrangement is primarily sensitive to axial
strain; however, it will also sense any unintentional bending strain. The compensating gag.:,
mounted on a sample of unstrained material identical to the test material, is located so that
its temperature and that of the specimen will be the same. In Ibis case,
Et
=
Ea + Eb + ET,
E2 = E T ,
EJ =
E4 =
0 ( a fixed resistor),
0 (a fixed resistor)
where
Ea
Eb
Er
Solution
=
the strain caused by axial loading,
= the strain caused by any bending component,
=
the strain caused by temperature changes
Substituting in Eq. ( 1 9) gives us
deu
F
=(Ea + Eb l
4
e;
If the bending strain is negligible,
de0
e;
=
Fea
4
and the bridge constant is unity. Note that any strains caused by temperature effects cancel.
EXAMPLE 7
The arrangement shown in Case G, Table 4, uses gages in each of the four bridge anns . Gages
I and 4 experience positive-bending strain components and gages 2 and 3 sense negative­
bcnding components. All gages would sense the same strains derived from axial load
and/or temperature should these be present. In addition, should the member be subjected
to an axial torque, gages I and 2 would sense like strains from this source, as would
gages 3 and 4. All gages would sense strain components of like magnitude from any
378
Strain and Stress: Measurement and Analysis
torque acting about the lon gi tudinal axis or the member; however, strains sensed by gages
I and 2 would be or opposite sign to those sensed by 3 and 4.
Solation
Substitution of all these effects into Eq. ( 19) yields
de0
e;
- =
(F)
-
4
(4Eb)
=
FEb
We see that only bending strai ns will be sensed and that the bridge constant is 4.
.16.1
Gages Connected In Series
Figure IS shows a load cell element using six gages. three connected in series in each or the
bridge arms I and 2. At first glance it might be thought that the three gages in series would
provide an output three times as great as that from a single gage under like conditions.
Such is not the case. for it will be recalled that it is the percentage change in resistance. or
dR/ R, that counts, not dR alone. It is true that the resistance change for one ann, in this
case, is three times what it would be for a single gage, but so also is the total resistance
three times as great. Therefore, the only advantage gained is that or averaging to eliminate
incorrect readings resulting from eccentric loading. The remaining two arms (not shown
in the figure) may be made up of either inactive strain gages or fixed resistors. The bridge
constant is I + 11.
Force
rn
I
FIGURE 15: Load cell em ploy ing three series-connected axial gages and three series­
connecled Poisson-ratio gages.
Strain and Stress: Measurement and Ana lysis
17
17 .1
SPECIAL· PROBLEMS
Cross Sensitivity
�
Strain gages are arranged with most of the strain-sensitive fi lament aligned with the sell$·_
live axis of the gage. However, unavoidably, a part of the grid is aligned transversely.
transverse ponion of the grid senses the strain in that direction and its effect is superim.
posed on the longitudinal output. This is known as cross sensitiviry. The error is Small
seldom exceeding 2 or 3%, and the overall accuracy of many applications does not wu:
rant accounting for it. For more detailed consideration the �eader is referred to [ I S), [ 16),
and [ 17).
1 7.2
Plastic Strains and the Postyield Gage
The average commercial strain gage will behave elastically t•:> strain magnitudes as high 85
3%. This represents a surprising performance when it is realized that the corresponding
umaxial elastic stress in steel would be almost
lbUi n. 2 (i f elastic conditions in the
steel were maintained). It is not very great, however, when viiewed by the engineer seeking
strain information beyond the yield point. When mild steel i:s the strained material, strains
as great as I S% may occur immediately following attainmen1L of the elastic limit before the
stress again begins to climb above the yield stress . Hence, the usable strain range of the
common resistance gage is quickly exceeded.
Gages known as posryield gages have been developed, extending the usable range
to approximately 10% to 20%. Grid material in very ductile condition is used, which is
literally caused to How with the strain in the test material. The primary problem, of course,
in developing an "elastic-plastic" grid is to obtain a gage factor that is the same under bOlh
conditions. Data reduction presents special problems, and for coverage of this aspect see
references ( 1 8) and [ 1 9 ) .
1,000,000
1 7.3
Fatigue Appllcatlons of Resistance Strain Gages
17.4
Cryogenic Temperature Appllcatlons
Strain gages are subject to fatigue failure in the same mamner as are other engineering
structures. The same factors are involved in determining their fatigue endurance. In general,
the vulnerable point is the discontinuity formed at the juncture of the grid proper and the
lead wire to which the user makes connection. Of course, as with any fatigue problem.
strain level is the most important factor in determining life.
lsoelastic grid material performs belier under fatigue conditions than does Constantan;
the carrier material is also an important factor. Figure 16 illustrates the effects or most or
the factors just discussed.
Strain measurement at extreme cryogenic temperatures must be done with an awareness
or both thermal and materials-related problems. As previousl.y discussed, resisiance strai n
gages are sensitive to temperature changes, and in this situation temperature compensation
is essential. Dummy gages, as discussed in Section I 0, may lbe added to the bridge circ uit
to cancel temperature effects on gage resistance. The dummy gage must be pl aced into
the same thermal environment as the measuring gage, and it must experience the same
temperature and noise level as the measuring gage. On the materials side or the problem.
adhesives and backings can become glass-hard and brillle at these temperatu res. Whereas
380
Strain and Stress: Measurement and Analysis
a�� .
;JI:
n1
5
-
1 0,000
Ji;
i
�
5000
6
-
4
:I,
.;
�
�..
c
i
(/)
1
3
1 000
' 2
01
II�
4
I
500
1 00
10
1�
1�
1�
1�
107
Reve rsed strain cycles
FIGURE 1 6: Relationship of endurance limit lo strain level for gages of various materials
and constructions (data from various sources including manufacturers' l iterature) .
the mechanical properties of certain grid materials are drastically cunailed, those of others
remain only slightly affected. Metal foil gages remain useful in cryogenic work. The
bonding agent, typically an epoxy, should ideally have a thennal expansion coefficient
similar to the gage back so as lo avoid thennally induced strains in the measurement (20).
17.5
High-Temperature Appllcatlons
Maximum continuous-use temperatures for polyimide or Kapton backed gages range from
about l oo•c to 250°C. Primary limiting factors are decomposition of cement and carrier
materials. At these temperatures grid materials present no particular problems. For appli­
cations at higher temperatures (to 1 000° C) some fonn of ceramic-base insulation must be
used. The grid may be of the strippable suppor1, free-element type with the bonding as
described next, or the gage may be of the "weldable" type.
Use of the free-element-type gage involves "constructing"lhe gage on the spot. Either
brushable or flame-sprayed ceramic bonding materials are used. Application of the fonner
consists of laying down an insulating coating upon which the free-element grid is secured
with more cement. The process demands considerable skill and carefully controlled baking
or curing-temperature cycling.
Flame spraying involves the use of a plasma-type oxyacetylene gun (2 1 ) . Molten
particles of ceramic are propelled onto the test surface and used as both the cementing
and insulating material for bonding the grid element to the test item. In both cases, leads
must be attached by spot-welding to provide the necessary high-temperature properties to
the connections. Lead-wire temperature-resistance variations may also present problems.
It is obvious that considerable technique must be developed to use either of these types
satisfactorily.
Strain and Stress: Measurement and Analysis
eal
A weldable strain gage consists of a res is ance element surrounded by a ceramic­
th . The gage is applied by spot-weldia
in sulat ion and encapsulated within a metal sh
edges of the assembly to the test member [22 ]. (A novel laser-based extensometer
:.::
usable
!en�
1 900° C or higher and having an overall accuracy of ±0.0002 in. over a 0.3-in . g
age
is described in [23 ] . )
1 7.6
Creep
Creep in the bond between gage and test surface is
a
factor sometimes ignored in strain­
gage work. This prob lem is approximately diametrically opposite to the fatigue problem in
oc
a
that i t is of i mportance only in static strain testing, pri m ari l y of the long-duration variety.
For example, residual stTcSses are
c asi on l ly determined by measuring the dimensional
relaxation as stressed material is removed. In this case,
eq a
the
g u
once and once only. The loading cycle cannot be repeated.
gage creep will result in direct errors
strain is applied to
u l to the ma nit de of the creep.
ev
be slow ly cycled, the creep w i l l appear as a hysteresis loop in the results.
a function of several things but is primarily determined by the strain l
used for bonding.
17.7
the gage
loadffeccan
Under these circumstances,
If the
This e
t is
el and the cement
Residual Stress Determination
ge l
Occasionally it is necessary to determine the residual stresses ex i sting in a structure or
h
machine element. These stresses are
rmi
ne ra l y developed during mechanical forming pro­
e at treatment. These stresses can be dete
ned by using the
strain-measuring techniques previou s l y described, although they ge neral ly destroy the sb11c­
ture being analyzed.
cesses, such as casting or
Consider the pressure vessel of Example
h
5. If i t is desired to esti mate the residual
stresses near a pressure-vessel nozzle due to weldi ng, 1he rectangular rosette may be applied
to the unpressurized vessel as shown. After t e various strain-gage lead wires are attached
to strain readout equipment, the region of pressure vessel containing the rosette is removed
(cut away) from the rest of the material, and the resulting change in strains from the gages
is recorded. Using these data (note the change in sign), the residual stresses ex ist i ng in the
unpress u rized vessel at this location may be e stim ated . (See Problem 30 for an example of
th is process.)
Most strain-gage manufacturers provide a s
pec a
i l strain rosette, whereby the strain­
gage elements are arranged in such a fashion that a single hole may be drilled, relieving the
stresses in the reg i on and thus eli m i n ating the need to comp le tely cut away the material.
18
ANAL REMARKS
In addition to being the key to experimental stress analysis, strain can be made an analog for
p ra
essentially any of the various mechanical inputs of interest to the engineer: force, torque,
m
displacement, pressure. tem e t ure , motion, and so on. For this reason strain gages are
very widely and successfully used as secondary transducers in measuring syste s of all
types. Their response characteristics are ex ce l len t, and they are liab le, relatively l inear.
re
and inexpensive. It is important, therefore, that the engineer concerned with ex perimenlal
work be well versed in the techniques of their use and application.
382
Strain and Stress: Measurement and Analysis
SUGGESTED READINGS
Dall y, J . W., and W. F. Ri ley. Experimental Stress Analysis. 3rd ed. New York: McGraw-Hill, 1 99 1 .
Doyle, J . F. Modem Experimental Stress Analysis. Ollches!er, England: John Wiley, 2004.
Reed. Strain Gage Users ' Handbook . Bethel , Conn.: Society for Experi·
mental Mechanics , 1992.
Hannah. R. L., and S. E.
Khan. A. S .. and X. Wang. Strain Meas,,rements and Stress Analysis. Upper Saddle River, NJ.:
Prenlice Hall. 2001 .
Murray, W. M., and W. R . M il ler. The Bonded Electrical Resisrance Strain Gage: A n lmmducticn.
New York: Oxford University Press , 1 992.
Doyle, J. F., and J. W. Phillips. Manual on Experimental Stress Analysis. 5th ed. Bethel. Conn.:
Society for Experimental Mechanics, 1989.
�.':'�� ;.
ci°?J.f'PROBLEMS
_;_�-l<i.r�:'
t.
2.
A simple
tension member with a diameter of 0.505 in. is subjected to an axial fon:e of
72 15 lbf. Strains of 1640 and -485 µ-strain are measured in the axial and trans\'erse
directions, respecti"Vely. Assuming elastic conditions, determine the values of Young's
modulus and Poisson "s ratio for the material. (Note: The diameter that is specified is
commonly considered a "standard" for cin:ularly sectioned metal specimens. Do you
know why?)
A single strain gage is mounted on a tensile member, as shown in Fig. 6. If the readout is
425 µ- s tra in , what is the axial stress (a) if the member is of steel, and (b) if the member
is of aluminum?
3.
A strain gage is centered along the length of a simply supported beam carry i ng a centrally
positioned , concentrated load. The beam is four gage lengths long. What correction
factor should be applied to care for the strain gradients if the purpose of the measurement
is 10 detecmine the maximum strain?
4.
Referring to the previous problem, what correction should be applied if the beam canies
a uniformly distributed load?
S.
Referring to Problem 3, what corrcction factor should be applied if, instead of bei ng
simply su p ported, the beam has built-in ends?
6.
A resistance·lype strain gage having a factorof2.00 ± 0.05 and a resistance of 1 2 1 ±2 n
is used in conjunction with an indicator having an uncertainty of ±2%. What maximum
uncertainty may be introduced by these tolerances? What probable uncertainty?
7.
Four strain gages are located a.� shown in Fig. 17. The gages are connected in a full
bridge (see Case G, Table 4). Their nominal resistances are 300 {2 and their gage factors
are 3.5. 1£ the bridge is powered with a regulated 5.6·V de soun:e, w hat will be the
voltage output •"Om:sponding to the maximum design load of 300/1 8 = 16.67 lbf?
8.
1\vo strain gages are mounted on a cantilever beam as shown in Fig. 7. If the total strain
readout is 620 µ-strain, w hat are the outer fiber stresses (a) if the member is of steel, and
(b) i f the member is of aluminum?
Strain and Stress: Measurement and Analysis
FIGURE 1 7 : Strain-gage/beam con figu ration dei;cribed in Problem 7.
9.
Assume a system configured as shown in Fig.
readouL
I 0, usi n:g a conventional osciUoscopc for
.
( a ) Make a list of variables that you feel will have a measurable effect on the o\'tnll
uncertai nty of the system. Indicate those that you would expect to change with
input magnitude and those that will be relativeli1 constant; see Eq. ( 1 1 ). Include
the u ncertainty due to limits of resolution of the readout method and note Iha!
some form of system calibration must be used, w ith its attendant uncellai111y.
( b ) Assign what you be l ieve to be reasonable u nci:rtain ties to each factor in your
l i s t and determine the overa ll uncertainty in th.e final readout. Finally, divide
the uncertaimies into two categories: those hav·ing a major effect on the overall
uncertainty and those of minor importance.
10.
A plastic spec imen is subjected to a biaxial
Uy
1 1.
= 605 lbf/in. 2 •
Calculate
stress con•dition for which u, = 1 380 and
are e, = 1 780 and Ey = 1 39.
Measured strains (in microstrain)
Poisson's ratio and Young ' s modulus.
Two identical strain gages are mou nted on a constant- m oment beam as shown in Fig. 1 8.
They are connected into a Wheatstone bridge as shown in Fig. 18(b). Wi th no load on
the beam the bridge is nulled with all anns having eq ual resistances. When the loads
are applied, a bridge output of 650 µ,V is measured. Determine the gage factor for the
gages on the bas i s of the following additional data: E
29.7 x IW lbf/in.2, Poisson's
ratio = 0.3. and the gage nominal resistance is 1 20 !2.
=
12.
A two-element strain roseue is mounted on a s i mple tensile specimen of steel. One gage
is a l ig ned in an axial direc tio n and the o t her in a transverse di rection . The gages arc
connected in adjacent arms of the bridge. If the to tal bridge readout (based on single-gage
20 x 1 0 10 Pa (29 x
ca l ibration) is 900 !-'-strain, what is the axial stress in pascal? E
106 lbf/in. 2 ) and 11 = 0.29.
=
13.
Each line in Table 5 represents a set of data correspondi1ng to a gi ven plane stress condi ­
tion. The first three colu mns are strains (in mic rostrain ) obtained using a three -elemenl
rectangular rosette. The final two items are material properties. For a selected sci of
data, determine the following:
( a ) The pri ncipal strains
( b) The princ i pal stresses
384
i[���-'-��0_2���_._��-'
Strain and Stress: Measurement and Analysis
.
,
(b)
e1 s
10V
FIGURE 1 8 : Detail o f arrangement described i n Problem 1 1 .
( c ) 1be maximum shear stress
( d ) Principal directions referred to the axis of gage a
Also, sketch the following:
( e ) Mohr's circles for stress
( r) Mohr's circles for strain
( g) An element similar to that shown in Fig. 1 4
Use units corresponding t o those given for E.
14.
Repeat Problem 1 3(a) through (g), assuming an equiangular rosette.
16.
H a rectangular-type rosette happens to be aligned such that elements a and c coincide
with the principal directions, then measured values of Ea and Ee will be £1 and £2 (or vice
versa). Show that under these circumstances the strain sensed by will be (£1 + e 2 )/2.
IS.
b
Devise a spreadsheet template ancVor a computer program IO evaluate the rectangular
strain rosette relationships in Table 3.
Strain and Stress: Measurement and Analysis
TABLE 5: Data for Problem 1 3
Ila
llb
- 320
1 585
1 250
- 1 020
2 10
470
-820
985
0
850
- 5
-5 1 0
E
E
2220
0
- 10 1 0
-210
E
E
E ·
17.
18.
19.
12
-£
E
Ile
10 I «J6 lbf/in.2
0 29 106 lbflin.22
425
15 r<>6 lbf/in.
-420 30 x 106 lbf/in.22
0 30 x 1 06 lbf/in.
680
x
x
x
-990
1440
-212
E
-E
-E
1.5 x 1010 Pa
20 x 1010 Pa
10 x 1010 Pa
E
E
E
11
0.29
0.3
0.28
0. 3
0.29
0.28
0.3
0.28
II
II
II
Devise a spreadsheet template and/or a computer program to evaluate the T-delta strain
rosette relationships i n Table 3.
Devise a spreadsheet template and/or a computer program to evaluate the equiangular
strain rosellc relationships in Table 3.
Values of 11 and E must be used i n equations for converting strains 10 stresses. For
steel, 11 = 0.3 and E = 30 x l o6 lbf/in. 2 are often assumed. In fact, however, for steel
11 may vary over the approximate range 0.27 10 0.32 and E may vary over a range of
about 28 x 1o6 10 32 x
lbf/in.2. Using the rectangular rosette strain values giw:n
in Example 5 in Section 1 5.2, analyze the effects of variations in assumed values 11
and E on the calculated pri nci pal strains, stresses, and di rections . It is suggested that
the spreadsheet te mplate (or program) wriuen for Problem 16 be used 10 minimize the
dru dge ry of number crunchi ng.
106
20.
21.
S how how strain gages may be mounted on a simple beam to sense temperature change
while being insensitive to variations in beam loading.
Pa and Poisson's ratio =
Two strain gages are mounted on a stee l shaft (£ = 20 x
0.29), as s hown in Case J, Table 4. The gage resistance is
n and F = 1 .23. When
a 250,000- Sl resistor is shunted across gage a 3.4-cm upward shift is recorded on the
face of the oscilloscope. When the shaft is torqued, a 5.7-cm shift is measured. For these
conditions, and assuming bending and axi al loading may be neglected,
1,
1oto
1 19
(a) Calculate the max i mu m torsional stress.
(b) What are the three pri nc i pal stresses on the shaft surface?
(c) Plot Mohr's circles for stress.
22.
(d) If a bending moment and/or axial load is present, how would the results be affected?
Strain gages A, 8, C, and D
g
are mounted on a plate subjected to a simple bendin
moment M and an axial load F, as shown in Fig. 19. Gages are to be i nserted into a
standard bridge (see Table 4) in order to accomplish the following:
386
Strain and Stress: Measurement and Analysis
F
F
+:
F
�--F-
M
$-
FIGURE 19: Arrangement of strain gages described in Problem 22.
(a) To sense bending only and, under this requirement, provide maximum bridge
output
(b) To sense axial stress only (eliminating bending stress)
each case, what will be the bridge constan� and will adjacent-arm temperature com­
pensation be accompli.•hed'!
In
23.
24.
25.
To determine the power transmitted by a 10 cm (3.94 in.) shaft, four strain gages are
mounted as shown in Case L, Table 4. They are connected as a four-arm bridge and the
output is fed to a chan recorder. Gage resistances are 1 1 8 '2 with a gage factor of 2. 1 . A
2 1 0,000 '2 calibration resistor may be shunted across one of the gages. Figures 20(a) and
(b) show the calibration and strain records, respectively. The chait speed is 1()0 mm/s.
The shaft is of steel with E = 20 x 1 0 10 Pa and Poisson's ratio = 0.3. Determine the
extreme and mean values of transmitted power in watts.
!
four axially aligned. identical strain gages an: equally spaced around a 1 -in. (3 1 .75mm)-diameter bar, as shown in Fig. 2 1 . The basic load on the bar is tensile; however,
because of a small load ecenlricity a bending moment also exists. If the strain readings
shown on the sketch are determined for the individual gages, what axial load and bending
moment must exist? Also determine the position of the neutral axis of bending.
four gages are mounted on a thin-walled cylindrical pressure vessel. Two of the gages
aligned circumferentially (these are gages I and 4 in the standard bridge, Table 4),
and the remaining gages 2 and 3 are aligned in the axial direction. (Note that this is
not necessarily an optimal configuration.) If the bridge output is 27.8 units when a
300,000· '2 resistor is shunted across gage I , and an output from the bridge of 47 units is
teeorded when the vessel is pressurized. what is the circumferential stress? Use F = 3.5,
R1 = 1 80 '2, E = 7 x 1010 Pa, and Poisson's ratio = 0.3. Assume that the conventional
2-to· I, circumferential-to-longitudinal stress ratio applies.
are
387
Strain and Stress: Measurement and Analysis
t
30 mm
Zero
(a)
28
mm
Zero
(b)
FIGURE 20: Recorder output for conditions of Problem 23.
1 (E1 • 989)
4
2
--++-���� + ����-+� ·
(£i! • 869)
(� = 959)
4
2
FIGURE 2 1 : Configuration of strain gages described in Probl em 24.
microstrain.
Values
of f
are
in
Strain and Stress: Measurement and Analysis
26.
27.
28.
Strain readouts from a rectangular strain rosette are Ea = 620, lib = -200, and Ee =
4 10 µ-strain. Assume that under the same conditions an equiangular roscne is mounted
and that its a element is aligned with the direction of the a element of the original
rectangular rosette. What readouts should be expected from the delta gage? Assume the
same gage factor and resistances for both roseues.
A strain gage having a resistance R1 = 120 Sl and a gage factor F = 2.0 is used in an
optimum ballast circuit. Whal is the maximum error over a range of 0 < E < 1500 µ­
strain relative 10 a "best" straight line referenced to E = O?
Analyze the effect of lead wire length and wire gage on the sensitivity or the following
strain gage circuits:
( a ) Ballast circuit
( b ) Circuit shown in Fig. 6
( c ) Circuit shown in Fig. 7
( d ) A four-arm bridge such as shown in Fig. 19
The following data may be useful if a quantitative analysis is being made.
Wire Size
A.W.G.*
Ohms per 1 000 ft
12
15
1 .62
3.25
6.5 1
1 0.35
26. 1 7
18
20
24
29.
JO.
31.
at 25°C
• American Wire Gage.
Analyze the uncertainly inherent in shunt calibration of strain-gage circuits.
mechanical engineering student wishes lo determine the internal pressure existing in a
diet soda can. She proceeds by carefully mounting a single-element strain gage aligned
in circumferential direction on the center or the soda can, as shown in Fig. 22. After
wiring the gage properly to a commercial strain indicator, she "pops" the Hip·lop lid,
which relieves the internal pressure. She notes that the strain indicator reads -400 µ­
strain. If the can body is made of aluminum with a thickness of 0.0 I 0 in. and a diameter
of 2.25 in., what was the original internal pressure of the scaled can?
A
Another student also performed the experiment described in Problem 30. Unfortunately,
he did not have access 10 the commercial strain indicator, and instead he had to construct
his own Wheatstone bridge circuiL His strain gage had an initial resistance of 120 Q
and a gage factor of 2.05. He used the single gage as one leg of the bridge, which he
powered with a 6- V battery. The bridge ou1put was fed to an amplifier (gain = I 000 ),
and the amp's output was read by a voltmeter. The student balanced the bridge circuit
before he opened the can. After the can was opened, the voltmeter indicated a voltage
of - 1 .57 V. What was the measured strain for his can?
Strain and Stress: Measurement and Analysis
Ci rcumferential
strain gage
AGURE 22: lnsuumented soda can .
REFERENCES
(1) Hetenyi, M. Ha11dbook of Experimental Stress Analysis. New York: John Wiley, 1950.
(2) Brookes-Smith. C. H. W., and J. A. Coils. Measurement of pressure, movement, accel­
eration and other mechanical quantities by electrostatic systems. J. Sci. Inst. (London)
1 4:36 1 . 1 939.
(3) Carter, B. C., J. F. Shannon, and J. R. Forshaw. Measurement of displacement and strain
by capacity mc:thods. Proc. Inst. Mecli. Eng. 152:215. 1945.
(4)
Langer, B . F. Design and application of a magnetic strain gage. SESA Proc. 1 (2):82.
1 943.
[SJ Langer, B. F. Measuremc:nt of torque transmitted by rotating shafts. J. Appl.
67(3):A.39, March 1 945 .
Mech.
(6) Khan, A. S., and X. Wang. Strai11 Measurements and Stress A11alysis. Upper Sadd le
River. N .J . : Prentice Hal l, 200 I .
(7) Thompson, K . On the electro-dynamic qualities of metals. Phil Trans. Roy. Soc. (l.Dn·
don) 146:649-75 1 , 1 856.
(8) Eato n, E. C. Rcsistanc.: strain gage measures stresses in conc rete . Eng. News
1 07:6 1 � 1 6, Oct. 1 93 1 .
Rec.
.
[9] Bloach, A . New methods for measuring mechanical stresses al higher freq uenc ies
Nature 1 36:223-224, Aug. 1 9. 1 935.
.
[10) Clark, D. S., and G. Datwyler. Stress-strain relations under ten sion impact loading
Proc. ASM 38:98- 1 1 1 . 1 938.
390
Strain and Stress: Measurement and Analysis
[ll) Mills, D., III. Strain gage waterproo fing methods and installation of gages on propeller
stnll of USS Saratoga. SESA Proc. 1 6( 1 ): 1 37, 1958.
f
[12) Frank. E. Series versus shunt bridge calibration. Instr. Automation 31 :648, 1 958.
[13] Campbell, W. R., and R. F. S uit, Jr. A transistorized AM-FM radio-link torque telemeter
for large rotati ng shafts. SESA Proc. 14(2):55, 1957.
[14] Baumberger, R., and F. Hines. Practical reduction formulas for use on bonded wire
strain gages in two-di mens ional stress fields. SESA Proc. 2( 1 ) : 1 33, 1944.
[15) Perry, C. C., and H. R. Lissner. The Strain Gage Primer. 2nd ed. New York: McGrawHill, 1962, p. 1 57.
[16] Meier, J. H. On the transverse sens i tivi ty of foil gages. Exp. Mech. 1: July 1 96 l .
(17} Wu, C . T. Transverse sensitivity o f bonded strain gages. Exp. Mech. 2 : 33 8, Nov. 1962.
[18) Plan, T. H . H. Reduction of strain rosettes in the plastic range. J. Aerospace Sci. 26:842,
December 1959.
[19) Ades, C. S. Reduction of strain rosettes in the plastic range. Exp. Mech. 2:345, Novem­
ber 1962.
[20) Timmerhaus, K. D., and T. M. Flynn. Cryogenic Process Engineering. New York:
Plenum, 1 989.
[21) Leszynski, S. W. The development of flame sprayed sensors. /SA J. 9:35, July 1962.
(22) Rastogi, V. • K. D. Ives, and W. A. Crawford. High-temperature strain gages for use n
sodium environments. Exp. Mech. 7:525, December 1 967.
(23) Kamie, A. J., and E. E. Day.A laser extensometer for measuri ng strain at incandescent
temperatures. Exp. Mech. 7:485, November 1967.
ANSWERS TO SELECTED PROBLEMS
2
8
11
20
23
v =
0.3
(a) u = 1 2,750 psi
= 9300 ps i
1 .68
t!. e / ei = (F /4)(2er)
Pmn = 53,800 watts
(a) u.
F
=
0
Measu rement of Pressu re
1
2
3
4
5
6
7
8
9
10
11
INTRODUCTION
STATIC AND DYNAMIC PRESSURES IN FLUIDS
PRESSURE-MEASURING TRANSDUCERS
MANOMETRY
BOURDON-TUBE GAGES
ELASTIC DIAPHRAGMS
ADDmONAL PRESSURE TRANSDUCERS
MEASUREMENT OF HIGH PRESSURES
MEASUREMENT OF LOW PRESSURES
DYNAMIC CHARACTERISTICS OF PRESSURE-MEASURING SYSTEMS
CALIBRATION METHODS
I
INTRODUCTION
Pressure is the nonnal force exerted by a medium, usually a Huid, on a unit area. In
engineering, pressure is most often expressed in pascal (I Pa = N/m2 ) or pounds-force
per square inch (lbf/in. 2 , or psi). 'JYpically, pressure is detected as a differential quantity,
that is, as the difference between an unknown pressure and a known reference pressure.
Atmospheric pressure is the most common reference, and the resulting pressure difference,
known as gage pressure, is of obvious importance in dctennining net loads on pressure
vessel and pipe walls. In other cases, the reference pressure is taken to be zero (a complete
absence of press u re), and the pressure measured is called absolute. In the English system
of units, gage and absolute pressure are distinguished by writing psig and psia, respectively.
Figure I illustrates these relationships.
Pressure is often expressed in units of hydrostatic force per unit area at the base of a
column of liquid, usually mercury or water. For example, standard atmospheric pressure
( 1 0 1 ,325 Pa or 14.696 psia) is approximately equal to the pressure exerted at the bottom of
a mercury column 760 mm (or 29.92 in.) in height. 1 Therefore, one often finds standard
atmospheric pressure specified as 760 mrnHg or 29.92 inHg, even though the fundamental
unil of pressure is neither millimeters nor inches. Pressure measurement using l iquid
columns is called manometry (see Section 4 ).
An absolute pressure less than atmospheric pressure is often referred to as a vacuum.
Vacuum is occasionally measured in terms of a negative gage pressure (so that - 7 psig
When the vacuum is nearly complete, however, small variations in
= 7 psi vacuum).
•n.e pn:ssure is equal 10 pgh, for p the liquid dcnsily. g the local value of the gravita1ional body force, and h
1he column height Since liquid density varies with temperature. the precise height ror I atm pressure depends on
both local gravily and ambient lcmperature.
Inc.
From Mechm1ical Measurrments. Sixth Ed i tion, Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V.
Copyright () 2007 by Pearson Education,
Published by Prentice Hall. All righis reserved.
393
Measurement of Pressure
!!
::J
�
Atmospheric
pressure
p-
Gage pressure
P1 (negaUve)
P,
0
Barometric
pressure
--------
------------ -
Absolute
pressure P1
Zero absolute
pressure
FIGURE I : Relations among absolute, gage, and barometric presSW"eS.
TABLE 1 :
Relation of Various Units of Pressu re to the Pascal [ I ). H20 at 4°C; Hg ll O"C.
1 microbar = 0. 1 Pa
I µmHg = 0. 1 333 Pa
1 N/m 2 = 1 Pa
1 mmH 2 0 = 9.807 Pa
1 mbar = 100 Pa
I mmHg = 133.3 Pa
1 torr = 133.3 Pa
2
I inH20
I kPa
I ftH20
1 i n Hg
I psi
I bar
1 atm
= 249 . 1 Pa
= 1000 Pa
= 2989 Pa
= 3386 Pa
= 6895 Pa
= t<>5 Pa
= 101 325 Pa
atmospheric pressu re can produce large errors in the measured gage pressure. Hence.
absolute pressure is always used to describe a high vacuum. The low absolute pressures of
a high vacuum are sometimes expressed in u n its of torr ( 1 t orr = I mmHg) or micrometerS
of mercury (µmHg).
. High pressures are o fte n written in unit� of atmospheres ( I atm = 1 .01 325 x llP Pa),
bar ( 1 bar = HP Pa), or megapascal ( I MPa = 1 06 Pa). Selected units of pressure
measuremen t are summ ari zed in Table I .
STATIC AND DYNAMIC PRESSURES IN FLUIDS
When a fluid is at rest, a small press u re sensor in it will read the same static pressure at 8
given position in the fluid no matter how it is oriented. In other words, at any particular
point in the fluid , the small surface experie nces the same pressure whether it faces upward
or d ow nward or left or righ t . Gravitational force can produce a vertical pressure gradient.
caus i ng a higher pressure at lower leve l s in the lluid, but at any particular level the pressure
on the small surface remains independent of its orientation.
.
When the fluid is in motion, a surface placed in it may experience not only the stauc
394
Measurement of Pressure
l!ll!l'li=i.'i��-
Flow
-
-
-
A
Pressure
transducers
8
FIGURE 2: Impact tube, A, and static-pressure tube, B. Tube A senses the total or stagna­
tion pressure.
pressure, but also a dynamic pressure. For example, if the surface is perpendicular to the
direction of flow, the fluid must come to rest at the surface. This stagnation of the flow
results in the conversion of kinetic energy into an additional pressu re on the surface, much
like the pressure you feel when standing in the wind. On the other hand, if the surface is
parallel to the How direction, the Huid is not stagnated and Hows along the surface without
creating any additional pressure. Thus, a pressure transducer's reading in a moving Huid
will depend on its orientation.
In Fig. 2, two small tubes each sample the pressure in an air duct. Pressure tap B
senses only the static pressure in the duct. Tube A, on the other hand, is aligned so that the
How impacts against its opening, and it senses the total or stagnation pressure. The static
pressure is identical to the pressure one would sense if moving along with the airstream.
The stagnation pressure can be defined as that which would be obtained if the stream were
brought to rest isenlropically. The difference between the stagnation and static pressures
results from the motion of the Huid and is called the velocity pressure or dynamic pressure:
Dynamic pressure
=
stagnation pressure - static pressure
This pressure difference can even be exploited for measurement of the Huid's velocity.
We see , therefore, that to obtain and interpret pressure measurements properly, How
conditions must be taken into account. Conversely, to interpret flow measurements properly,
the pressure conditions must be considered.
Sound Pressure
Sound waves propagate in an elastic medium as longitudinal pressure variations (along
the path of propagation), with press ure H uctuating above and below the static pressure. The
instantaneous difference between the pressure at any point and the time-average pressure
there is called the sound pressure. Because sound pressures are normally relatively small,
they are often expressed in units of m ic robar ( I µ.bar = 10- t Pa). Measurement of sound
pressure is accomplished with microphones and related apparatus.
395
Measurement of Pressure
3
PRESSURE-MEASURING TRANSDUCERS
Pressure measurement most often involves converting a pressure difference into a
fi
and t hen measuring that force. In some cases, the force may be measured dilectJ on:e
c ompari n g it to the weight of an object or of a column of liquid . In o� casay
force may be used to produce a deflection in an elastic mem ber, such as a Cllned 'tube
or a d i aphragm . This deflection, in tum, may be measured either mechanically or by
sccondafl:' electrical tran ucer, such � a strain gage or ind ucti�•e or capacitative sensor.
01her devices, a pressure · md uced stram may change the electnocal properties of the elas
tic
member itself, as when a piezoelectric material generates a ch arge in response to a load.
Many ot her me1hods of pressure measurement have been devisec� particularly in connection
with vacuum systems.
�
�
�
Pressure, as force per unit area, must ultimately be relall!d back to the staodanls of
length, mass, and time which define force and area . There is no separate standard for
pressure . The most common way to connect pressure to the standards is 10 use a calibrated
mass subjected to a known value of gravity to create a force that is supported by the unknown
pressure as applied to a carefully measured area. For example, the piston force balance or
dead-wei gh t tester (Fig. 3) produces a constant pressure that may be used to calibrate other
press ure gages. Masses sitti ng atop a piston are supported by the pressure of a fluid below.
If the piston's area is known, the pressure is calculated easi ly using the known masses and
the local value of gravity. These devices are commonly usedl by standards laboratories
for high-accuracy calibration of other pressure sensors. When properly applied, they are
accurate to belier than 100 parts per million (2). Similarly hi gh accuracy can be obtained
by ba lanc i ng a pressure force against the hydrostatic pressure at the base of a column of
liquid whose height is known; this is called manometry.
Weights
FIGURE 3: Dead-weight tester.
Measurement of Pressure
When then pressure to be measured varies rapidly in time (a dynamic pressure mea­
surement), the transducer used must have sufficiently high frequency response to track the
signal. In general, only electromechanical transducers are adequate for dynamic pressure
measurements, and then only if the associated elastic members are light enough to respond
rapidly to changing conditions. In addition to the response of transducer itself, we must take
account of the responsiveness of any connecting tubing or chambers between the transducer
and the point at which pressure is to be measured.
As an example, suppose that a diaphragm-type transducer were to be used for mea­
suring the pressu re at a specific point on an aircraft skin. In such an application, it may be
undesirable 10 place the diaphragm Hush with the aircraft surface. Possibly the size of the
diaphragm is IOO great in comparison with the pressure gradients existing; or perhaps Hush
mounting would disturb the surface lo IOO great a degree ; or it may be necessary to mount
the pickup internally to protect ii from large temperature variations. In such cases, the
pressure would be conducted to the sensing element of the pickup through a passageway,
and a small space or cavity would exist over the diaphragm. The passageway and cavity
become, in essence, an integral part of the transducer, and the mass, elasticity, and damping
properties of the passage and chamber contribute to the overall response of the system. It
is obvious that it would be insufficient to consider only the transducer characteristics in
assessing the frequency response. This issue is discussed further in Section 10.
International conferences on pressure mea5urementare held periodically by the national
standards laboratories of several dozen nations. The proceedings of those conferences
should be consulted for detailed information on pressure measurement al the highest levels
of accuracy [3, 4).
4
MANOMETRY
Manometry refers to the measurement of pressure by comparison to the hydrostatic pressure
produced by a column of liquid. The manometer is one of the mosl elementary measuring
devices imaginable. It is simple, inexpensive, and relatively free from error, and yel ii may
be arra nged to almost any degree of sensitivity. Its major disadvantages lie in its pressure
ranges and in its poor dynamic response. II is nol very practical for measuring pressures
greater than, say, 200 kPa (30 psi), and it is incapable of following any but slowly changing
pressures.
A simple well-type manometer is shown in Fig. 4. A force-equilibrium expression
for the net liquid column is
(P1a A
-
"2,, A) =
Ahp
(�)
(I)
.( )
( l a)
or
( Pia
-
"2,, ) = PJ
= hp .!..
8c
Measurement of Pressure
Fluid
FIGURE 4: Well-type manometer.
where
Pia and Pia = ihe applied absolute pressures,
=
P.i = ihe pressure difference or differential pressure,
p
the density of the fluid (mass/volume),
h = the net column height, or "head,"
g = the gravitational body force, and
Be = the dimensional constant
In practice, pressure P"lo is commonly atmospheric and
(2)
where
Pig = the gage pressure at point 1
Perhaps it would be w ise at this point 10 make sure we understand the units to be used.
In simpli fied form the preceding equations may be written as
P.i = hp
( t)
Substituting units in the right-hand side of the equation, we have, for the SI System,
(m)(kglm3 )(mls2 )(N · s2 /kg · m) = N/m 2 = Pa
398
(2a)
Measurement of Pressure
Using the English system of units, we have
(ft)(lbmtft3 )(ft/s 2 )(1bf .
s2 /lbm . ft) = lbftft2
EXAMPLE 1
Calculate the pressure at the base of a column of water 1 m (3.281 ft) in height if the local
gravity acceleration is 9. 75 mts2 (3 I .99 ft/s2 ) and the temperature is 200C (68°F).
Solation Assume that the density of water at 20°C is 998.2 kgtm3 (62.32 lbm/ft3 ). Using
SI units, we have
Psi = (1)(998.2)(9.75/ 1 )
=
9732 Pa
Using English unils, we have
Piing = (3.28 1 ) (62.32)(3 1 .99/32. 1 7) = 203.3 lbflft2 = 1 .412 psi
Because the fluid density is involved, accurate work will require consideration of
temperature variation of density: The manometer possesses a cenain amount of temperature
sensitivity.
When the pressure Pia is atmospheric and the absolute pressure Pia is made to be zero
(as by sealing and evacuating the top of the tube in Fig. 4 ), we obtain the ordinary barometer.
In this case the fluid has traditionally been mercury. In recent years, health regulations have
strongly discouraged the use of mercury in many settings, so that the traditional mercury
barometer has become relatively uncommon.
Figure 5 illustrates the function of the U-tube manometer. Pressures are applied to
both legs of the U, and the manometer fluid is displaced until force eq uilibrium is attained.
Pressures Pia and Pia are transmitted to the manometer legs through some fluid of density
Pt , while the manometer fluid has some greater density Pm . In general, we See thai
Pi a
-
Pi,,,
= h (Pm
-
p, )
(t)
(3)
In many cases the density difference between the two fluids is great enough that the lesser
density may be neglected (Pm » Pt )-when air is the transmitting fluid and water is the
measuring fluid, for instance. In that case, we have what might be called a simple U-tube
manometer, and Eq. (3) reduces to
(3a)
EXAMPLE 2
Suppose the manometer fluids in Fig. 5 are water and mercury. This situation might occur
when a manometer is used to measure the differential pressure across a venturi meter through
which water is flowing. We will consider both systems of units used in this book along with
the following data:
399
Measurement of Pressure
h
Manometer
ftuld with
density� Pm
FIGURE 5: U-tube manometer.
h = 1 0 in. or
i ft (0.254 m),
Density of water = 62.38 lbm/ft3 (999.2 kg/m3 ),
Specific gravities of H2 0 and Hg = 1 and 1 3.6, respectively,
Standard gravity applies (32. 1 74 ftls2 and 9.80665 m/s2 )
Determine the differential press ure .
Solution
In the English system of units,
Pia - Pi.a
(5 )
= 6
( 1 3.6 - 1 ) (62.38)
( 32. 1 74 )
_
32 174
= 655 lbf/ft2 = 4.55 psi
For the SI system of units, we have
Pia
-
Pi.a
= (0.254)( 1 3.6 -
1 )(999.2)
= 3 1 ,360 Pa = 3 1 .4 kPa
(9.80665 )
--1
It is left for the reader to show that the two answers represent lhe same physical quantil)'
and that the unit balance is proper in each case.
In general, a simple U-tube manometer will have a greater pressure range when a more
dense measuring fluid is used and a greater sensitivity (change in height per unit change 111
Measurement of Pressure
A = cross-sectional area
9
Fluid
FIGURE 6: Inclined-type manometer.
pressure) when a less dense fluid is used. From Eq. (3a),
Se
.. .
ns1uv1ty
h
=-=
t.. P
1
--­
(3b)
Pm(g/gc )
Greater sensitivity may also be obtained through a displacement amplification scheme,
two of which are shown in Figs. 6 and 7. For the single inclined leg (Fig. 6), in which
h = L sin8,
Pm (L sin 8)g
Pl a =
+ Pia
(3c )
gc
so that
Se
L
.. .
ns1Uv1ty = - = ----­
t.. P
Pm sin 8 (g /gc)
P + ,dP
p
Y
IP""...._"i'l u
p,
o --M-14-P2
ir.11;ai1ri'tirfaCi
T"
b
Fluid level
'1P applied
Initial ftuld
level
h
Final interface
d
FIGURE 7: Two-fluid manometer with reservoirs.
(3d)
Measurement of Pressure
In the case of the two-Ouid type mano meter (Fig. 7 ),
Sensitivity =
...!
!.._ =
t!.P
1
[ (d/D) 2 (P2 + Pt ) + (/>2
-
P 1 ))(g/gc)
(4)
When the reservoir diameters arc l arge and the Ouid densities are similar, the sens itivity can
be su bstantial (a micromanometer). In c o mpari s o n 10 the simple U-tube manometer, lhe
deftection amplification, M, equals
(4a)
where p,. = the density of the Ouid in the s i mp le manometer a nd f>2 > Pl . An extensive
survey of mi cromanometers has been given by Brombacher (5).
For h igh -acc u racy measwements, special attention must be given to the measurement
of the l iq u id level in the manometer. The most common means of locating the liquid
surface is by direc t visual comparison to an adj acent scale. With appropriate lighting and
lens arrangements, it is possible to obtain accuracies of several micrometers . For higher
resolution, nonvisual tec h niques are used, including capacitative detection, interferometry,
and ul tras ound [6].
5
BOURDON-TUBE GAGES
B ourdon-lube gages, like other elastic transducers, operate on the principle that the deftec­
tion or deformation accompanying a balance of pressure and elastic forces may be used as
a meas ure of pressure. A tube, norm a l l y of o val section, is in i tial ly coiled into a circular
arc of radius R, as shown in Fi g . 8. The i ncl uded angle of the arc is usuall y less than 360°;
however, in some cases, when increased se nsitiv ity is des i red , the tube may be fanned into
a helix of several turns .
As a pressure is a pplied to the tube, the oval section lends to round out, becoming
more circular in section. The inner and o uter arc l engths w i ll re m ai n approxim ate ly equal
to their original lengths, and hence the only recourse is for the tube to uncoil. In the s imple
pressure gage, the movement of the end of the tube is c ommun i c ated through linkage and
gearing to a pointer whose movement over a scale becomes a measure of presswe . In other
forms, the end of the tube may be linked to a p os i tio n transducer, such as an LVDT, in
order to track the displacement. The mechanics of Bourdon-tube action were the subject of
many analyt i c al studies pri or to the development of finite element computation [7], but the
primary observation is that the lip d is place ment i ncreases nearly linearly with the pressure.
6
Bou rd on -t u be gages are available for a very wide range of pressures . Full-scale
readings on commercially available gages range from 100 kPa or less to 1 50 MPa or more.
Accuracy is typ i cal ly between 0.5 and 2 % of the full sc a le reading. Bourdon-tube gages are
gene rally useful only when the pressure is static or sl owly changing. Friction and backlash
in the li n kage or gearing may cause hysteresis in t he readings when the direction of pressure
variati on c hanges .
ELASTIC DIAPHRAGMS
Many dynamic pressure - m easuring devices use an elastic diaphragm as the primary pressure
transducer. Such diaphragms may be either Hat or corrugated; the Hat type [Fig. 9(a)) is
402
Measurement of Pressure
FIGURE 8: Basic Bourdon tube.
often used in conjunction with elecuical secondary uansducers whose sensitivity enables
detection of very small diaphragm deflections, whereas the corrugated type [Fig. 9(b)) is
particularly useful when larger deflections are required, perhaps for driving mechanical
linkages.
Diaphragm displacement may be transmiued by mechanical means to some form of
indicator, perhaps a pointer and scale as is used in the familiar aneroid barometer. For engi­
neering measurements, particularly w·hen dynamic results are required, diaphragm motion
is usually sensed by some form of elecuical secondary transducer, whose principle of oper­
ation may be resistive, capacitive, inductive, piezoelectric or piezoresistive, among other
possibilities.
Diaphragm design for pressure transducers generally involves all the following require­
ments 10 some degree:
I. Dimensions and total load must be compatible with physical properties of the mate­
rial used.
2. Flexibility, and thus sensitivity to pressure change, must provide diaphragm deflec­
tions that match the input range of the secondary transducer.
3. Volume of displacement should be minimized to provide reasonable dynamic response.
4. Natural frequency of the diaphragm should be sufficiently high to provide satisfactory
frequency response.
5. Output should be linear.
403
Measurement of Pressure
Section A-A
E��;�
(a)
(b)
Section B--8
FIGURE 9: (a) Flat diaphragm, (b) corrugated diaphragm.
6.1
Flat Metal Diaphragms
Deflection of Hat metal d iaphragms is limited either by stress rc'(! ui rements or by devia·
tion from l inearity. A general rule is that the maximum deflec�ion that can be tolerated
while mai n taining a linear pres sure-d isp lace ment relation is about 30% of the d iaphragm
thic knes s.
In certain cases secondary transducers fC"IUire physical connc:ction with the diaphragm
at its ce n ter. This is ge nera l l y 1rue when mechanical l inkages are u1sed and is also necessary
for cenain types of electrical secondary transducers. In additi on , auxiliary spring force is
sometimes introduced to increase the diaphragm deflection const ant . These requirements
make necessary some form of boss or re i n forcem ent at the cente'r of the diaphragm face,
which reduces a di aphragm flexibility. When a central connect ion is made, a concentrated
force will nonna lly be appl ied . In general , therefore, the diaphragm may be simultane·
ously subjected to two de fl ect i on forces, the distributed pressure load and a central concen­
. tra ted fo rce .
An u ndesi rab le characteristic of simp le Hat diaphragms that is often encountered is a
nonlinearity referred to as oil cann in g. The tenn is derived from the action of the bottom
of a simple oil can when it is pressed . A slight unintentional di m pl i n g in the assembly of a
flat-diaphragm pressure pickup is d i fficu lt to eliminate unless special precautions are taken.
D i aph ragm design is covered in detail in reference [8 ].
6.2
Corrugated Metal Diaphragms
Corrugated diaphragms are n orm ally used in larger diameters than the Hat types. Corru­
gati ons permit increased linear deflections and reduced stresses. Since the larger size and
deflection red uce the dy namic response of the corrugated diaphra�:ms as compared with the
H at type, they are more commonly used in static applications.
Measurement of Pressure
Two corrugated diaphragms are often joined at their edges to provide what is referred
to as a pressure capsule. This is the type commonly used in aneroid barometers.
Similarly, metal bellows are sometimes used as pressure-sensing elements. Bellows
are generally useful for pre."5 ure ranges from about 3 kPa to 10 MPa full scale. Hysteresis
and zero shift are somewhat greater problems with this type of element than with most of
the others.
6.3
Semiconductor Diaphragms
Diaphragms can be micromachined directly onto silicon chips. producing semiconductor
diaphragm pressure sensors. Semiconductor diaphragms are usually instrumented with
directly embedded piezoresistive or capacitative sensors that track their deflection (9).
Bridge circuitry, amplification, temperature compensation, and other signal conditioning
can be provided directly on the chip, using standard integrated circuit technologies. Some
models are quite inexpensive. In others, the silicon clement may be isolated from the mea­
sured environment by a steel diaphragm to which it is coupled, providing a fairly rugged
transducer. Because these sensors are rather small, with diaphragms of a few square mil­
limeters, their basic frequency response can be very high, exceeding I 00 kHz, although
packaging requirements may lower the achievable value substantially. A wide range of
commercial products now incorporate semiconductor diaphragm pressure sensors, includ­
ing the engine control systems of most automobiles and battery-powered handheld pressw-e
transducers.
Silicon-diaphragm pressure transducers are sometimes simply called "piezoresistive"
transducers. Transducers with full scale ranges of as high as 100 MPa or as liule as 150 Pa
are on the market.
7
ADDmONAL PRESSURE TRANSDUCERS
A wide variety of pressure transducers are commercially available. Many are based on
diaphragm deflection, and most electromechanical transducer principles have been applied
to diaphragm-based pressure pickups. Other types of elastic deflection are also used to
convert pressure 10 strain or displacement. The following examples are only representative
of the possible variations.
7.1
Strain Gages and Flat Diaphragms
An obvious approach is simply to apply strain gages directly to a diaphragm surface and
calibrate the measured strain in terms of pressure. One drawback of this method is the
small physical area available for mounting the gages; for this reason, gages with short gage
lengths or of custom design must be used
Special spiral grids may used in constructing the strain gage. Grids are mounted in the
central area of the diaphragm, with the elements in tension. Ordinary strain gages may also
be used by mounting them [ 10). When pressure is applied to the side opposite the gages, the
central gage is subject lo tension while the outer gage senses compression. The two gages
may be used in adjacent bridge arms, thereby adding their individual outputs and simul­
taneously providing temperature compensation. Some commercial transducers employ
four-gage bridges made by depositing thick-film resistors directly onto the diaphragm. For
new designs, the effect of gages on diaphragm stiffness and mass should be taken into
account.
.
Measurement of Pressure
Pressure
Section A-A
FIGURE
1 0: Loca1ion of strain gages on Hat diaphragm.
Other designs involve connecling a bending beam to lhe center of the diaphragm by
means of a rod. The strain gages are mounted on the beam. The rod serves to provide
lhennal isolation of 1he gages from the diaphragm.
S1rain-gage based pressure transducers are available in full-scale ranges from roughly
IO kPa to 100 MPa.
7.2
Inductive Transducers
Variable inductance is somelimes used as a Conn of secondary transduction with a diaphragm.
Figure 1 1 illustrates one arrangemcm of this sort. Flexing of the diaphragm due to applied
press ure causes ii 10 move 1oward one pole piece and away from the other, 1hereby aller­
ing lhe relative induclances. An inductive bridge circuil may be used, as shown. Variable
inductance or variable reluctance transducers are fairly rugged and provide good sensitivity.
As a resull of lhe need for ac excitalion and lhe surrounding coils, lhey are relalively bulky
lransducers and perhaps bes1 suited for laboratory installations. Inductive transducers are
available wi1h f\111-scale ranges as low as 20 Pa and as high as 70 MPa.
7.3
Piezoelectric Transducers
Piezoeleclric transducers are typically constructed by placing several quartz columns behind
a metal diaphragm with a compressive preloading. Aexion of the diaphragm stresses the
quartz columns and creales a piezoelectric charge on them. By detection of this charge.
the pressure is sensed. Because piezoelectric charge dissipates fairly quickly, these devices
are bes1 sui1ed for dynamic pressure measuremenis. The high stiffness of these transducers
Measurement of Pressure
Differential
pressure connections
Signal
output
FIGURE 1 1 : Differential pressure cell with variable inductance secondary transducer.
gives them very high natural frequencies, with values of 1 50 kHz being representative. On
the low-frequency end, piezoelectric transducers are not usually suitable for cycle periods
of more than a few minutes' duration. The sensitivity of these transducers is usually high.
A charge-amplifier is used to convert the piezoelectric charge to voltage, but commer­
cial devices may incorporate this type of signal conditioning within the transducer housing.
Piezoelectric pressure transducers have full-scale ranges from I 00 kPa to 1 GPa. They have
a variety of applications, including measurements in internal combustion engine cylinders,
hydraulic systems, and ballistic devices [ 1 1 ). Low-pressure piezoelectric sensors, capable
of resolving 0. 1 Pa pressure variations, are used as microphones.
7.4
Capacitative Transducers
Capacitative pressure transducers use a Hexible diaphragm as one pi ate of a capacitor.
The change in capacitance caused by the diaphgram displacement is detected using either
a capacitance bridge or the change in frequency of an electrical oscillator. Capacitative
transducers may have extremely high sensitivity, and they are thus very useful for high­
resolution vacuum pressure measurements. Full-scale ranges from about 100 Pa to about
1 0 MPa are available. Drawbacks of capacitative sensors are temperature sensitivity and
long-term drift [ 12, 13).
Capacitative diaphragms a re also the basis of th e condenser microphone.
407
Measurement of Pressure
12:
FIGURE
Flattened-tube pressure cell that employs resistance strain gages as secondary
transducers.
7.5
Strain-Gage Pressure Cells
Any form of closed container will be strained when pressurized. Sensing the resulting strain
with an appropriate secondary transducer, such as a resistance strain gage, will provide a
measure of the applied pressure. The term pressure cell is sometimes applied to this type or
pressure-sensing device, and various forms of elastic contaim:rs or cells have been devised.
For low pressures, a pinched tube may be used (Fig. 1 2 ) . This arrangement supplies a
bending action as the tube tends to round out. Gages may be placed diametrically opposite
on the flattened faces, as shown, with two unstressed tem1perature-compensating gages
mounted elsewhere. This arrangement completes the electrical bridge.
13.
Probably the simplest form of strain-gage pressure transducer is a cylindrical tube
such as that shown in Fig.
In this application two activ•e gages mounted in the hoop
direction may be used for pressure sensing, along with two temtperature-compensating gages
mounted in an unstrained location. Temperature-compensating gages are shown mounted
on a separate disk fastened to the end of the cell. Design r1elationships may be found in
most mechanical design texts.
'
The sensitivity or a pair of circumferentially mounted strain gages (Fig.
factor F is expressed by the relationship2
6. R
2FRd 2
"""P; = -E
-
1 3) with gage
[ 2 d2 ]
-
D2
-
II
2Equation (5) is based on the l.aDll! equations ror heavy-wall piasun: cy linders.
(5)
See
Problem 27.
Measurement of Pressure
Sensing gages
Threads
FIGURE 13: Cylindrical-type press ure cell.
where
AR = the strain-gage resistance change,
R = the nominal gage resistance,
P; = the internal pressure,
d
=
the inside diameter of the cylinder,
D = the outside diameter of the cylinder,
E = Young's modulus,
11 = Poisson's ratio
The bridge constant, 2, appears because two circumferential gages arc assumed. If a single
strain-sensitive gage is to be used, the sensitivity will be one-half that given by Eq. ( 5 ). Of
course, these relations are true only if elastic conditions are maintained and if the gages are
located so as to be unaffected by end restraints.
Improved frequency response may be obtained for a cell of this type by minimizing
the internal volume. Thi s may be accomplished by use of a solid "filler" such as a plug,
which will reduce the ftow into and out of the cell with pressure variation.
Figure 14 shows the electrical circu itry used for a transducer of this type. Gage M
is a modulus gage, used to compensate for variation in Young's modulus with temperature.
The calibration and output resistors are adjusted to provide predetermined
bridge resistance
·
and calibration.
Measurement of Pressure
Output resistance ad)usiment
FIGURE 14: Strain-gage circuitry for pressure cells employing a modulus gage.
8
MEASUREMENT OF HIGH PRESSURES
The high-pressure range has been defined as beginning at about I MPa (about 10 atm) and
extending upward to the limit of present techniques, which is on the order of 100 GPa (about
lo6 atm) [ 1 4]. Various conventional pressure-measuring devices, such as piezoelectric
transducers for dynamic measurements and Bourdon-tube gages for static measurements,
may be used at pressures as high as 500 MPa to I GPa. Bourdon tubes for such pressures
are nearly round in section and have a high ratio of wall thickness to diameter. They are,
therefore , quite stiff, and the deHection per tum is small. For this reason, high-pressure
Bourdon tubes are often made with a
8.1
number of turns.
Electrical Resistance Pressure Gages
Very high press ures may be measured by electrical resistance gages, which make use of the
resistance change brought about by direct application of pressu re to the electrical conductor
itself. lbc sensing elemenl consists of a loosely wound coil of relatively fine wire. The
length and cross-section of the wire affect its electrical resistance, and both dimensions
vary with applied pressure at a rate determined by the bulk modulus of the material. The
electrical resistance change may thus be calibrated against the applied pressure.
Figure 15 shows a bulk modulus gage in section. In this particular gage, the sensing
element does ·not actually contact the process medium but is separ!lted therefrom by a
kerosene-filled bellows. One end of the sensing coil is connected to a central tenninal, as
shown. while the other end is grounded, thereby completing the necessary electrical circuit.
Assume:
dR
R
=
dl
L
41 0
_
2
dD
D
+
dp
p
(6)
Measurement of Pressure
Kersoene-lilled
bellows
Terminal
Sensing element
FIGURE 1 5 :
Section through a bulk-modulus pressure gage.
where
R = the electrical resistance,
l
D
=
=
the length of the conductor,
the wire diameter,
p = the electrical resistiv i ty
The wire will be subject to a biaxial slress condition because the ends, in providing
electrical continuity, w i ll generally not be subject to pressure. Assuming that ux = uy =
- P and u, = 0, we may write
(7)
and
Ez = T = E
dl
Combining the above relations gives
2vP
(7a)
us
dR
R
2P
E
dp
p
- = -+­
(8)
If t he resistivity were i ndependent of pressure, so that dp = 0, this would yield a linear
relationship between resistance change and pressure. In practice, the resistance is belier
represen ted by a second-order re l ation sh i p :
dR
= ko P - kt P 2
R
where ko i s known as the pressure c oeffic ie n t of resistance.
411
(9 )
Measurement of Pressure
Special alloys are nonnally used for resistance gages, particularly manganm,
.
is 84% copper, 12% manganese, and 4% nickel. For manganin, approximate
5
coeffic ents in
(9) are ko = 2 x 1 0- /MPa and kt '°'. 2.5 x 10- t ofMPal.
_.
coeffictent ko 1 s somewhat sens111ve to temperature, which can complicate measuranen
L
In the range to 20 to 50°C, for example, /co for manganin increases by 0.02% per
of temperature increase. A variety of other alloys have been ·examined for use in e
resistance press u re gages, although most have either a sma1ller pressure coefficient than
manganin or a greater temperature sensitivity. For examp:le, gold-2. 1% chromium has
ko = I x 10- 5 /MPa and a resistance decrease of 0.08% per degree kelvin in the same
temperature range. Despite these somewhat poorer numbers, gold�hromi um has found
application in high-pressure hydrogen environments [ 14 ).
�
�·
values�
The�
�
�
9
MEASUREMENT OF LOW PRESSURES
9.1
The Mcleod Gage
Atmospheric pressure serves as a convenient reference datum, and, in general, pressures
below atmospheric may be called low press ures or vacuum5'. We know, of course, lha1 a
positive magnitude of absolute pressure exists at all times, even in a vacuum. It is impossible
to reach an absolute pressure of zero.
In the older literature, absolute pressure in vacuum sy stems was often expressed in
micrometers of mercury (µmHg}, equivalent to 0. 133 Pa. Units of torr (equal to I mmHg
or 133 Pa) were also common. Today's practice is to use the SI unit pascal.
Two basic methods are used for measuring low pressure: ( I ) direct measurement
based on a displacement caused by the action of a pressure force, and (2) indirect or
inferential methods wherein pressure is detennined through the measurement of certain
other pressure-controlled properties, including volume and thermal conductivity. Devices
i ncluded in the first category encompass most of those discussed in the earlier sections
of this chapter, including manometers, Bourdon gages, and the various diaphragm-based
electronic transducers. Since these have been discussed in the preceding pages, they need
not be discussed funher here except to say that their use i'� generally limited to lowest
absolute pressures in the range of I to 100 Pa. For measurement of lower pressures, one of
the inferential methods is nonnally required.
Operation of the McLeod gage is based on Boyle's law for the isothermal compression of
a gas:
P1 =
P2 V2
Vt
--
( 10)
where P1 and f'2 are pressures al initial and final conditions, mspectively, and Vi and V2 are
volumes at corresponding conditions. By compressing a known volume of the low-pressure
gas to a higher pressure and measuring the resulting volume and pressure, one can calculate
the initial pressure.
Figure 16 illustrates the basic construction and operation of one type of McLeod
gage. Measurement is made as follows. The unknown pressul'C source is connected at point
A, and the mercury level is adjusted to fill the volume reprei;ented by the darker shading.
Under these conditions the unknown pressure fills the bulb B and capillary C. Mercury
is then forced out of the reservoir D, up i nto the bulb and reference column E. When the
Measurement of Pressure
A
(Pressure source)
+
Relerenc;e
/;
0
1
2
3
-
� c.=
-
+
Barometer
column 30' + Hg
D
t
FIGURE
16:
McLeod vacuum gage.
mercury level reaches the cutoff point F, a known volume of gas is trapped in the bulb and
capillary. The mercuiy level is then further raised until it reaches a zero reference point
in E. Under these conditions the volume remaining in the capillaiy is read directly from
the scale, and the difference in heights of the two columns is the measure of the trapped
pressure. The initial pressure may then be calculated by use of Boyle's law.
The pressure of gases containing vapors cannot normally be measured with a McLeod
gage, for the reason that the compression will cause condensation. By use of instrumenlS
of different ranges, a total pressure range of from about O.ot Pa to 10 kPa may be measured
with this type of gage.
9.2
Thermal Conductivity Gages
The temperature of a given wire through which an electric current is flowing will depend
on three factors: the magnitude of the current, the resistivity, and the rate at which the heat
generated in the wire by resistive losses is conveyed to the surrounding environment. The
latter will be largely dependent on the conductivity of the surrounding media. As the density
of a given medium is reduced below a cenain level, its conductivity will also decrease and
Measurement of Pressure
Measumg eel
Compensating cell
Power supply
FIGURE 1 7 : The Pirani thermal conductivity gage.
the wire will become hotter for a given current flow. The decrease in gas conductivity
with pressure begins at about 1 30 Pa. Below about 1 Pa, heat loss by thennal radiation is
dominant, and the gas conductivity effect becomes indetectably small.
This phenomenon is the basis for two different forms of gages for measuring low
pressures. Both use a heated filament but differ in the means for measuring the tempenuure
of lhe wire. A single platinum filament enclosed in a chamber is used by the Pirani gage. As
the surrounding pressure changes, the filament temperature, and hence its resislance, also
changes. The resistance change is measured by use of a resislance bridge that is calibrated
in terms of pressure, as shown in Fi g . 1 7 . A compensating cell is used to minimize variations
caused by ambient temperature changes.
A second gage also depending on thermal conductivity is the thermocouple gage.
In this case the filament te mpera tures are measured directly by means of thermocouples
welded to them. Filaments and thermocouples are arranged in two chambers, as shown
schematically in Fig.
When conditions in both the measuring and reference chambers
are the same no thermocouple current will How. When the pressure in the measuring
chamber is altered, changed conductivity will cause a change in temperature, which will
then be indicated by a thermocouple current.
18.
In both cases lhe gages must be calibraled for a definite press urized medium, for the
conductivity is also dependent on gas composition. As noted ear lier, gages of these types
are useful in the range from about I to I 00 Pa.
9.3
Ionization Gages
For measurement of extremely low pressures--<lown to 10- 6 Pa-an ionization gage may
be used. The m ax imum pressure for which an ioni zation gage may be used is about 0. 1 Pa
( I µ;mHg). An ionization cell for pressure measurement is very similar to the old-style triode
electronic tube. II possesses a heated filamen t, a positively biased grid, and a negatively
biased plate in an envelope evacuated by the pressure to be measured. The grid draws
electrons from the heated filament, and collision between them and gas molecules causes
414
Measurement of Pressure
Sealed reference
cell
To potentiometer
/
Thermocouples
<
To pressure
source
FHaments
\
IJ""=:s:ii,,__...
Calibrated
resistances
To voltage
regWltor
FIGURE 1 8: Thermocouple gage.
ioni7.ation of the molecules. The positively charged molecules are then attracted to the
plate of the tube, causing a current How in the external circuit, which is a function of the
gas pressure.
Disadvantages of the heated-filament ionization gage are that ( I ) excessive pressure
(0. 1 to 0.2 Pa) will cause rapid deterioration of the filament and a short life; and (2) the
electron bombardment is a function of filament temperature, and therefore careful control of
filament current is required. Another form ofionization gage minimizes these disadvantages
by substituting a radioactive source of alpha particles for the heated filament; these gages
operate in the much higher-pressure range from 0.0 1 Pa up to atmospheric pressure [ 15).
DYNAMIC CHARACTERISTICS OF PRESSURE-MEASURING SYSTEMS
Basic pressure-measuring transducers are driven, damped, spring-mass systems whose iso­
are theoretically similar to the generalized systems. In appli­
cation, however, the actual dy n am ic characteristics or the complete pressure-measuring
system are usually controlled more by factors extraneous to the basic transducer than by the
transducer characteristics alone. In other words, overall dynamic performance is dctennined
less by the transducer than by the manner in which it is inserted into the complete system.
lated dynamic characteristics
When the transducer is used to measure a dynamic air or gas pressure, the compress­
ibility of the gas in any connecting lines or volumes can introduce natural frequencies that
41 5
Measurement of Pressure
are substantially lower than that of lhe transducer itself. 'Mien liquid pressures are mea­
sured, lhe effective moving mass of lhe system will necessarily include some poni
on of
the liquid mass, often lowering lhe system's .natural frequency considerably. In botb cases,
the damping of the system may be greater lhan that of lhe transducer itself. Co
llllecliag
tubing and unavoidable cavities in lhe pneumatic or hyd1raulic circuitiy lhus chaqge the
dynamic characteristics of lhe measurement system, causing differences between lllCaSunld
and applied pressures.
Much theoretical work has been done in an attempt to predict lhese effects (16-19).
Each application, however, must be weighed on its own individual merits; for Ibis rasoa
only a general summary of some of lhe ractors involved is practical in this discussion.
In general, efforts should be made to place the pressure transducer as close as possible
to the pressure to be measured, and, when this is not possible, dynamic calibration of the
complete system should be made ralher than relying upon the dynamic calibration of the
transducer alone.
10.1
Gas-Fiiled Systems
In many applications, it is necessary to transmit the presirure through some fonn of )185sageway or connecting tube. This is particularly true when lhe transducer is bulky or when
the environment to be measured is harsh. Figure 19 illustnm:s typical cases.
If the pressurized medium is a gas, such as air, acoustical resonances may occur in the
same manner in which the air in an organ pipe resonates. If sympalhetic driving frequencies
are present, nodes and antinodes will occur, as shown in die figure. A node, characterized
by a point of z.ero air motion, will occur at the blocked end (assuming lhat the displacement
of the pressure-sensing element, such as a diaphragm, is negligible). Maximum presswe
variation takes place at Ibis point. Maximum oscillatory motion will occur at the antinodes,
and the distance between adjacent nodes and antinodes equals one-fourlh lhe wavelength
FIGURE 19: (a) Gas-filled pressure measuring system, (b) gas-filled pressure-measuring
system wilh cavity.
Measurement of Pressure
of the reson ating freq ue ncy Theoretical resonant frequencies may be determi ned from
the relation
f = �(2n - 1 )
(11)
4L
.
where
f = the resonance frequencies (i nc l udi ng both fundamental and
harmonics), in henz,
c = th e
velocity of sound in the pressurized medium, in mis,
L = the length of the connecting tube, in m,
n = any positive integer (It will be no ted from the equation
that only odd harmonics occur.)
I n many cases a cavity is required at the transducer end to adapt the instrument to
the tubing, as shown in Fig. 19(b). If we assume that the medium is a gas, and that the
containing system, including the transducer, is relatively stiff compared with the gas, we
have what is known as a Helmholtz resonator. The the mass of gas in the tubing together
with the compressibility of that in the cavity form a spring mass system having an acoustical
resonance whose fundamental frequency may be expressed by the relation (20]
-
f
=
where
2:/ V(L + :ma/2)
( 12
)
a = the cross-sectional area of the connecting tube. in m2,
V = the net i nternal volume of the cavity, excluding the
volume of the tube, i n m3
This equation is accurate so long as the tu bi ng vo l u me, a L, is smal l relative to the cavity
volume, V. When the tubi ng volume is not small, the following approximation may i ns tead
be used:
f
=
2: I V(L :aL/2)
( 1 3)
More comph:te equations for long pneumatic tubes are give n by Andersen [ 19).
Liquid-Filled Systems
When a pressure measuri n g system is lilled with liquid rather than a gas, a co nsiderabl y
different situation is presented. The li qu id becomes a major part of the total moving mass,
thereby becoming a si gn ilicant factor in determ i n i n g the natural frequency of the system.
If a single degree of freedom is assumed
-
,
( 14)
Measurement of Pressure
where
f = the natural frequency, in hertz,
m = the equivalent moving mass, in leg, = m1 + m2 ,
m 1 = the mass of moving transducer elements, in kg,
m2 = the equivalent mass of the liquid column, in kg,
k, = the effective transducer stiffness, in N/m
)
By simplified analysis, White (2 1 ) has determined the following approximate relation
for the effective mass of the liquid column:
(
4
A 2
m2 = 3 pal ;
(IS)
where
p = the ftuid density, in kg/ml ,
a = the sectional area of the tube, in m2 ,
l = the length of the tube, in m,
A = the effective area of the transducer-sensing element, in m2
area
It will be noted that A is the effective
, which is not necessarily equal to the actual
diaphragm or bellows area but may be defined by the relation
A=
where
AV
Ay
( 16)
AV = the volume change accompanying sensing-<:lement deftection, in ml ,
Ay = the significant displacement of the sensing element, in m
Likewise, the transducer stiffness, k, , may be defined in terms of the pressure change, A P
( Pa), required t o produce the volume change, AV:
AP · A
k _
' - AV/A
( 17)
By substitution of Eq. ( 15) into Eq. ( 1 4), we have
f
= 2ir
I
k,
m t + � pal(A/a)2
( 1 8)
In many cases the equivalent mass of the liquid, mz, is of considerably greater magni·
tude than m 1 , and the latter may be ignored without introducing an appreciable discrepancy.
By so doing, and writing a in tenns of the tube diameter, we get
a = ir D2/4 ,
I = !!.. /
D = tubing 1.0., in m,
Jk,
SA irpl
41 8
( 19)
Measurement of Pressure
As mentioned before, pressure transducers involve spring-restrained masses in the
same manner as do seismic accelerometers, and therefore good frequency response is obtain­
able only in a frequency range well below the natural frequency of the measuring system
itself. For this reason it is desirable that the pressure-measuring system have as high a
natural frequency as is consistent with required sensitivity and installation requirements.
lnspeciion of Eq. ( 1 9) indicates that the diameter of the connecting tube should be as large
as practical and that its length should be minimized.
In addition, it has been shown that optimum performance for systems of this general
type requires damping in rather definite amounts. White (2 1 ) gives the following relation
for the damping ratio � of a system of the sort being discussed:
411' L11(A/a)2
(20)
� = ---.;r;m
411'L 11(A/a) 2
where 11
(21 )
= the viscosity of the fluid. If we ignore m 1 and insert a = 11' D2 /4, we may write
as
the equation
�=
CALIBRATION METHODS
1 611A
02
J
3L
11'k,p
(22)
Static calibration of pressure gages presents no particular problems unless the upper pressure
limits arc unusually high. The familiar dead-weight tester (Fig. 3) may be used to supply
accurate reference pressures with which transducer outputs may be compared. Depend­
ing upon the specific design, testers of this type are useful to pressures exceeding 1 GPa
( 1 50,000 psi).
Although static calibration is desirable, transducers used for dynamic measurement
should also receive some form of dynamic calibration, so as to determine the transducer's
frequency response. Since pressure transducers may often be modelled as second-order
(spring-mass-damper) systems, dynamic calibrations may be used to find the transducer's
natural frequency, damping ratio, and static amplitude for use in determining its amplifica­
tion ratio as a function of frequency, that is, to find the transducer's transfer function (22, 23).
Dynamic calibration problems consist of ( 1) obtaining a satisfactory source of pres­
sure, either periodic or pulsed, and (2) reliably determining the true pressure-time relation
produced by such a source. These two problems will be discussed in the next few paragraphs.
Dynamic pressure sources may be periodic (steadily oscillating) or aperiodic (lran­
sienl). Some sources of dynamic pressure are as follows:
I.
Periodic pressure sources
(a)
(b)
Piston and chamber
Rolating valve
Measurement of Pressure
(c) Siren disk
(d) Acoustic resonator
n. A period ic pressure sources
(a) Qu ick- release valve
(b) Closed combustion bomb
(c) Shock tube
When a periodic source is used, the freque ncy of the so0urce may be varied so as to as
to measure the transducer's frequency response directly. Wh0en aperiodic sources are used,
additi onal calculations will be required.
1 1 .1
Periodic Pressure Sources
Most periodic pressure sources are designed for gaseous media, and they are generally
l i m i ted to relatively low frequencies and amplitudes, particularly if nearly sinusoidal fluc­
tuations are desired. At higher ampl itudes or frequenc ies , the pressure variations tend
toward a sawtooth form owing to the nonlinear behavior of compressible gas flows (23, 24).
One source of steady-state periodic calibration pressure is simply an ordinary pisroo
and cylinder arrangement, shown schematically in Fig. 20. If the piston stroke is fixed,
pressure amplitude may be varied by adjusting the cylinder volume. For a fixed volume, a
known repeatable pressure variation can be ge n erated at a single frequ ency. Amplitude and
frequency ranges wi l l depend on the mechanical design; however, peak pressures to 7 MPa
and frequencies as high as 100 to 1000 Hz have been reported (22, 25).
A method very similar to this has been used for microphone cal ibration . In this case
req uired pressure amplitudes are quite small, and instead of the piston being driven with a
mech an ical linkage, an electromagnetic sy ste m is used (26]. Similar approaches have used
�=:::l.!ill..
li .._
A
To pressure transducer
bo�ing calibrated
To pressure calibration
standard
FIGURE 20: Schematic diagram of a piston and cy l i nder peri odic pressure source.
420
Measurement of Pressure
vibration test shakers as a source of piston motion or of periodic inenial loading, the latter
method being adaptable to liquid media (23).
Several pressure supplies have been based upon rotating valves that periodically
switch the pressure sent to a transducer between high- and low-pressure supplies. The
pressure signal is approximately rectangular in such systems if acoustic resonance can be
avoided in the gas-containing volumes. Recent designs have been reported to work at up to
100 kPa and I kHz (27, 28).
Figure 21 illustrates another method for obtaining a steady-state periodic pressure.
A source of this type has bee n used to 3000 Hz with amplitudes to 7 kPa (29). A variation
of Ibis method uses a motor-driven siren-type disk having a series of holes drilled in it so
as alternately to vent a pressure source to atmosphere and then shut it off (30) . Sirens have
bee n used at frequencies to I kHz and pressures to 200 kPa [22).
Steady-state sinusoidal pressure generators consisting of an acoustically driven res­
onant system (sec Section 10. 1 ) have also been used. lYpical arrangements place a loud­
speaker at one end of a tube whose opposite end is closed by a movable piston; the piston
position is adjusted to give a half-wavelenglh separation from !he loudspeaker. These
systems are useable at frequencies up to a few kHz and amplitudes up to a few kPa [ 17, 23 J.
All the methods suggested here simply supply sources of pressure variation, but,
in lhemselves, they do not provide means for determining pressure amplitudes or time
characteristics. They are useful, therefore, for comparing a pressure transducer having
unknown characteristics to one whose performance is known.
1 1 .2
Aperiodic Pressure Sources
Periodic sources used to determine dynamic characteristics of pressure transducers are
limited by the amplitude and frequency that can be produced. High amplitudes and steady­
state frequencies are difficult to obtain simultaneously. For this reason, it is necessary to
resort to some form of step change in pressure to determine the transducer's high-amplitude
and high-frequency performance. Typically, the transducer's response to a step-changed
pressure is recorded, and Laplace transform calculations are then used to determine the
Connected to adjustable
pressure supply
el=i::ilfllill---
A
To pressure
standard
To pressure transducer
being calibrated
Variable-speed
sine-wave cam
FIGURE 2 1 : Jet and cam steady-state pressure source.
Measurement of Pressure
transducer's dynamic characteristics.
Various methods are used to produce the necessary pulse. One of the simplest is
to
use a fast-acting valve between a source of pressure and the transducer. These devices
be configured so that the transducer is held in a high-pressure chamber which is
vented to atmosphere or so that the transducer is kept in a small chamber at tow
which is abruptly opened to a large chamber at high pressure. Rise ti mes, from o to 90%
of final pressure, of l ms or less have been reported (22-24).
Another source of stepped-pressure is the closed combustion bomb, in which a pres­
sure generator such as a dynamite cap is eKploded. Peak pressure is contro lled by net internal
volume, and pressure steps as high as 5 MPa in 0.3 ms have been obtained (17, 22, 27 ).
Undoubtedly the shock tube provides the nearest thing to a transient pressure stan­
dard (24, 3 1 ) . Construction of a shock tube is quite simple: h consists of a long IUbe,
closed at both ends, separated into two chambers by a diaphragm, as shown in Fig. 22. A
pressure differential is built up across the diaphragm, and the diaphragm is burst, cilher by
the pressure differential itself or by means of an eKternally controlled mechanism or cutter.
Rupturing the diaphragm causes a press u re discontinuity, or shock wave, to travel at very
high speed into the region of lower pressure and a rarefaction wave to travel through the
chamber of initially higher press ure. The reduced pressure wave is reflected from lhe end
of the chamber and follows the stepped pressure down the tube at a velocity that is higher
because it is added to the velocity already possessed by the gas particles from the pre5SIR
step. Figure 23 illustrates the sequence of events immediately following the bursting of
the diaphragm.
A relationship between pressures and shock-wave velocity may be eKpressed as fol­
lows [32):
2k
P1
2
- = I + -- (Mo - I )
(23)
k+I
Po
min:?
�
where
k
P1 = the intermediate transient pressure,
Po = the lower initial pressure,
Mo
=
=
the ratio of specific heats.
the shock Mach number (shock speed divided
by sound speed in region of low pressure)
We see, then, thal if the gas properties are known, measurement of the propagation
velocity will be sufficient to determine the magnitude of the pressure pulse. Propagation
t;- -:if::::-::Jt:--2>;
Pressure
Diaphragm-piercing
High-pressure end
)'
Diaphragm
Low-pressure end
FIGURE 22: Basic shock tube.
422
--- -�
-
·
Measurement of Pressure
length of lube
(a) Before diaphragm ruptures
J
I
I
Rarefaction
I!
1
-
(b) After Nplure of diaphragm
-+- P,
-
-
(c) After reftection of rarefaction
AGURE 23: Pressure sequence in a shock lube before and immedialely after diaphragm is
ruplured. Abscissa represents longitudinal axis of lube.
velocily may be determined from information supplied by accuralely posilioned pressure
transducers in lhe wall of lhe lube. By lhis means, a known transienl pressure pulse may
be applied to a pressure transducer or lo a complete pressure- m'easuring system simply by
mounling lhe transducer in lhe wall of lhe shock lube. The response characlerislics, as
delermined in lhis manner, may then be used lo calculate lhe response or transfer function
of lhe device or syslem over a spectrum of frequencies.
The rise lime of a shock lube corresponds to lhe time for lhe thin shock wave lo
pass by and is generally well below I microsecond. The test duration is the lenglh of time
lhat lhe inlcnnediate pressure, Pi . is suslained at the transducer. It is a function of lhe
length of lhe shock tube and is lypically some number of milliseconds. Shock tubes are
lhus used for calibrations in the frequency range above several hundred hertz. For lower
frequency calibrations, a fast-acting valve is preferred (22, 24 ) . Shock tubes have been used
for calibrations al up to 20 MPa.
SUGGESTED READINGS
ASME PTC l 9.2- 1 987. lnslrumenls and Apparatus: Part 2. Pressull
American Society of Mechanical Engineers, 1 987.
Meas�ment.
New York:
Benedict, R. P. Fundamentals o/Temperru�. PllSSUll and Flow Measullments, 3rd ed. New York:
Wiley-Inierscience. 1 984.
423
Measurement of Pressure
Fowles. G. Flow, �I and Pressure Measurement in the Waler Industry. Oxford, U.K.: BulleiworthHeineman, 1 993.
Gautschi. G. H. Piet.oekctric Sensorics. Berlin: Springer-Verlag, 2002.
Kovacs, G. T. A. Micromachined Transducers Soun:ebook. New York: McGraw-Hill, 1998.
Peggs. G. N. (ed.).
ers, 1 983.
High-Pressure Measurement Techniques. L1Jndon: Applied
Science Publish­
PROBLEMS
l.
2.
3.
4.
Slandard atmospheric pressu re is 1 .0 1 325 x IOS Pa. Wh1a1 are the equivalents in (a) new­
tons per square meter. (b) pounds-fon:e per square foou, (c) meters of water, (d) inches
of oil, with 0.89 specific gravity, (e) millibars, (f) micmmelers, and (g) torr?
Determine the factors for converting pressure in pascals to "head" in (a) meters of Wiiier;
(b) centimeters of mercury.
The following are some commonly encountered pressu1res (approximate). Convert each
to the SI units, Pa or kPa: (a) automobile lire pressure of 32 psig; (b) household watu
pressure of 120 psia; (c) regulation football pressure of 13 psig.
rust in
SI units and then in Eng l ish units,
( a ) Write expressions relating the height ofa Huid co.fumn in terms of a reduced gage
pressure (a vacuum).
( b ) Under Slandard conditions of atmosphere and gra'lily, what is the maximum height
10 which water may be raised by suction alone?
( c ) Under similar conditions. 10 what height may a column of mercury be raised?
5.
The common mercury barometer may be formed by sealling the upper end of a rube (e.g.,
Fig. 4 ). inverting it and filling it with mercury, then righting the rube into a mercury-filled
reservoir. A vacuum is formed over the column and the lheight of the column is governed
primarily by the pressure (air pressure) applied at the base. Under these conditions. is a
true zero absolute pressure (a complete vacuum) formed over the column? Investigate the
vapor pressure of mercury and detennine the degree of error introduced if it is ignored.
6.
Rewrite Eq. (3) in terms of specific gravities.
7.
Write a few sentences explaining the mechanics of "suc:tion."
8.
Write a few sentences explaining the operation of a syp•hon.
9.
Solve Problem 9 u sing English units.
424
10.
Measurement of Pressure
Figure 24 illustrates a manome1er installation . Write an expression for detennining the
static pressure in the conduit in 1erms of h 1 . h2, and the other pertinent parameters.
Conduit fluid (density = p2)
atmosphlric
Tubing
Ambient
pressure
h:z
--r
h,
Manometer
fluid (density = p)
FIGURE 24: Manometer arrangeme nt referred to in Problem 10.
1 1.
For the cond itions shown in Fig. 24, if the manomeler Huid is Hg and the conduit Huid
is H?O (both al 20"C), hi = 1 8.4 cm (7.24 in ), and h2 = 0.7 {II (2.30 ft), what pressure
exists in the conduit? Solve u sing SI units.
.
12.
Solve Problem 1 1 using E ngli sh units.
13.
Express the ratio of sensitivities of an i nc li ned manometer (Fig. 6) to that of a simple
manometer in terms of the angle 9. For an i ncl i ned manometer six times more sensitive
than a si mple manometer, what shou ld be the angle of inclination?
14.
Veri fy Eqs. (4) and (4a).
JS.
Derive an expression for the two·Huid manome te r in Fig. 7, substituting reservoirs of
d iameters 01 and Di for the like-sized reservoirs shown in the figure.
Measurement of Pressure
16.
A two-ftuid manometer as shown in Fig. 7 uses a combination of kerosene (specific gray.
ity, 0.80) and alcohol-diluled water (specific gravity, 0.83). Also, d = in . (6.35 mm)
and D = 2 in. (50.8 mm). What amplification ratio is obtained with this arrange111en1 as
compared to a simple water manometer? What error would
introduced if the ratio of
diameters was ignored?
!
be
17.
18.
Confirm Eq. (Jc).
Note that Eq. (Jc) is based on a moving datum-namely, the liquid level in the reservoir.
Derive an equation for the differential pressure based on the movement of the liquid in
the inclined column only. (Note 1ha1 a practical solution would be to make provi sion for
adjusting the reservoir level to an index or "zero" line.)
19.
Figure 13 shows a cylindrical pressure cell using two sensing strain gages. If !he cylinder
may be assumed to
"thin wall," then
be
Off =
and
For this case, show that
Pd
21
Pd
GL = 41
l!i.R
P
=
[
FRd i - .!'.
2
Et
]
where
F = gage factor,
R
=
gage resistance.
E = Young's modulus, and
20.
v
= Poisson's ratio
For a circulardiaphragm ofthe type and loading shown in Fig. 9(a), the maximum normal
stress occurs in the radial direction at the outer boundary, expressed as follows (33 1:
a, = � (7)2
p
Greatest linear deflection occurs al the center and is equal 10
I:':
.... =
(2-) ( Pa4 )
16
where
P
=
£1 3
( I _ vz)
pR.5sure,
a = radius,
E = Young's modulus, and
v = Poisson's ratio
426
Measurement of Pressure
For a
in. (6.35 mm), E = 30 x 1<>6 psi (20.68 x 107 lcPa), and " 0.3, and for a
design stress or9 x IO" psi (6.2 x !OS JcPa), what maximum deftection may be expected
·
ir P = 300 psi (2.07 x HP kPa)?
=!
=
21.
Solve Problem 20 using SI units and reconcile the two answers.
22.
A pressure transducer is constructed rrom a steel tube having a nominal diameter or
15 mm (0.59 in.) and a wall thickness of 2 mm (0.0787 in.).
( a ) Ir the design stress is limited to 2.75 x Io' Pa (3.99 x IO" psi), what maximum
pressure may be applied to the transducer?
( b ) For the maximum pressure calculated in part (a), determine the circumferential and
longitudinal strains that should be expected. Use E = 20 x 10 10 Pa(29 x 106 psi)
and v = 0.3.
23.
For so-called heavy-wall cylinders, a simplified assumption leads to error. The following,
more complex relations, often referred to as the Lame equations, must be used:
P( 0 2 + d2 )
02 d2 on the inner surface.
2 P d2
UH =
02 _ d2 on the outer surface,
P d2
UL = 02 - d2 '
u, = - P on the inner surface
uH =
All
are
•
_
principal stresses.
( a ) For a design stress of 2.75 x J08 Pa (3.99 x 1 04 psi), d = 2 cm (0.787 in.), and
D ·= 5 cm ( 1 .968 in.), what is the maximum pressure that may be applied'
( b ) If the maximum pressure is applied, what circumferential and longitudinal strains
should be expected on the outer surface? Use 0.3 for Poisson's ratio and 20 x
J0 10 Pa for Young's modulus.
24.
25.
Solve Problem 23 using English units.
Derive Eq. (5). Note that this equation is based on the Lame equations for heavy-wall
pressure vessels. Sec Problem 23.
427
26.
Confinn Eq. (8).
27.
The speed
Measurement of Pressure
of sound in an ideal gas. c, may be expressed by the relation [32)
where
c = .ffifi'
k = lhe mioof specific heats ( 1 .4 for air),
R = lhe gas constant (287 Jlkg·K for air), and
T = absolute temperature (in kelvin)
Using the above equation and Eq. ( 1 1 ), de1ermine the change in frequency in pen:an,
conesponding to a 1empera111re change from lO"C to 40"C.
28.
A pressure-measuring system inwl..,. a -in.-diameler lll be, 24 in. long, connecting
a pressu re source to a U'8JISducer. Al the transducer end there is a cylindrical cavity
in. in diameter and in. long. Proper perfonnance requires that the lnquency or
app lied pressures be such as to avoid RSOnance. Calcula11e the resonance frequency or
!
}
!
the system.
Assume that the 24-in. connecting rube used in Problem 28 is reduced to zero length.
What will be the resonance f'Rquency? [Use Eq. ( 1 2), letting L = 0.)
29.
A Helmholtz resonator consists of a spherical cavity to which a circularly sectioned tube
is attached. It may be considered as approximating the tu be and cavity of Problem 28.
The resonance frequency of a HelmbollZ resonator may be estimated by the relation (32)
JO.
f=
c fa
2rr •(vi
The symbols have the same mr.aning as in Eq. ( 1 2). Use this equation to estimace the
resonance frequency of lhe system described in Proble m 28.
REFERENCES
[1] Taylor, B. N. Guide to tht Use oftht International System of Units (SI). NIST Spe­
cial Publication 8 1 1 . Gaithmbllrg, Md.: National lnstitul•e of Standards and Technol­
ogy, 1995.
(2) Benedict, R. P. Fundamentals o[Ttmptrature, Pressure a.'Ul Flow Measurements. 3rd
ed. New York: Wiley-lnterscience, 1 984.
(3) Proceedings of CCM third international conference: Pre:;sure metrology from ultra­
high vacuum 10 very high pressURS -10-7 lo 1 09 Pa. Merrologia 36(6), 1999.
[4) Proceedings of CCM second international seminar: Pressure mctro logy from I kPa 10
I GPa. Metrologia 30(6), 1993194.
[SJ Brombacher, W. G.
Survey ofMicrollllJlm
lO eters. NBS M o nograph 1 14. Washington,
D.C.: U.S. Government Printing Office, 1970.
(6) Tilford, C. R. Three and a half centuries later-The modern art of the manometer.
Metrologia 30:545-552, 1993194.
Measurement of Pressure
(7) Kardos, G. Bourdon Tubes and Bourdon Tube Gages: An Annotaled Bibliography. New
York: American Socie ty of Mechanical Engi neers , 1 978.
(8) Di Giovanni, M. Flat and Corroga1ed Diaphragm Design Handbook. New York: Mar­
cel Dekker, 1982.
[9] Kovacs, G. T. A.
Hill, 1998.
(10]
Micromachined Transducers Sourcebook.
New York: McGraw­
Wenk, E. Jr. A diaphragm-type gage for measuri ng low pressures i n flu ids. SESA Proc.
8(2):90, 1 95 1 .
(11] G autschi , G . H.
Piezoelectric Sensorics.
Berlin: S pri nger- Ve rlag, 2002, Chap. 8.
(12] Adams, E. D. High-resolution capacitative pressu re gages. Rev. Sci. Instr. 64(3):60161 1 , 1 993.
[13) Miiller, A. P. Measurement performance of high-accuracy low-pressure transducers.
Metrologia 36: 6 1 7--62 1 , 1 998.
[14] Peggs, G. N. (ed.). High Pressure Me"asurement Techniques. London: Applied Science
Publishers, 1983.
(15] ASME PTCl9.2- 1987. Instruments and ApParatus: Part 2. Pressure
New York: American Society of Mechanical Engineers, 1987.
Measurement.
(16) lberall, A. S. A tten uation of oscillatory pressures in instrument lines.
45:85, July 1 950.
NBS J. Res.
( 17) Hyl kema , C. G., and R. B. Bowersox. Experimental and mathematical techniques for
determining the dy namic response of pressure gages. /SA Proc. 8: 1 15, 1953, and ISA
J. I :27, February 1 954.
Basic Engr. 84:547-552, 1962.
of Pneumatic Systems. New York: John Wiley,
[ 18) Brown, F. T. The transient response of fluid lines. J.
[19) Andersen, R. C. Analysis and Design
1967, Sect. 4.3.
(20) Lord Rayleigh.
1945, p. 1 88.
The Theory of Sound, vol. II. 2nd ed. New York:
Dover Publications,
.
(21] White , G. Liquid filled pressure gage systems . Instrument Notes 1. Los Angeles :
Statham Laboratories, January-February 1 949.
[22) Bean, V. E. Dynamic pressure metro l og y. Metrologia 30:737-74 1 , 1 993/94.
[23) Hjelmgren, J. Dynamic Measurement of Pressure-A Uterature Survey. SP Report
2002:34. Boris, Sweden: SP Swedish National Testing and Research Institute, 2002.
[24] Damion, J. P. Means of dynamic calibration for press ure transducers.
30:743-746, 1993/94.
Metrologia
[25] Taback, I. The res ponse of pressure measuring systems to oscillating pressure. NACA
Tech. Note 1819: February 1 949.
[26] Bad maie ff, A. Techniques of microphone cal i brat ion . Audio Eng. 38: Dec. 1 954.
[27] Sc hweppe, J. L., L. C., Eichberger, D. F. M uster, E. L. Michaels, and G. F. Paskusz.
Methods for the Dynamic Calibration of Pressure Transducers. National B ureau of
Standards Monograph 67. Washington, D.C.: U.S. Department of Commerce, 1963.
Measurement of Pressure
Kobota, T., and A. Ooiwa, Square-wave press ure generator using novel rotating val
ve.
1999.
[28)
Metrologia 36:637--{i4(),
[29) Patterson, J. L. A miniature elecUical pressure gage utilizing a stretched ftatdiapltragm'
NACA Tech. Note 2659:
April 1 952.
[30) Meyer, R. D. Dynamic pressure transmitter calibrator. Rev. Sci. Inst 1 7: 1 99
,
May 1 946.
[31) Bean, V. E., W. J. Bowers, W. S. Hurst, and G. J. Rosasco, Development of a pri­
mary standard for the measuremen t of dynamic pressure and temperature. Metro/ogia
30:747-750, 1993194.
[32)
[33)
Thompson,
P. A. Compressible-Fluid Dynamics. New York: McGraw-Hill, 1972.
Roark, R. J. Formulasfor Stress and Strain. New York: McGraw-Hill, 1965, p. 217 .
ANSWERS TO SELECTED PROBLEMS
1 1 P2 - Pi = 1 6.65 kPa
13 0 = 9 . 6°
20 Ymax = 3.4 x 10- 3 in.
23
EH
= 322 µ. strain; SL
29 f = 3400 Hz
30 / = 3 1 0 Hz
= 76 µ. strain
Measurement of Fl u i d Flow
1
2
3
4
5
6
7
8
9
10
11
INTRODUCTION
FLOW CHARACTERISTICS
OBSTRUCTION METERS
OBSTRUCTION METERS FOR COMPRESSIBLE FLUIDS
ADDITIONAL FLOWMETERS
CALIBRATION OF FLOWMETERS
MEASUREMENTS OF FLUID VELOCITIES
PRESSURE PROBES
THERMAL ANEMOMETRY
DOPPLER-SHIFT MEASUREMENTS
FLOW VISUALIZATION
INTRODUCTION
Fluid flow encompasses a wide range of si1uations. The flowing medium
m ay
be a liquid.
a gas, a granular solid, or any combinalion !hereof. The flow may be laminar or 1urbulen1,
sleady slale or 1ransien1. The desired measuremenl may be lhe velocily al a poin� lhe ralc
of flow through a channel, or simply a picture of lhe enlire flow field. Each of lhcse faclo rs
affects lhe seleclion
of an appropriale measurement technique, and many different me1hods
have been developed for the various situations. This chap1er, therefore, will prcscnl o nl y
an oulline of some of the more imponanl aspecls of lhe general topic.
The mosl direct way to measure flow rate is 10 caplure and record the volume
or
mass lhat flows during a fixed time inlerval (a primary measurement of flow rale). More
often, some other quantity, such as a pressure difference or mechanical response, is used to
infer lhe flow rate
(a seconda ry measurement). We may also make a distinclion be1ween
Flowmeters delermine volume or mass now rales (e.g.,
Howmelers and velocity sensors.
liters per minule or kilograms per second) through tubes and �hannels, whereas velocity­
sensing probes measure fluid speed (e.g., mclers per second) at a po i n l in lhe now. Although
velocily-sensing probes can be used as building blocks for flowmelers, 1he converse is rarel y
lrUe. In addilion,ftow-visualization techniques are sometimes employed to obtain an i mage
of lhe overall flow field.
A categorizalion of How-measurement melhods is as
follows:
I. Primary or quantily melhods
(a) Weigh! tanks and so on
(b)
Volume ·1anks gradualed cylinders, bell provers, and so on
From Mechanical Measu�ments, Sixth Edilion, Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V.
Copyright Cl 2007 by Pearson Education, Inc. Published by Prentice Hall. All rights reserved.
431
Measurement of F l u id Flow
2. Flowmeters
(a) Obstruction meters (respond i ng to pressure differentials)
i. Venturi meters
ii. Flow nozzles
iii. Orifices
iv.
Variable-area meters
(b) Volume flowmeters (responding to volumetric flow rates)
Turbine and prope l ler meters
Electromagnetic flowmeters (liquids only)
iii. Vortex shedd ing meters
iv. Ultrasonic fl ow meters
v. Positive-displacement meters
i.
ii.
-
(c) Mass flowmeters ( respondi ng to mass flow rates)
i.
ii.
iii.
3.
Coriolis meters
Critical How venturi meters
Thermal mass flow anemometers
Vel oc i ty probes
(a) Pressure probes
i. Total pre ss u re and Pilot-static tubes
ii.
Di rect io n sen si ng probes
-
(b) Hot-wire and hot-film anemometers
(c) Doppler-shift methods
i.
ii.
Laser-Doppler anemometer
Ultrasonic-Doppler anemometer (liquids only)
(d) Partic le- image ve loc i metry
4. Flow-visualization techniques
(a)
(b)
(c)
(d)
Smoke trails and smoke wires (gases)
Dye injection, chemical preci pitates, particle tra•cers ( liqu ids)
Hydrogen bubble technique (liquids)
Laser-induced fluorescence
(e) Refractive-index change: interferometry, schlie�en, shadowgraph
Measurement of Fluid Flow
The preceding outline does not exhaust the list of flow-measuring methods, but it does
attempt to include the most common types. Obstruction meters are probably those most
often used in industrial practice. Application of some of the methods listed is so obvious
that only passing note will be made of them. This is particularly true of quanlity methods.
Weight tanks are especially useful for steady-state calibration of liquid flowmeters, and no
particular problems are connected with their use .
FLOW CHARACTERISTICS
We may measure the flow through a duct or pipe using its mass flow rate, m, perhaps in
kg/s, or its volume flow rate, Q, perhaps expressed in m3/s. These two quantities are related
through the density of the Huid, p in kglm 3 , by
I
/
•·
t
f.1
.;.· :
·.'
,i,-.:.:�.\:
�s/
.. ,
.
.
We may also define the average velocity,
area of the duct, A:
( I)
V, of the fluid in· the duct using the cross-sectional
·
.n
Q
V = - = -
pA
A
(2)
When fluids move through uniform conduits at very low velocities, the motions of
individual particles are generally along lines paralleling the conduit walls. The particle
velocity is greatest at the center and zero al the wall, with the velocity distribution as shown
in Fig. l(a). Such a flow is called laminar.
As the flow rau: is increased, a point is reached where the particle motion becomes
turbulent, showing unsteady, random vonices throughout the pipe. In this case, we think of
the lime-average velocity distribution, which has the appearance shown in Fig. I (b ). The
approximate velocity at which this change occurs is called the critical velocity.
c:::11:: :: ::]
(a )
(b)
AGURE I : Velocity distribution for (a) laminar flow in a pipe or tube and (b) turbulent
flow in a pipe or tube.
Measurement of Fluid Flow
Experiments have shown that the critical velocity is a function of several factors that
may be put in a dimensionless fonn called the Reynolds number, Reo , 1 as follows:
Reo =
pVD
-­
µ.
(3)
where
D = the diameter of the pipe (or hydraulic diameter if the pipe
is not circular),2
p = the density of the Huid,
V = the average velocity of the Huid,
µ. = the dynamic viscosity of the Huid
The critical Reynolds number for pipes is usually between 2 1 00 and 4000. Below this
range, the How will be laminar. Above this range, it will be turbulent
The volume flow rate, Q, through a pipe or duct is just the integral of the velocity
distribution, V (x , y) , over the cross-sectional
A:
Q =
l
area
V (x, y) dA
(4)
Aowmeters measure Q and/or V, while velocity probes measure V (x , y ) . The output of a
velocity probe can be integrated to obtain Q .
Changes i n fluid velocity o r elevation produce changes i n pressure. For example,
if an incompress ible fluid flows from a section of large area at point I into a section of
smaller area at poi nt 2 (Fig. 2), its average velocity must increase, according to Eq. (2). The
corresponding pressure change is given by Bernoulli's eq uation for incompressible ftow3 as
(5 )
FIGURE 2: Section through a restriction in a pipe or tube.
1The unilS for dynamic viscosily, µ,, arc kg/m · s or lbf . sifl2, depending on Jhc syslem of uniJs used. We see
dJa! althou1h the fonn or Eq. ('.!) .. written is most common. inclusion of lie is requir<d to obtain a proper unit
balance in the English engineering syslem.
2The subsaip1 D is used 10 indicaJe nominal pipe diameler. When a Reynolds number is based. for example.
on lhc lhroal diameter of a vcn1uri or an orifice, a lowercase d is commonly used (e.g., Red).
l Bcmoulli's equation applies to steady lossless How along a streamline, expressed in 1crms of local velocity.
II may be applied approximalely in terms of the average velocity in a due� so long as the losses are negligible.
434
Measurement of Fluid Flow
e
wh re
P = pressure
N/m2 (or Pa)
kg/m3
,
= densi ty ,
V = linear veloci ty ,
Z = elevatio n ,
g = acceleration due to gravity ,
8c = dimensional constant
p
mis
m
9.807 m/s2
l kg . m/N . s2
lbflrt2
lbm/ft3
ft/s
ft
32. 17 ft/s2
32. 17 lbm . ft/lbf . s2
As wrinen here, the relationship assumes that there is no mechanical work done
on or by the fluid and that there is no heat transferred to or from the Huid as ii passes
between
i nts l and 2. This equation provides the basis for evaluati ng the operation of
H ow - m
uri ng d v ices gene rally classified as obstruction meters and of veloci ty sensors
classified as pressure probes.
po
eas
3
e
OBSTRUcrtON METERS
Figure 3 shows three commoi:i forms ofobstruction meters: the venturi tube, the flow nozzle,
and the orifice. In c h case, the basic meter acts as an obstacle placed in the path of the
ftowi ng fluid, causing localized changes in velocity. In conjunction with the velocity change,
the pressure will c ang as illustrated in lhe figure. At points of maximum restriction, hence
maximum velocity, minimum pressures are found. The difference between this minimum
pressure and the upstream pressure is measured so as to determine the velocity.
A certain rtion of the pressure drop through an obstruct ion meter is irrecoverable
owing to dissipation of kinetic energy; therefore, the output pressure will always be less
ea
h e,
po
than the input pressure. As the figure indicates, the venturi, with its guided reexpansion, is
the most efficient. In contrast, losses of about 30%-40% of the differential pressure occur
through the orifice meter.
3.1
Obstruction Meters for Incompressible Flow
For incompressible fluids, with Eq.
(2),
P1 = P2 = P
h
d
d
and
m = pA 1 V1 = pA2 V2
w ere poi nts 1 and 2 are as in icate in Fig .
(A2/A 1 ) V2 in to Eq. (5), we obtai n
Pi
-
p
Pi
=
Vi - V12
2gc
3. If we lei Z 1 = Zz and su bs t i tute V1
=
Vi
2gc
[I _ ( A2 )2]
A1
(6)
which treats the How as ideal (without any pressure losses). Solving Eq. (6) for Vi. we may
compute the mass flow rate:
m;dcaJ = pA2 V2
=
[ ,/I A2 ]
-
(A2/A 1 > 2
,/2g,p( P1
-
/>i)
(7)
For a given m t r, A 1 and A 2 are established values, and it is often convenient to cal­
c ulate a velocity of approach factor, E:
ee
(7 a )
435
Measurement of Fluid Flow
JP
(b)
-- - - - - -----
­
=====!
JP
(c)
FIGURE 3: (a) A
�
�------ --=$-
venturi tube, (b) a flow nozzle, and (c) an orifice ftowmeter.
For circular sections, lhe area = rr(diameter)2 /4; hence
E=
where
fJ
1
---
�
=D
d
and
d
= the smaller diameter,
D = lhe larger diameter
(7b)
Measurement of Fluid Flow
To account for losses through the obstruction meter, the discharge coejJicienl, C, is
introduced:
(7c )
The discharge coefficient, C, is occasionally combined with the velocity of approach factor,
E, to define the flow coefficient, K.
K = CE =
c
_
__
(7d)
�
ci nt K is used simply as a matter of convenience.
The flow coe ffi e
Therefore,
we
may write
mac1ua1 = CEA1,/2gcp( P1
= KA2 ,/Zgcp(P1
and, with Q..,,ual =
-
-
m-.a1/p,
"2)
(8a)
f'i)
(8b)
(8c)
3.2
Venturi Tube Characteristics
international
standard is available for three types of venturi tube flow meters [ I ). The
standard includes detailed pecifica ion for their
Dimensions common 10
these
are indicated in Fi g . 4.
An
designs
s
t s
construction.
Divergent ooUet
Li ;? D or L1 � CM + 250 mm)
L1 = d
" = 0.5d
z = 0.50
1
•
+1
-----
a1 = 21 ° ± 1 °
7° S a2 S 1 5°
FIGURE 4: Reconunended proportions of standard venturi tubes. Specifications for the transition
radii (R1 , R2 . RJ) and
pressu re tap diameters, c!, are given in the ISO standard ( 1 ).
the
Measurement of Fluid Flow
TABLE 1: Discharge CoefficienlS for Slandard Venturi
Range
Type
100 mm ::; D ::; 800 mm
0.3 :::: fJ :::: 0.75
2.5 x !OS :::: Reo :::: 2 x 106
As cast
Machined
Rough welded sheet iron
50 mm ::; D :::: 250 mm
0.4 :::: fJ :::: 0.75
2 x ! OS :::: Reo :::: I x 106
200 mm :::: D :::: 1200 mm
0.4 :::: fJ :::: 0.7
2 x ioS :::: Reo :::: 2 x 106
Tubes [ 1 )
c
0.984 ± 0.7%
0 .995 ± ! %
0.985 ± 1 .5%
Standard venturi tubes are hig h-efficiency devices with discharge coefficients very
near the ideal value of C = I . T le I gives the discharge coeffic ie nlS for the tluee standard
designs, together with their uncertainties and the s peci fied range of operation for each
design. OulSide the specified range of operation, the discharge coefficient may differ by a
few percent.
ab
3.3
ra
Flow-Nozzle Characteristics
Figure 5 illu st tes two types of flow nozzle, with the dimensions set by the relevant intema·
tional slandard [2 ]. The approach curve must be proporti oned to prevent separat ion between
the flow and the wall, and the parallel section is used to ensure that the flow fills the throat.
The discharge coefficient may be calculaled to an uncertainty of 2% with the fol l owing
relationship, as given in the international standard:
c
= 0.9965 - 6.53
r;t;;
(9)
This equation applies for 0.2 :::: fJ :::: 0.8 and 1 04 :::: Reo :::: 107•
The usual range of discharge coefficients is shown in Fig. 6. Observe that the flow
nozzle has greater losses than the venturi, especially at low Reynolds numbers.
3.4
Orifice Characteristics
The primary variables in the use of flat-plate orifices are the ratio of orifice to pipe diameter
and the pressure lap locations. Figure 1 illustrates typical orifice insiallations. Three lap
locations are indicated: ( I ) flange taps, (2) " I D" and D" taps, and (3) comer taps. These
are al l shown in composite fashion in Fig. 1; however. onl y one set woul d be used for a
given instal l at ion . As for the other obstruction meters, an international s1andard specifies
the proper dimensions and positioning of the orifice meter (3).
As fluid flows through an orifice, the necess ary transverse velocity components
imparted to the ft uid as it approaches the obstruction carry through to the downstream
si de. As a result, the mi ni mu m stream section occurs not in the plane of the orifice, but
"!
438
Measurement of Fluid Flow
(a) High rdo 0.25 s f3 s o.a
(b) Low ratio 0.2 s f3 s o.s
FIGURE S: Dimensional relations for standard long-radius Dow nozzles [2]: (a) high ratio
0.25 :s fJ :s 0.8; (b) low ratio 0.2 :s fJ :s O.S,3 mm :s t2 :S O. I S D , r � 3 mm for
D
>
6S
min; else t
�
2 mm .
439
Measurement of Fluid Flow
1 .00
0.98
c 0.96
0.94
0.92
1 0•
4
2
FIGURE 6: R
6
8
1115
Red
2
4
6
8 1o'
ange of discharge coefficients for long-radius ftow nozzles.
j- O and % taps -J
r- 0
I
c::=::::=:::l l.
� '
-+---..+-- Flange taps
� � �·L:=====i
T
--.. - o - - - d -
_j_
Conier taps
FIGURE 7:
Locations of pressure taps for use with orifice meters; l = 25.4 mm.
Measurement of Fluid Flow
Vena conllBcta. Tiie ftow profile section of
minimum dimension
(a)
0.8
fJ
0.6
0.4
0.2
o ._�_._�_.��-'-�-'-�-'-��.._
0.2
0.4
0.6
0.8
1 .0
1 .2
Pipe diameters from upstream face of orifice
(b)
FIGURE 8: (a) Diagram illustrating vena conlracta location for an orifice; (b) guide for
locating vena contracla as measured from orifice face.
somewhat downstream, as shown in Fig. 8(a). The term vena conlracta is applied to the
location and conditions of this minimum stream dimension. This is also the location of
minimum pressure. A guide for the location of the vena contracta is given in Fig. 8(b ).
The formation of the vena contracta is sensitive to the shape of the orifice, which must
therefore be manufactured with care 10 obtain accurate measurements. The downstream
face of the orifice must be beveled at an angle of approximately 45° if the plate thickness
is greater than 2% of the pipe diameter. Similarly, in order to dissipate the effects of pipe
bends or valves, substantial lengths of straight pipe may be required upstream of the orifice
(many tens of diameters, typically); and sufficient straight length is required downstream
10 avoid interference with the vena contracta. The standard should be consulted for further
details [3].
The discharge coefficient of an orifice meter with comer lappings may be calculated
from the following equation (3. 4]:
C
=
0.5961 + 0.026 1 /j2 - 0. 2 1 6/j 8 + 0.0005 2 1
( to6p)0.7
Reo
-
35
+ (0.01 88 + 0.0063A)/j ·
Ol
(�) .
Reo
( 10)
Measurement of Fluid Flow
in which
A=
oa
( 19, 000/J ) .
Reo
00.)
This rather formidable result should be coded (into a spreadsheet, say) if repetitive
c:.lcu·
lations are to be made. The equation applies for
0. 1 � tJ � 0.75
{5,000
000, 1000
and
Re o :'!
1 6•
ooop 2
5
for tJ � 0. 6
for tJ > 0.56
( I Ob)
with d :'! 1 2.5 mm and 7 1 . 1 2 nun � D �
nun . The uncertainty of Eq. (10) is 0.St.
for 0.2 � tJ � 0.6 and Re o :'! 1 0,
rising to as much as 1 .25% outside these ranges.
For other tapping arrangements or smaller values of D, additional terms enter Eq. ( 10),
amounting to a few percent. Figure 9 shows the orifice discharge coefficients given by
Eq. ( 1 0) as a function of Reynolds number for several values of /J.
An orifice plate is vulnerable to damage caused by pressure surges, entrained debris.
and the like. An estimate of the maximum stress due to differential pressure may be found
from the following. The relationship is adapted from a rather co.mplex equation (5) and
assumes Poisson's ratio = 0.3 (as is typical of steels).
FD2 t:.P
tTnw = --,2--
( 1 1)
0.64
0.63
"l
0
.s!
.II
0.62
..
§
0.61
i5
0 . 60
..
E!'
..
£
0.1
0.59
o.se
1 04
1115
1 oS
Reynolds number, Re0
107
y
FIGURE 9: Range of discharge coefficients for Hat-plate orifices with comer tappings as given b
Eq. ( 10) for several values of tJ.
442
where
Measurement of Fluid Flow
umu
= max i m um nonnal stress (radial direction at the clamped edge),
t = plate thickness,
l!.P = differential pressure across the plate,
F = a factor, the value of which may be estimated from the following :
fJ
0.2
0.3
0.4
0.5
0.6
0.7
0.8
F
0. 1 8
0. 17
0. 1 5
0. 1 2
0.09
0.06
0.04
According to the standard, the plate thickness should lie in the range 0 .005 0 !:: t !:: O.OSD.
EXAMPLE 1
An orifice meter with corner taps is placed into a horizontal 200-mm-diameter (7.874 in.)
line that carries 30 Us (475.5 gal lo ns per min.) of water. If the throat diameter is 1 20 mm
(4.724 in.), what differential pressure may be expected across the pressure taps? The water
femperalure is 20"C (68°F).
Solve the problem (a) using SI units and {b) usi ng the English engineering system.
Solulion
(a) For water at 20°C, the density and dynamic viscosity are
3
= 998.2 kg/m
µ. = 1 0.05 x 10-4 Pa .
p
The areas of the pipe, A i . and the ori fice , Ai, are
A t = 0.03 142 mi
and
s
A i = 0.0 1 1 3 1 m
2
and the diameter ratio and ve loc i ty of approach factor are
fJ
=
d
o
=
1 20
200
= 0·600
1
E = -- = 1 .072
,/ 1 - p4
The velocity in the pipe, Vi , is
Vi =
Q
=
A1
The Reynolds number is
Re o =
p Vi D
µ.
=
30 x 10- 3
0.03 142
= 0.9548 m/s
(998.2)(0. 9548)(0 .200)
10.05 x I 0 -4
443
=
1 . 897 x l<>5
of Fluid Flow
Substituting for /J and Re0 in Eqs. ( I Oa) and ( 10), we 6nd 1\ = 0. 1055 and
Measurement
C
=
� o.?
0.5961 + 0.0261 (0.6)2 - 0 .216(0.6)8 + 0.0005 21 ( La6 (0.6
. 1.897 x I )
) 0.3 0.
1
+ (0.0188 + 0.0063(0. 1055) ) (0. 6) 3
6084
·5 ( 1 .897<>6x LOS
"
=
so that K = CE
remembering that
= (0.6084)(1.072) = 0.6522.
Be = l in the SI System:
P1
-
Pz
=
=
(b) For water al 68°F,
Finally, we may rearrange
(JL
2Be . )
2 )2 (.!...
[ (KA0.6522)(0.0l
30 10-1 ] 2 [ '998.2 ]
l 31) . 2(1)
Eq.
(Be),
x
= 8255 Pa = 8.26 kPa
62.32 Lbmlft3 and µ. = 2.099 x m- s lbf . s/ ft2
In addition, Q = 475.5 gpm = 1 .060 ft 3 /s, A t = 0.3382 fl:!, A1 = 0. 1217 ft2 , and
4
4. 72 =
fJ = 7 . 874 0·600
1
E = r.--7J = 1.072
v L - fJ4
p
=
The velocity in the pipe, Vi , is
V1
1.060 = 3. 1 34 fl/s
= o.3382
Be in this case:
p V1 D
(62. 32)(3. 134 )(7. 874/ 12) = 1 . 898 x l a5
=
Re o =
8c /J.
(32. 17)(2 . 099 x 10-S)
fJ
As before, substitution of and Reo into Eqs. (10a) and (10) yields C = 0.6084 and
K = 0.6522. Finally,
The Reynolds number must include
-
Pz
=
((1 .060)/(0.6522)(0. 1217))2 ((62:.32)/2(32. 17))
1 72.7 lbf/ft2 = 1 .203 lbf/in.2
See Problem 6 for. the case o f a vertical pipe.
P1
=
Measurement of Fluid Flow
3.5
,
Relative Merits of the Venturi, Flow Nozzle, and Orifice
Good pressure recovery and resistance to a brasion are the primary advantages of the venturi.
They are o ffset however, by considerably greater cost and space requirements than with
the orifice and nozzle. The orifice is inexpensive and may often be installed between
existing pipe flanges. However, its pressure recovery is poor, and it is especially susceptib le
to inaccuracies resulting from wear and abrasion. It is also quite sensitive to upstream
flow disturbances requiring e i ther flow conditioning or significant lengths of s trai gh t pipe
upstream. The flow nozzle possesses the advantages of the venturi, except that it has lower
pressure recovery, and ii has the added advantage of shorter physical length. II is expensive
compared with the orifice, is relatively difficult lo install properly, and is the least accurate
of the three meters.
A major disadvantage of these meters is that the pressure drop varies as the square
of the flow rate [Eqs. (8)]. This means that if these meters are to be used over a wide
range of flow rates, pressure-measuring equipme nt of very wide range will be reqµired. In
genera l, i£ the pressure range is accommodated, aC:curacy al low flow rates will be poor:
The small pressure readings in that range will be limited by pressure transducer resolution.
One solution would be to use two (or more) pressure-measuring systems: one for low flow
rates and another for h ig h rates.
4
,
OBSTRUCTION METERS FOR COMPRESSIBLE FLUIDS
When compressible fluids flow through obstruction meters of the types discussed in Sec­
tion 3, the density does not remain constant during the process; that is, P t -F Pl · The usual
practice is lo base the energy relation, Eq. (5), on the density at cond i tion I (Fig. 2) and 10
introduce a dimensionless expansion/actor; Y, into Eq. (8a) as follows:
( 12)
The value of Y is less than or equal 10 unity, with Y = I correspond i ng to i ncompressi b le
flow.
The expansion factor, Y, may be determined theoretically for gases fl owi ng through
nozzles and venturis and experimentally for gases in orifice meters. For nozzles and venturi s
Y may be calculated from this isentropic-ftow relationship [ I ) :
y
=
in which
[( Pi ) 2/k ( - ) ( k
Pi
k
I
I _ ( f'i / Pi )ck - 1 )/t
I
( P2 / P1 )
)(
I
_
p'
I - f14 (/'2 / P1 ) 2 / k
)]
,
1 /2
( 1 2a)
k = specific heat at constant pressure
speci fic heat at constant volume
Approximate values of the specific heat ratio, k, are gi ve n for various gases in Table 2. For
orifice meters, the following empirical relation should be used [3]:
y
=
[
1 - 0.35 1 + 0.256/14 + 0. 93 JJ 8
]
[1 - (�) ]
l/k
.
( 1 2b)
Measurement of Fluid Flow
TABLE 2: Approximate Specific Heat Ratios at 20°C
Gas
AI, He, Kr, Xe
Air, CO, H2 , N2 . 0i
Cff.t, C02
Specific Heat Ratio, k
1 .67
1 .4
1.3
Th e uncertainty o f this correlation, i n percent, i s 3 . 5 ( 1 - Pi/ Pi ) when Pi! P1 :?: 0.75. In
both of these equations, absolute pressures must be used.
Expansion factors, Y, for venturis and nozzles with k = 1 .4 are shown plotted against
pressure ratio in Fig. l O(a). Similar values for orifice meters are given in Fig. lO(b). From
these graphs, we see that compressibility become significant when ( P1 - Pi)/ P1 :=::; 0.0 1 .
1 .00
0.95
0.90
0.85
y
0.80
0.75
l---1----1--�1--�1-----1�,..._:>..d-,..._---rl...,.:>...,.-=..,;.
o;;-;
.20.4
-"
0.5
0.6
0.65
--1----1--i---�i---4-'>,,_..,._._.._.-I 0.7
0.70 10.75
o.n5
0.65 1---1---1----1---�1---+---+-,..._-"I 0.8
0.82
0
t____J'---_
---1.__...J.__
...
-L.__J____J�----1.--�
0.05
0.10
0.15
0.20
(P1 - P�)IP,
0.25
0.30
0.35
0.40
(a)
FIGURE 1 0: Expansion factors for k = 1 .4: (a) venturis and nozzles [6] ; (b) ori ficc meters.
446
Measurement of Fluid Flow
).. 0.960
I
i 0.940 1----+---+---1"<::"""-""<:"""';:-'l"'<:::"""ea-..:::--1
0.1
I
w
0.920
M
1----+---<>---�---<
0.6
0.65
0.7
0.75
0.25
0.900 >------�--<
0.00
(P1 - P2YP,
0.05
0.10
0.15
0.20
(b)
FIGURE 10: Co111i11ued
EXAMPLE 2
An orifice with fJ = 0.6 is used to measure flow of 30°C air through a 0.25 m diameter
circular-sectioned duel. Estimate the flow rate if the differential pressure between vena
contracta taps is 80 kPa and the upstream absolute pressure P1 is 400 kPa.
Solution Assume that the density and dynamic viscosity is at 1 atrn. The density, but not
the viscosity, must be corrected for the higher pressure in the duct:
Pl
=
=
Patmos
(
p,Pi
almos
)
l
1 . 14((400 x 103 )/( 1 0 1 . 325 x 1 03 )] kg/m
= 4.50 kg/m l
1L 1 =
1 .85 x 10- s Pa · s
Using the given information, we compute
E
=
I
r.--oJ
v l - /J4
= 1 .072,
1
A 1 = 0.04909 m , and
2
A1 = 0.0 1 767 m
447
Measurement of Fluid Flow
We may find Y from Eq. { 12b) with k
Y
[
=
1 .4:
=
8
4
1 - 0.35 1 + 0.256(0.6) + 0.93(0.6)
=
0.94 1 1
][
1 -
(400� ) ]
-
80 1 / 1 .4
a
To determine C from Eq. ( 1 0) requires the Reynolds number, which cannot be com
pulcd until we determine 1he How rate; however, the terms involving the Reynolds number
are smal l for high Reynolds numbers, so for now we may es m .te C by neglecting them:
2
8
c � 0.596 1 + 0.0261 (0.6) - 0.2 1 6(0.6) = 0.6019
ti
Using Eq. ( 1 2), and noting that gc
,;.
1 in the SI System, w e have:
=
(0.60 1 9)( 1 .012>co.0 1161)(0.94 1 1
=
­
= 9. 105 kgls
h/20 J <4.50J<SO x
1
03
)
The value of Reo may now be calculated and the accuracy e>f our estimate of C deter­
mined.
V1
Re o
=
=
=
-Pl A 1
,;.
=
P1 V1 D1
9. 105
04 ()1) = 4 1 . 2 1 m/s
9 )
(4 . 50)(0.
JL l
(4.50)(4 1 .2 1 )(0.25)
s
1 .85 x 6lo2.506 x 10
Rec omput i ng C using al l the terms in Eq. ( 1 0) yields 0.6045, which is within 0.5% of the
value we estimated previously. Greater refinement is not warranted.
4.1
Choked Flow and the Critical Flow Venturi Meter
One particular issue necessitates caution when considering the How of compressible Huids.
The pressure difference across a constriction (such as 1he throat of a nozzle) drives How
through it, as shown in Fig. 3. As flow rates are increased, p erhaps by decreasing the
downstream pressure, the How velocity through the constric1ion eventually reaches the
speed of sound (Mach number = 1 .0). W hen this occ urs, downstream press ure changes
cannot propagate lhrough the nozzle to affect the upstream How. Any further decrease
in pressure downstream of the constriction has no inHuence on the mass How rate. The
condition of Mach nu mber equal to unity at a constriction is called chokedflow (7].
The critical
raJio at which choked flow occurs ma:y be expressed in terms of
the upstream total pressure. P, 1 and the throat pressure, Pz, as
pressure
!:!_
P1 1
=
(-z-)k/(k -1)
k + l
( 1 3)
Measurement of Fluid Flow
(see Section 8 for a discu.�sion of total pressure). So long as the pressure ratio Pi/ P1 t is
greater than the value given in Eq. ( 1 3), the ftow may be predicted by Eq. ( 1 2). However,
when the pressure ratio given by Eq. ( 13) is reached, the ftow is choked and the mass ftow
rate cannot be increased by lowering the pressure ratio further. For air, k =
and this
ratio is 0.528.
Choked ftow conditions can be used for very precise measurement of mass llow rate.
Specifically, once sonic ftow is reached i n the throat of a venturi nozzle, the mass How rate
of a perfect gas is given by
1 . 4,
,;,
=
2 __1 1 .jk ( -- ) (k+l)/2(k-ll
__
.,fTIT;j .
CA p,
2
k+ I
( 1 4)
for C the dischargecoefficientofthe nou.le in question, A2 the throat area, R the gas constant
for the particular gas, and T, t the total temperature. For ftow-metering applications, toroidal
or cylindrical throat venturi nou.les are used [ 8]. The discharge coefficients of properly
designed nozzles are about 0.99, with a weak dependence on Reynolds number. Additional
corrections may be applied to account for the fact that real gases are not perfect.
Critical ftow venturi meters have the advantages of very high precision (with uncer­
tainties potentially well below 1% ), of a linear relationship between mass llow rate and
upstream pressure, and of independence from downstream pressure. They have the dis­
advantage of relatively l imited range. They have primarily been used for laboratory and
calibration applications.
5
ADDITIONAL FLOWMETERS
The preceding discussion covers ftowmeters that directly detect ftow-induced pressure dif­
ferentials. Many other physical phenomena can be adapted to the measurement of ftow
rate, most of which produce a response proponional to either volume ftow rate or mass ft ow
rate. Several of these devices are of high accuracy, especially the belier turbine, Coriolis,
and positive displacement meters. With the exception of the Coriolis meter, the ftowmeters
discussed in this section have a linear response to ftow rate. This means that their resolution
is the same at both high and low ftow rates, which is an advantage over the obstruction
meters. Table 3 summarizes the characteristics of various common llowmeters.
5.1
Turbine Meters
The familiar anemometer used by weather stations to measure wind velocity is a simple
form of free-steam turbine meter. Somewhat similar rotating-wheel ftowmeters have long
bee n used by civil engineers to measure water ftow in rivers and streams ( 1 0). Both the
cup-type rotors and the propeller types are used for this purpose. In each case the number
of turns of the wheel per unit time is counted and used as a measure of the ftow rate.
Figure 1 1 illustrates a typical adaptation of these methods to the measurement of ftow
in tubes and pipes. Rotor motion is sensed by a variable reluctance-type pickup coil. A
permanent magnet is encased in the rotor body, and each time a rotor blade passes the pole
of the coil, change in permeability of the magnetic circuit produces a voltage pulse which
is received by the meter's signal processing electronics. The turbine blades are designed
so as to make the rotation rate linearly proponional to the volume llow rate over some
Characteristics of Common Flowmeters [4, 9). Wider Ranges Are Available for Some Meters If Lower Accuracy ls Acceptable.
Specialized Designs May Allow Broader Application of Some Types of Aowmeter
TABLE 3:
Type
Differential Pressure Sensing
Venturi
Aow nozzle
Orifice
Variable area
Gae
Liquid
Slurry
Dirty
Accuracy
Range
y
y
y
y
y
y
y
y
y
y
y
M
M
M
VL
S: I
S: I
S: I
10 : I
y
y
l!i. P
Coat
M
M
MIH
H
I.JM
M
L
M
I.JM
L
MIH
L
I.JM
M
Volwne Flow Ratt Sensing
§
Turbine
Electromagnetic
Vonex
y
Ultrasonic
Positive displacement
y
y
y
y
y
H .
y
y
y
M
M
M
H
10
30
100
10
20
10
:
:
:
:
:
:
1 (1)
l (g)
I
I
I
1 (1)
M/H
M
M/H
M/H
Measurement of Fluid Flow
A: Turbine rotor
B: Bearing support and
A
straightening vanes
reluctance
pickup (see Secllon 6.12)
C: Variable
AGURE 1 1 :
Turbine Howmeter.
range of flow. Low accuracy models may use the rotor 10 drive a wonn-gear connected to
a mechanical councer; these types are sometimes used as water or gas meters.
Turbine meters can be among the most accurate of How metering devices. High
accuracy models may have uncertainties of 0.25% to 0.5% for liquid How and 0.25% to
1 .5% for gas flow. Bearing wear makes periodic recalibration necessary. Turbine meters
lose accuracy al low flow rates, which limits their range of operation. Maximum 10 mi ni mum
flow rates are about I 0: I for liquids and about 30: 1 for gases. These meters are less accurate
in unsteady flow conditions.
-
5.2
Electroma gnetic Flowmeters
Electromagnetic flowmelers are based on Faraday's law of induc�-d voltage for a conductor
moving through a magnetic field, expressed by the relation
e =
SBDV
x
10- 4
( 1 5)
where
the induced voltage, in volts,
the sensitivity coefficient for the particular meter,
B = the magnetic ftux density. in gauss,
D = the length of the conductor, in m,
V = the velocity of the conductor, in mis
e =
S
=
The basic ftowmeter arrangement is as shown in Fig. 1 2. The flowing medium is
passed through a pipe, a short section of which is subjected 10 a transverse magnetic Hux.
The ftuid itself acts as the conductor having dimension D equal 10 pipe diameter and
velocity V roughly equal lo the average ftuid velocity. Fluid motion relative 10 the field
causes a voltage lo be induced proportional 10 the Ouid velocity. This emf is detected by
electrodes placed in the conduit walls. An alternating magnetic Hux is usually used, with
451
Measurement of Fluid Flow
(a)
•
+
(b)
FIGURE 1 2 : (a) Schematic arrangement of an electromagnetic Oowmeter; (b) section
showing electrodes and magnetic field.
both sine-wave and square-wave excitation being common. The output is detected and
processed by appropriate circuitry. For a uniform magnetic ftux and a uniform velocity, the
sensitivity S = I ; in practice, neither B nor V is uniform, and the value of S is determined
by calibration [ 1 1 ].
For most applications, the ftuid need be only slightly electrically conductive (com·
parable to tap water), and the conduit must be lined with a nonconducting material. The
electrodes are placed flush with the i nner conduit surfaces and make direct contaet with the
Rowing ftuid. With purely sinusoidal excitation, eddy currents and induced voltages may
cause drift in the signal at zero ftow rate. By the use of either square-wave excitation or
signal conditioning circuitry, most modern meters provide acc urate accurate output down
to zero flow.
Commercial electromagnetic flowmeters have rated accuracies of0.5% to 2% . They
are particularly useful for corrosive liquids and slurries, due to the absence of moving parts.
and they are useful for homogeneous gas-liquid flows at low void fraction. The calibration
is generally not sensitive to the particular liquid being metered.
5.3
Corlolls Flowmeters
Coriolis flowmeters measure the mass flow rate by detecting the Coriolis force on a section
of the tube carryi ng the fluid when the tube is oscillated. This force is proportional to
452
Measurement of Fluid Flow
the product of the mass How rate and the frequency of oscillation. The force is generally
detected from the ben.ding or Hexion that it causes in the tube.
Coriolis meters can be very acc urate, although the accuracy varies over the range of
the meter. Values of 0.4% to I % are typical. The range of How rates spanned by a specific
meter may be 100: I . Coriolis meters are suitable for both gases and liquids, and they are
not directly sensitive to viscosity, density, or temperature.
Vortex
Shedding Flowmeters
Vortex shedding meters are based on the fact that when a blulT body is placed in a stream,
vortices are alternately formed, first to one side of the obstruction and then to the other
(Fig. 13). The frequency of formation, f in Hz, is a function of flow rate:
I = (�)
v
( 1 6)
where
St = the Strouhal number,
V
= the flow velocity, in mis,
D = the dimension of the obstruction transverse to the flow direction, in m
The Strouhal number is approximately constant over a broad range of Reynolds numbers,
giving a vortex shedding frequency that is proportional to velocity [ 12). The value of St
may be regarded as a calibration constant for the meter.
Various schemes are used to sense the frequency of vortex formation. The obstructing
body may be mounted on an elastic support and the support oscillation sensed by one of a
number of means. Heated thermistors downstream, with one to each side of the obstruction,
and the Hexing of diaphragms have been used to detect the vortices. Another technique
makes use of an ultrasonic beam that is amplitude modulated by pulses. In another scheme,
a differential piezoelectric pressure transducer is implanted in the rear of the bluff body.
Commercially available vortex shedding meters are accurate to 0.5% to 1 .5% of
How rate. They are used for both gas and liquid Hows. Thecalibration is relatively insensitive
to viscosity, although the Reynolds number based on pipe diameter should generally be
above 2 x 1 04 •
FIGURE 1 3 : Vortex shedding caused by bluff body in flow stream.
Measurement of Fluid Flow
FIGURE 14: Ultrasonic flowmeter.
5.5
Ultrasonic Flowmeters
Ultrasound refers to acoustic waves at frequencies wel l above human hearing. Ultrasound
can be used for time-of-travel measuremenlS of mean How velocity. A pair of piezoelectric
or magnetostrictive transducers are located on the outside of a conduit a few inches apart
(Fig. 14 ). One serves as a sound source and the other serves as the pickup. As the sound
wave travels from the source to the receiver, its ordinary velocity in the stationary fluid, c,
will be either increased or decreased due to the fluid velocity V. For example, if the sound
wave crosses the pipe at an angle 8 relative to the flow direction. then the effective velocity
of the wave is c ± V cos 8, depending on whether the wave moves upstream or downstream.
Since a wave travels more slowly in the upstream direction than in the downstream direction,
the flow velocity can be determined from the difference in travel time or the relative phase
shift between upstream and downstream waves. To obtain both upstream and downstream
waves, the function of the two ultrasonic transducers is reversed periodically.
The flow rate is obtained by multiplying the measured velocity V by the pipe's cross­
sectional area, as if V were uniform over the cross section. This assumption is appropriate
for turbulent flow.
In flow-metering applications in gases, ultrasonic frequencies may range from tens
to hundreds of kilohertz. In liquids, where the sound speed is greater, the frequency is
higher, from hundreds of kilohertz to several megahertz, so as to keep the wavelengths
short. Particles-and bubbles in particular--<:an auenuate the sound waves, interfering
with the transit time measurement; thus, this type of ultrasonic meter is best suited to clean,
single-phase fluids. Ultrasonic flowmeters have typical accuracies ranging from a fraction
of a percent to about ±5% [ 1 1 ) .
5.6
Positive Displacement Flowmeters
Positive displacement meters have many forms and variations. Common exam ples are the
water and gas meters used by suppliers to establish charges for services. Basically, dis·
Measurement of Fluid Flow
placement meters are roiating hydraul ic or pneumatic chambers whose cycles of motion are
recorded by some form of counter. The volume displaced on each cycle is known with great
accuracy, allowing either the volume How rate or the total volume passed to be detennined.
Only such energy from the stream is absorbed as is necessary to overcome the friction in
the device, and this is manifested by a pressure drop between inlet and outlet. Many of the
configurations used for pumps have been appl ied to metering. These include reciprocating
and oscillating pistons, sliding-vane arrangements, various types of gears and rotors, the
nutating (or nodding) disk, helical screw devices, Hexing diaphragms, and so forth.
Posi tive displacemen t meters have the advaniages of high accuracy (from 0.2% to
2%), a very wide range of How rate (perhaps 100 : I i n a given meter), the ability to retain
acc uracy at very low rates or during on/off How conditions, and a general insensitivity to the
viscosity or velocity profile of the Huid. These meters have the disadvan tage of requiring
very c lean Huids (since particulates may cause wear or jamming), of the potential to create
blocked lines (should they jam), and of i nducing How pulsations.
5.7
The Variable-Area Meter
The variable-area Howmeter is shown in cross section in Fig. 15. This instrument is also
known by the trade name Rotameter. 1\vo parts are essential, the Hoat and the tapered tube
in which the Hoat is free to move. The termfloat is somewhat a misnomer in that it must
Pipe
�t
Glass
w
t�
FIGURE 1 5: Vari able - area
455
Oowmeter.
Measurement of Fluid Flow
be heavier than the liquid it disp laces . As fluid flows u p ward thro1ugh the tube, � fO!tles
act on the float: a downward gravity force, and u pward pressure, and viscous drag fortes
For a gi ven flow rate, the float assumes a position in the tube where the forces acting ·
it are in equilibrium. The total drag is dependent on flow rate and the annular
the float and the tube, and, for a gi ve n tube taper, the flo t 's position will be determined by
the flow rate alone. A basic equation
for the variab le-area meter has been developed in the
·
following
a
form [ 1 3) :
Q = AwC
[2gvf (Pf
-
A JPw
Pw )
]
area�
l /2
(17)
whe re
Q = the volumetric rate of flow,
vI
g
= the volume of the float ,
=
acceleratio n due to
pf = the float density,
gravity
,
p., = the liquid density,
A I = the area of the float,
C = the d isc harge coefficient,
Aw = the area of the annular orifice
=
G)
D = th e di
am
[(D + by)
2
-
d2 ] ,
eter o f the lube when the float is a t the
zero position,
b = the change in lube diameter per unit chan ge in height ,
a
d = the maximum diameter of the float,
y = the height of the float above zero pos i tion
Norma lly, the values of D, b, and d will be selected to produce an essentially linear variat ion
of A ., w i t h y. Thus, the flow rate is a l inear function of the reading.
Cert in di sadvan tages of the variable-area meter are that: the meter must be installed
in a vertical position; the float may not be visible w he n opaque fl u1 ids are used; i t cannot be
used with liquids carry i ng large percentages of solids in suspensi on; for high pressures or
tempe rat ures , it is expensive; il usually does not prov ide electronic readout; the calibration
is affected by Hu id density; and it i s not te rri b l y accurate. Advantages inc l ude the following:
there is a uniform flow scale over the range of the instrument, wi th the pressure loss fixed
at all How rates; the capacity may be changed with relative ease by ch g ing float and/or
tu be ; many corrosive fluids may be h and led without com pl ication ; the cond i tion of flow
is readi l y visible; and model s are avai lable for ve ry low flow rates. Variable-area meters
are typi cal ly accurate to no better than
of full scal e and may have accuracies of only
1 0% or so.
an
2%
6
CALIBRATION OF FLOWMETERS
Facilities for producing standardized fl ows are required for flowmeter calibration.
Fluid at
known rates of flow must be passed through the meter and the rate compared with the meter
456
Measurement of Fluid Flow
readoul. When the basic How input is determined through measurement of time and either
weight (mass How) or linear dimensions (volumetric How). the procedure may be called
primary calibration. After receiving a primary calibration, a meter may then be used as
a secondary standard for standardizing other meters through comparative calibration. In
general, higher accuracy may be obtained for liquid calibrations than for gas calibrations,
owing to the low densities and handling difficulties associated with gases [ 1 1 ).
Primary calibration is usually carried out at a constant How rate by measuring the
tolal How for a predetermined period of time. Primary calibration in terms of mass is
commonly accomplished by means of a weigh tank, in which the liquid is collected and
weighed. Although the latter method is nonnally used only for liquids, with proper facilities
it may also be used for calibration with gases. Volumetric displacement of a liquid may be
measured in terms of the liquid level in a carefully measured tank or container. For gases,
at moderate rates, volume may be determined through use of a bell prover, which consists
of an invened bell that creates a gas chamber above a liquid; the lower rim of the bell lies
beneath the liquid surface (see Fig. 1 6). Gas is pumped into the chamber through a pipe
leading to the Howmetcr being calibrated; as the gas flows in, the bell rises funher above
the liquid, maintaining a constant pressure within the chamber. The bell's displacement
provides the measure of volume [ 14 ).
Figure 17 illustrates a method obviating the requirement for direct mass or volume
measuremenl A standpipe of known capacity (diameter) is used as a collector. We see
that the pressure or head at the base is the analog of the mass or volume as it is collected.
0
Pulley
Counterweight
Flowmeter
-
FIGURE 16: The bell prover for gas How calibrations.
457
Measurement of Fluid Flow
Alternate
ftow path
(constant
back
pressure)
Flow source
Meter being
calibrated
Standp1pe
r-
-
Rate
Pressure
valve
cell
FIGURE 17: Standpipe e mpl oyed for flowmeter calibration.
Calibrations wilhin 0. 1
%
for meters handli ng 20 to 300,000 kglh have been reported ( 15).
The accuracy of lhis approach will clearly depend upon accurate knowledge of lhe liquid's
densi ty.
Secondary cal i brati on may be acco mpl i s hed by simply plac i ng a secondary standard in
series with the meter to be c al i b rated and compari n g !heir respec t ive readouts overlhedesired
range of flow rates. Hig h- accu racy meters, s uc h as tu rb i ne meters and positive displacement
meters, are generally used as secondary s ta ndards. It is clear !hat this procedure requires
careful consideration of meter i nstal l ations, m in i mizing interactions or olher disturbances
such as might be caused by nearby line obstructions, like elbows or tees. Flow strai ghteners
may be placed upstream of the standard meter for this purpose. Standard meters are often
used in pairs , in order that dri ft in one of t he standard meters will appear as a difference
between them.
For obstruction meters that are designed to meet an I SO or other standard, a cali bralion
may be done by measuring the relevant dimensions of the meter and using !he corresponding
equations for the discharge coefficient or flow coeffi c ient as given in the governi ng standard.
This approach shou ld lead to uncertainties of to 2%.
Changes in the viscosi ty and density of the fluid be i ng metered will affect lhe calibra­
t ion of a meter, i nsofar as they will affect lhe Reynolds n umber and olher parameters !hat
may i n fl uence the meter's per formance. These properties depe nd not on ly on lhe particular
fluid, bu t also on its temperature and pressure. Liquid viscosity is especially sensitive to
temperature. In general, cal ibration should be done using the specific fluid to be metered
at lhe expected operating temperature and pressure. Corrections may be applied to move
a cal i bratio n to other conditions or fluids, provided lhat one knows how (or whether) lhese
propenies affect the meter's response.
1%
7
MEASUREMENTS OF FLUID VELOCITIES
l p
l
vl
Often velocity per se is
Flow ra te is ge nera ly roportiona to some flow e ocity ; hence, by measuring the velocity,
a measure of flow rate is obtained.
desired, eilher
458
the velocity oflhe
Measurement of Fluid Flow
TAB LE 4: Velocity Measurement Techniques. Response and Resolution Are Representative Values.
Frequency Response in Panicular Will Differ Depending Upon Probe Design, Operating Conditions,
and Ancillary Hardware; for example, Panicle Image Velocimeter (PIV) Resolution Can Reach
O. l µm when Combined with Microscopy.
Frequency Spatial
Response Resolution
(Hz)
(mm)
Device
Cost
Pilot tube
Hot-wire anemometer
Laser-Doppler anemometer
Ultrasonic-Doppler anemometer
Particle image velocimeter
< 0. 1
Low
Moderate 1 0,000
High
100
Moderate
High
30
5
I
0. 1
10
1
Reversing
Flow?
No
No
Yes
Yes
Yes
fluid itself or the velocity relative to a fluid. An example of the latter is an aircraft moving
through the air ( 1 61. The following sections deal with measurement of the absolute and
relative veltJcities of fluids. The instruments considered measure local velocity-at one
point-rather than a spatially averaged velocity. We shall refer to the local velocity as V
for simplicity of notation.
Fluid velocities may vary in time, either as a result of unsteadiness in the flow system
or as the result of turbulence. In the case of turbulent flow, the velocity fluctuates in
time about a mean value. The frequency of these fluctuations may reach some number of
kilohertz, depending upon the flow conditions. The velocity to be measured may be either
a time-average velocity or an instantaneous velocity. For the lauer, of course, we must
consider the frequency response of the velocity sensor.
Table 4 lists several common techniques for measuring fluid velocity, together with
typical frequency response and spatial resolution; the ability to measure a flow that reverses
its direction is also noted. We shall discuss these devices in the sections that follow.
8
PRESSURE PROBES
A common reason to measure pressure at some point in a fluid is to determine flow conditions
at that point. The flowing medium may be gaseous or liquid in a symmetrical conduit or
pipe or in a more complex configuration such as a jet engine or compressor.
Point measurement of pressure is accomplished by the use of tubesjoining the location
in question with some form of pressure transducer. A pressure probe is intended insofar as
possible to obtain a reliable and interpretable indication of the pressure at the measurement
point. Therein lies a difficulty, however, for the mere presence of the probe will alter, to
some extent, the pressure being measured.
Many different types of press ure probes are used, with the selection depending on the
information required, space available, pressure gradients, and constancy of flow magnitude
and direction. Basically, pressure probes measure one or both of two different pressures
(Fig. 1 8). Specifically,
P,
=
P, + P.
( 1 8)
Measurement of Fluid Flow
-
Static
Total
pressure, P1
pressure, 1'1
Differential pressum
transducer,
senses P. = P1 - P,,
FIGURE 1 8 : Total and static pressure probes.
where
P1 = the total pressure (orten called the
P, = the static pressure, and
8.1
Pu = the velocity pressure
stagnation
pressure),
Incompressible Fluids
Referring to Eq. (5) for incompressible fluids, we may write
2
P1 = P, + 2Ee
pV
where
p = the fluid density,
V = the fluid velocity, and
Ee = the dimensional conversion constant for English units (g,, = I in SI units)
Solving for velocity, we obtain
6.P
V =
g
2
c ( P, - P, )
p
=
/2gc(flP)
p
( 1 9)
= P, - P, . From this we see that velocity may be detenni ned simply by
where
measuring the difference between the total and static pressures.
When velocity is used to measure flow rate, consideration must be given 10 the velocity
distribution across the channel or conduit. A mean may be found by traversing the area 10
determine the velocity profile, from which the average may be calcula11ed, or a multiplication
constant may be determined by calibration (see Problem 26).
Measurement of Fluid Flow
8.2
Compressible Fluids
For high speeds, the compressibility of gases must be accounted for in de1ermining the
velocity from pressure measurements. At the probe lip, an isenttopic compression of the
gas occurs as 1he pressure changes from P, to P, , provided 1hat flow is subsonic. If the
flow is supersonic, a shock wave will be produced ahead of the probe. Under subsonic
conditions, the equations of isenttopic flow (7) lead to the following relationship between
velocity and pressure:
v
2
( --) ( ) [(
k
k - 1
p'
Ps
-
I +
-)
t, p <k- 1 )/k
P,
]
- I 8c
( 1 9a)
where
k
=
p,
=
the ratio of specific heats (see Table 2), and
the density of the flowing gas (the static density)
This equation should be used for when 1he flow speed is berwcen 30 and 100% of the speed
of sound, that is, for Mach numbers between 0.3 and I . For supersonic flow, an additional
correction is required to account for the shock wave. The speed of sound in an ideal gas
is given by c = ,,/k RT, where R is the gas constanl for lhe gas in question and T is the
absolute 1emperature.
8.3
Total-Pressure Probes
Obtaining a measure of total or stagnation pressure is usually somewhat easier than getting
good measure of static pressu re, except in cases such as a jet of air issuing into an open room,
when a barometer reading provides the static pressure. The simple Pitot tube (named for
Henri Pilot) is usually adequate for determining slagnation pressure. More often, however,
the Pilot tube is combined with static openings, constructed as shown in Fig. 19. This
is known as a Pilot-static tube, or sometimes as a Prandtl-Pitot tube. For steady-How
conditions, a simple differential manometer, often of the inclined type, suffices for pressure
measurement, and P, - P, is determined directly. When variable conditions exist. some
form of pressure transducer, such as one of the diaphragm types, may be used. Of course,
care must be exercised in providing adequate response, particularly in the connecting tubing.
A major problem in the use of an ordinary Pilot-static tube is to obtain proper alignment
of the tube with flow direction. The angle formed between the probe axis and the How
streamline at the pressu re opening is called the yaw angle. This angle should be zero, but
in many situations it may not be constant: The flow may be fixed neither in magnitude
nor in direction. In such cases, yaw sen.�itivity is very important. The Pilot-static tube is
particularly sensitive to yaw, as shown in Fig. 20. Although sensitivity is influenced by
orientation of both stagnation and static openings, the latter probably has the greater eITect.
461
Measurement of Fluid Flow
Total pressure,
p1
�I
equallyspaceddia.*
Static pressure, P,
a holes 0.04
free from burrs
fe in. dia
t
I
I
±
Section A-A
�==���
f 1 .T
5 1n. dla.
L
22
1
I
on .
.
s
f in . -I
a dia.
'
t
s
s
5 in. �
FIGURE 19: A Pitot-static tube.
1
6
dia.
----.tM
The Kiel tube, designed lo measure total or tag atio n pressure only (there are no
static openings), is shown in Fig. 2 1 . It consists of an impact tube surrounded by what is
essentially a venturi. The curve demo trates the stri king insensitivity of this type of probe
to variations in yaw. Modifications of the Kiel tube make use of a cylindrical duct, beveled
at each end, rather than the streamlined venturi. This appears to have little effect on the
performance and
the co ns truc t ion much less expensive.
8.4
makes
n
n
Static-Pressure Probes
Static-pressure probes have
been used in many different forms ( 1 7). Ideally, the simple
opening with ax.is normal to How direction should be satisfactory. However, slight burrs
or yaw introduce appreciable e rrors. As mentioned previously, in many situations the yaw
angle may be continually changing. For these reasons, special static-pressure probes may be
used. Figure
22
shows several probes of this type and the corresponding yaw sens itivities.
As mentioned earlier, the mere presence of the probe in a pressure- How situation alters
the parameters t o
be measured.
Probes interact with other probes, with their own supports.
and with duct or conduit walls. Such interaction is primarily a function of geometry and
relative dimensional pro portions ; it is also a function of Mach number. Much work has
been conducted in this area (see, e.g . , [ 1 8,
462
1 9)).
Measurement of Fluid Flow
+2
!
'ii­
�
a.
v..? -
0
r--....
·1
� -2
y
I/
-
"
/
,_Total
'
I\
v
1/1
Slalic head
head
'
'
\
\
\Dynamic head
\
--8
4
0
20:
12
8
Yaw, degrees
16
20
FIGURE
Yaw sensitivity of a standard Pilot-static tube. (Counesy: The Airfto Instru­
ment Company, Glastonbury, Connecticut)
� T�-I+
o
i·
0.1 60"
+l
Kiel probe (approxlmate dimensions)
I
0.019·
( (
-0.2
--0.3
-80
' .....
r---.. .......
--40
\ \
l
-
Pitch
Yaw
-20
0
20
Yaw and pitch angle
40
60
80
FIGURE 2 1 : Kiel-type total-pressure tube and plot of yaw sensitivity. (Courtesy: The
Airflo Instrument Company, Glastonbury, Connecticut)
463
Measurement of Fluid Flow
ti -...
...
-20
-15
-10
Yaw
--ll!!l
�!!l!ll!lla•lli
Prandll tube
-5
0
5
angle, II, degrees
1 0·
15
FIGURE 22: Angular characteristics of certain static-pressure-sensing elements. (Courtesy:
Instrument Society of America, Research Triang le Park, North Carolina)
8.5
Direction-Sensing Probes
Figure 23 il lu strates two fonn s of direction-sensing or yaw -angl·e probes . Each of these
probes uses two impact tubes. In each case the probe is p laced transverse to flow and
is rotatable around its axis. The angular position of the probe i:s then adj us ted until the
pressures sensed by the openings are equal. When this i s the cas.!, the flow direction will
correspond to the bisector of the angle between the openings. Probes are also available
with a third open ing midway between the other two. The additiCJnal hole, when properly
aligned, senses maximum s tag n ation pressure.
EXAMPLE 3
A Pitot-static tube is used to determine the velocity of air al ihe center of a pipe. Static
pressure is 1 24 kPa ( 1 8 psia), the air temperature is 26.7°C (80°F), and a di fferential pressure
of96.5 mm of water (3.8 in.) is measured. What is the air velocity7 Perfonn the calculations
usi ng: (a) the SI system of units; and (b) the English sy stem .
Solution
(a) Using the SI system of units, we find from Table I. that the density of air is
kglm3 at standard atmospheric pressure of 1 0 1 .325 kPa. At 124 kPa,
P26.7 = 1 . 153
P26.7 = ( 1 24/ 1 0 1 .325) x 1 . 1 5 J
I mm H20 = 9.807 Pa (from Table I ),
t.P = 96.5 x 9.807
=
= 946.4 Pa,
1 .4 1 1 kglm3 ,
P, = P, + t.P = 1 24, 000 + 946.4 = 1 24.946 kPa,
k = 1 .4
464
Measurement of Fluid Flow
-10
0
50
-5
Yllw angle, 6, degrees
10
15
FIGURE 2 3 : Special direction-sensing elemenlS and their yaw characteristics. (Courtesy:
Ins1rument Sociery of America, Research Triangle Park, North Carolina)
Using Eq. ( 1 9a) wilh gc = 1, we have
V=
<0 4 4
j
2(1 .4/0.4)( 1 24, 000/ 1 .4 1 1 )( ( 1 24, 946/ 1 24, 000) . / 1 . l - 1 )
= 36.57 mis
(b) Using 1he English engineering system of units,
3
0.0735 lbm/fl al a pressure of 14.7 psia. At 1 8 psia,
we
x
1
find from Table 1 1ha1 P80
P80 = (0.0735)( 1 8/ 1 4.7) = 0.0900 lbmlft3 ,
l in. H2 0 = 5.202 lbf/ft2
t:.P = 3.8 x 5.202 = 19.77 lbf/ft2 ,
P, = 18 x 144 = 2592 lbf/ft2 •
P,
=
P, + t:. P
k = 1 .4
=
2592 + 1 9.77 = 26 1 2 lbf/ft2 ,
SubslilUting in Eq. (l9a) gives us
v =
=
jz
0 4 l 4)
( l .4/0.4)(2592/0.090)((26 1 2/2592)< . / . - l) x 32. 1 7
1 1 9 fl/s (or 36.4 mis)
Close scrutiny of the arilhmetical manipulations required in lhis example clearly
shows lhat calculation errors of considerable size may easily resull from lhe fact thal P, and
465
Measurement of Fluid Flow
P, are qu ite common ly of very nearly the same magnitudes: Calculation of the ratio must
be quite precise.
To determine !he effect of neglecting compressibility, we may substitute the values
for part (a) of thi s example directly into Eq. ( 1 9), obtaining
v =
J2 x
I x (946.4) / 1 .4 1 1 = 36.63 mis
Essentially the same answer is obtained. In this case, the flow is nearly incompressible
because the Mach number is low: The sound speed in air at this temperature is about
347 mis. At higher flow speeds, the agreeme nt would be much worse. (Problem 24 also
bears on thi s matter.)
9
THERMAL ANEMOMETRY
A heated object in a moving stream l oses heat at a rate that increases with the fluid velocity. If
the object is electrically heated at a known power, i t will reach a temperature determined by
the rate of cooling. Thus, its temperature will be a measure of the velocity. Conversely, the
heating power may be controlled by a feedback system to hold the temperature constanL
In that case, heating power is a measure of velocity. These relations are the basis for
thermal anemometry.
The most commonly used thermal velocity probes are !he hot-wire and hot-film
anemometers. The hot-wire anemometer consists of a fine wire supponed by two larger­
diameter prongs; an electric current heats the wire to a temperature well above the fluid
temperature (Fig. 24 ). Ty p ically, the wire is 4 to 10 µ.m in diameter, is I mm in length,
and is made of platinum or tu ngsten . These fine wires are extremely fragile, so hot-wire
probes are used only in clean gas flows. In liquids, or i n rugged gas-flow applications, the
:S Is::::ii:i T T
Ceramic
Ceramic
Ceramic
___ ,____
lnconel
�-----------· ---------,
�
r
�
-·------------------·--·
..------
Position of
fine wire
�
lnconel wire
:
(a)
iE:: i! : x:f T f
Cerarric
Position of fine wire
T
Ceramic
Ceramic
Ceramic
lnconel
Ceramic
i
---�------------- · ··------.
:___,_�-------�------------!
(b)
FIGURE 24: 1Wo forms of hot-wire anemometer probes: (a) wire mounted normal to
probe axi s , (b) wire mounted parallel to probe axis.
Measurement of Fluid Flow
hot-film probe is used instead. Here, a quanz fiber is suspended between the prongs, and
a platinum film coaled onto the fiber sunace provides the electrically healed element. The
fiber, with a diameter of 25 lo 150 µ.m, has much greater mechanical s1rength than fine wire.
Hot-wire and hot-film probes are most often operated at constant temperature using
a feedback-controlled bridge (Fig. 25). The probe fonns one leg of a voltage-sensitive
deftection bridge. Curren! ftowing 1hrough 1he bridge provides heating power to the wire.
The resistance of !he wire is a function of temperature, and any increase in How velocity
tends to lower the wire's temperature, reducing its resistance and causing bridge imbalance.
The vollage imbalance drives a feedback amplifier, which increases the voltage and current
supplied to the bridge; the added current increases the heating power, thus raising the
wire temperature and resistance and resioring bridge balance. The voltage supplied to the
bridge also serves as lhe circuit oulput. High-quality bridges are available commercially;
alternatively, acceptable bridges can be built at minimal cost (20].
The relation between flow speed and bridge outpul is obtained by equating the elec­
trical power to the heat loss. The heating power in watts is
.
Eleclncal power
-� ( --)
= (
=
e
eo
R,.,
2
Feedback amplifier
•a =
=
R.,
R0 + Rw
e0
)2
1
-
Rw
Rw
( Ro + Rw) 2
.:l e + 9boi..co
>----->-----o +
Output
••
RGURE 25: ConslanHemperature-anemometer bridge circuit.
467
Measurement of Fluid Flow
where
the voltage across lite wire (V),
ew
=
Rw
= lite wire resistance
R0
= lite upper leg's resistance (Q), and
e0
= lite output voltage (Q)
(V),
Heat loss from the wire is mainly by convection to the fluid: Thermal radiation
is
negligible, and conduction 10 lite supporting prongs may be accounted for by calibration.
The rate of heat loss in waits is given by
Rate of heat loss = Awh(Tw - Ti >
where
Aw
= lite wire surface area
T., = lite
Tt
h
= lite ftuid temperature
=
(m2),
wire temperature (K),
(K), and
convective heat transfer coefficient (W/rn2 K)
The heat transfer coefficient
for small-diameter wires is given by
h
=
A + B /PV
where
A
and
B
= constants lhat depend on the wire diamet<:r, lite fluid,
and lite temperatures;
p = density of the ftuid (kg/m3 ); and
V
= velocity of lite fluid approaching the wire: (mis)
Upon setting lite electrical power equal
2 ((Ro + Rw)2
e0
=
Rw
10 lite rate of heal loss and solving, we obtain
(T., - Tt )Aw
)(
A+B
(20)
J�)
pV
Since lite bridge holds lite wire temperature and resistance constant, the resistances and
temperatures may be lumped with
A
and
B
into new constants, C and D:
(20a)
This result is usually called
King 's law (2 1 ).
Hot wires and hot films must be calibrated before use.
comparison to
a
Typically, a side-by-side
secondary standard, such as a Pilot tube, is made over a range
speeds. The results
are used to fit the values of C and D.
or ftow
By us i n g a p air of hot-wire o r hot-film probes oriented a l an angle t o one another,
two components of the flow velocity vector may be measured. Since the wires are typically
oriented al
±45° to the probe body, such an arrangement is called! an X-wire probe.
Measurement of Fluid Flow
Significant complications arise if the fluid temperature varies. Small changes in T1
may be corrected for by placing a line-wire resistance thermometer (a cold-wire probe)
adjacent to the hot wire, so as to measure the temperature difference in Eq. (20). Larger
temperature changes necessitate a further modification of Eq. (20), in which A and B
become functions of temperature [22, 23).
The primary value of the hot-wire anemometer lies in its high frequency response and
excellent spatial resolution. The frequency response of a hot wire can easily reach I 0 kHz;
if loss of spatial resolution [24) is unimportant, hot wires can be applied at frequencies
several times higher. On the other hand, owing to their relatively high cost and inherent
fragility, hot wires are usually justifiable only when their fast response is essential to the
measurement at hand. The most common situation requiring high-frequency response is
the measurement of the fluctuating velocity in turbulent flows.
Themwl ftowmeters are also available. These devices measure flow rates in tubes or
pipes. In one form, a section of a pipe wall is electrically heated; the resulting increase in
fluid temperature, downstream, is proportional to the mass-How rate in the pipe. Some auto­
motive fuel-injection systems use a type of thermal flowmeter as part of the engine control
system. A hot-film sensor is located in the intake manifold; this sensor is of relatively rugged
construction, lowering its frequency response in favor of improved reliability. The hot-film
signal identifies p V; when the latter is mul tiplied by the manifold cross section, the total
mass-flow rate is obtained. A temperature sensor is incorporated to compensate for changes
in environmental conditions. The measured mass-flow rate enables a microprocessor to set
the fuel injectors for the correct fuel mixture.
Hot-wire and hot-film anemometers have been studied extensively, and a large body
of literature is available on their performance and use (23, 25).
10
DOPPLER-SHIFT MEASUREMENTS
When light or sound waves of a given frequency are scattered off of particles in a moving
fluid, they undergo a frequency change or Doppler shift. The Doppler shift of the scat­
tered waves is proportional to the speed of the scauering particle. Thus, by measuring
the frequency difference between the scattered and unscallered waves, the particle or How
speed may be found. The wave sources most often used in fluid velocimetry are laser light
and ultrasound.
The Doppler shift is responsible for the familiar change in pitch as a moving source
of sound passes, such as that heard in a siren or car horn. The size of the Doppler shift in
a wave scallered from a moving particle depends on the particle's direction relative to the
incident wave, as we l l as the observer's position (Fig . 26). A calculation shows that the
frequency shift observed is (26):
6f
=
(2;)
cos tJ sin
(i)
(2 1 )
lndicenl wave, f
Measurement of Fluid Flow
111
s
-,;=
V
A/
I
I
I
I
I
I
I
I
I
I
I
/Bisector
Q
� Observer
AGURE 26: When an incident wave of frequency f is scauered from a particle moving al
speed V, the observer sees a scauered wave of frequency f + Af, where Af is the Doppler
frequency shift (Eq. 2 1 ).
where
A/ = lhe Doppler frequency shift (Hz),
V = the particle velocity (mis).
.\
=
the wavelength of the original wave before scallering (m),
{J = the angle between the velocity vector and the bisector of the angle SPQ,
a =
the angle between the observer and the axis of the incoming wave
For the purpose of flow measurement, the important aspect of the Doppler shift is its
proportionality to the particle velocity. If we can measure the shift, we can find the particle
speed; and if the particle moves with the flow, this speed should equal the fluid speed.
Because laser light and ultrasonic waves have relatively high frequencies, the Doppler
shift is only a small fraction of the original wave's frequency. For example, the fractional
frequency change in scauercd laser light may be only I / I OS. The Doppler frequency is
usually resolved by heterodyning the scattered wave with an unshifted reference wave lo
produce a measurable beat frequency.
10.1
Laser-Doppler Anemometry
The original laser-Doppler anemometer (LOA) used separate scauering and reference
beams to create an optical heterodyne at a pholodetector [Fig. 27(a)]. The light scattered by
particles in the flow interfered with the light from the reference beam to produce beats al a
frequency of one-half the Doppler shift. Unfortunately, these systems were fairly difficult
to align because the intensity of the reference beam must nearly equal that of the sc•uered
light in order to achieve an acceptable heterodyne. Consequently, reference beam systems
have largely given way to the differential Doppler approach shown in Fig. 27(b).
The differential Doppler system splits the laser into two equal intensity beams, which
are focused into an intersection point. A particle passing through the intersection scatterS
470
Measurement of Fluid Flow
Test section
window
0
Flow
with particles
(a)
0
0
Test section
window
Scattered light
Laser
0
Beam
splitter
Flow
with particles
(b)
FIGURE 27:
Laser-Doppler optical systems: (a) reference-beam arrangement;
(b) differential-Doppler arrangement.
l ight from both beams, and this light is collected by a photomultiplier tube (PM11 . The
Doppler shift for eac h beam is equal and opposite, by virtue of their different angles, but
the intensities of the two scaltcred waves are now identical. The resulting beat frequency
al the detector is equal in magni tude to lhe Doppler shift of lhe beams. Figure 28 shows a
commercial LOA lransmiuer/receiver arrangement. ·
The signal detected b y a differential LOA may also be interpreled i n terms of the
light and dark interfennce fringes prod uced at the beam crossover point (see Fig. 28). The
distance between these fringes may be shown to be
6=
>.
---
2 sin(9/2)
471
(22)
Measurement of Fluid Flow
i Particle flow
Laser velocimetry optics
0
Cdlection angle
-� �
Photodetector
Signal wire
oulput
Waist
�
t�
...
.5
Interference
lringe pattem
Beam crossover
nme -
Scatter Intensity piol
(particle signal)
FIGURE 28: LDA transmitter and receiver packages. (Courtesy: David Carr, Acrometrics
Inc., Sunnyvale, California)
where
li
A.
9
= the fringe spacing (m),
= the laser wavelength (m),
= the angle between the two bearns
A small particle crossing the fringe pattern produces a burst of sGattered light whose intensity
varies as the particle crosses each fringe (Fig. 28). The frequf:ncy of this Doppler burst is
just the particle velocity divided by the fringe spacing:
Yx (2 Vx) . ( 2 )
lo = -;; = T
where
sm
9
(23)
ID = the Doppler-burst frequency (Hz).
Yx
= the particle velocity in the direction nonnal I•> the fringes (mis)
Nole that the Doppler burst frequency depends only on the ve:locity component normal to
the plane of the fringes. Also note that the burst frequency is independent of the position
of the photomultiplier tube.
The PMT signal is processed digitally to find the frequency, /0, and thus the particle
velocity,
One common scheme identifies the Doppler frequency . by calculating the
Yx.
472
Measurement of Fluid Flow
Fourier transform of the burst signal, and others use digital time-corre lation techniques [ 27 J.
beam
The
intersection volume can be quite small, depe nding on the focusing optics.
These probe volumes are elliptical, with a major axis of O. I to I mm and fringe spac i ngs mea­
sured i n micrometers. 1be scattering particles are usually seeded into the flow. For l iq uids ,
. natural impurities may provide acceptable seed particles; if not, addi n g small pol ys tyrene
spheres or even a little milk will work. In gases, an aerosol of nonvolatile oil can be used.
For good signal quality, the scattering particles should ge nerally be of d iameter smaller than
the fri nge spacing; seed particle diameters of about I µ.m are common for gas flows.
Particles in the flow cross the probe volume at random time intervals, and, as a res u l l,
the data con si st of a sequence of individual veloc i ty measurements that must usually be
analyzed statistically. An additional consequence i s that the freque nc y response of LDA
systems depends primarily on the rate al which particles cross the intersection poi n t, rather
than on the optical or electronic configuration. Frequency responses can reach hundreds of
hertz with good seeding conditions.
.
The actu al i ntensity of the scattered l i ght depends upon the an g le from which the
scattering particle is viewed as well as the ratio of the laser wavelength to particle diameter
and the particle's index of refraction [28]. 1be strongest signals are obtained when the
scattering particle ' s diameter is several times the wavelength and when the probe volume
is vi ewed using a small collection angle (see Fig. 28). However, the collection angle may
be increased up to 1 80" i f space is limited and weaker signals can be tolerated.
Finally, it should be noted that ordi n ary LDA cannot distingui sh the fl ow direction :
Positive and negat ive values of Vx will
produce the same Doppler shift. This difficulty
additional, known frequency shift to one of the laser beams.
This causes the fringe pattern to move at the shift frequency, so that a stationary particle
would scatter l ight at the shift frequency. For a m ov i ng part ic le. the shift frequency is
increased or decreased by the amount of the Doppler shi ft. The measured burst frequency
no longer passes through zero when the flow direction changes; instead, it becomes larger
or smaller relative to the shift frequency. With freq ue ncy shift, LDA becomes one of the few
velocimeters capable of measuring flow reversals. Fu rthermore, if additional pairs of laser
beam s are used, LDA can measure two or three components of velocity simultaneously.
is overcome by apply ing an
Commercial LDA sy stems are very expensive, and their use is just ifi ed only when
local and nonintrusive measurements are absolutely i mperative. Ifintrusion can be tolerated
and if flow reversal or flowborne particulates are not an issue, thermal ane mometers are a
more economical al tern ative .
10.2
Ultrasonic-Doppler Anemometry
Ultrasonic waves in the range of tens of kilohertz to several megahertz have been applied
The ultrasonic tra nsm itter and receiver may be
piezoelectric and are often designed to be clamped to the outside of a pipe.
to Doppler flow measurements in fluids.
Like LDA, ultrasonic-Doppler anemometry requires that particles be present in the
flow. In industrial settings , fluids are often clean, and del i berate seed i ng may very unde­
sirable. As a result, ultrasonic-Doppler systems have found greatest application lo the
measureme n t of slurries and dirty liquids which already include particulates. Ultrasonic­
Doppler flowmeters are accu rate no better than 2%, and they can be q u ite susceptible to
misinstallation errors [ 1 1 ].
473
Measurement of Fluid Flow
Because ultrasonic beams tend to be relatively large (measured in centimeters per.
.
haps), the spatial resolution of ultrasonic Doppler systems is poor. One conseq
that ultrasonic-Doppler lacks laser Doppler's ability to measure the Ouctuating velociti
of turbulent Oow, which are generally of much smaller scale.
uei:ce
11
=
FLOW VISUALIZATION
Many Oows are so complex that the designer may have difficulty predicting their form.
An experimental visualization of the How field then becomes an integral part of the design
process. For example, an understanding of the Oow past an automobile is essential 10
determining drag and improving fuel efficiency, particularly for the Oow toward the rear of
the car and in its wake. Normally, a scale model of the proposed auto body design is tested
in a wind tunnel. Smoke trails introduced upstream of the car are used to make the Oow
visible as it passes the model. The designer can then identify regions of separated Oow and
adjust the body contour to reduce the drag.
The advantage, of course, is that Oow visualization can illustrate an entire Oow field
whereas velocity probes yield information at only a single point. Regions of separation
recirculation, and pressu re loss may be identified without detailed (and more expensive)
velocity measurements or calculations.
Techniques of Oow visualization are widely varied Some methods introduce a visible
material, such as particles or a dye, into the Oow. In other cases, density variations (and
thus refractive index variations) in the Ouid itself may be rendered visible . The following
lislS a few of the common techniques. Further discussion may be found in the Suggested
Readings for this chapter.
:
wire visualization: A thin steel wire (-0. 1 mm) is coated with oil and placed
in an air How. An electric pulse resistively heats the wire, causing the oil to fOOJ)
smoke. The thin line of smoke is carried with the Oow, showing the Ouid pathlines.
1. Smoke
2.
Hydrogen bubble visualization: A very fine wire, often platinum of 25 to 100 µm
diameter, is placed in a water How. A second, Hat electrode is placed nearby in the
Oow. When de power of about 100 V is applied to the wire, a current passes through
the water. causing electrolysis at the wire surface. Tiny hydrogen bubbles are created.
These bubbles are too small to experience much buoyancy, and they instead follow
the water as a visible marker (29).
3. Panicu/ate tracer visualization: ReOective or colored particles make the How pattern
visible. In liquids, particles of near-liquid density are preferred; diameters of 25 to
200 µm are typical. Examples are polystyrene spheres, hollow glass spheres, fish
scales, and aluminum or magnesium Hakes. In gases, very-small-diameter particles
must be used, since larger particles tend to settle out of the How. Smokes of hydro­
carbon oils or titanium dioxide (which are composed of particles 0.01 to 0.5 µ m
diameter) have been used, as have oil droplelS, various hollow spheres, and even
helium-filled soap bubbles [30, 3 1 ).
4. Dye injectio11: A colored dye is bled into a liquid flow through a small hole or holes
in the surface of a test object. The dye track shows the path taken by the liquid as it
passes the object.
S. Chemical indicators: A chemical change is produced, often electrolytically, to cause
a solution to change color or to cause formation of a fine colloidal precipitate. For
474
Measurement of Fluid Flow
example, a hydrogen-bubble electrical arrangement may al ternatively be used to cause
a th ymol - b lue solution to change color from orange to blue.
6. laser-induced fluorescence: Fl uorescence is the tendency of some molecules to
absorb light of one color (or frequency ) and reemit light of a di ffere nt color (a lower
freque ncy ) . Fluorescent dyes are added to water, as in regular dye i njection, but a now
thin sheet of laser ligh t is used to excite the dye in a specific plane of lhe flow. The
resul ti ng lluorescence provides visualization of the flow in that plane alone. 'fypical
dyes for use w ith a blue- gree n argon -ion laser include rhodamine (fluoresces dark
red or yel l ow ) and fl uoresceine (fluoresces green). Laser-induced lluorescence can
be adapted to gas flows as well [321.
7. Refrac1ive-i11dex-change visualizations: A ll uid 's refractive index changes with its
de ns i ty (or
lemperature). Variations in refractive index will dellect or phase-sh i ft light
pas sing throug h a fluid; and with an appropriate optical arrangement, these effects can
be made visible. For example, the abrupt density change al a shock wave deflects light
and can be made to appear as a thin shadow in a photograph (a shadowgraph). Smaller
de nsity changes in compressible Hows or temperature gradients in buoyancy-driven
flows may also be visualized (and quantilalively measured) using refractive-index
methods such as the schlief'l!n technique and holographic inteiferometry [33].
Digital image processi ng is a useful adjunct to many of the preceding methods. For
exam ple, a time series of digitized images can be processed to trace the time history of
particle motions, y ie lding a whole-field velocity measurement. This technique i s know n as
particle-image velocimetry, or PIV [3 1 , 34) . PIV systems are commerc ially avai l able.
SUGGESTED READINGS
H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea. laser Doppler and Phase Doppler Mea­
sul'f!menr Techniques. Berlin: Springer-Verlag, 2003.
Baker, R. C. An l111roJuctory Guide lo Flow Measumnenl (ASME Edition). New York : ASME
Press. 2003.
Baker, R. C.
Flow Meusuromenr Handbook .
Cambridge, U.K.: Cambridge University Press, 2000.
Bruun, H. H. Hot- \Vire Anemometry: Principles and Signal Analysis. Oxford. U.K.: Oxford Univer·
sity Press.
1 995.
Goldstei n. R. J. Fluid Mechanics Measurements. 2nd ed. Washington. D.C.: Taylor and Francis,
Lynn worth, L. C. Ultrasonic Measul'f!menls for Prrx:ess Control: Theory. Techniques.
tions. San Diego: Academic Press,
1 989.
1996. .
and Applica­
Merzkirch. W. Flow Visualiltltion. 2nd ed. Orlando, Fla.: Academic Press, 1987.
Miller, R. W.
Flow Measuroment Engineering Handbook.
3rd ed. New York: McGraw-Hill, 1996.
Raffcl, M. , C. E. Willert, and J. Kompenhans. Particle Image Vtlocimetry.
Verlag, 1 998.
Yang, W.-J. (ed.)
cis,
2001 .
Handbook of Flow
Berlin: Springer-
Visualization. 2nd ed. Washington, D.C.: Taylor and Fran­
475
Measurement of Fluid
Flow
PROBLEMS
I.
2.
Check Eq. (5) for a balance of units using (a) the SI system of units and (b ) the English
system.
A section ofho rizontally oriented pipe gradual ly tapei:s fror� a diameter of I 6 cm (6.3 in.)
_
lo 8 cm (3. 15 m.)
over a length of 3 m (9.84 ft). Oil having a specific gravity of 0.85
flows at 0.05 m3/s ( 1 .77 ft3/s). Assuming no energy lo:;s,
should exist across the tapered section? Check unit balanc:e.
what
pressure differential
Problem 2 using Eng lish units.
3.
Solve
4.
2 is oriented into a vc'rtical position with the larger
at the lowest point, what differential pressure across the section should be
found? Solve (a) using SI units and (b) using English uni l:s. (c) Check the two ariswers
for equivalency. (d) Would there be a difference if the smaller diameter is at the low·
est position?
If the conduit described in Problem
diameter
S.
Show that Eq. (Sc) may be wriuen as follows:
= the differential pressure across the meter, measured in the "head" of lhe
Rowing fluid. Check the unit balance.
where h
6.
When an obstruction meter is placed in a vertical run of pi pe (as opposed 10 a horizontal
run), what precautions must be made in measuri ng the differential pressure?
7.
Prepare a spreadsheet template for solving
8.
A machined venturi meter is placed in a horizontal run of 8-in., schedule 40 commercial
9.
Solve Problem
venturi tube proble ms .
pipe (10. = 7.98 1 in. (202.7 mm)] for the purpose of metering medium heating oil as
it is pumped into a large storage tank. The throat diameter of the venturi is 6.00 _in.
( 1 52.4 mm). If the differential pressure is held 10 2.54 ps i ( 1 75 kPa) for one-half hour.
how many gal lons (liters) of oil should have been pumped? The oil temperature is 60"F
( 1 5.6°C) and its specific gravity is 0.86.
8 using SI units.
476
Measurement of Fluid
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
Flow
Water at I S°C and 6SO kPa llows through a IS x 10 cm ( I S-em pipe and 10-em throat
diameter) as-east venturi tube. A di fferential pressure of 2S kPa is measured. Calculate
the flow rate (a) in kg/min and (b) in m1 /h.
A rough-welded venturi meter with a 40-em ( I S.7S-in.) d iameter throat is used to meter
IS°C (S9°F) air in a 60-em (23.62-in.) duel If the differential pressure is measured to
be 84 mm (3.3 1 in.) o f water and the upstream pressure (absolute) is 12S kPa (18. 1 3
psi), what is the fl ow rate i n kgls? I n m 1/s?
Solve Problem 1 1 using Engl ish u n i ts.
Prepare a spreadsheet template to be used for solving comer-tapped orifice meter prob­
lems.
Use the spreadsheet template prepared in answer to Problem 1 3 to check the calcu lations
in Example
l(a).
A comer-tapped orifice meter is used to measure the ftow of kerosene in a 7S-mm (2.9S­
in.) diameter line. If {J = 0.4, what differential pressure may be expected for a ftow rate
of 1 .3 m 1tmin (47.67 ft1fmin) when the temperature of the fl uid is IO"C (SO"F)?
Solve Problem I S using English units.
A comer-tapped orifice meter is used to measure the flow of 60°F water in a 4-in. l.D.
pipe. Prepare a p lot of differential pressure readout (inches of H20) versus {J over a
range 0.2 < {J < 0.7S if the flow rate is fixed at 230 gaVmi n . (Note: A spreadsheet
solution is recommended.)
A comer-tapped orifice meter is to be used in a 4- i n . ID pipe carry i ng water whose
temperature may vary between 40-F and 1 20-F. The range of How rate is from I SO to
600 gaVmin. The differential pressure across the orifice will, among other th ings, depend
on the value of {J. To minimize losses, it is desirable to use as large an orifice diameter
as is feasib le.
Investigate and speci fy the type of differential pressure sensor to be used . Con­
sidering sensitivity limits, determine the largest prac tical value of {J that wi l l salisfy
your specifications. Write a statement e xp lai ning the selections you have made. It is
suggested that a spreadsheet be used.
If the steel
plate used for a
k ·in. thick orifice plate in a 5-in.-diameter pipe has a yield
-
strength of 45,000 lbflin. 2 and {J
distortion may be expected.
=
O.S, estimate the di ffere n ti a l pressure above which
Direct secondary calibration is used to determi ne the ftow coefficient for an orifice to
meter the flow of ni trogen. For an orifice inlet pressure of 2S psi a ( 172 kPa), the ftow
rate determ i ned by the primary meter is 9 lbmlmin ( 1 9.8 kgfm) at 68°F (20-C). If the
differential pressure across the orifice being calibrated is 3. 1 in. (7 .87 cm) of water, the
conduit diameter is 4 in. ( 1 0. 16 cm), and fJ = 0.S, what is the value of (K x Y)? (Use
p = 0.0726 lbmfft1 at 68°F and standard pressure . ) Solve using En g lish uni ts.
Solve Problem 20 using SI units.
Using Eq. ( 13), plot Pi/ P1 1 versus k over a range of k
=
I to 1 .4.
A Pi lot-static tube is used to measure the veloc ity of 20-C (68°F) water ftowing in an
open channel. If a differential pressure of 6 cm (2 . 36 in.) of H2 0 i s measured, what is
the corresponding How velocity"? Check the result by usi ng English units and comparing
your answer to the SI resu lt .
477
Measurement of Fluid Flow
Pilot-static lube is used to measure lhe ve locity of an aircraft. If lhe air lelll)Jenluie
and pressure are 5°C (4 1°F) and 90 kPa ( 1 3.2 psia) , respectively, what is the ain:ral\
ve loc i l y in km/h if lhe differential pressure is 450 mm ( 17-5 in.) of waler? (a)
SolYe
using Eq. ( 1 9), lhen (b) using Eq. ( 1 9a).
24.
A
25.
Solve Prob lem 24 using English units.
26.
The ve loci ty profile for turbulenl How in a smoolh pipe is sometimes given as (35]
= lhe pipe diameter, r = lhe radial coordinale from lhe cenier of the pipe, and
in val ue from aboul 6 to 10, depending on lhe Reynolds number. For n = 8,
de1ennine lhe value of r al which a Pitot tube should be placed to provide the velocity
v... such lhat Q = A v••.
where D
11 ranges
27.
Like a Pi tot- stati c tube, a hot-wire probe may also suffer from angular misalignment
Figure 29 de fi nes lhree possible angles lhat the probe may have relative to ilS
desired alignmenL Describe separately what error is incurred when each angle is nonzero.
Give qualitalive answers and assume lhat the angles remain less lhan 45°.
errors.
.,, c.b
z
v
D;)
+f . .
"'
y
= Roll angle
I/I • Pitch angle
<I> = Yaw angle
8
FIGURE 29: Angular orientati on of a hot-wire probe.
478
Measurement of Fluid Flow
REF ERENCES
(II ISO 5 167-4:2003 Measurement offluid flow by means of differential devices inserted
in a circular cross-section running full-Part 4: Venturi tubes. Geneva: International
Organization for Standatdization, 2003.
(2) ISO 5 1 67-3:2003 Measurement offluidflow by means of differential devices inserted
in a circular cross-section running full-Part 3: Nozzles and Venturi nozzles. Geneva:
International Organization for Standardization, 2003.
(3) ISO 5 167-2:2003 Measurement offluidflow by means of differential devices inserted
in a circular cross-section running full-Part 2: Orifice plates. Geneva: International
Organization for Standardization, 2003.
(4) Baker, R. C. An Introductory Guide to Flow Measurement (ASME Edition). New York:
ASME Press, 2003, App. 4.3.
[SJ Roark. R. J. Formulas for Stress and Strain. 4 th ed. New York: McGraw-Hill, 1965,
p. 22 1 , Case 1 7 .
(6) AS M E PTC 1 9.5, Application-Pt. II o f Fluid Meters: Interim Supplement on Instru­
ments and Apparatus, 1972, p. 232.
(7) Thompson, P. A. Compressible-Fluid Dynamics. New York: McGraw-Hill, 1 972.
(8) ISO 9300: 1995 Measurement of gas flow by means of critical flow Venturi nou;les.
Geneva: International Organization for Standardization, 1 995.
(9) Miller, R. W. Flow Measurement Engineering Handbook. 3rd ed. New York: McGraw­
Hill, 1 996.
( 10) Nagler. F. A. Use of current meters for precise measurement of How. ASME Trans.
57:59. 1935.
(11) Baker, R. C. Flow Measurement Handbook. Cambridge, U.K.: Cambridge University
Press. 2000 .
( 12) Lienhard I V. I. H., and J. H. Lienhard V. A Heat Transfer Textbook. 3rd ed. Cambridge,
Mass.: Phlogiston Press, 2003, Section 7 .6.
(13) Schoenborn, E. M., and A. P. Colburn. The How mechanism and performance of the
rotameter. Trans. A/ChE 35(3):359, 1 939.
[14) Wright, J. D., and G. E. Mattingly. NIST Calibration Services for Gas Flow Meters:
Pisto11 Provers and Bell Prover Facilities. Gaithersburg, Md.: National Institute of
Standards and Technology, Special Publication 250-98, 1 998.
( 15] Jarre t , F. H. Standpipes simplify Howmeter calibration. Control Eng. 1 :37, Decem­
ber 1 954.
( 16) Gracey, W. Measurement of aircraft speed and altitude. NASA Reference Pub. I 046 :
1980.
[ 17) Gracey, W. Measurement of static pressure on aircraft. NACA Tech. Note 4184:
November 1 957.
[18) Gettelman, C. C., and L. N. Krause. Considerations entering into the selection of
probes for pressl!re measurement in jet eng i nes. ISA Proc. 7 : 1 34, 1 952.
( 19) Krause, L. N., and C. C. Genelman. Effect of interaction among probes, supports,
duel walls and jet boundaries on pressure measurements in ducts and jets. /SA Proc.
7: 1 38, 1 952.
479
Measurement of Fluid Flow
[20) ltsweire, E. C., and K. N. Helland. A high-perfonnance low-cost constant-lemperallft
hot-wire anemometer. J. Phys. E: Sci. lnstrum. 1 6:549-553, 11983.
R
(21) King, L. V. On the convection of heat from small cylinders iin a stream of fluid, with
applications to hot-wire anemometry. Phil. Trans. oy. Soc. (London) 2 1 4, 1 4, Ser.
A:373-432, 1 9 1 4 .
(22) Lienhard V, J. H. The decay of turbulence in thennally stratified flow. Doctoral dis­
sertation, University of California, San Diego, 1 988, Chapter 3.
[23) Bruun, H. H . Hot- Wire Anemometry:
Oltford University Press, 1995.
Principles and Signal Ana/y.ris. Oxford, U.K.:
(24) Wy ngaard , J. C. Measurement of small-scale turbulence structure with hOI wires. J.
Phys. E: ScUnstrum.
1 : 1 105-1 108, 1 968.
[25) Freymuth, P. Bibliography of Thermal Anemometry. 2nd ed. St. Paul, Minn.: TSI,
Inc., 1 �3.
[26) Drain, L. E. The Laser Doppler Technique. New York: John Wiley, 1980, Chapter 3.
[27) H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea. Laser Doppler and Phase
Doppler Measurement Techniques. Berlin: Springer-Verlag, 2003.
[28) Van de Hulst, H. C. Light Scattering by Small Particles. Ne:w York: Dover Publica­
tions, 1 98 1 .
[29) Geller, E . W. A n electrochemical method o f visualizing the boundary layer. J.
Sci. 22:869-870, 1 955.
Aero.
[30) Merzkirch, W. Flow Visualization. 2nd ed. Orlando, Fla.: Academic Press, 1 987, p. 46.
[31) RalTel, M., C. E. Willert, and J. Kompenhans.
Springer-Verlag, 1998, pp. 1 3-22.
Particle Image Velocimetry.
Berlin:
[32) Hansen, R. K., and J. M . Seitzman. Planar Fluorescence Imaging in Gases, in Yang,
W.-J. (ed.) Handbook of Flow Visualization. 2nd ed. Washington, D.C.: Taylor and
Francis, 200 I .
[33) Merzkirch, W. Flow Visualization. 2nd ed . Orlando, Fla.: Academic Press, 1987,
Chap. 3.
[34) Adrian, R. J. Particle-imaging techniques for experimental llu id mec han ics . Annu. Rev.
Fluid Mech. 23:26 1-304, 199 1 .
[JS) Fox, R. W., and A . T. McDonald. ltrtroduction to Fluid Mechanics. 5th ed. New York:
John Wiley, 1 998, pp. 353-354.
Measurement of Fluid Flow
TABLE
1:
Temperature
degrees C (F)
5 (4 1 )
1 0 (50)
15 (59)
20 (68)
25 (77)
30 (86)
35 (95)
40 ( 104)
45 ( 1 1 3 )
50 ( 1 22)
ANSWERS TO SELECTED PROBLEMS
2
19
23
l> P = 39.35 kPa
l> P
= 530 psi
V = l .085 mis
Propenies of Water: SI System
Absolute Viscosity
Pa s
·
15. 1 9 x 10- 4
1 3.08
1 1 .40
10.05
8.937
8.007
7.225
6.560
5.988
5.494
Density
kg/m3
1000
999.7
999 . 1
998.2
997.0
995.7
994 . 1
992.2
990.3
988 . I
Tem peratu re M easu rements
2
3
4
S
6
7
8
9
10
11
12
INTRODUCTION
USE OF THERMAL EXPANSION
PRESSURE THERMOMETERS
THERMORESISTIVE ELEMENTS
THERMOCOUPLES
SEMICONDUCTOR-JUNCTION TEMPERATURE S ENS ORS
THE LINEAR QUARTZ THERMOMETER
PYROMETRY
OTHER METHODS OF TEMPERAT\JRE IN D ICATIO N
TEMPERAT\JRE MEASUREMENT ERRORS
M EASUREMENT O F HEAT FLUX
CALIB RATION OF TEMPERAT\JRE-MEASURING DEVICES
INTRODUCTION
Temperature change is usually measured by observing the change in a temperature-dependent
physical property. Unlike the direct comparison of other fundamental physical quantities
to a cali brated s tan dard
(as
mass is measured by comparison to the Internat ional Prototype
Ki logram) , direct comparison of an unknown temperature to a reference temperature is
relatively difficult. I n formal thermodynamics, this comparison is made by connecting a
Carnot engine between two systems at different temperatures. In practical thermometry,
temperature is instead gauged by its effect on quantities such as volume, pressure. electrical
resistance, or r.idiated energy.
The International Practical Temperature scale (ITS-90). assigns values of temperature
to a few highly reproducible states of matter, such as certain freezing points and triple
poinl�. These defined reference temperatures then provide calibration points for various
special thermometers, which are used to interpolate between the reference points. The
interpolating thermometers undergo a change in some other physical property, such
as
pressure or electrical resistance, as their temperature changes, and the value o f that property
is used to infer the corresponding temperature. In a sense, temperature itselfis never directly
sampled in practical thermometry.
In this chapter, we survey a selection of common temperature-sensing techniques.
These thermometers are
based on changes in a broad range of physical properties, among
which are the following:
From Mecl1anicul Measuremen1s, Sixth Edition, Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V.
Copyright O 2007 by Pearson Education, Inc. Published by Prentice Hall . All rights reserved.
483
Temperature Measurements
I. Changes in phy sical dimensions
(a) Liquid-in-glass thermometers
(b) Bimetallic elements
2. Changes in gas pressure or vapor pressure
(a) Constant-volume gas thermometers
(b) Pressure thermometers (gas, vapor, and l iq uid filled)
3. Changes in electrical properties
(a) Resistance thermometers (RID, PRT)
(b) Thermistors
(c) Thermocouples
(d) Semiconductor-junction sensors
4. Changes in emitted thennal radiation
(a) Thermal and photon sensors
(b) Total-radiation pyrometers
(c) Optical a nd two-color pyrometers
(d) Infrared pyrometers
s. Changes in chemic al phase
(a) Fusible indicators
(b) Liquid crystals
(c) Temperature-re fere nce (fixed-point) cells
Of these methods, electrica l sensors are perhaps the most broadly used. particularly
when automatic or remote recording is desired or when temperature sensors are incorpo­
rated into control systems. Bimetallic elemen ts are used in vario us low- acc uracy, low-cost
applications . Radi an t sensors are used for noncontact temperatu re sen sing , either in high­
temperature applications like com bus tors or for infrared sens i ng at lower temperatures;
since radiant sensors are opti cal in nature, they are al so adaptab lle to whole-field tempera­
ture measurement-so-called thermal i magi ng. The familiar l iqu id - in-glass thermometer
c onti n ues to appear in both l aboratory and household situations, primarily because of its
ease of use and low cost. Changes in c hem ica l ph ase are somewhat less often applied in
engi neering work.
Table I outli nes approximate ranges and u nce rtai n ties of various temperatu re-measuring
devices. The values listed in the table are o n l y approximate, and many untabulated factors
will cause dev iations from the values listed. Among those factors are the precise form of
electrical s ignal conditioning employed, the inHuence of man u facturer's and/or laboratory
calibration techniques, and ded icated efforts to extend the operati ng range of a particu­
lar sensor.
484
Temperature Measurements
TABLE 1: Characteristics of Various Temperature-Measuring Elements and Devices (Data
from Various Sources)
Type
Uqaill In Glass
Mercury filled
Pressuriud mercury
Alcohol
BinutaJ
Pnssure Systems
Gas {laboratory)
Gas {induslrial)
Useful Range•
32o•c
Umlts of
Uncertainty•
-37 to
-35 to 600° F
o. 1•c
-37 to 6S0°C
-lS 10 1200°F
o. 1°c
-75 to ! 30°C
- 100 to 200° F
o.s•c
-65 to 4 30°C
-80 10 800° F
o.s 10 1 2•c
oo•c
-210 10 t
-450 to 2 1 2°F
-270 10 760°C
-450 10 1400° F
0.2°F
0.2°F
1°F
Comments
Low cost. Remote reading
not practical.
Lower limit of
mercury-filled
thermometers
dctennined by freezing
point of mercury.
Upper limit determined by
boiling poinL
Rugged. Inexpensive.
I to 20°F
0.002 to o.2•c
o.oos to 0.5°F
o.s
10 2%
o f full scale
Very accurate. Quite
fragile. Not easily used.
Used as an interpolating
s1andard for ITS-90
(see Section 7).
Bourdon pressure gage
used for readout
Rugged, with wide
range.
c
Liquid (except mercury)
-90 10 310°c
- 1 25 10 700°F
1°
2°F
Liquid {mercury)
-37 to 630°C
-35 to 1 200°F
0.5 10 2%
of full scale
Vapor pressu re
-7S to 340°C
- 1 00 to 6S0°F
10 2%
of full scale
485
o.s
Relative elevations of
readout and sensing
bulb are critical.
Smallest bulb. Up lo
3 m { I 0 fl) capillary.
Same as above.
Fast response. Nonlinear.
Lowest cost
Temperature Measurements
TABLE
Type
Thtrmocoupks
General
1: continued
Useful Range•
Umlts of
Uncertainty*
-210 to noo•c
-454 to 4200°F
::1:1 10 2•c
::1:2 to 4°F
Comments
Extreme ranges.
USllly
ll
inexpensi"W:. Small
size.
Type 8: Pt/30% Rh (+)
vs. Pt/6% Rh ( )
-
870 to 11oo•c
1600 to 3 I00°F
±0.5'*1*
Not ror reducing
atmos phele . Use
nonmetallic sheath.
Prefened to types s ml
R abow 1200°C.
Type E: Cluomelt (+)
vs. Constantant ( -J
-250 to 900°C
-420 to 1650°F
±0.5%*
Type J: Fe (+) vs.
Constantant ( - )
O to 760°C
32 to l400°F
::l:0.75%+
Type K: Chromelt (+)
-250 to 1260°C
-420 to 2300°F
Most atmosphen:s.
Popular and
inexpensive.
Calibrations less
consislent.
vs. Alumelt (-)
Type N: Nicrosilt
Highest OUlpUt or common
thermocouples. Not for
reducing or 111CWm
atmosphere. 8e5I
common type for
cryogenics.
::l:0.75%+
Oxidizing or inen
atmosphere. Less
oxidation than types E.
J, and T. Mosl linear
(+)
vs. Nisilt (-)
-250 to 1260°c
-420 to 2300°F
::1:0.75%*
type.
Popular.
Higher slability and
oxidation
resistanCC
lhan types E. J, K. and
T above I OOO"C.
Oxidizing or inert
atmosphere.
Type R: PV1 3% Rh (+)
vs.
Pt ( -)
Type S: Pt/10% Rh (+l
vs. Pt (-)
-5o to 143o•c
-50 to 2700°F
-so to 1480°C
-so to 2100°F
for reducing
::l:0.25%+
Not
::l:0.25%*
Not for reducinl
aunosphel'CS- Use
nonmetallic sheath-
atmosphere. Most
siable iype. Use
heathnonmetallic s
Temperature Measurements
TABLE 1:
Type
'fype T: Cu ( +) vs.
Constantant (-)
Useful Range•
-200 to 31o•c
-330 to 700°F
continued
Umlta of
Uncertainty•
:l::0 . 7S'll·t
Comments
All atmospheres. Stable.
The high thermal
conductivity may cause
errors.
WIS% Re (+) vs.
W/26-. Re (-)
RWlance
Platinum
Nickel
400
1o 231o•c
150 lo 4200°F
-260 1o 980°c
-43S 1o 1soo°F
± t .0%*
0.02 1o o.2•c
0.04 to 0.4°F
-180 to 320°C
-300 to 600°F
0.3°C
o.s•F
Thermistor (doped
germanium)
-273 to -173°C
459 to -280°F
o.03°C
o.os°F
-
487
High repeatability.
Nonlinear. Produces
g ate resi stance
change per degree than
does Pt. Sensor can be
as far as I SOO m
(SOOO ft) from readout
re r
to
- I SO lo 600° F
- 100 31s•c
i
High repeatabil ty. Linear.
Used as an interpolating
device for JTS-90 (see
Seclion 7). Sensor can
be used as far as I SOO m
(SOOO ft) from readout
oxide)
Thermistor (metal
No standards. Not for
oxidizing or reducing
atmospheres. Highest
temperature limit of all
lhennocouples.
Negative temperature
coefficient Highly
nonlinear. Less stable
than metal types.
High repeatability.
Nonlinear. Negative
temperature coefficient.
Cryogenic sensor.
Temperature Measurements
TABLE
Type
Useful
1bennis1or
Semiconductor
Junction
Diode (silicon, GaAIAs)
Linear integrated circuit
continued
Umlts of'
Uncertainty•
o.os•c
-212 10 50°C
-458 lo 1 25°F
(calbon-glass)
o.1°F
- 272 to so•c
-457 to 125°F
o.o5•c
0.1°F
-so to
o.5°C
15o•c
-60 to 300° F
J °F
Comments
High rcpealabilily.
No nlinear. Negative
tempera111re coelficienL
Cry9genic sensor.
Nonlinear. High accuncy
requires calibralion.
Cryogenic sensor.
Inexpensive. Linear.
Easily integrated into
electronics. Limited
temperature range.
Pyrometers
Tuial radiation
Spectral
Range•
1:
band
Disappearing 6Jament
200 10 2000°c
400 10 3600° F
250 10 3000° C
500 lO 5400° F
800 10 4200°C
1 500 to 7600° F
•Approximate values. Actual values depend on
Rugged. Leasl complex
O.S to J % !1
More sensitive
0.5 to 2%'1
Requires manual
lype.
lhan lolal
radiation type.
manipulation
operator.
-1s to 2soo0c
- 1 00 10 4500°F
Infrared
0.75 10 2% §
0.5 10 2%·1
by
Particularly useful ror low
temperatures.
many factors such llS sheathing and insulation, physical
size of sensor or thermocouple wire gage, purity of materials, calibration employed, etc. "fypes such
as thermocouples and resisiance thermometers require add iticmal signal-conditioning apparatus;
values given
arc for sensors cmly. Unsuiiable ranges of ceriain thermocouple types are omiued.
tTrade names of alloys.
*1n higher ranges.
Percentages rcrer to temperature in
I For measurement of blackbodies with e =
error may be much larger.
•c.
I . For surfaces of lower or poorly known emissiv ity. lhe
Temperature Measurements
2
z.1
USE OF THERMAL EXPANSION
Liquid-in-Glass Thermometers
The ordinary thermometer is an example of the liquid-in-glass type (Fig. 1 ). Its essential
elements are a relatively large bulb at the lower end, a capillary tube with scale, and a liquid
filling both the bulb and a ponion of the capillary. In addi tion , an expansion chamber is
generally incorporated at the upper end to serve as a safety reservoir when the intended
temperature range is exceeded.
As the temperature is raised, the greater expansion of the liquid compared with that
of the glass causes it to rise in the capillary or stem of the thermometer, and the height
of rise is used as a measure of the temperature. The volume enclosed in the stem above
the liquid may either contain a vacuum or be filled with air or another gas. For the higher
temperature ranges, an inen gas at a carefully controlled initial pressure is introduced in this
volume, thereby raising the boiling point of the liquid and increasing the total useful range.
In addition, it is claimed that such pressure minimizes the potential for column separation.
Main
scale
Auxlllary
scale
[
{
- lnvnersion line
- Contraction chamber
- Bulb
AGURE 1: Liquid-in-glass thermometer.
Temperature Measurements
High -grade liquid-in-glass thermomelers may include several additional fealures An
immersion line may be inscribed on the thermometer to indicate the depth lo which it s
(d
be submerged inlo the measured enviroment. A contraction chamber may be provided to
shorten the overall length of capi l lary needed and to prevent bu bbles from being formed •.nt
lhe bulb when lhe thermomeler is cooled. Finally, an auxiliary scale may be provided fi 0
checking calibration points outside the main range of the thermomeler, such as O" C or
Several desirable properties for the liquid used in a glass thermometer � as follows:
�
100"�
1. The lemperature--d imension rel ation shi p should be linear, permiui ng a linear ins1ru­
men1 scale.
2. The liquid should have as large a coefficienl of expansion as poss ible. For this reason,
alco hol is better lhan mercury. Its larger expansion makes possible larger capil lary
bores and hence prov ides easier reading.
3. The liquid should accommodate a reasonable temperalure range without change of
phase. Mercury is limiled at the low-temperature end by its freezing point, -37.97"F
(or -38.87°C), and spiri ts are limited al the high-lemperature end by their boil­
ing points.
4. The liquid should be clearly visible when drawn into a line lhread. Mercury is
obviously acceptable in this regard, whereas alcohol is usable only if dye is added.
5. Preferably, the liquid should nol adhere to the capillary wal ls. When rapid temperature
drops occur, any film remaining on the wall of the tube will cause a reading thal is
100 l ow. In thi s respecl, mercury is better than alcohol.
Within its te m perature range, mercury is u ndou b1edl y lhe best liq u id for l iq uid- i n·
glass thermometers and is generally used in lhe higher-grade instruments. Alcohol is usually
satisfaclory. Other liquids are also used, primarily for the pu rpose of ex lending the usefu l
ranges to lower lem peratu res .
2.2
Calibration and Stem Correction
High-grade liquid-in-glass t hermomelers are made w ilh lhe scale etched direclly on lhe
thermome 1er slem, thereby maki n g ii mech an i cal l y impossible lo shift the scale relalive to
the slem. The care wilh which !he scale is laid oul de pend s on the intended accuracy of lhe
i nstru men t (and to a large exten1 governs its cost). The process of eslabl i sh i ng
marks
from wh ic h a scale is determined is known as pointing, and 1wo or more marks or points are
required. In spile of contrary intenlions, a particu lar thermometer will exhibit some degree
of n onl i nearily. Th is may be caused by nonlinear 1e mpera1ure--di me ns ion characlcristics
of liquid or g l ass or by the nonu ni form ity of the bore of !he column. In the si mplest case.
two poinls may be eslablished, such as the freezing and boiling points of water, and equal
divisions used 10 in 1crpo la1e ( and e xtrapo lale ) lhe com plele scale. For a more accurate
scale, addilional po i n ls-someti mes as many as live-are used . Calibration points for lhis
purpose are ob1ained through use of known phase-eq u il i br i um 1empera1ures, as discussed
be11ch
in Section 12.
Grealest sensi 1iv i ty to temperatu re is al the bul b , where the largest volume of l iquid
is conlained; however, all p ortio ns of a glass th ermometer are tempera1urc-sensitive. With
temperature vari at io n , !he slem and upper bulb will also change dimensions. thereby allering
lhe avai l ab le l iqu id space an d hence the thermometer read in g. Thus, if max im um accuracy
Temperature Measurements
is to be attained, it is necessary to prescribe how a glass thennometer is to be subjected to the
temperature. Greatest control is obtained when the thennometer' s total measuring length (to
within a few divisions of the top of the liquid column) is immersed in a unifonn temperature
bath. Often this is not possible, especially when the medium is liquid. A common practice,
therefore, is to calibrate the thennometer for a given partial immersion, with the proper
depth of immersion indicated by a line scribed around the stem (the immersion line). This
technique does not ensure absolute uniformity because the upper portion of the stem is
still exposed to the ambient temperature. Thennometer accuracy is prescribed only for the
specified partial immersion and a specified ambient temperature. Thennometers are called
total immersion thermometers in the Conner case and partial immersion thennometers in
the latter case.
If some part of the stem is at a condition different from that used for calibration, an
estimate of the correct reading may be obtained from the following relation [ I , 2):
(I)
where
the correct temperature,
the observed temperature reading,
Ti = stem temperature specified for calibration of a partial
immersion thennometer or the bath teinperature of a total
immersion thermometer,
T2 = the temperature of emergent stem (this may be detennined
by attaching a second thermometer to the stem of the main
thermometer or approximated by the ambient temperature
surrounding the emergent stem),
k the differential expansion coefficient between liquid and glass (for
mercury thermometers, commonly used values are 0.00009 for the
Fahrenheit scale and 0.000 1 6 for the Celsius scale),
n = number of scale degrees equivalent to the length of the emergent stem
T =
robs =
=
The value n is determined as follows: For a total immersion thermometer, n should be the
number of scale degrees between the point of emergence and the top of the liquid column.
For the partial immersion thennometer, n should be the number of scale degrees that fit
in the distance between the scribed calibration immersion line and the top of the liquid
column.
Another factor influencing liquid-in-glass thennometer calibration is a variation in
lhe applied pressure, particularly in pressure applied to the bu l b. The resulting elastic
defonnation causes displacement of the column and hence an incorrect reading. Normal
variation in atmospheric pressure is not usually of importance, except for the most precise
work. However, if the thermometer is subjected to system pressures of higher values.
considerable error may be introduced.
491
Temperature Measurements
2.3
Blmetal Temperature-Sensing Elements
When tw� metal strips ha�ing different thermal expansion coeifficients
_
a change m temperature will cause a free defteclion of the assennbly [3).
� 1ini7.C;d ·�
Sut:li �Sin·.'
�
fonn the basis for control devices such as the analog home heating sysrem
Th">'. are a_I� used to some extent �or tempe�ture ��nxnt
the . laaer
c:ist, the
to a sunple helical spring
As the temperature changes, the free end of the helix rotates (the diameter oflhe helix ei....;
sensmg stnp 1s commonly wrapped mto a helical form, smulair
�
increasing or decreasing due to the differential action). The 1:otational motion
indicated by the movement of a pointer over a circular scale.
is diiecdy
Thermometers with bimetallic temperature-sensitive elements are often used because
of their ruggedness, their ease of reading, their low cost,
panicular fonn.
3
(See Problem 4.)
and the convenience of their
PRESSURE THERMOMETERS
Figure 2 i l lustrates a simple constant-volume gas thermometer.
Gas, usually hydrogen or
mercury column, B, is adjusted so that mfemice point
C is maintained. In this manner, a constant volume of gas is held in the bul b and adjoining
capillary. Mercury column h is a measure of the gas pressure llllld can be calibrated in 1e1ms
helium, is contained in bulb A. A
of temperature.
In this fonn the apparatus is fragile, difficult to use, ancll restricted to the labontory.
It does, however, illustrate the working principle of a group of practical instruments called
pressure thermometen.
Figure 3 shows the essentials of the practical pressure thermometer. The necessary
B, press ure-sensing gage C, and some sortoflilling medium. Pressure
gas filled, or vaporfilleci, depending on whether the
filling medium is completely liquid, completely gaseous, or a combination of a liquid and
its vapor. A pri mary advantage of these thermometers is than they can provide sufficient
force output to pennit the direct driving of recording and controlling devices. The pressure­
type temperature-sensing system is usually less costly lhan 011her systems. Tubes as long
as I 00 m (330 ft) may be used successfully.
pans are bulb A, tube
thermometers
are
called liqu idfilled,
B
--r
J
h
_
__
FIGURE 2 : Skelch illustrating the essentials of a constant- volume gas thermomeler.
492
Temperature Measurements
FIGURE 3:
thermometer.
Schematic diagram showing lhe operation of a practical pressure
Expansion (or contraction) of bulb A and the contained Huid or gas, caused by tem­
perature change, alters the volume and pressure in lhe syslem. In lhe case of lhe liquid-filled
sys1em, the sensing device C acls primarily as a differenlial volume indicator, wilh lhe vol­
ume increment serving
an analog of temperature. For lhe gas- or vapor-filled systems,
lhe sensing device serves primarily as a pressure indicator, wilh the pressure providing lhe
measure of temperature. In bolh cases, of course, both pressure and volume change.
as
Ideally the tube or capillary should serve simply as a connecting link between the
bulb and lhe indicalor. When liquid- or gas-filled systems ace used, the tube and ils filling
are also temperalure·sensitive, and any difference from calibration conditions along the tube
introduces output error. This error is reduced by increasing the ratio of bulb volume to lube
volume. Unfortuna1ely, increasing bulb size reduces lhe time response of a sys1em, which
may inlroduce 'problems of anOlher nature. On lhe other hand, reducing tube size, wilhin
reason, does not degrade response particularly because, in any case, flow rale is negligible.
Another source of error lhal should not be overlooked is any pressure gradienl resulting
from difference in elevation of bulb and indicalor nol accounted for by calibration.
Temperature along lhe tube is not a factor for vapor-pressure systems, however, so
long as a free liquid surface exisls in lhe bulb. In lhis case, Dalton's law of vapors applies,
which states that if both' phases (liquid and vapor) ace present, only one pressure is possible
for a given 1empera1ure. This is an important advantage of lhe vapor-pressure sys1em. In
many cases, !hough, lhe tube in this type of syslem will be filled wilh liquid, and hence the
sys1em is susceptible 10 error caused by eh!vation difference.
493
Temperature Measurements
4
THERMORESISTIVE ELEMENTS
The electrical resistance of most materials \laries with temperature; this supplies a trouble­
some extraneous input to the output of strain gages. It can only follow that this relation
which proves so worrisome when unwanted, should be the basis for a good method
temperature measurement.
of
Historically, resistance elements sensitive to temperature were first made
metals
generally considered to be good conductors of electricity. Examples are nickel, copper,
platinum, and silver. A temperature-measuring device using an element of this type is
commonly referred to as a resistance thermometer, or a resistance temperature detector
abbreviated RTD. Of more recent origin are elements made from semiconducti ng materi:
als having large resistance coefficients. Such materials are usually some combination
metallic oxides of cobalt, manganese, and nickel. These devices are called thermistors.
of
of
One important difference between these two kinds of material is that, whereas the
resistance change in the RTD is small and positive (increasing temperature causes increased
resistance), that of the thennistor is relatively large and usually negative. In addition, the
RTD has a nearly linear temperature-resistance relation, whereas that of the thermistor is
nonlinear. Still another important difference lies in the temperature ranges over which each
may be used. The practical operating range for the thermistor lies between approximately
- IOO"C and 300°C ( - 150°F to 575°F). The range for the resistance thermometer is
much greater, being from about -260°C 10 1 000° C ( -435° F 10 I 800°F). Finally, the metal
resistance elements are more time-stable than the semiconductor oxides; hence they provide
better reproducibility with lower hysteresis .
4.1
Resistance Thermometers
of
Evidence the importance and accuracy of the resistance thermometer may be obtained by
recalling that the International Temperature Scale of 1990 specifies a platinum resistance
thermometer as the interpolation standard over the range from -259.35°C 10 96 1 .78°C.
Certain properties are desirable in a material used for resistance thermometer ele­
ments. The material should have a resistivity permiuing fabrication in convenient sizes
without excessive bulk, which would degrade time response. In addition, its thennal coeffi­
cient of resistivity should be high and as constant as possible, thereby providing an approx­
imately linear output of reasonable magnitude. The material should be corrosion resistant
and should not undergo phase changes in the temperature range of interest. Finally. it should
be available in a condition providing reproducible and consistent results. In regard to this
last requirement, it has been found that lo produce precision resistance thermometers, great
care must be exercised in minimizing residual strains, which requires careful heat treatment
subsequent to forming.
As is generally the case in such matters, no material is universally acceptable for
resistance-thermometer elements. Platinum is undoubtedly ihe material most commonly
used, although others sucti as nickel, copper, tungsten, silver, and iron have also been
employed. The specific choice normally depends upon which compromises may be accepted.
Temperature Measurements
The temperature-resistance relation of an RTD must be determined experimentally.
For most metals, the result can be accurately represented as
R(T)
=
[
Ro I + A(T - To) + B(T
-
To> 2
]
(2)
where
R(T) = the resistance at temperature T,
Ro = the resistance at a reference temperature To,
A and B
= temperature coefficients of resistance depending on material
Over a limited temperature interval (perhaps 50 K for platinum). a linear approxima­
tion to the resistance variation may be quite acceptable,
R(T) = Ro [ ! + A(T - To > J
(2a)
but for the highest accuracy, a polynomial fit is required [4].
The resistance element is most often a metal wire wrapped around an electrically insu­
lating support of glass, ceramic, or mica. The latter may have a variety of configurations,
ranging from a s imple flat strip. as shown in Fig. 4, to intricate "bird-cage" arrangements (5).
The mounted element is then provided with a protective enclosure. When permanent instal­
lations are made and when additional protection from corrosion or mechanical abuse is
required, a well or socket may be used, such as shown i n Fig. 5.
More recently, thin films of metal-glass slurry have been used as resistance elements.
These films are deposited onto a ceramic substrate and laser trimmed. Film RTDs are less
expensive than the wire RTDs and have a larger resistance for a given size; however, they
are also somewhat less sta,ble [6). Resistance elements similar in construction to foil strain
gages are available as well. The resistance grid is deposited onto a supporting film, such as
Kapton, which may then be cemented to a surface. These sensors are generally designed to
have low strain sensitivity and high temperature sensitivity.
FIGURE 4: Section illustrating the construction of a simple RTD.
Temperature Measurements
Connecting head
AGURE 5: Inslallalion assembly for an indus1rial-1ype re!:istance thermometer.
Table 2 describes characteristics of several lypical commercially available resistance
thermometers.
4.2
Instrumentation for Resistance Thermometry
Some form of electrical bridge is normally used to measure the resistance change in the
RTD. However, particular auention must be given to the manner in which the thermometer
is connected
into the bridge. Leads or some length appropriate to the situation are required,
and any resistance change therein due lo any cause, including temperature, may be credited
to the thermometer element. It is desirable, therefore, that the lead resistance be kept as low
as possible relative to the RTD resistance. In addition, some modification may be made to
the circuit so as to compensate for variations in lead-wire resistance.
Figures 6(a), (b), and (c) illustrate three different bridge arrangements used to min­
imize lead error. Inspection of the diagrams indicates that arms AD and
DC each contain
the same lead lengths. Therefore, if the leads have identical properties to begin with and
are subject to like ambient conditions, the effects they introduce will cancel.
In each
case the battery and vohmeter may be interchanged without affecting balance. When the
Siemens arrangement is used , however, no current will be cairried by the center
balance.
This may be considered an advantage.
lead
at
The Callender arrangemenl is useful
when thermometers are used in both arms AD and DC to provide an output proportional
TABLE 2: 'I)'pical Properties of Resistance-Thermometer ·Elements
-
-- -----Y-em pemure
�
.....
Type of
E lament
case
Material
Platinum
(labo1810ry)
Pyrex glass
-
Platinum
(industrial)
Stainless steel
-200 10
Platinum (film)
Ceramic coaling
Rhodium-iron
Alumina
glass
Copper
Brass
Nickel
Brass
*'J'Ypical values.
and
Temperature Ranga,
•c <·F>
1 90 to S40 ( - 3 1 0 to 1000)
Raalamr-, a
coefftclent,
A, Ol(a . •c>
(approx.)
Limits of
Error" , K
RaeponM,t
•
25 at 00C
0.00385
1 25 (-325 10 260)*
- 18 to S40 (0 10 1000) 1
100 11 0°c
0.00385
±1
±2
I O IO 30
IO IO 30
-SO to 600 (-60 to I.JOO)
100 11 o•c
0.00385
±0.3
-1
-272 10 200 (-458 to 390)
21 at o•c
0.0037
±0.04
-75 10 120 (- 100 to 250)
I O at 25°C
0.0038
±0.S
20 10 60
O to 120 (32 to 250)
1 00 at 200C
0.0067
±0.3
20 I0 60
IOO at OOC
±0.01
tnme required 10 detect 90'll> of any temperatun: change in water moving at 30 cm/s. The lower value is for the thermometer case only, when:as the higher
value is for the thermometer in a protective well. Actual values vary considerably with sensor paclcaging and flow conditions. Respon&e in gases will be
· much sl0wer than in liquids; n:sponse will be faster at higher flow speeds.
*Low range.
lffigh range.
Temperature Measurements
(a) Siemens lead
anangmant
(b) Callander lead
arrangmant
(c) Four�aad
arrangmant
__.,. ;
-i=O
Current
source
R( T)
v...
-i= O
(d) Four-wire constant-currant circuit
FIGUR E 6: Four methods for compensating for lead resistance.
t o temperature differential between the two thermometers. Th e four-lead arrangement is
used in the same way as the one with three leads. Provision is made, however, for using
any combination of three, thereby pennitting checking for unequal lead resistance. By
averaging readings, more accurate results are possible. Some fonn of this arrangement is
used where highest accuracies are desired.
The general practice is to use the bridge in the null-balance fonn, but the deflection
bridge may also be used . In general, the null-balance arrangement is limited to measurement
of static or slowly changing temperatures, whereas !he deflection bridge is used for more
rapidly changing inputs. Dynamic changes are most conveniently recorded rather lhan
simply indicated, and for this purpose either the self-balancing or the deflection types may
be used, depending on time rate of temperature change.
When a resistance bridge is used for measurement, current will necessarily flow
heating of the
through each bridge arm . An error may, therefore, be introduced by
i2
R
Temperature Measuremenu
resistance lhcnnomcler. For resistance lhennometers, such an error will in general be small
because lhe gross effects in individual arms will be largely balanced by similar effects in
lhe other arms. An estimale of lhe overall error resulting from ohmic heating may be had
by making readings at different cunent values and exttapolating to zero current
Figure 6(d) shows a four-wire constant-current circuit for RID resistance measure­
ment. In lhis case, lhe current source holds i constant. irrespective of changes in either
lead or sensor resistance. The output voltage, V..,., is read wilh a high-input-impedance
meter, so that no current is drawn through lhe output leads and no voltage drop occurs along
them. Thus, the output voltage is a linear function of sensor resistance, V001
i R(T),
and it is independent of lhe lead resistance. However, because this circuit is essentially a
ballast-type circuit, it lacks a bridge circuit's sensitivity to small resistance changes. Also,
ohmic heating effects are still present.
=
43
Thermistors
The lhennistor is a lhennally sensitive variable resistor made of a ceramic-like semicon­
ducting material. Unlike metal resis(IJICe thermometers, thermistors generally respond to
an increase of lemperature with a decrease in resistance. This happens because increasing
lemperalUre usually tiiakes more charge carriers available in a semiconductor. Figure 7
shows typical lemperature-resistance relations for thcnnistors in relation to that of a typical
RID.
107
1 o'
1 115
10"
g
c
1 a3
1 <>2
.,; 101
� 1oO
j 11r1
i
Ul
\
\
\ \.
"' i'.,,_ Grade 1 lharmlstor
,
�
.......
.......
Gl!lda 2 thermistor� � ......
-.....;:
1 0-2
1 0-3
1 o-4
Platinum
1 0-S
1 0-S
1 0-7
-1 00
0
1 00
200
Temperature, •c
300
400
FIGURE 7: Typical lhennistor temperature-resistance relations.
Temperature Measuremenu
Thennistors are often composed of oxides of manganese, 1lickel, and cobalt in for­
mulations having resistivities of 100 to 450,000 Q · cm. In cryo11:enic applications, doped
gennanium and carbon-impregnated glass are used. Thenniston; are available in various
fonns, suc h as shown in Fig. 8. Some types are packaged for specific applications, such
as air and waler lemperature sensors for aulomobile engines or as surface mountable chips
for printed c ircu i t boards. Table 3 l ists some typical properties o.f commercially available
·
thennistors.
The temperature-resistance function for a thermistor is given by the relationship
(3)
where
R
Ro
{J
=
the resistance at any temperature T, in K,,
= the resistance at reference temperature
=
Tei, in K,
a constant, in K
The constant {J depends on the thermistor material; for meull-oxide thennistors, {J is
·
t y pical l y in the range of 1000 lo 5000 K.
Disk type
with leads
washer lype
---<a--.._
Bead type
�'Contact
spri�l
Fiber
insulator
Thermlslor
(washer)
pin
Heavy
Washer lhermlst0< assembly
.. ,Ji -+-Copper tlA>e
Disk
thermislOr
Lead washer wilh
tin rolled on
Disk lhermiistor asserOy
Ol
FIGURE 8: Various thennistor forms commercially available.
Temperature Measurements
Represe ntalive Metal-Oxide Thennistor Specificalions.
Resislallees Is Available in Any Specific Shape
TABLE 3:
Res ista nce
A Wide Range of
Approximate Maximum
Continuous
Temperature, •c
Type
(see Rg. 8)
At o•c
At 2S°C
Bead
Glass-coaled bead
1 65 kQ
60 kQ
25 kQ
8.8 kQ
3 . 1 kQ
l .3 kQ
Washer
28.3 Q
IO Q
4. 1 Q
1 50
Washer
3270 Q
IOOO Q
360 Q
1 50
150
At so•c
300
Rod
1 03 kQ
3 1 .5 kQ
I l .3 kO
Rod
327 kQ
IOO kQ
36 k0
1 50
Disk
283 0
100 0
40.7 0
1 25
When a lhennistor is used in an electrical circuil, current nonnally flows lhrough
it, resulting in ohmic heating. The temperalure of the thermistor is lhen raised, by an
amount depending on the resistance to heat dissipation. For a given configuration and
a given ambient lemperature, a specific thermistor temperature w i ll be obtained together
with a specific electrical resistance. Through proper application of thermal analysis and
electrical circuit analysis, thermistors may thus be used for both measurement and control
of temperature. In addition, they are quite useful for compensating electrical circuitry
for changing am bient temperature-largely because the decreasing electric resistance of
the thermistor is in contrast to increasing resistance of other most electrical components
when temperature rises. Also, thermistors can facilitate time-delay actions through proper
balancing of electrical and heal transfer conditions.
Figure 9 illustrates typical lhennislor self-healing response charac1eris1ics. Of course,
the environment (the heal transfer condition) is a major factor in an aclual application.
Thennistors can be qui1e small (a few millimeters in diameter), so their response to changes
in ambient temperature may potentially be very rapid.
The inherently high se ns i t i vity possessed by thermistors pennits the use of very simple
electrical circuitry for temperature measurement. Ordinary oh mmeters may be used wilhin
the limits of accuracy of the me1er itself. More often one of the various forms of resistance
bridge is used, eilhcr in the null-balance fonn or as a deflec tion bridge. Simple ballast
circuits are also usable. In some cases, special linearizing circuits arc used to obtain an
output voltage that varies linearly with temperature .
Through use of the thermistor's temperature-resistance characteristics alone, or in
conjunction with controlled heat transfer, thennistors have been used for measurement
of many quantities, including pressure, liquid level, and power. They arc also used for
temperature control, timing (through use of their delay characteristics in combination with
relays), overload protectors, warning devices, and so on.
501
Temperature Measurements
I 1
50
40
/
I
10
0
E = 80 V
,,,,....
70 V
v
60 V
I /
f1
/
I
) �0
0
i..-
so v
/
/
2
,,,,.
40 V
�
---
4
3
Time,
--
5
30 V
.....--
6
7
8
s
AGURE 9: fypical current-time relations for thermistors.
Thermistors can also be made to have large positive temperature coefficients. These
PTC sensors can be made from semiconductor oxides having barium titanate as the main
component; they can also be made with heavily doped silicon. PTC sensors show an
enormous increase of resistance with increasing temperature; this resistance change can be
tailored to occur abruptly at a given temperature, which makes PTC thermistors useful as
temperature-controlled switching elemenls. In conjunction with ohmic self-heating, they
are also applied as current-limiting devices.
5
THERMOCOUPLES
In 1 82 1 , T. J. Seebec k discovered that an electric potenlial occuis when 1wo different
metals are joined inlo a loop and the two junctions are held at different tempera1ures (71.
Subsequently, this potential was shown 10 be caused by an electromotive force present in
any conduc1or experiencing a temperature gradient (Fig . IO). This Seebeck emf is a vol1age
difference between lhe two ends of the conductor thal depends on the lemperature difference
of the ends and a ma1erial property called the Seebeck coefficient, u :
£(T2 ) - E(Tt )
=
1 T1
T1
u (T) dT
(4 )
If lhe ends of the wire have the same tempcralure, no emf occurs-even if the middle of
1he wire is honer or colder.
When wires of two different materials, A and B, are connected as shown in Fig. 1 1 .
the emf that occurs depends on the temperatures of free ends of the two wires and the
lemperalure of the junction be1ween the two wires. In particular, if the two free ends haVC
a temperature T..t and the junction has a temperature Tm , th e voltage difference beiween
Temperature Measurements
,- - - - - - - - ,
I
I
I
I
r - - - - - - - -,
:
I
1
I r,
I
I
I
I
£( T,)
I
I
I
I
I
l_ _ _ _ _ _ _ _ ..J
I
I
T2
£( T, )
I
I
l_ _ _ _ _ _ _ _ ..J
FIGURE 1 0: Seebec k emf between the ends of a wire of varying temperature.
lhe free ends is
E=
=
(T• flA dT
lrrcr
+
{T""
Jr.
as dT
[£A (T,. ) - £A (Tn:r)) + [£s (T..,r) - Es (T., ) )
= [£A ( Tm ) - Es ( T,. ) J - [£A (T..,r) - Es ( T..,r) ]
lf we define the relative Seebeck enif
of materials A and 8 as £As (T) "' £A ( T > - Es (T),
·
· then
(5)
For any given pair of materials, the relative Seebeck emf can be measured and tabu­
lated as a function of temperature. Moreover, if the composition of the wires is carefully
controlled, this emf is highly reproducible and provides a reliable means of temperature
measuremenL For example, in the circuit of Fig. 1 1 , if the temperature Tn:r is known, a
measuremen t of E allows detennination of the unknown temperature T,. using tabled values
of EAs ( T) and Eq. (5).
r - - - - - - - -,
I
I
8
:
A
r-------1
I
I
E
I
r,..
________
_J
:
T,.
l_ _ _ _ _ _ _ _ _J
FIGURE 1 1 : Net voltage when different conductors are connected.
503
Temperature Measurements
Figure 1 1 is the simplest type of IMrmocouple circui L Note particularly
that such
circuits always involve junctions al two temperatures . In general, one junction senses the
unknown temperature; this one we shall call the hot or m.�asuring junction. 11te oilier
j unction(s) will usually be maintained at a known temperat1ure; these we shall refer 10
15
cold or reference junctions.
Apan from the Seebec k e ffec t, two
other thermoelecuic phenomena are known, the
(8) and the Thomson effect (9). B oth are associated with the ftow of electrical
current. The Peltier effect causes heating or cooling at the j u nction of two metals when a
Peltier effect
curre nt flows through it. The Thomson effect causes heating or cooling of a nonisothenna1
conductor through which a current ftows.
The Peltier and Thomson effects are usually negligible in thermocouple thennometry
because electrical current is deliberately minimized in thermocouple circuits.
Current flow
i s undesirable because it causes resistive voltage drops that produce errors in the circuit emf.
Thennocouples should
be used
in an open circuit configuration , and their emf's should
measured only with high-input-impedance devices.
5.1
be
Application Laws for Thennocouples
thermocouple behavior can be prov•en by integrati ng the Seebeck
emf around the circuits described [l 0).
Law of intermediate metals. Insertion of an interm1:diate metal into a ther­
The following laws for
mocouple circuit will not affect the net emf, provided that the two junctions
introduced by the third metal are at identical temperatures.
from Eq. (4), since a wire whose ends have the same temperature
Ti ) produces no net emf. Applications of the law are s:hown in Fig. 1 2. In part (a) of
the figure, a third metal C is introduced within the circuit. I.f the two new junctions r and
s have the same temperature, wire C will create no additional potential and the net voltage
E of the circuit will be unchanged. Pan (b) of the figure shows that the third metal may
even be placed in at the measuring junction, so long as the junctions r and s are both at
temperature Tm . Th is makes possible the use ofjoining mater.ials, such as soft or hard solder,
in fabricating the thermocouple junctions. In addition, a lhe:rmocouple may be embedded
directly into the surface or interior of either a conductor or 11onconduc1or without altering
This law follows directly
( T2
=
the thermocouple's usefulness.
The following law may
be proven using Eq. (5).
Law of intermediate temperatures. If a simple thermocouple circuit (Fig. 1 1 )
develops an e m f Er,, r1 when its junctions are al temperatures Tm = Ti and
T,.r = Ti , and it develops an em f Er3.r2 when its junctions are at temperatures
Tm = TJ and Trer = Ti , then it will develop an emf Er:1.r, = (Er,.r, + E1,.1, )
when its j u nc t i ons are at temperatures Tm = TJ and T.-er = T, .
This law makes ii possible to correct for reference junction s whose temperatures may be
known but not controllable. It also makes possible the use of thermocouple tables based
on a standard reference temperature (usually
Tref = 0°C) in those cases when the reference
junctions are not at the standard temperature.
5.2
Thermocouple Materials and Construction
.
Theoretically, any two unlike conducting materials could be used to form a thermocouple
In practice, of course, cenain materials and combinations aire belier than others, owi ng 10
504
Temperature Measurements
r-------1
l
(a)
I
I
i
l
c
8
E
I
I
I
l
T,..
l
r - - - - ----,
(b)
I
I
I
I
+
FIGURE
1 2:
'---
A
8
E
I
I
I
I
T..,
I
I
._ _ _ _ _ _ _ _ _.
,- - - - -
,
.,____.____
.
s
I
L ------------1
._ _ _ _ _ _ _ _ _.
I
I
I
I
I
I
8
+ o---+--+-I
A
--
- --
- -,
I
I
!
I
I
I
- -'
r---- ----,
I
I
r .
I
I
I
I
Tm
I
I
I
I
I
s
I
I
._ _ _ _ _ _ _ _ _.
c
Diagrams illustrating the law of intermediate metals.
higher or more smoothly varying voltage, belier resistance to oxidation, higher temperature
limits, or other desirable attributes. Table I lists the most common types and some of their
characteristics. The letter types are ANSI standard thermocouples.
Thermocouple wire is available in spools or as part of a complete probe assembly.
When using spooled wire, the thermocouples may be prepared by twisting the two wires
together and brazing, or preferably welding, the junction (Fig. 1 3). The wire may be sold
with a TeHonjacket, or, for higher temperatures, a glass-fiber jacket may be used. The jacket
of the negative thermoelement is color-coded red for all thermocouple types. When bare
wires are used, electrical separation of the leads can be maintained with ceramic separating
clements (Fig. 1 4).
Thermocouples are often housed in a closed-ended tube, either to form a more rugged
probe or as protection against high temperatures or corrosive environments. Probe sheaths
are commonly made from stainless steel, inconel, or a hard-fired ceramic. The thermocouple
may be separated from the sheath by a mineral-oxide powder or a ceramic insulator. Many
different probe configurations are commercially available. Figure S shows a section through
one typical protective tube.
In certain instances, it is desirable to know precisely where within a thermocouple
junction the indicated temperature occurs. This is especially important when either the size
of the junction or the temperature gradient through it is large. This effective location is
indicated in Figs. 1:3 and 14 as the point j.
Usually, a heavier wire size is needed when the operating temperature of the thermo­
couple increases. This helps minimize the effect of corrosion on the thermocouple voltage.
ASTM (American Society for Testing and Materials) provides standards for wire gauge
505
Temperature Measurements
:
For gas, eleclrl\:, and arc welding
;�
,...__
...--
For resistance welding, large wires
For forming noble-metal wil'llS
electllc arc welding
for
FIGURE 13: Common forms of thermocouple construction.
as a function of operating temperature ( 10). As the wire size increases, however, so does
the mass of the measuring junction; hence, the t i me required to change the junction's tem­
perature increases. Protective tubes also reduce thermocouple response markedly. Some
compromise between time response and durability is usually required.
�
c::=========================
Bare etemen,
Element with bead insulators
Element with dOIAlle-bore insulators
Element with ceramic·tubilg Insulators
FIGURE 14: Methods of separating thermocouple leads.
506
Temperature Measurements
5.3
Values of the Thermocouple EMF
The magnitude of the thermoelectric emf is actually quite small, generally a value in milli·
volts. The output varies among the different types, always increasing with the temperature,
as shown in Fig. 1 :5 . A few numerical values of voltage are given in Table 4. More detailed
data for type K thermocouples is given in Table :5. In each case, the voltage E is obtained
from Eq. (5) with Trcr = 0°C.
Full tables of thcnnocouple emf for the letter-designated types are published by the
National Institute of Standards and Technology [ I I ) . Those tables list the emf to five or
six digits at increments of I 0C over the full temperature range for each thermocouple type;
they also provide polynomial functions for the voltage as a function of temperature. The
tables are adopted as standards by ANSI and ASTM.
For purposes of data reduction or computer data processing, some of the NIST poly­
nomials are given in Tables 6 and 7. Table 6 lists the exact NIST reference functions for E
as a function of Tm . Table 7 lists NIST's approximate inverse functions (Tm as a function of
E) and their maximum errors. Polynomials for other thermocouple types and temperature
ranges are given in reference [ I I).
70
60
50
>
g
"'
40
30
.s
{
'S
Q.
8
20
10
0
Measuring junction temperature, Tm (0C)
FIGURE 1 5: 'Thermocouple voltage versus temperature for reference junctions at 0°C. Voltages are
shown only for the recommended range of use.
507
Temperature Measurements
TABLE 4: Thermocouple Voltage E in Millivolts versus Temperature Tm for Reference Junc tions at
Tref = O"C. Values Are Limited to the Recommended Range of Use [ 1 1 )
Thermocouple Type
Temperature
· c ("
F)
-200 (-328)
-150 (-238)
-100 (-148)
-50 (-58)
0 (32)
50 ( 1 22)
100 (212)
150 (302)
200 (392)
300 (572)
400 (752)
600 (l l l2)
800 (1472)
1000 ( 1 832)
1200 (2 192)
1400 (2552)
5.4
Chromel vs.
Constantan
E
-8.825
-7.279
-5.237
-2.787
0.000
3.048
6.3 19
9.789
13.42 1
2 1 .036
28.946
45.093
61.017
Chromel vs.
Pt/10% Rh
Iron vs.
Copper vs.
v s. Platinum Constantan
Alumel
Constantan
K
J
s
T
-5.891
-4.9 1 3
-3.554
1 .889
-
0.000
2.585
0.000
2.023
4.096
5.269
8.010
10.779
16.327
6. 138
8. 1 39
1 2 .209
2 1 .848
1 6.397
33. 102
24.906
33.275
4 1 .276
48.838
-
5 . 603
-4.648
-0.236
0.000
0.299
0.646
1 .029
1 .44 1
2.323
3 .259
5.239
7.345
9.587
1 1 .95 1
1 4.373
-3.379
1.81 9
0.000
2.036
4.279
6.704
-
9.288
1 4 . 8 62
At a given temperawre, type E thennocouples have the highest output voltage, but
even this is less than 70 millivolts. The temperature sensitivity of thennocouples is also
relatively low. For example, Table 4 shows the type E volltage 10 increase from 6.3 1 9 lo
13.421 mV between 100°C and 200°C. The average change per degree Celsius is only
71 µ.V! Because of these factors, thennocouples require accurate and sensitive voltage
measurement, and in practice cannot resolve temperature clhanges less than about 0. 1 •c.
Measurement
of Thermocouple EMF
Historically, thermocouple emf was measured using a voltaige-balancing potentiometer and
a fixed voltage reforence, the standard cell [I OJ. Today, a high-quality digital voltmeter is
sufficient for all but the most exacting measurements. In fact, digital voltmeter circuitry is
ofien combined with a microprocessor and LED 10 produc:e a thennocouple thermometer
that directly displays the measuring-junction temperature.
A very simple thermocouple measuring arrangement is shown in Fig. 16. Here, the
measuring junction senses a temperature Tm . The reference junctions are located where
the thermocouple wires meet the input tenninals of the vcollmeter. If we assume that the
meter's two input terminals have the same temperature. Tree. this circuit is identical to that
in Fig. I I and the voltage detected will be E = &A a ( Tm ) -- £11B ( Tn:r) .
508
Temperature Measurements
TABLE 5: Vollage E in Millivolts versus Temperature Tm for Type K Thennocouples Having
Reference Junctions at Trer = o•c [ 1 1 I
Type K
·c
-200
- 1 75
0
5
10
15
20
-5.89 1
-5.8 1 3
-5.354
-5.730
-5.550
-5.029
- 3.997
-5.642
-5. 1 4 1
-4.542
-3.852
- 150
- 1 25
-5.454
-4.9 1 3
-4.276
- 100
-75
-50
-3.554
-2.755
- 1 . 889
-3.400
-2.587
- 1 .709
-3.243
-2.4 1 6
- 1 .527
-3.083
-2.243
- 1 .343
-2.920
-2.067
- 1 . 156
-25
-0.968
-0.778
-0.586
-0.392
-0. 197
0
25
50
75
0.000
1 .000
2.023
3.059
0. 198
1 .203
2.230
3.267
0.397
1 .407
2.437
3.474
0.597
1.612
2.644
3.682
0.798
1 .817
2.85 1
3.889
1 00
1 25
150
4.096
5 . 1 24
6. 138
1 75
7. 140
4.303
5.328
6.340
7.340
4.509
5.532
6.540
7.540
4.7 1 5
5.73 5
6.74 1
7. 7 39
4.920
5.937
6.94 1
7.939
200
8. 1 39
9. 1 4 1
8.338
9.343
10.357
8.739
8.940
9. 747
9 . 950
1 1.382
8.539
9.545
10.56 1
1 1 .588
10.766
1 1 .795
1 0.97 1
12.624
1 3.665
1 2.832
13. 8 75
13.040
14.7 1 3
1 5 .764
14.923
1 5 . 1 33
16. 1 86
-4.793
-4. 138
225
250
275
10. 153
1 1 . 176
300
1 2.209
325
350
1 3 . 248
1 2.4 1 6
1 3.457
14.293
15.343
14.503
1 5.554
375
-5.250
-4.669
1 5.975
-4.4 1 1
-3.705
1 2.002
14.084
To make use of the voltage reading, T,.r must be measured using a temperature
sensor located at the meter's input tenninals. In digital thermocouple thennometers, this
second sensor might be an integrated-circuit temperature sensor (Section 6) or a calibrated
thennistor (Section 4.3) coupled to the microprocessor. In old potentiometers, it was simply
a good liquid-in-glass thennometer. The reader may well ask why the thennocouple is
needed at all-why not just use the second sensor measure Tm ? General answers to this
question include the very broad temperature range of thennocouples, their ruggedness, their
small size, and their potentially fast time response.
509
TABLE 6: Polynomial Expansion for 1bennocouple Output E in Millivolts as a Function of Measuring Junction Temperature Tm in Degrees
= ao + a 1 Tm + a 2T� + · · · + a. T,:: . For Type K, add ao exp(a1 (T
,. - 126.9686)2 } 10 the polynomial. Reference junctions at
o•c 1 1 1 1
Celsius: E
E
Type
Tempeniture
Range
�
0
ao
<1 1
"2
....
..,
<13
"6
a7
•8
"9
.. . .
<1 10
a 12
<1 1 3
-21ooc to o•c
O"C to 1 000" C
0.000 000 000 0
0.000 000 000 0
5.866 550 870 8 x 10- 2
x 10- S
5.866 550 871 O x 1 0-2
4.S41 097 7 1 2 4
-7.799 804 868 6
)( 10- 7
-2.580 016 084 3 )( 1 0 - 8
-S.94S 258 305 7 x 10- I O
-9.32 1
40S 866 7
x 1 0- 1 2
I x 1 0- 16
- 1 .028 760 5S3 4 x 10- I J
-8.037 0 1 2 362
-4.397 949 739
x 10- 1 1
0
- 1 .64 1 477 635 5 x 1 0-2
- 3.967 36 1 95 1 6 x 1 0 - 23
-5 .582 7 3 2 872 1 x 10- 26
I
-3.465 71!4 20 1 3
x
10 - 29
K
J
x 1 0-s
2 x 10-8
4.S03 227 SSS 2
2.890 840 n1
-3.30S 689 665 2 )( 1 0- 1 0
6.502 44 0 327 0 x 1 0 - 1 3
- 1 .9 1 9 749 sso 4 )( 10- 16
- 1 .253 660 049 7 )(
1 0- 1 8
2. 148 92 1 756 9 )( 1 0-2 1
- 1 .438 804 1 78 2 )( 10- 24
3 .596 089 948
I
x 10-28
-210"C to 7IO"C
1 1 8 781 S x
ox
106 S72 0 )(
0.000 000 000 0
5.038
3.047 S83 693
-8.S68
1 0-s
1 0-2
10-8
1 .322 8 1 9 529 5 )( 10 - 1 0
- 1 .705 29S 833 7 )( 1 0- 1 3
2 .094 809 069 7 )( 10- 16
- 1 .2S3 839 S33 6 x
10-19
1 .563 172 S69 7 )( 10-23
T
O"C to 1 372°C
- 1 .760 04 1 368 6 )(
3.892 120 497 5 x
I .SSS an 003 2
-9.94S 1S9 287 4
1 0-2
10- 2
10- S
x 1 0 -8
x
3 . 1 84 094 S 1 1 ' 9 )( 1 0- 1 0
9 x 10- 1 3
5.607 SOS 90S 9 x 1 0 - 1 6
- 3 .202 o n ooo 3 " 1 0- 1 9
9 . 7 1 S 1 1 4 7 1 5 2 x 10-23
- 1 .210 472 127 5 )( 10-26
-S.607 2 84 488
ao
a1
=
=
1 . 1 8597 6 x 1 0 - 1
- 1 . 1 8343
2 x 10-4
o•c to 400" C
0.000 000 000 0
3.874 8 1 0 636 4
3.329 222 788 0
x
x
1 0-2
10-S
2.061 824 340 4 )( 10-7
-2. 1 88 22S
684 6
)( 1 0 -9
10 - 1 1
)(
-3.08 1 S7S 877 2 x
4.S47 9 1 3 529 0 x
1 .099 688 092 8
-2.7S l 290 167
3
1 0- 1 4
1 0- 1 7
x 10-20
TABLE 7: Polynomial Expansion of Measuring Junction Temperature Tm in Degrees Celsius
Millivolts: Tm
=
Type
co + c 1 E
+ ci E2 +
· · ·
+ Cn E" . Reference Junctions at O"C [ 1 1 )
as a Function of Thermocouple Output
K
E
Temperature
Range
EMF Range
UI
....
....
-:ZOOO C to O"C
o•c to 1 ooo• c
-2oo•c to ll"C
O"C to !ClO"C
SOO"C to 1 372°C
-8.825 mV to 0.0 mV
o.o mv to 76.373 mV
-5.891 mV to 0.0 mV
0.0 mV to 20.644 mV
20.644 mV to 54.888 mV
0.000 000 0
co
<i
-4.35 1 497 0
<4
-9.2SO 287 I x io- 2
,,
C)
cs
<6
'i
<8
c9
Maximum Error
1 .697 728 8 x 1 0 1
x
- 1 .585 969 7 x
io- 1
1 0- 1
-2.608 431 4 x 1 0- 2
io- 3
- 3 .403 403 0 x io-•
-4. 1 36 019 9 x
- 1 . 1 56 489 0
x
:i:0.03°C
1 0 -5
0.000 000 0
1 .705 703 s
101
10- 1
6.543 SSS S x 10- l
-7.356 274 9 x 10- s
- 2 .330
1 75 9
x
- 1 .789 600
x
I x 1 0-6
x io-9
8.403 6 1 6 s x 1 0-8
- 1 .373 587 9
1 .062 982 3 x 1 0- 1 1
-3.244 708 7
x I 0- 1 4
:1:0.02°c
0.000 000 0
2 .5 1 7 3 46 2
- 1 . 1 66 287 8
x
- 1 .083 363 8
-8.977 354 0
x
-3.734 237 7 x
-8.663 264 3
x
- l .04S OS9 8
x
-5. 1 92 057 7 x
:i:0.04°C
101
10- 1
10- 1
to-2
10-2
1 04
0.000 000 0
2 . S OB 3 5 5
7 .860 106
-2.503 1 3 1
x 101
x 10-2
x 10- 1
8.3 1 S 270 x 10- 2
- 1 .228 034 x 1 0 - 2
9.804 036
x
10-4
1 .057 734
x
x
I0-6
I0- 8
-4.4 1 3 030 x to-s
- l .OS2 1SS
:1:0.os•c
- 1 .3 1 8 058
4.830 222
- 1 .646 031
x 1 <>2
x
1 01
73 1 x io- 2
x io-•
8.802 193 x 10-6
S.464
-9.650 7 1 5
-3. 1 1 0 8 1 0 )( 10- 8
±0.06°C
E
in
��·��·: ��:.i.��;."D-),i��.i-�··
Type
,.....,....
R8111111
�
N
EMF Rmge
co
••
.,
•4
<2
<5
q
.,
..
"'
Muinwm Enor
TABLE 7: continued
"
N
O"C to 7IO"C
IOD"C to 1300"C
s
T
2511"C to 1 200"C
-200"C to O"C
O"C to 400" C
o.o mv to 42.919 mV
20.113 mv to 47.513 mv
1 .874 mV to 1 1 .llliO mv
-1.1113 mv to o.o mv
0.0 mV to 20.972 mV
1 .978 425 IC
-2.001 204 IC
l .036 969 "
-2.s49 687 "
3.58S 153 "
-5.344 285 IC
5.Q99 890 "
3.300 943 IC
-3.915 1S9 IC
9.ass 391 IC
- 1 .274 37 1 IC
7. 767 On "
1.466 298 863 IC
- 1 .534 7 13 402 IC
3.145 945 973
-4.163 2S7 839 "
3.117 963 771 IC
- 1 .291 637 500 IC
2.183 475 0l7 IC
- 1 .447 379 51 1 IC
6.21 1 272 125 "
2.594 91!1 2 IC IO I
-2.1 31 696 7 IC 10-1
7.901 1Mi!1 2 " 10-1
4.252 m 1 " 10- 1
2.592 IOO IC 101
-7.602 961 " 10-1
4.637 791 IC 10-2
0.000 000
101
10-I
10-2
10-4
10-6
10-1
10-10
:t0.Cl4"C
1 .972 485 IC IOI
101
10-I
io-3
10-4
10-7
:t0.04°C
1.291 Sf11 177 IC 101
:l:0.01°C
1112
101
10- 1
10-2
10-3
10-5
10-7
10-'
0.000 000 0
1 . 330 447 3 IC 10- 1
2.o24 144 6 " 10-2
l .2M 817 l 1C 10-s
:l:O.Cl4"C
0.000 000
-2. 16S 394 IC 10-J
6.o48 144 x 10-S
-7.293 422 IC 10-7
:l:0.03°C
Temperature Measurements
1il mperature sensor for deteminlng Trt1
8
2.759 mV
T
,.,
A
Measuring junction
Olgltal 1.oltmeter
FIGURE 1 6: Simple arrangement for measuring thermocouple emf.
EXAMPLE 1
Suppose that the arrangement shown in Fig. 16 uses a type K thermocouple (chromel­
alumel) and has a reference junction temperature of 20°C. If the voltmeter reads 2.759 m V,
what is the measuring junction temperature?
Solution Because the readout is for reference junctions at 20°C while the TC tables are
referenced to O"C, we must use the law of intermediate temperatures to correct the displayed
emf, as follows:
Er• .o = Er.,.20 £20.0
+
Here, Er,..o and Er., .20 are the emf's for the unknown temperature referenced to 0 and
20°C, respectively, and £20. 0 is the emf for 20°C referenced to o•c.
From Table 5, we read £20.0 = 0.798 mV; hence,
Er,.. o = 2.759 + 0 798 = 3.557 mV
.
Interpolation of the tabulated values yields T,,, = 87°C .
To illustrate the use of the polynomial relationships in Tables 6 and 7, let us check the value
obtained in Example I . What temperature does the equation give for a type K thermocouple
with reference junctions at o•c and an emf of 3.557 mV?
T,. =0.000 + (2.508355
=86.96°C
x
1
10 )
x
(3.557) + (7.860106 x 10- 2 ) x (3.557) 2 +
· · ·
°C
Therm<X.-ouple wire is relatively expensive compared to most common materials, such
as ordinary copper. To. help reduce costs, thermocouple circuits are often constructed using
cheaper lead wires between the thermocouple and the recording instrument. An arrangement
of this type is shown in Fig. 17. As before, we shall assume that the input terminals of the
voltmeter are at the same temperature, Tmeter· The reference junctions of this circuit are
where the thermocouple wires meet the copper leads. In order foc the lead wire arrangement
to work, these two junctions must also be at a common temperature, T,.r.
Temperature Measurements
A
I
I
I
I
I
I
r..,
I
l_ _ _ _ _ J
Cu
I
+
'-----1-o E
Vohmeter at r_
FIGURE 17: Extension wires used in a thennocouple circuit.
The output of the circuit is easy to calculate by summing the Seebeck emrs of each
wire:
E = [Ecu(T..,r) - t'cu C Tmeter)) + [E.t (T., ) - E.t (T..,r)J
+ [Es ( T,.r) - Es (T., ) ] + [Ecu ( Tme1erl - t'cu (T,.r))
=EAs(T., ) - l'.tB(T,.,r)
(6)
In other words, because the two copper leads have the same endpoint temperatures (T,.r
and Tm-J. their Seebeck emf's cancel. The circuit's output voltage is identical to that for
the circuits in Figs. 1 1 and 16.
For laboratory work, the reference junction temperature is usually carefully con­
trolled. A very common arrangement places the reference junctions in an ice-water mixture
held in a Dewar flask, as shown for two cases in Fig . 1 8 . The ice bath remains at O"C (32°F)
while the ice melts. The voltage of both circuits shown is
Ice bath references can be very accurate when they are used carefully; however, if the ice
at bouom of the Dewar melts, the remaining water can warm to as much as 4°C while ice
still floats at the top of the Dewar. This may create a significant error. (See Section 12 for
further discussion of ice baths.)
EXAMPLE 2
During an experi ment, a thermocouple circuit has i ts re ference junctions i n an ice bath and
The experiment lasts several
hours and much of the ice in the Dewar melts, leaving the reference junctions in 4°C water
at the bottom of the Dewar. If the experimenter nevertheless assumes that the junctions are
at 0°C, what will be the error in his measurement?
its measuring junction in a duct carry ing wann air at 40°C.
51 4
Temperature Measurements
(8)
Digital wltmetar -.....
10
-
===-2.235 mV
]
F
QQ=OO
=OO
�OO
=El=El=
OO=oo
�
l.=:=====�+J
Fe
r
_
Cu
Cu
r,.,
Reference junctions
In Ice bath
(b)
Digital wltmeter
gggggg EIEIEIEIEIEIEIEI
Tm•er
Fe
Fe
Constantan
r,.,
Reference junction
in ice bath
FIGURE 18: Systems using an ice bath to fix the reference junction temperature.
51 5
Temperature Measurements
Solution The actual output voltage will be E40,4. According to the law of intenneclilte
temperatures, E40,o = E40,4 + E4,0. Thus ,
E40,4 = E4o.o - E4.0
= 1 .6 1 2 - 0 . 1 5 8 = 1 .454 mV
from Tu.ble 5. The experimenter is assuming this voltage represents
which corresponds to Tm
Er,..o = 1 .454 mV
= 36. 1 °C. Thus, he will report a te mperature of 36. l "C rather
caused a -3.9"C
than the com:ct temperature of 40°C. The melting of the ice balh has
error.
EXAMPLE 3
nitrogen is in
temperalUre of
77 K. A warm copper block is dropped into the Dewar, where it cools. The block has a type
K thennocouple embedded at its center, and the ends of the thermocouple connect to copper
lead wires outside the block in the nitrogen (Fig. 19). How does this circuit work, and what
is the block temperature when the output of the thermocouple ·circuit is 0.579 my?
A Dewar flask contains liquid nitrogen. Because the upper s urface of the
contact with warmer air, the nitrogen tends to remain near its boiling point
Solution
The reference junctions are those where the copper wire joins the thermocouple
wire. The liquid nitrogen acts as a fixed-lemperature bath holding the reference junctions
77 K = - l 96°C while the copper block cools.
To find lhe temperature for an output of 0.579 mV, we: must correct for refeR11Ce
junctions that are not at 0°C, as we did in previous examples:
at
Er,..- t96 = Er,.,o
+ Eo. - 196
= Er• . o - .E- 196,0
Nole thal the reference and measuring junctions were interchanged in the last term, changing
its sign . 1 From Table
5, we find E - 196,0 = -5.829 mV. Thus,
Er,..o = 0.579 + (-5.829) = -5.250
and from lhe table we read Tm =
5.5
mV
- 1 65°C.
Electronic Instruments
As previously mentioned, digital circuits can
be used to build e lectronic thermocouple ther­
mometers. These devices connect directly to the thermocouple wires or
probe using no
external reference junction control. A microprocessor converts the voltage of the thermo­
couple wires to a displayed temperature, using, for example, programmed NIST polynomial
equations. The refercnce junction voltage, at the thennometer's input tenninals, must also
be accounted for.
1 Specifically, Eo, - 1 96 = £oA1(0) - £oA1 ( - l 96) = - l£oA1 ( - 1 96) - £oA1(0JI
51 6
=
- E - 196.0 ·
Temperature Measurements
..
E
Cu
Cu
Copper block
Liquid nitrogen
FIGURE 19: -Copper block with thennocouple in liquid nitrogen.
Reference junction compensation in electronic thennometers may be accomplished in
several ways. A calibrated temperature sensor may be incorporated at the input tenninals to
the thennometer and linked to the microprocessor. Alternatively, an electronic circuit may be
used to mimic an icepoint reference junction. Such electronic icepoints produce a stable emf
equivalent to that of a 0°C junction for the particular thermocouple type [ 12). Integrated cir­
cuits including cold-junction compensation and amplification aie also available at low cost.
1)'pical specifications for a digital thermocouple thermometer are as follows:
Range
Resolution
Response time
Input impedance
Selectable display scale
- I OO to lOOO°C
0. 1 or 1°c
Less than 2 s
l OO MQ
•c or °F
These thermometers arc available as handheld, panel-mounted, or benchtop units. They are
often restricted to a single wire type, owing to the incorporation of specific temperature�mf
and ice-point relationships.
Temperature Measurements
5.6
Thennopiles and Thermocouples Connected in Parallel
Thermocouples may be connected electrically in series or in parallel, as shown in Fig. 20.
When connected in series, the combination is called a thennopi/e, whereas the paqlld.
connected arrangement has no particular name.
The total output from n thermocouples connected 10 form a thermopile [F'"ig. 20(a))
will be equal 10 the sums of the individual cmf's, and if the thermocou ples are identical,
the total output will equal n limes the emf of a single thermocouple. For example, the
thermopile in Fig. 20(a) has five thermocouples and an output of 5[£AB( T1 ) - £.u(Tl)).
The purpose of using a thermopile rather than a single thermocouple, of c ourse, is to obtain
a more sensitive element.
The thermocouples in a thermopile should usually be clustered together as closely as
possible, so as lo measure the temperature at only a single point. On the other hand, the
junctions must remain electrically separated in order to avoid short-circuiting the individual
thermocouple emf's. The fabrication of compact thennopiles has bee n greatly improved
by modem circuit-manufacturing techniques that deposit fine-featured metal films onto
(a)
-
,
_�--''---<II
�
-�,'--�����-_-_-_-_-_
- - - -
-
+
E
2
r,
3
4
5
(b)
I_ _ _ _ _
l
e..;:r
;: ----------=�
= -§lil-...-+ -
E
1
_ _ _ _ _
I
I
I
I
I
I
I
I
I
1
FIGURE 20: (a) Series-connected thermocouples forming a thermopile; (b) patallel·
connected thermocouples.
51 8
Temperature Measurements
suilable substrates. Such thin-film thennopiles are used in heat flux gauges (Section 1 1 )
and pyrometers (Section 8).
Parallel connection provides an averaging of the thennocouples ' temperatures, which
is advantageous in certain cases. This arrangement is not usually referred to as a thermopile.
Similarly, a series-connected thermopile with junctions spread over an area can provide a
spatially averaged temperature.
6
SEMICONDUCTOR-JUNCTION TEMPERATURE SENSORS
The junc tio n between differently doped regions of a semiconductor has a voltage-current
curve that depends strongly on te m peratu re (see Section 1 5). This dependence has been
harnessed in two types of temperature sensors: diode sensors and monolithic integrated­
c i rcui t sensors [Fig. 2 1 (a)). Like many semiconductor sensors , these devices have maximum
operati ng temperatures of 100 to 1 500C. Both types can be small , having dimensions of a
few millimeters.
'
Semiconductor diode sensors, when properly calibrated, are the more accurate. The
diode is powered with a fixed forward current of about 10 µ.A, and the resulting forward
voltage is measured with a four-wire constant-cu rrent circuit [the diode replaces the resistor
in Fig. 6(d)). The diode's forward voltage is a dec reas ing function of temperature, known
from the calibration [Fig . 2 1 (b)). Diode sensors can be accurate to about 50 mK for
temperatures between 1 .4 K and 300 K [ 13). l)'pical temperature-sensing diodes are made
from either silicon or gallium-aluminum-arsenide, and they are often applied in cryogenic
tem peratu re measurements. Precision diodes are rel ati vely expe nsive .
Monolithic integrated-circuit devices use silicon transistors to generate an outpu t
current proportional to absolute temperature. A modest voltage (4 to 30 V) is applied to
the sensor and the current through the circuit is monitored with an ammeter [Fig. 2 1 (c)).
One suc h sensor is the Ana log Devices AD590, which produces a c urre nt in microarnperes
numerically equal to the absolute temperat ure in kelvin (e.g., 298 µA at 298 K or 25°C).
Because the device is a current source, it s su scepti bility to voltage noise and lead-wire errors
is minimal. IC temperature sensors are inexpensive (a few dollars). They are applied as
sensors for control c i rc u i ts, as temperature-compensation elemen ts in precision electronics ,
and even as electro nic icepoints for thermocouple circuits. Accuracy is about 0.5°C.
1
THE LINEAR QUARTZ THERMOMETER
The relationship between temperature and the reson a ti ng frequency of a quartz crystal has
long been recognized. In general , the re l at i onsh i p is n on l i near, and for many appl icati o ns
very considerable effort has been expended in allempts to minimize t he frequency drift
caused by temperature variation. Hammond ( 1 4 ) discovered a new crystal orientation
called the "LC" or "linear cut," which provides a temperature-frequency relation ship of
1000 Hz/°C with a deviation from the best straigh t line of less than 0.05% over a range of
-40°C to 230°C (-40°F to 446°F). This l i nearity may be compared with a value of 055%
for the plati n u m - resi sta nce thennometer.
The nominal resonator frequency is 28 MHz. and the sensor output is compared to
a re fere nce frequency of 28.208 MHz supp l ied by a reference oscillator. The frequency
d i fterenc e is detected, converted to pulses, and passed to an electronic counter, which
prov ides a digital display of the tem pe rature magnitude. Various probes are avai l able, all
with time cons ta nts of I s. Resolution is dependent on repetitive readou t rate, with a value of
0.000 1 °C attainable in 1 0 s. R eadou ts as fast as four per second may be obtained. Absolute
51 9
Temperature Measurements
�
sensor
(a)
1.8
1.6
Average slOpe - .._
-26 mVIK
1 .4
�
1.2
'\.
•
� 1.0
g
I
{l
\.
0.8
'
�r-....
� .......
0.6
'\.
\
0
.....
�
'\.
10
I'-
['....,
20
30
50
60
10
!'-.
.......
Average slope ..... .....
-2.3 mV/K
0.4
40
0.2
......
...... ..._
� .......
0.0
0 20 40 60 80 1 00
300
2 IO
19 mperature
(b)
(K)
400
(c )
:Z t : (a) Semiconductor junction sensors. (Courtes:y: Lake Shore Cryotronics,
Inc., and Analog Devices, lni:.); (b) diode forward voltage ver.;us temperature (Lake Shon:
Standard Curve 1 0); (c) typical AD590 measuring ciccuit.
FIGURE
520
Temperature Measurements
B
accuracy is rated at ±0.040°C. Remote sensing to 3000 m is possible.
PYROMETRY
The term pyrometry is derived from the Greek words pyros, meaning "fire," and metron,
meaning ''to measure." Literally, the tenn means general temperature measurement. How­
ever, in engineering usage, the word has historically referred to the measurement of tem­
peratures in the range extending upward from about 500°C ("'=' 1 000° F). Although cenain
thermocouples and resistance-type thennometers can be used above 500°C, pyromelly nor­
mally implies thennal-radiation measurement of temperature without contacting the object
being measured.
Electromagnetic radiation extends over a wide range of wavelengths (or frequencies),
as illustrated in Fig. 22. Pyrometry is based on sampling the energies in cenain bandwidths
of this spectrum. At any given wavelength, a body radiates energy of an intensity that
depends on the body's temperature. By evaluating the emitted energy at known wavelengths,
the temperature of the body can be found.
Pyrometers are essentially photodetectors designed specifically for temperature mea­
surement Like ordinary photodetectors (Section 16), pyrometers are of two general types:
thermal tktectors and photon detectors. Thennal detectors are based on the temperatwe
rise produced when the energy radiated from a body is focused onto a target, heating it. The
target temperature may be sensed with a thermopile, a pyroelectric element, or a thennistor
Gamma and cosmic rays
10"4
10""'
X rays
104
Ullraviolet
Visible
100
...
J:
�..
"
er
!
IL
1112
1 0•
106
1o'
Infrared
E
"-
j
Short radio waves
.!!
I
Commercial TV and FM radio
AM raclo
1010
Long radio waves
1 012
10 1 4
l(Hz) x A (µm) •3 x 1 0 1 • p.mfs = 3 x 108 mts
FIGURE 22: The electromagnetic radiation spectrum.
521
Temperature Measurements
or RTD. Photon detectors usually use semiconductors of either the pho!oconducti'VC or ph
todiode type. In those devices, the sensor responds directly to the intensity of radiated
1
by a corresponding change in its resistance or in its junction curren t or voltage.
Pyrometers may also be classified by the set of wavelengths measured. A to
tal.
radiation pyrometer absorbs energy at all wavelengths or, at least, over a very broad rang
of wavelengths. A spectral-band pyrometer (or optical pyrometer) measures radialede
idered
over a narrow band of wavelengths; the band will often be narrow enough to be cons
1
single wavelength. A wide-band pyrometer uses a broader range of wavelengths, usually in
order to obtain a stronger signal. The infrared pyrometer is a type of wide- band pyrometer
used for measurements near room temperature, where radiation is weak and mainly on
infrared wavelengths. The two-color pyrometer compares the radiated energy at two specific
wavelengths in order to determine the temperature.
Because infrared pyrometry can be used at or below room temperature, it overturns
the traditional perception of radiation pyrometry as a strictly high-temperature technique. In
fact, the Greek meaning of pyrometry, mentioned before, is no longer so inappropriate. Irre­
spective of the temperature range, however, radiation pyrometers retain the distinguishing
feature of finding an object's temperature without directly contacting it.
Ii;
�
8.1
Radiation Pyrometry Theory
All bodies at temperatures above absolute zero radiate energy. Not only do they radiate
or emit energy, but they also receive and absorb it from other sources. We all know that
when a piece of steel is heated to about 550°C it begins to glow (i.e., we become aware of
visible light being radiated from its surface). As the temperature is raised, the light becomes
brighter or more intense. In addition, the color changes from a dull red, through orange
to yellow, finally approaching an almost white light at the melting temperature ( 1430°C
to 1 5400C).
We know, therefore, that through the range of temperatures from approximately 550°C
to 1540°C, energy in the form of visible light is radiated from the steel. We can also sense
that at temperatures below 55QoC and almost down to room temperature, the piece of steel is
still radiating energy or heat in the form of infrared radiation, for if the mass is large enough
we can feel the heat even though we are not touching the steel. We know, then, that energy
is radiated through cenain temperature ranges because our senses provide the necessary
information. Although our senses are not as acute at lower temperatures, on occasion we
can actually "feel" the presence of cold walls in a room because heat is being radiated from
our body to the walls. Energy transmission of this son does not require an intervening
medium for conveyance; in fact, intervening substances may interfere with transmission.
The energy of which we are speaking is transmitted as electromagnetic waves or
photons traveling at the speed of light. As our discussion of hot steel shows, the wavelength
of this radiation depends upon the temperature of the radiating substance. It also depends
upon the physical propenies of the substance. Let us consider these propenies funher.
Radiation striking the surface of a material is partially absorbed, partially reftected,
and partially transmitted. These ponions are measured in terms of absorptivity (a), reflec­
tivity (p ) . and transmissivity ( r ), where
(7)
a+p+r = I
For an ideal reflector, a condition approached by a highly metal polished surface, P
522
�
I.
Temperature Measurements
In many cases, gases represent substances of high transmissivity, for which r -+ 1; for
opaque materials, on lhe olher hand, r = 0. A small opening into a large cavity approaches
an ideal absorber, or black body, for which a ..... I ; !his is because a photon entering lhe
cavity is very unlikely to be reflected back out lhe opening.
A body in radiative equilibrium wilh its surroundings emits as much energy as it
absorbs. It follows, lherefore, !hat a good absorber is also a good radiator, and it may be
concluded that the ideal radiator is one for which the val ue of a is equal to unity. In olher
words, a black body is both a perfect absorber and a perfect emitter of radiation. When we
refer to emitted radiation as distinguished from absorption, the tcnn emissivity (E) is used
ralher than absorptivity (a). However, the two are direc t ly related by Kirchlwff's law,
E=a
In general, each of lhe propenies a, p, r, and E is a function of wavelenglh, temperature,
and the angle wilh which radiation approaches or leaves the surface. Fonunatcly, for angles
wilhi n about 500 of lhe normal to the surface, lhe angular dependence is weak enough to
be ignored . Pyrometers are generally used wilhin Ibis range of angles. For opaque bodies,
wilh r = 0, Kirchhoff's law and Eq. 7 show that a and p can be detennined if E is known.
Table 8 lists values of emissivities for cenain materials, averaged over all wavelengths.2
As we mentioned earlier, lhe radiated color changes with increasing temperature.
Change in color, of course, corresponds to change in wavelength, and the wavelength of
maximum radiation decreases wilh an increase i n temperature. A decrease in wavelength
shifts the color from lhe reds toward the yellows. Steel al 540°C has a deep red color. At
8 15°C the color is a bright red, and at l 200°C the color appears white. The corresponding
radiant energy maximums occur at wavelengths of 3.5, 2.6, and I . 9 µm, respectively.
If we should heat an ideal radiator to various temperatures and detennine the relative
intensities at each wavelength , we would obtain the energy-distribution curves shown in
Fig. 23. Not only is the radiation intensity of a higher-temperature body increased, but
the wavelength of maximum emission is also shifted toward shoner wavelengths. The
intensity distribution, or spectrum, for an ideal radiator (black body) may be expressed as
follows [ 1 5 ):
(8)
l
Kin:hhoff's law is srric1ly valid only at any panicular wavelength. When a is averaged over all wavele�lhs,
lhe value may di!fer fiom lhe wavelength-averaged e if lhe body is absort>ing radiation from a much holler or
colder body ( 1 5). In lhe parlance or radiation lheory, values at a single wavelength an: called sptctral •'Glues.
whereas !hose averaged o-er all wavelengths an: called total values.
523
Temperature Measurements
TABLE B: Total Emissivity for Certain Surfac•es (15)
Temperature, •c
Surface
Polished silver
Polished aluminum
Platinum wire
Heavily oxided aluminum
Rusted iron
Rolled sheet steel
Roofing paper
Plaster
Rough red brick
Rough concrete
Smooth glass
Water,
:::_0. 1
mm deep
200
200-600
40-1370
90-540
40
40
40
40-260
40
40
40
40
Black body
Emlsalvity
0.01--0.04
0.04--0.06
0.04--0. 19
0.20--0.33
0.61--0.85
0.66
0.9 1
0.92 .
0.93
0.94
0.94
0.96
l.00
1 0-<
0.1
FIGURE
0.3
1 .0
23: Radiation intensity as a function
or black body, wi th s = I .
3.0
Wllll8tanglh, A (µm)
1 0.0
of wavelength and temperature for an ideal
524
Temperature Measurements
where
Eu
= the energy emitted by a blackbody at wavelength .\,
in W/m2
·
µ.m,
T = the absolute temperature, in K,
.\ = the wavelength, in µ.m,
C1
C2
=
374 . 1 8 MW . µ.m4 /m2 ,
= 1 4388 µ.m · K
The wavelength of peak i nte nsity for a particular temperature is given by the
Wien displace­
menl law:
Amax T
=
(8a)
2897.8 µ.m · K
For a nonideal body, the intensity distribution must be multiplied by the value of the
emissivi ty, e(.\), that is appropriate to the wavelength considered
EA
e(.\)C1
- e(.\) E b )., - ,\S (eCz/AT - l )
( 8b)
where E,. denotes the spectrum of a nonideal body.
These relations are the basis for spectral-band and two-color pyrometers. Optical
fi l ters arc used to eliminate all but the wavelengths of interest, whose intensities are then
measured. The body's temperature is calculated from the measured intensity, using Eq. (8b)
e(.\). Note that u ncertain ty in e (.\ ) leads directly to uncertainty in the
and the value of
measured temperature.
For total-radiation pyrometers, no filters are used and radiation from all wavelengths
is sampled. Thus, the detector receives energy from the source at a rate proportional to the
total radiant energy, q, emined by the surface, as given by the Stefan-Boltvnann law:
(9)
where
q = radiant heat ffux emitted by the source, in W/m 2 ,
e (.\) = the emi ssivity of source at wavelength .\,
e = the emissivi ty of source averaged over all wavelengths,
T = absolute temperature of source, in K,
cr = the Stefan-Boltzmann constant, 5.6704
x
10- 8 W/m 2 K4
q is focu sed onto the detector, whose temperature rise is measured. With
a knowledge of the source's emissivity, e, the source temperature can be calculated from
Eq. (9). A calibration test is required to establish the relationship between the detector's
temperature rise and q .
The radiation
525
Temperature Measurements
Particular attention must be given to the optical system of a radiation pyrome
te
appropriate optical glasses must be selected 10 pass the necessary range of wa
ve1
Pyrex glass may be used for the range of 0.3 to 2.7 µm, fused silica for 0. 3 to 3 .8 µm,
:.:
and
calcium fluoride for0.3 to IO µm. Thus, while Pyrex glass may be used for high -tem
perature
m��rement based on short wavelengths, ii is practically opaque to the long-wavelen
gth
rad1ation oft ow-temperature sources, say, those below 550°C. By choosing calcium fluoricl
e
and adding appropriate filters, a radiation pyrometer may instead be made to sense o
nly
longer infrared wavelengths (2 to IO µm, for instance).
Although radiation pyrometers may theoretically be used at any reasonable distance
from a targel, some practical limi1ations should be mentioned. First, the size of target
will largely detenni ne lhe degree of temperature averaging, and in general, the greater the
distance from the source, the greater the averag ing. Second, the nature of the intervening
atmosphere will have a decided effect on the pyrometer reading. If smoke or dust is present,
or if certain gases or solids, even though they may appear to be transparent, are in the path,
considerable energy absorption may occur. This problem will be particularly troublesome if
such absorbents are not constant but vary with time. Third, heat radiated from surrounding
bodies may be reflected from the measured object into the pyrometer, particularly if the
measured body has a low emissivity. For example, if the steel discussed previously is
located inside a furnace, radiation from the walls of the furnace may be reflected from
the steel to the pyrometer. In other cases, reflected sunlight may cause errors. For these
reasons, minimum practical distance is advisable, along wi1h careful selection of pyrometer
sighting arrangements.
For all pyrometers, calibration is essential 10 accoun1ing for the effects of lhe optical
system, the detector response, and, when it is unknown, the source emissivity.
8.2
Total-Radiation Pyrometry
Figure 24 shows, in simplified form, a thermal-detector total-radiation pyrometer. F..ssential
parts of the device consist of- some light-direcling means, shown here as baffles but which
are more often lenses, and an approximate blackbody receiver with means for sensing
temperature. Although the sensing element may be any of 1he types discussed earlier in
this chapter. it is generally some form of thermopile or pyroeleclric sensor; occasionally, a
thermistor or gas-pressure thermometer is used. A balance is quickly established between
the energy absorbed by lhe receiver and that dissipated by conduction through the leads
radiation emitted to the surroundings. The receiver equilibrium temperature then becomes
the measure of source temperature, with the scale established by calibration.
and
Figure 25 shows a sectional view of a commercially available pyrometer. Although
total-radiation pyrometry is primarily used for temperalures above 550°C, the pyrometer
shown is selected to illustrate an i nstrument sensitive to very low-level radiation (50"C to
375°C). The arrangement, however, is typical of general radiation pyrometry practice. A
lens-and-mirror system is used to focus the radiant energy onto a thermopile. The ther­
mocouple reference temperature is supplied by maintaining the assembly at constant tern·
perature through use of a heater controlled by a resistance thermometer. In many cases
compensation is obtained lhrough use of temperature-compensating resistors in the electri­
cal circuit.
As previously noted, a knowledge of the radiating body's emissivity is required in
order to find the temperature. Handbook data may be used i f accurate values are available
526
Temperature Measurements
FIGURE 24: A s impli fied form of total-radiation pyrometer.
for the particular surface being measured. Alternatively a calibration may be done by com­
paring the pyrometer readout with that of some standardized device, such as a thennocouple,
attached to the radiating source. Often a single poi nt calibration suffices. Mechanisms for
,
-
Resistance
lhermometer
windings
calcium
flourlde _..J,....-+-t-�- 1
lens
Thermopile
mirror
FIGURE 25: Section through a commercially available low-temperature, total radiation
pyrometer (Courtesy: Honeywell, Inc., Process Control Division, Ft. Washington, Penn­
sylvania).
527
Temperature Measurements
adju sti ng a total-radiation pyrometer's calibration include the following:
I.
Microprocessor- based adj ustmen t of the sensing circuitry or display;
2. Variable aperture area at the thermopile or lens;
3. Movable metal plug screwed into the thennopile housi ng adjacent to the hot junction
which acts as an adjustible heat sink;
4. Movable concave mirror reftecting varying amounts of enurgy back to the thermopile
of a lens-type pyrometer (see Fig. 25).
The error in temperature due to error in the estimated emissivi ty !for a total radiation pyrom.
etcr is given by ( 1 6]
( 10)
For example, a 10% overesti mate of " leads to a 2.5% u nderes�imate of the absolute tem­
perature .
8.3
Spectral-Band Pyrometry
Spectral-band, or optical, pyrometers measure radiant intensity ut only one or two specific
wavelengths , which are isolated by use of appropriate fil ters. The intensity is found either
by usi ng the output of a calibrated thermal or photon detector or by visual comparison
to a calibrated source. The temperature depe ndence of the intensity d i sbibution, Eq. (8),
provides the necessary relation between measured i nten sity and 11emperature.
In typical systems, a wavele ngth of 0.65 µm to 0.85 µm is s1:l ec ted by an optical filter.
The rad iatio n is detected by a silicon ph otodiode whose output i:; processed eleclronically
(Fig. 26). These devices are ge neral ly used with high temperature sources, above about
T
Display
Sampled
area
_l_
Signal
processor
Detector
FIGURE 26: Sc hematic diagram of a spectral-band pyrometer. Light rays are shown leaving
on edge of the sampled area 10 illustrate that the field stop limits the image size whereas
the aperture stop limits the amou nt of light collected. Similar rays may be drawn from any
point in the sampled area.
528
Temperature Measurements
SOO"C. The source emissivity must be known at the wavelength used , and the appropriate
value of e(;I.) is input to the pyrometer's microprocessor.
These devices are less sensitive to uncertainties in the emissivity than total radiation
pyrometers. The relationship of temperature error and emissivity error is ( 16)
--:r = - c2 7
AT
;l.T A e
( 1 1)
for C2/;l.T > 2. Since typical values of C2/AT are 10 or more, a l 0% overestimate
of e leads to I% or smaller underestimate of the absolute temperature. In addition, the
radiation emitted at a single, shon wavelength varies more rapidly with temperature than
the total radiation, making spectral-band pyrometers more sensitive to temperature change
than total-radiation pyrometers.
The disappearing filament pyrometer is an example of the visual comparison type
(Fig. 27). The intensity of an electrically heated filament is varied to match the source
intensity at a particular wavelength. In use, the pyrometer is sighted al the unknown
temperature source at a distance such that the objective lens focuses the source in the
plane of the lamp filament The eyepiece is then adjusted so that the filament and the source
appear superimposed and in focus to the observer. In general, the filament will appear
either hotter than or colder than thC unknown soun:e, as shown in Fig. 28. When the battery
current is adjusted, the filament (or any prescribed ponion such as the tip) may be made to
disappear, as indicated in Fig. 28(c). The current indicated by the milliammeter to obtain
this condition may then be used as the temperature readout. A red filter is generally used
to obtain approximately monochromatic conditions, and an absorption filter is used so that
the filament may be operated at reduced intensity, thereby prolonging its life. The filaments
are typically tungsten and can be calibrated to high accuracy. The disappearing filament
pyrometer has, however, largely been replaced by electronic spectral-band pyrometers.
Objectiw lens
Lamp
Eyepiece
�
Obserwr
��
!'!
!
g_
::J
tfl
E
Red lllter
�
Battery
Millianmeter
AGURE 27: Schematic diagram of a disappearing filament pyrometer.
529
Temperature Measurements
(a) Filament too hot
(b) Filament too cold
(c) Filament and source
at same �,.
FIGURE 28: Appearance offilament when (a) filament temperature is too high, (b) filament
temperature is too low, and (c) filament temperature is correct.
Two-color pyrometry is an adaptation of spectral-band pyrometry that mi nimiz:es the
inftuence of the emissivity. Specifically, two-color pyrometers measure the source intensity
at two adjacent wavelengths, At and A 2 . I f the wavelengths are close and the emissivity is
not too rapidly varying, then the emissivity will be nearly the same for each wavelength.
Hence, the ratio of measured intensities depends only on temperature:
E(A 1 ) E,., "" E>.,
E>.,
e(A2 ) E,.,
(
A2
= A.
)
s (e C2 />-2T
(e C2/>-1T
--
1)
I)
1\vo-color pyrometery is particularly useful for situations in which the source emissiv­
ity varies in time, such as during the processing of steel and aluminum [ 1 7). The two-color
pyrometer can also be used with small objects that do not fill the instrument's field of
view, since only the ratio of intensities is required. Color-ratio pyrometry is the defining
temperature standard for ITS-90 in the range above 1234.93 K (4).
8.4
Infrared Pyrometry
The infrared region begins al a wavelength of about 0. 75 1tm-where the visible region
ends-and extends upward to wavelengths of about I 000 1tm. Infrared pyrometry is simply
an adaptation of spectral-band pyrometry to sensing a range of infrared wavelengths. The
benefit of infrared sensing is found in Wien 's displacement law (Fig. 23). which shows that
the peak radiant intensity of low-temperature bodies occurs in the infrared. For example, a
body at 25°C (298 K) radiates at a peak wavelength of 9.7 1tm. Thus, infrared deteetion is
essential to radiant measurements of near-room-temperature objects.
Common infrared detectors use optical filters to isolate some portion of the interval
between 2 and 14 1tm. This interval corresponds to peak radiant temperatures between
about 200 and 1 400 K. However, this interval is also dictated by the need lo avoid infrared
absorption by air and water vapor between the source and the detector: atmospheric absorp­
tion is al a minimum in several bands between 2 and 5 µm and in the interval from 8 lo
14 1tm. Most commercial infrared pyrometers are centered on one of these transmitting
bandwidths. Specialized devices, such as satellite-based far-infrared detectors, may respond
to wavelengths up tu 100 µm; however, detectors for such long wavelengths generally must
be operated at cryogenic temperatures.
Detectors for infrared pyrometers are often thermopile heat ftux sensors of the Gar­
don gage type, as described in Section 1 1 . Pyroelectric materials are also quite useful as
530
Temperature Measurements
broadband IR (infrared) sensors. These materials have an intrinsic electrostatic polariza­
tion that decreases with an increase in temperature. Pyroelectric elements, then, act as
thennal detec tors that respond to a change in temperature by developi ng a charge, in much
the same way that piezoelectric materials respond to strain. Pyroelectric materials include
ferroelectric crystals, such as tri g l i sine sulfate (TGS) and lithium tantalate (LiTaO:J), and
some organic polymer films, such as polyvinylidene Ouoride (PVDF) ( 1 8).
Like piezoelectric charge, pyroe lect ric charge d issi pates in time. To obtain meas11re­
ments of steady temperatures, a rotating "chopper" may be used to interrupt the radiat ion
periodically, producing a square-wave signal whose peak amplitude is related to the tar­
get temperature ( 1 9). Electrical sig na l cond i tioni ng requ irements are otherwi se similar to
those of piezoelectric devices. Apart from their use in temperature measurement, pyroelec­
tric sensors are commonly used i n motion sensors for light switches and burglar alarms.
Focusing optics for infrared applications are complicated by the pooc infrared trans­
missivity of ordinary glasses. Infrared lenses and windows are sometimes made from
calcium Ouoride or germanium, which have high IR transmissivi ty. Mirrors are usually of
the first-surface type, having the reOective coating on top rather than beneath a layer of
glass [ I 9). When low temperature levels are to be detected, background radiation from the
pyrometer's own mirrors, lenses, and windows may be comparable to that from the source.
Here, rotating c h oppers are ag ai n useful in periodically removing the �ource radiation, so
that the background sign al may be identified and subtracted
As with other pyrometers, IR pyrometers depend upon a knowledge of the emissivity . .
Equation ( 1 1 ) may be used to estimate uncertainties for these devices. Errors in temperature
due to errors in emissi vi ty tend to be larger for infrared pyrometers, owing to the l onger
wavelengths involved.
8.5
Thermal Imaging
One important app l icati on ofinfrared pyrometry is to whole-field tem peratu re mea.�urement,
called thermal imaging or infrared thermography. Infrared thermography is used in medical
imaging, in tes t i ng buildings for heat leakage, in satellite surveys, in ni ght ti me surveillance,
and in measuring temperature distributions in electronic equipmen t . Many devices operate
by scanning the image across a single, cooled photon detector. More recent desig ns take
advantage of the CCD (c h arge-cou pled diode) array technology deve loped for digital video
cameras. Here, a two-dimensional array of detectors is posi tioned behind a came ra lens
to record the thermal image of the field viewed. At present, arrays of photoconductive
sensors arc u su ally employed. Surprisingly, photon detectors (suc h as silicon diodes) are
not currently used in IR arrays; this is because the candidate materials either lack broadband
se nsi ti vity above 1 µm ( li ke silicon) or are difficult to fabricate as detector arrays.
Uncertainties in the emissivity can be a particular problem for thermal imaging sys­
tems because the emissivity may vary across the field of view.
9
OTHER METHODS OF TEMPERATURE INDICATION
One method of temperature measurement given in the introduction to thi s chapter has not
bee n referred to in the intervening pages. It is the application of changes chemical state or
phase. Several techn iq ues based on this pri nciple should be menti oned [20).
531
Temperature Measurements
Seger cones have long been used in the ceramic industry as a means of Checki
a
temperatures. These devices are simply small cones made of an oxide and glass. When g
predetermined temperature is reached, the tip of the cone softens and curls over,
providing the indication that the temperature has been reacho!d. Seger cones are made in a
standard series covering a range from 600°C to 2000°C.
Somewhat similar temperature-level indicators are available in the forms ofcrayonlike
sticks, lacquer, and pill-like pellets. Each may he calibrated at temperature intervals through
a range of about 50°C to I 1 00°C. The crayon or lacquer is stroked or brushed on the pan
whose temperature is to be indicated. After the lacquer dries, 1it and the crayon marks appear
dull and chalky. When the calibration temperature is reached, the marks become liquid and
shiny. The pellets are used in a similar manner, except that they simply melt and assume a
shiny liquidlike appearance as the stated temperature is reach•::d. By using cray ons, lacquer
,
or pellets covering various"temperatures within a range, the maximum temperature attained
during a test may he rather closely determined.
Liquid crystals are perhaps the most colorful of tempierature indicators. The liquid
crystal is a meso-phase state of certain organic compounds that shares properties of both
liquids and crystals. As temperature increases past a threshold, liquid crystals successively
scatter reds , yellows, greens, blues, and violets until an upper threshold temperature is
reached. By changing the crystal composition, the entire colcor change can made occur over
an interval of as much as 50°C or as little as l •c. Liquid crystals are useful from roughly
- 30°C to 1 20°C; they may resolve temperature changes as small as 0. 1°C.
Liquid crystals are available commercially in various temperature ranges and pack­
agings. They may he encapsulated into tiny pellets which are suspended into a liquid slurry
for later use. They may also he coated in a thin film over a blackened plastic or paper sheet
that makes the scattered light more visible; the sheet is then affixed to a surface whose tem­
perature distribution is to he observed. High-accuracy meas1llrements using liquid crystals
usually employ a digital camera to record the color pattern on a surface. The digital image
is analyzed on a computer to determine the temperature disu·ibution.
the�
10
TEMPERATURE MEASUREMENT ERRORS
The number of potential sources of error associated with temperature measurement is unlim­
ited. However, several are significant enough to warrant special note. These will be dis­
cussed in the next several pages.
10.1
Errors Associated with Convection, Radiation, and c:onduction
Any temperature-measuring element senses temperature because heat is transferred between
the surroundings and the element until some kind of equilibiium condition is reached.
As an example, consider the use of a thermocouple tc• measure the gas temperature
in a furnace. When a thermocouple probe is inserted through1 the wall of a furnace (assume
it to he gas-or coal fired), heat is transferred to it from the immersing gases by convection.
Heat also reaches the probe through radiation from the furnace walls and from incandescent
solids such as a fuel bed or those carried along by the swirlin1� gases. Finally, heat will flow
from the probe by conduction through any connecting leads or supports. The temperature
indicated by the probe therefore will be a function of all thE:se environmental factors, and
consideration must be given to their effects to interpret or control the results intelligently.
532
Temperature Measurements
Convection Effects
When the temperature of a gas or liquid is to be measured, we rely on heat flow by
convection to bring the probe to the fluid temperature. The rate of heat transfer between
the fluid and the probe is described by the following relationship: .
Q
=
where
h A (Tf - T,),
( 1 2)
Q = the heat transferred, in W,
h
A
Tt
= the heat transfer coefficient, in W/ml K,
=
=
the surface area of the probe, in m l,
the fluid temperature, in K, and
1i = the probe temperature, in K
The heat transfer coefficient, h , may be predicted using using correlations or equations
appropriate to the particular situation at hand ( 15). However, several factors should be
noted. Heat transfer coefficients are generally higher in flowing fluids than in fluids at
rest, and they rise as the fluid velocity increases. Heat transfer coefficients in liquids are
typically one to two orders of magnitude greater than in gases, other things being the same.
In addition, heat transfer coefficients are generally higher for small objects than for larger
ones. Representative ranges of h are as follows, for gases and nonmetallic liquids:
5-25 W/ml K
1�200 W/m l K
5�2000 W/m2 K
1 00- 10,000 W/ml K
Gases at rest
Flowing gases
Liquids at rest
Flowing liquids
EXAMPLE 4
A 5-mm-diameter thermistor sits in a 4 mis air flow. It shows a resistance of 40 kQ,
corresponding to a temperature of 30.0"C. If the sensing current is I mA and the heal
transfer coefficient is h = 1 10 W/ml K, what is the air temperature?
Solution
The electrical power dissipated in the thermistor must be removed by convection.
Hence,
Q =
so
h A( Tt1ierm - Tair)
=
l
i R
that
Tair =
Tlhenn
;lR
h (rr 0 ) ( 1 0- 3 ) 2 (40
- --2
= 30.0 -
= 25.4°c
x
103 )
( l lO)rr(0.005) 2
The error, 4.6 K, may be reduced by lowering the current to the thennistor.
533
Temperature Measurements
Radiation EITects
Radiation between the probe and any source or si n k of different � .
function of the difference in the fourth powers of the absolute temperatures. It
a
true, therefore, 1ha1 radiation becomes an i ncreas i ng ly i mportant source of tCDlpelature
as the temperatures increase. As discussed in Section 8. 1 , radi ant-heat transfer is also
function of the emissivities of the objects involved. For th i s reason, a bright, shiny
less affected by t hermal radiation than is one tarnished or covered with soot.
Calculation of radiation heal exchange generally req u ires consideration of the geo­
metric configuration of the objects involved. One simple case is particularly important,
however: that of a small object exchanging radiation with a large isothermal enclosure (IS).
In this case, we may estimate the radiation heat transfer as follows:
is�
�
probe�
(13)
where
Q
=
the heat transferred from the probe, in W,
e = the emissivity of the probe,
A = the surface area of the probe. in m2 ,
a = the Stefan-Boltzmann constant, 5.6704
T,
x
10-8 W/m2K4,
= the temperature of the enclosure; in K, and
T, = the probe temperature, in K
This equation will apply when the surface area of the probe is small compared to the surface
area of its enclosure.
EXAMPLE 5
A thermocouple is located within a large exhaust duct . The thermocouple has an emis­
sivity of e = 0.5. The walls of the duct have a tem pera ture of 400 K. The thermocou­
ple reads 470 K. If the heat transfer coefficient between the air and the thermocouple is
h = 80 W/m 2 K, what is the air temperature"!
Solution The thermocouple gains heat from the air heat by convection and loses heat by
radiation to the the walls. Thus.
Qconv = Q ,;u1
h A (T3;, - 7ic) = eAa(1;� - T�111)
Hence,
Tair
=
=
h(T,c - Twanl + 7ic
0 . 5(5.6704 x 10-8) ((47
£ (]
4
4
80
= 478.2 K
534
0)4 - (400)4 ) ) + 470
Temperature Measurements
Radiation error may be largely eliminated through the introduction of radiation shield­
ing. This consists of placing reftective barriers around the probe, which prevent the probe
from "seeing" the radiant source or sink, as the case may be. For low-temperature work,
such shields may simply be made of sheet metal appropriately formed to provide the nec­
essary protection. Aluminum is a common choice, because its very low emissivity sharply
limits radiation heat transfer. At higher temperatures , metal or ceramic sleeves or tubes
may be used. In applications where gas temperatures are desired, however, care must be
exeicised in placing radiation shields so as not to cause stagnation of ftow around the probe.
As pointed out earlier, desirable convection transfer is a function of gas velocity.
Consideration of these factors led quite naturally to the development of an aspirated
high-temperature probe known as the high-velocity thermocouple (HVT) (2 1 ) . Figure 29
illustrates an aspirated probe with several types of tips. Gas is induced through the end,
over the temperature-sensing element, and either is exhausted to the exterior or, if it will not
alter process or measurement functions, may be returned to the source. A renewable shield
provides radiation protection for the element, and through use of aspiration, convective
transfer is enhanced. Gas mass-ftux past the element should be not less than 20 kg/m2 s for
maximum effectiveness.
When a single shield is used, as shown in Fig. 29(b), the shield temperature is largely
controlled by convective transfer from the aspirated gas through iL Its exterior, however, is
subject to thermal-radiation effects, and thus its equilibrium temperature, and hence that of
the sensing element, will still be somewhat influenced by radiation. Maximum shielding
may be obtai ned through use of multiple shields, as shown in the lower two sections of
Fig. 29(b). Thermocouples using multiple shielding are known as multiple high-velocity
thermocouples (MHVTs) (2 1 ]. The effectiveness of both the HVT and MHVT relative to
a bare thermocouple is illustrated in Fig . 30.
Our discussion of radiation effects has been centered largely on the high-temperature
application of thermocouples. It should of course be clear that the principles involved
apply to any tem perature- measuring system or situation. Radiation may introduce errors
at low temperatures as well as at high ones, and it will present similar problems to all
types of sens i ng elements. When the fl uids are liquid rather than gaseous, the problem is
essentially eli mi nated because most liquids, and to some degree water vapor in air, act as
strong absorbers of thermal radiation.
Conduction Effects
.
In general, if heat is conducted through the region of a temperature sensor, temperature
gradients will be present. These gradients may lead to differences between the sensor's
reading and the temperature that is desired.
Consider a temperature-measuring probe that is placed into a warm fluid. This probe
will require a mechanical support, and, in genera l some connection must be made to an
external readout or recording device. These supports and connections provide paths through
which heat may be conducted to cooler walls or surroundings. Ideally, convection between
the fluid and the probe will hold the probe at the Huid's temperature. But if heat is steadily
cond uc ted out of lhe probe by the suppons. the sensing element may remain at a temperature
somewhat below the ftuid. In the language of heat transfer, such a probe is said to behave
as a heat conducting "fin" [ 1 5 ) .
Such fin effects can be minimized b y introducing a nonconductive separator into the
probe's s upporti ng structure (a so-called thermal break) or by having a sufficiently long
535
Thermocouple 2 hOla Insulator
(a)
(b)
"°"'lain protection tuba
0.200" OD, 0.150" IO, 1 2" Lg
Oetall ol Thermocouple Assembly
Temperature Measurements
\,'-
sI!!
�
2500 1---;---;---t---i:---::---:M==-t
2000
J!!
! 1500
� 1000
1----+---+---+-+-,E----1�--1
1----+--T-J<'-4--+---i---I
1---+--->t<--I---+---�
500
1000
Observed
1500
2000
gas t�ature, °F
2500
3000
FIGURE 30: Graphical representation of the effectiveness of the high-velocity thenno­
couple (Counesy: Babcock & Wilcox, Barberton, Ohio)
probe. Assuming that the sensing element of the probe (a thennocouple junction, say) is at
the tip of the probe and that the base of the probe is attached directly to a wall, fin conduction
errors will be miminal if the length of the probe satifies the following relationship:
L -> 5 '/{kA;
hi'
where
L
= the probe length from sensor to wall , in m,
P
=
k
=
( 1 4)
h = the heat transfer coefficient between probe and fluid, in W/m2 K,
Ac =
the perimeter of the probe (e.g., :rr times its diameter), in m,
the cross-sectional area of the probe, in m2 , and
the thennal conductivity of the probe, in W/m K
·
In the case of a probe whose cross section includes several different materials, such as a
stainless steel jacket with a powered filler inside, the product kAc should be computed for
each pan of the cross section, and the results added to together to get an effective value of k Ac
for the probe. For example, in the case just described, (kA c)effoai"" = (kAclstee1 + (k Adfiller·
The perimeter is based only on the outside surface of the probe, in contact with the fluid.
Conduction effects are also of concern for surface temperature measurements. If a
thennocouple is affixed to a hot surface and the lead wires of the thermocouple extend into
537
Temperature Measurements
cooler air, then heat may be conducted out of the surface through the w ires of the lhermo­
couple. The result may be to cool the surface in the vicinity of the thermocouple, .,.._; ftft
a measurement error (22). This kind of error can be especially sigificant if lbe �
�
a poor conductor of heat or if the thermocouple is separated from the surface by a
of low-conductivity cement One way to minimize such errors is to run the thennoeoup1e
w ires along the surface for some distance away from the location of measuremenL The
cooling effect will thus be moved away from the junction to the poi nt where the leads depart
from the surface.
Similar problems can arise if a thermocouple is embedded within an object at some
distance below its surface with the aim of measuring the surface temperature. If heat ftows
through the surface, then a temperature gradient is present in the object, which leads 10
a difference between the surface temperature and that
' registered by the embedded sensor.
Such errors are greater for nonconductive materials or for high rates of heat transfer.
..;
10.2
Measurement of Temperature in Rapidly Moving Gas
When a temperature probe is placed in a stream of gas, the Row will be partially stopped
by the presence of the probe·. The lost kinetic energy will be converted to heat, which will
have some bearing on the indicated temperature. 1\vo "ideal" states may be defi ned for
such a condition. A static or true state would be that observed by instruments moving with
the stream, and a stagnation or total state would be that obtained if the gas were brought
to rest and its kinetic energy completely converted to heat, resulting in a temperature rise.
A fixed probe inserted into the moving stream will indicate conditions lying between the
two states. For exhaust gases from internal combustion engines, we find that temperature
differences between the two states may be as great as 200°C [23].
An expression relating total and static temperature for a movi ng gas may be written
as follows (24 ):
y2
T, - T, = 2Cp
(15)
This relation may also be written
T,
- = 1
T,
1
+ - (k - l ) M2
2
(!Sa)
where
T,
=
the total or stagnation temperature, in K,
V
=
the velocity of How, in mis,
T, = the static or true temperature, in K,
cp = the specific heat at constant pressure, in J/kg · K,
k = the ratio of specific heats,
M
= the Mach number, V/(sound speed)
The effectiveness of a probe in bringing about kinetic energy conversion is described
by the ratio
T; - T,
r = --T, -
T,
538
( 1 6)
Temperature Measurements
where
T; = the temperature indicated by the probe, in K,
r
=
the recovery factor, which is proportional
to the energy conversion
If r = I , the probe would measure the stagnation temperature, and if r = 0, it would measure
the static or true temperature. Experiment has shown that for a given instrument used under
adiabatic conditions, the recovery factor is essentially a constant and is a function of the
probe configuration. II changes l iule with composition, temperature, pressure, or velocity
of the flowing gas (24]. In practice, however, heal losses due to thermal radiation and
heat conduction will cause some probes to become more sensitive to variations in flow
speed (25] .
Combining Eqs. ( 1 5 ) and ( 1 6), we obtain
, y2
T, = T; - �
( 1 7)
p
or
T, = T; +
( 1 - r) V 2
----
2cp
( 17a)
The recovery factor, r, for a given probe may be determined experimentally (23, 25).
However, this approach does not generally provide sufficient information to determine either
the true or the stagnation temperature. Inspection of Eqs. ( 17) and ( 1 7a) indicates that in
addition to knowing the indicated temperature T; and the recovery factor r, we must know
the stream velocity and the specific heat of the gas. When these values are known, the
relations yield the desired temperatures directly. In many cases, however, it is particularly
difficult to determine the flow velocity, and further theoretical consideration of the situation
is required.
For sonic velocities, with M = I ,
( 1 8)
where
4'
=
k+I
-2-+-r(_k__
_I_)
( 1 9)
One solution to the problem or temperature measurement in high-velocity gases has been to
make the measurement at Mach I , through use of an instrument called a sonic-flow pyrom­
eter. Such a device is shown in Fig. 3 1 . The basic instrument comprises a temperature­
sensing element (thermocouple) located at the throat of a nozzle. Gas whose temperature
is to be measured is aspirated (or pressurized by the process) through the nozzle to produce
critical or sonic velocity at the nozzle throat. Under these conditions, Eqs. ( IS) and ( 1 9)
apply, and in this manner determination of How velocity need not be made. It is still nec­
essary to know the ratio of specific heats, but these can usually be determined or estimated
with sufficient accuracy.
539
'
Temperature Mea surements
\� juJe
Radiation
Bl\
: t �·59
�
I
Thermocouple
lnsluator
FIGURE 3 1 : Schematic diagram of a sonic-flow pyrometer. (Co1urtesy: National Institute
of Standards and Technology (24))
10.3
Temperature Element Response
An ideal temperature transducer would faithfully respond to fluc:tuating inputs regardless
of the time rate of temperature change; however, the ideal is ncit realized in practice. A
time lag exists between cause and effect, and the system seldom, if ever, actually indicates
true temperature input. Figure 32 illustrates dramatically the rn8j!:nitude of errors that may
resul t from poor response.
3000
.L
.. �L...��L2��....4��-1
�L_��-2L.�
6��-18�·�__!10��-112��-:'14
O
(Ignition)
Time, s
FIGURE 32: Temperature- ti me record made from two thermoco1uples of different size and
location du ri ng the stan i ng cycle of a large jet engine. (Courtesy: Instrument Society
of America (26))
Temperature Measurements
The time lag is detennined by the particular heat transfer circumstances that apply,
and the complexity of the situation depends
to
a large extent on the relative imponance
of the convective, conductive, and radiative components. If we assume that radiation and
conduction are minimized by design and application, we may equate the energy absorbed
by the probe per unit time to the rate of heat transfer by convection [ 15 ):
me (
d:: )
= h A ( Tg - Tp)
(20)
or
r
dt
(dTp )
+ Tp = T1
(20a)
where
Tp
T1
= the temperature of the probe, in
K.
= the temperature of the surrounding fluid, in
c = the
K,
specific heat capacity of the probe, in J/kg · K,
m = the mass of the probe, in kg,
t = the timt;, in s,
h = the convective heat-transfer coefficient, in W/m2
•
K,
A = the surface area of the probe exposed to fluid, in m2 ,
r
= me/ hA = the time constant, in s
If the probe is initially at a temperature
T,,.
when it is put in contact with the fluid, we may
=
{T
lrPO
write Eq. (20a) as follows:
[' dt
lo
Solving gives us
f
..:!!.e_
_
T, - Tp
(20b)
If we let
then
(2 1 )
This relation describes the response when a probe a t temperature TPO is suddenly exposed
to a fluid temperature
T,.
This would be approximated if the probe were quickly inserted
through the wall of a furnace or immersed suddenly in a liquid bath.
The quantity
r
will be recognized as the time constant or characteristic time
probe, or the time in seconds required for 63.2% of the maximum possible change T8
for the
-
TPO ·
Temperature Measurements
Obviously, r should be as small as possible, and inspection shows, as should be expected,
that this condition corresponds to low mass, low heat capacity, high transfer coefficient, 8lld
large area. Probes with low time constant provide fast response, and vice versa.
In response to a steady sinusoidal variation in fluid temperature of angular frequency
temperature will oscillate with reduced amplitude and will lag in phase and
ti me. To obtain faithful response, we would want wr « I .
''" the probe
It must be remembered that the time constant for a given probe i s not detennined by
the probe alone. The convec ti ve heat transfer coefficient is also dependent on the character
of the fluid How. For this reason, a given probe may show different time constants when
su bjected to different conditions. For example, a particular probe will usually respond much
faster in a liquid How than in a gas How.
Although practical probe response characteristics m ay, in many cases, be closely
approximated by the application of Eq. (21 ), in many other cases the response is complicaled
by the presence of other objects that absorb or transfer heat. The correspond ing temperature
response may be characterized by multiple time constants.
For example, the case of the common thermometer in a well or a thermocouple or
resistance thermometer in a protective sheath (Fig. 5) may be better approximared by a
two-time-constant model (27, 28). Both probe and jacket will have characteristic time
constants. Let us analyze this situation as follows. We wil l assume that a probe-jacket
assembly (Fig. 33) at temperature T1 is suddenly inserted into a medium at temperature T2 •
In the manner of Eq. (20), we may write two relationships, as follows:
(22)
1j
Jacket
Jackel temperabJ111 =
Probe
Probe temperabJre = Tp
Al I < 0, all temperatures equal
temperature of lhe
surrounding medium = T1
At I � o, temperature of
surrounding medium = T2
FIGURE 33: Temperature probe in jacket subjected to a step change in temperature.
542
Temperature Measurements
and
(22a)
where subscripts j and p refer lo the protective jacket and the probe, respectively. The
relationships may be rewrinen as
(23)
and
(23a)
Simplification may be obtained if we assume that the last term in Eq. (23) may be
neglected. This assumption will be legitimate if
hpAp
h J· A J-
«
l
When this assumption is made, Eqs. (23) and (23a) may be combined, to yield
2
dTp
d Tp
rjrp dif" + (rj + rp) dt + Tp = T2
The solution to lhis equation is
T2 - Tp
where
T2 - Tt
=
...E...._
ll.Tmax
=
(--)
'
{ - I
e -•/{ rp
-
(-1-)1
{ -
(24)
e -•l<p
(25)
fl. T = the momentary difference between lhe actual and indicated temperatures,
0. Tmax = the difference between the temperature of the medium and lhe probe
temperature at I
' = "!1.
= 0,
rp
Characteristics for various values of { are shown in Fig. 34. It is seen that for { = 0,
Eq. (25) reverts lo Eq. (2 1 ). In addition, as lhe time constant for the jacket is increased, lhe
overall lag is increased, as one would suspect it should be.
1 0.4
Compensation for Temperature Element Response
Time lag in electrical temperature-sensing elements, such as thermocouples and resistance
thermometers, may be compensated approximately by use of digital signal processing tech­
niques or by the introduction of appropriate electrical networks. The digital approach
involves taking a Fourier transform of a measured time-series of temperature and per­
forming an appropriate convolution lo correct for the time-lag (29, 30). This melhod is
543
Temperature Measurements
1 .0
0.8
...M...
.:! r.,,..
0.6
0.4
0.2
2
3
4
5
Vt,.
8
---9
11
10
FIGURE 34: Two-time constant problem: Plot or /!J. T /l!J.Tmu. versus t/rp for various ratios
or � = Tj /fp .
very appropriate to measurements in turbulent fluid flow, where high frequency response is
needed and for which digital signal processing is almost univers:11ly applied.
The electrical technique involves selecting a type of filter whose electrical-time char­
acteristics complement those of the sensing element [3 1 , 32]. Fig:ure 35 illu strates a simple
form of such a compensator. In the example illustrated, therrnoo:ouple response drops off
with increased input frequency (as shown in terms of multiples of time-<:onstant recipro­
cals). By proper choice or resistors and capacitance, satisfactory combined response may
be extended approximately 100 times.
Both approaches require that the sensor's time constant be: accurately known under
the conditions in which it is used. In general, however, the best practice is to minimize the
sensor's time constant so that compensation will not be required.
11
MEASUREMENT OF HEAT FLUX
Heal flux is the rate or heat flow per unit area . The coltllnon units are W/m 2 or Btu/h · ft2 •
We can write an expression for heat flux as follows:
dT
q = -k­
dx
where
q = heat flux, in W/ml,
k = the thermal conductivity of the material, in 'WIm
T = temperature, in K,
(26)
·
K,
x = materi al dimension in the direction of heal llow, in m
Knowledge of heat flux rather than temperature is of particular value in designing
systems to avoid excessive temperatures. Examples might involve supersonic aircraft, gas
Temperature Measurements
10
0.001
: Q �· :
't s R.c
�--�---�-�
0.01
0.1
10
1 00
Frequency, multiples of 1/t
FIGURE 35: Curves illustrating compensati ng action of a simple
Instrument
RC network. (Courtesy:
Society, of America)
turbine blades, combustor walls
in rocket motors, and similar situations where heat loads
are of concern.
ftux gages are of several fonns (33], of wh ic h three have particular importance:
type (Fig. 36); the foil or membrane type (Fig. 37), which is also known as the
Gardon gage; and the thin-film layered type (Fig. 38).
As shown in Fig. 36, the essentials of the slug-type meter include a concentrated
Heal
the
slug
FIGURE 36: Section through a slug-type heat ftux sensor.
Temperature Measurements
FIGURE 37: Section through a foil- or membrane-type heat flux sensor (Gardon gage).
mass or slug that is thennally insulated from its surroundings and a temperature sensor,
commonly a thennocouple. As heat flows in, the thennal isolation of the slug produces a
temperature differential between the slug and its surroundings. The governing relation is
q A iap
where
= me
dT
dt + Q1ou
(27)
Atop = the surface area of the top of the slug, in m2 ,
m = the mass of the slug, in kg,
c = the specific heat capacity of the slug, in J/kg
T = the slug temperature, in K,
r = the time, in s,
•
K,
Q1oss = the rate of heat loss through the thennal insulation, in W
Slug temperature is measured by the sensor and, through calibration, its derivative is the
analog of flux. The slug should have a high thennal conductivity so that its temperature
FIGURE 38: Thin-lilm layered heat-flux gage (vertical scale exaggerated).
.
Temperature Measurements
will be unifonn, and Q1oss must be mini mized A primary disadvantage of the gage is that
it is not useful for steady-state conditions. This gage is also called a thennal capacitance
calorimeter.
Construction of the Gardon gage [34), or asymptotic calorimeter, is shown in Fig.
37. It consists of an embedded copper heat sink, a thin membrane of constantan, and
an integral thennocouple. The nature of the qinslruction provides two copper-constantan
thennocouple junctions, one at the center of the membrane and the other at the interface
between the membrane and the heat sink. Thennocouple output, therefore, is a function of
the temperature differential between the center and the periphery of the membrane. This, in
t um is a function of the rate of heat flow from the membrane into the sink. The governing
relationship is
,
q =4
( :Z) •H
= Ce
(28)
where
t = the membrane thickness. in m,
k
,
= the thennal conductivity of the membrane material, in W/m · K,
R = the membrane radius in m,
t::. T = the temperature difference between the center and the edge, in K
C = a cal i bration constant, in W/m2 • K · mV,
e = the output of the thennocouple, in m V
With microfabrication technology, it is possible to create a large number of thermo­
couples, connected as a thennopile, on a membrane just a few millimeters in diameter.
Gages of this type can be packaged as a TO-can. They are very widely used as sensors in
infrared heat detectors (see Section 8).
The layered gage is shown in Fig. 38. Here temperature sensors are attached to the
upper and lower surfaces of a thin, thennally resisting layer. The heat flux is obtained
directly from the measured temperature difference by approximating Eq. (26):
dT
t::. T
q = -k dx "' k T
where
(29)
k = the thennal c onducti vi ty of the resisting film, in W/m · K,
t::. T
=
the temperature difference between the upper and lower
e
surface, in K,
o = th
thickness of the resisting film, in m
The temperature sensors are usually either RTDs or thermocouples. Because the temperature
di fference across the thin barri er is very small, a differential thermopile may be used to
obtain t::. T. As many as I 00 or more thermocouple junctions are onnected in series, with
successive junc tions located on opposite sides of the film.
c
547
Temperature Measurements
Layered gages became capable of high - frequency respo1Bsc only wi th the ad
of microfabrication technology, which made i t possible to deplsi t thin-fi lm tempera
sen sors (0. 1 to 0.5 µm thick) onto relatively thin (1 to 75 µm) ·thermal barriers . In s
ome
applications, the sensors are deposited onto each side of a Kapton film (35] ; i n other des
igns,
the sensors and the thermal barrier are sequentially sputtered onto a s upporting cerami
c
s u bstrate [ 36]. Th i s technology i s co n ti nui ng to evolve rapidly.
::
Calibration of heat flux meters can i nvo lve radiant, conductive, and convective heat
transfer and generally depends upon both the type of gage and its specific application [33].
One common approach uses thermal radiation from a
ature to heat the gage with a known heat flux (37).
12
black body so urce at known temper­
CALIBRATION OF TEMPERATURE-MEASURING DEVICES
If a meas u rement is to be meaningfu l , the measuring procedure a nd apparatus must be prov­
able.
This statem ent is true for all areas of measuremen t, but for some reason the impn:ssion
seems prevale n t that it is less true for temperature- measuri ng s ystems than for other sys­
tems . For example, it is generally tho ught that the on ly limitation in the use of thermocouple
tables is in satisfying the requirement for metal combination indiicated in the table heading.
Mercury-in-glass scale divisions and resistance- thermometer characteristics are commonly
accepted w i thou t question. And it is assumed that once proved , the calibrations
will hold
·
i ndefi n itely.
Of course, we know that
dependent
these ideas are i ncorrect .
Tht:rmocouple output is very
on puri ty of elementary metals and consistency
and homogeneit y of alloys.
Alloy s of supposedly like characteristics but manufactured by different companies may have
te m peratu re-em f relations sufficiently at variance to require di fferent tables. In
add ition,
alter thermoco u ple outputs. Resistance-tht:rmometer stabil ity is very
freedom from residual strains in the element, and comparat ive results from
ag i n g with use will
dependent on
like elements require
very careful use and control of the metallurgy of the materials.
Any calibration of a te m perature measuri n g system must be traceable
to the Interna­
1 990 (ITS-90). Direct calibration on this scale uses the fixed
phy sical condi tion s ( mel ti ng po i n ts, triple points, etc.). S uch cal i brat ions are done almost
excl us i ve ly at standards laboratories. Thermometers so cal ib1 rated are primary standard
thermometers, and other th ermometers are calibrated by some t:ype of comparison to them.
tional Temperatu re Scale of
In
general
practice, therm ometers are c al ibrated by puttii ng the thermometer
known-temperature enclos ure and reading
its outp ut . The
into a
enclosure temperature, in tum.
is known either because another calibrated thermometer is i n it or becau se the enclosure is
held at a
known of freezing or me lti ng
point temperature.
Kn ow n -tem perature enclosures are of two general types, one of whic h , the fixed­
p oi nt enclosure, has already been mentioned. A fixed-point endosure might house a quan­
tity of a material whose freezi ng point is known; often, t h is is a material whose tem­
perature is ass ign ed by ITS-90. Glass is used for the contaiiner material for the lower
tem perat u re s and graphite is used for the higher tem perat ures . I ntegral heating coils are
e mployed. To use the fixed-point cell, the temperature sensor to be calibrated is placed
in a well extend i ng into the center of the container. The heater is then turned on, and
the temperature carried above the melti ng poi n t of the reference substance and held un� l
_ 15
melti ng is completed. The cell is then permitted to cool , and when the freezing point
548
Temperature Measurements
reached, the temperature stabilizes and remains constant at the specified value as long as
liquid and solid are both present. This period may persist from several minutes to several
days. depending on the particular cell. Accuracies of approximately 0. 1 K may be easily
auained and, if great care is exercised, accuracies of beuer than 1 mK may be reached (4).
The other class of calibration enclosures are the variable-temperature enclosures,
which include furnaces for high-temperature work, stirred liquid baths for use from about
-50 10 600" C , and cryostats for very-low-temperature work (38). These are more com­
monly used than fixed-point cells, for reasons of lower cost and greater flexibility. With
a variable-temperature enclosure, it is possible to obtain calibration data at a number of
adjacent temperatures, which may then be used to adjust the coefficients of an appropriate
output-10-temperalure equation [e.g., Eq. (2).)
The simplest possible calibration enclosure is the icepoint bath, which consists of
crushed ice and water held in an insulated container, such as a Dewar flask or an expanded­
polystyrene box. The ice-point provides a convenient way to check the calibration of any
temperature sensor whose range includes O"C. Properly done, an icebath can produce a
temperalure of 0°C to an accuracy of better than 10 m K. The most important guidelines for
malcing an icepoint are the following ( 17).
•
Clean water should be used to make the ice and for the water that is added. Distilled
water is best. Salts or organic contaminants should be carefully avoided, as these
may depress the freezing point temperature. For low-accuracy work, tap water may
be sufficient, although it should have an electrical resistivity above O.S MO · m.
•
The ice should be shaved or crushed to pieces no larger than S mm diameter.
•
The ice should be tightly packed into the container and then filled with water. The
mixture must make good contact with the sensor.
•
Liquid that accumulates at the bottom of the container should be siphoned off peri­
odically.
In addition to the primary fixed points established by lTS-90 and the icepoint, numer­
ous secondary fixed points have been tabulated (see, for example, [39)). Some. examples
include lhe sublimation point of carbon dioxide or dry ice (- 78.5°C), the triple poi nt of
n-docosane (43.9°C), the boiling point of water ( 100°C), and the freezing point of lead
(327 .5°C). Fixed-point cells based on both primary and secondary materials are commer­
cial! y avai table.
Cal ibration of liquid-in-glass thennometers is discussed in detail in (40).
SUGGESTED READINGS
ASME 1 9.3- 1 974 (R 1 998) Ttmperature Measurrtment.
New York: American Society of Mechanical
Engineers, 1 998.
serie s MNL 12. Philadelphia:
4th ed. ASTM manual
American Socic1y for Tusti ng and Materials, 1 993.
ASTM. Manual on the Use of Thermocouples in Temperature MeasuremelU.
B entley, R. E. (ed.) Handboo/c o/Tempera1ure Measuremen1. Vols. 1-3. Singapore: Springer-Verlag,
1 998.
Bums. G. W., and M. G. Scroger. Tempera1ure·Elec1romo1ive Fon:e Reference Functions and Tables
for Leiter-Designated Thermocouple Types Based on 1he ITS-90. NIST Monograph 1 75. Washing­
lon. D.C.: U.S. Departmenl of Commerce, Nalional lns1i1u1e of Slandards and Technology, 1993.
(Supen:edes NBS Monograph 1 25)
Temperature Measurements
and H. D. Chang. Measwement of Tempera111re and Hat Tramra:
Chapter 16 of Rollscnow, w. �·· J. P. Hartnell, and Y. I. Cho (eds.), Handboolr. of Heai T-.fcr:
Goldsrein, R. J., P. H. Chen,
3rd ed. New York: McGraw-Hill, 1 998.
Lawton, 8., and 0. Klingenberg.
Tnuuienl Temperature
Oxford University Pre&s, 1 996.
Lienhard IV, J. H., and J.
Phlogiston Press , 2003.
H. Lienhard V. A
Hall
in
Engineering
and Sdentt.
New Yort:
Transfer Tatbook. 3rd ed. Cambridge, Mas. :
Mangum, B. W., and G. T. Furukawa. GuideUnes for RealWng IM lmemarional TemperatwrJ Salk
of 1990 (ITS-90). NIST Technical Nore 1 265. Washington, D.C.: U.S. Depanmen1 of Conunen:e,
National lnslitule of Standards and Technology, 1990.
McGee. T. D. Principles and Methods a/Temperature MeaJurement. New York: Jolut Wiley, 1988.
Michalski. L., K.
Wiley. 1 99 1 .
Eckersdorf, and
J. McGhee.
Tempemture Measurement.
New York: John
Nicholas, J . V., and D . R . While.
Traceable Temperalures: A n lnttr>duction to Temperature M-.
ment and Calibmrion. 2nd ed. New York: John Wiley, 200 1 .
Preston-Thomas,
H . The
International Temperallft
Scale o f
1 990 (ffS-90).
Me1rolo1ia 27:3-10.
1990 (with coneclions in Metmlogia 21: 107, 1990).
PROBLEMS
I.
At what remper:ature readings do the Celsius and Fahrenheit scales coincide?
,
2.
The remperature indicated by a "total immersion" mercwy-in-glass thermometer is 70"C
( I 58°F).
Actual immersion is to the s•c
(4 1 "F) mark.
What correction should be applied
ID account for the partial immersion? Assume ambient remperatun: is 20"C (68°F).
3.
The uncertainty of a thermometer is stated ID
range is -20"C ID
be ± I % of "full scale."
thermometer's range.
4.
be used ID derenninc
ftat at remperatun: To:
The following relation [3) may
bimetal strip that is initially
r-
t{3( 1
where
t
m
If !he thennomerer
I 20°C, plot the uncertainty as a percentage of !he reading over lhe
the radius of curvature,
r, of a
+ m)l + ( I + mn )[m 2 + ( l /mn))}
6(ai - a 1 )( T - To) ( I + m)2
= !he combined thickness of !he two strips,
= !he ratio of thicknesses of low- to high-expansion components,
n = !he ratio of Young's modulus values of low-ID
high-expansion components,
a1 and a2 = coefficients of linearexpansion, with a 1 < a1.
T
= !he temperature, in
a1 and a2
(a )
Devise a spreadsheet temp
•c or •f, depending on the units for
liite ID solve for r, using the preceding equation.
550
Temperature Measurements
S.
6.
Search the literahlle for the range of values for the constants A and B i n Eq. (2) for
commonly used resi stance thermometer materials. (See the Suggested Readi ng s)
1be circuit shown i n
Fig . 39 is used to drive a p lati num
whose temperature response is given
resistance
thermometer, R(T),
by Eq. (2a). If the temperature varies over a range
of t;.T = ±10 K, show that the current i through the sensor i s approximatel y constant
and that the output voltage is approximately gi ven by e0 = (A · e; /2 1 0 1 ) t;.T, where A
is the temperatu re coefficient from Eq. (2a).
+ 9;
1 000 R0
Bo
+
FIGURE 39: Circuit for Problem 6.
7.
used by the various automobile manufacturers for measuri ng
displaying engine temperatures. What accuracies do you think are obtained by the
Investigate the techniques
and
various systems?
8.
Devise a sim ple
thermistor calibration facility consisting of a variable- te mperatu re envi­
accurate resistance-measuring means that avoids ohmic heati ng of the ele­
ment. and a reliable temperature-measuring system to be used as the "standard." Cali­
brate several the rmistors and evaluate their degree of adherence lo Eqs. (3). (Avoi d the
ron ment, an
problems implied in Problem 1 2.)
9.
Prepare spreadsheet 1cmpla1es for Eq. (3) 10 solve
(a ) for R when T. To. Ro. and fl are given ;
( b ) for {J when T. To, R, and Ro are g iven .
551
Temperature Measurements
10.
The following are data for the calibration of a thennistor.
Temperature,
78
76
72.5
68
65
61
58
54
50.5
47.5
°F
Resistance, �:n
3. 1 6
3.23
3.89
4.24
4.47
�76
5.3 1
5.77
6.37
6.80
For each line of dala, calculate the value of /J, using Eq. (3), and To and Ro
corresponding to the values of 68°F (20°C) . Some spread in the results will be found;
however, use the average of the calculated va lues as the magnitude of fJ. (Use of the
spreadsheets prepared in answer to the previous problem is n:commended.)
1 1.
12.
Write Eq. (3) using the value of fJ found in answer to Problem 10, and plot the result
over the range of dllla. Spot-<: heck several points.
A small insulated box is consuucted for the purpose of obtaining temperature calibration
d8la for thermistors. Provision is made for mounti ng a thermistor within the box and
bringing suitab le leads out for connection to a commercial Wheatstone bridge . The bulb
of a slandardized mercury-in-glass thermometer is insened into the box for the purpose
of determining reference temperatures . A small heating element (a miniature soldering
iron tip) is used as a heat source.
After the heater is turned on, thermistor resistances andl thennometer readings are
periodically made as the temperature rises from ambient 10 a maxi mu m. The healer is
the n turned off and further data are taken as the temperature !falls.
It is quickly noted. however, that there is a very consi·derable discrepancy in the
"heating" resistance-temperature relationship compared with the corresponding "cool­
ing" data. Why should this have been expected? Criticize the design of the arrangement
described above when used for the stated purpose. How wou1ld you make a simple lab­
oratory setup for obtaining reasonably accurate calibration d ata for a thermistor over a
temperature range of, say, 80°F to 400°F?
13.
14.
IS.
16.
17.
18.
Prove the law of inte rmed i ate metals for the situation shown in Fig. 1 2(a) using Eq. (4).
Prove the law of intermediate temperatures using Eq. (5).
Show that the output of the circuit in Fig. 1 8(b) is £ = £F<Co (T,. )
-
£Feeo (O"C).
An ice-bath reference junction is used w i t h a copper-constantan thermocouple. For
four different conditions, millivolt ou tpu ts are read as follows: -4.334, 0.00, 8. 1 33.
and 1 1 . I 30. What are the respective junction tem pe ratu res (a) in degrees C and (b) in
degree s F?
Chromel-alumel thermocouples are used for measuring the 1en1peratures at various poiDIS
in an air conditioning uniL A referencejunction temperature of 22.8°C i s recorded. Jftbe
following e m f oulpUts are suppl ied by the various couples, ckterrnine the corresponding
temperatures: - 1 .689, - 1 . 1 08, -O. I J 3, and 3. 1 85 mV.
A K-type thermocouple circuit (Fig. 40) has its reference junction in liquid niuogen 81
I at m pressure (77 K). The output voltage is measured to be 1 .340 mV.
(a) What is the tem pera ture of the measuring junction?
552
Temperature Measurements
(b) What would the output voltage
room temperature air (20"C)?
be i f the measuring junction were now placed
in
(c) What would the output voltage be if the reference j unction were placed in room
temperature air (20°C) while the measuring j unction remained at the temperature
of part (a)?
Alumel
E
Alumel
Chrome!
77K
FIGURE 40: Circuit for Problem 18.
19.
20.
21.
Using the equation for E versus T for type K thennocouples given in Table 6, spot­
check at least five points in Table 5 to satisfy yourself that
. !he tabulated and calculated
values agree.
Select a lhermocouple type and write a computer program or spreadsheet that calculates
measuringjunction temperature as a function of circuit emf, assuming that the reference
ju nction temperature is O"C. If you will be performing experiments with thermocouples,
select the thermocouple type used in your lab. Take the appropriate eq uation from Table 7.
Write a spreadsheet (or other computer program) that calculates thennocouple outputs
using the equations in Tables 6 and 7.
( a ) Write the software to find temperature as a function of emf for each type in Table 7.
( b ) Write the software to find emf as a function of temperature for each type in Table 6.
Spot-check your results against the data in Table 4.
22.
Use appropriate equations from Table 6 to calculate the emf's that are expected for the
following situations, assuming a c irc u i t such as that shown in Fig. 16. If you wrote a
thermocouple spreadsheet, you may use it in your calculations.
Temperature at the
Reference Junction,
TC Type
·c
E
E
J
K
0
20
0
80
T
1 00
553
Temperature at the
Measuring Junction,
·c
500
750
650
1 1 50
3 15
Temperature Measuremenu
23.
Use appropriate equations from Tables 6 and 7 10 calculate the temperature 11 the mea­
suring junction for each of the following situations. Assume a cin:uil such as !hat shown
in Fig. 1 6. If you wrote a thermocouple spreadsheet, you may use ii in your calculations.
Temperature at the
Reference Junction,
TC Type
·c
emf, mV
E
E
J
0
90
15
280
0
700
60.63
-4.36
1 6.30
- 1 1 .78
29.79
-8. 14
J
K
K
24.
type N thermocouple has its reference junctions al O"C. Plot the measuring junction
temperature as a function of emf over the range 0.0 mV to 45.0 mV, using equations from
Table 7. If you have written a thermocouple spreadsheet, you may use it to do the job.
A
i_-
I
i�-1>- ------1I
e1
,...----A1u_
m__
-._'
-+
1 __c
____ ...
_u
__..
'Alumel
Chrome!
!
L _ _ _J
�
T�,
+
-
E
!
I
L _ _ _ _ _ _ _ _j
I
Tm111r
o•c
AGURE 4 1 : Circuit for Problem 25.
25.
Consider the thermocouple circuit shown in Fig. 4 1 . Copper extension wires are con­
nected to the thermocouple circuit. Suppose that the extension wire junctions are also
at 0°C. Show that the output voltage is unchanged when the second piece of alumel
(marked • ) is eliminated and the copper lead is directly connected 10 the chromel wire.
26.
The temperature difference between two points on a heat exchanger is desired. The mea­
suring and reference junctions of a chromel-alumel thermocouple are embedded within
the inlet and outlet tubes, I and 2, respectively. and an emf of 0.381 m V is read. Why
does this reading provide insufficient data to determine the differential temper.nure accu·
rately? What additional information must be obtained before the answer may be found?
27.
Show that the output voltage of the thermopile in Fig. 20(a) is 5(£..is(Ti ) - £..is ( Tz)].
28.
thermopile is constructed from three type K thermocouples (Fig. 42). The measuring
junctions each have different temperatures (T1 , T2, and T3), and the reference junctions
are all at O"C.
A
( a ) Whal is the output emf, E, of the thermopile when T1 = 20"C, Tz = 25"C, and
T3 = 30°C?
( b) If the user of the thermopile were to assume that all 1hn.-e temperatures were equal,
·
what temperature would he or she calculate? How does this compare 10 the actual
average of the three temperatures?
554
Temperature Measurements
r.
-=
r - - - - -,
===:::=----;-o
+
E
:I
I
I
I
I
I
I
ere:
I
I
r,
L. _ _ _ _ _ _,
FIGURE 42: Circuit for Problem 28.
29.
( c ) IrinSlead, T1 = 800°C, T2 = IOOO"C, and T3 = 1200°C, what temperature would
the user calculate? How does this compare to the true average temperature?
Thermocouple wire is generally manufactured to meet ASTM tolerance.� that vary by
wire type. For commercial-grade type E thermocouples between 0°C and 900" C , this
tolerance i s :!:: I. 7"C or :!:0.5% (whichever is greater). The tolerance is a bound on the
bias enor of the thermocouple's temperature as calculated from the measured emf using
the NIST standards. The actual siu of the bias enor for any specific thermocouple is
unknown. To measure temperature to an accuracy better than the standard tolerance, the
thennocouple must first be calibrated.
(a)
A type E thermocouple probe using an electronic icepoint is calibrated by compar­
ison to a platinum resistance thermometer (PRT). Both sensors are placed into
a well-stirred temperature-controlled liquid bath. The following readings
obtained.
PAT Temperature, •c
Thermocouple emf, mV
25.0
35. I
50.2
63.5
74. 9
1 .526
2. 146
3.09 1
3.94 1
4.68 1
are
Use these data to find an equation for the thermocouple's temperature as a
runction of its emr.
Temperature Measurements
( b ) S u ppose that the calibrated thermocouple is now used to •neaswe a temiieralwe.
The thermocoupl e circuit has the same electronic icepoint but now uses a lower­
quality voltmeter. Discuss the u ncenai nty in that me&SWl:ment and estimate its
si ze (at 95% confidence). You may assume the following uncertainties in the
equi pment used.
Calibration Equipment
PRT
Voltmeter
Liqu id bath
±0.05°C (bias, 95%>
±0. 00 1 mV (bias, 95%)
temperature fluctuations are
small and random
Electronic ice poi n t
±0.25°C (bias, 95%)
±0.005 mV (precfaion, 95%)
Thermocouple Equipment
Voltmeter
30.
To eliminate high-frequency electrical noise, a low- pass filter is connected to a type K
thermocouple c ircuit, as shown in Fig. 43. "The ou tput is displa:1ed on an oscilloscope
having 1 MQ input impedance. The thermocouple circuit's output is steady (i .e., a de
voltage), and the filter's cut-off frequency is known to be appropriate.
1 8 MQ
Al
Al
Low-pass filter
fc= 0.09 Hz
nK
FIGURE 43: Circuit for Problem 30.
When the voltages arc read from the scope and convened to temperatures, the
results arc c learly wrong: The apparent value of Tm is close to 77 K, when it shou ld be
much higher. E x pl a i n why, and describe a better filter.
31.
32.
Plot the error i n measured temperature as a fu nc tion of te mper.ature for the following
types of pyrometers, assuming a I 0% error in emissivity: a to�il radiation pyrometer.
a spectral-band pryometer operating at 0.9 µm; and an infrared pyrometer sensitive to
wave lengths from 8 to 14 µm. Use a range of 200°C to 2000° C.
Referri ng to Eq. (I 5a). plot the ratio of static 10 total temperatun:os versus Mach number
over the range o f M = 0 to 3 and for k = 1 . 3 , 1 .4, and 1 .5.
556
Temperature Measurements
33.
34.
Referring to Eqs. ( 1 8) and ( 1 9), plot the ratio of indicated to static temperatures versus
= 1 . 3, 1 .4, and 1 .5.
the recovery factor over the range of r = 0 to I , for k
The following temperature-time data were recorded after a probe was placed in a warm
environment:
(a )
( b)
{c)
( d)
{e)
35.
36.
37.
Time, s
Temperature, •c
0
4
8
12
20
30
40
50
20
83
123
1 52
1 82
1 94
201
203
Plot the data points.
From the plot, determine a time constant for the system.
Write a response equation assuming first-order process.
Calculate sufficient points to plot the theoretical curve.
Decide, on the basis of the plot, whether or not the process may be considered a
single time·constant first-order type.
A "two time-constant" temperature transducer has time constants in the ratio { = 4/ 1 ,
where rp = 1 .5 s . I f the transducer, initially at a temperature o f 80°C, i s suddenly
immersed in a 500"C environment, what will be the temperature indicated after 3 s?
If the transducer in Problem 35 is initially at 500°C and is suddenly immersed in an 80°C
environment, what temperature will be indicated after 3 s"!
The perfonnance of a temperature-measuring system approximates that dictated by two
time-constant theory: rp = 10 s and ri = 25 s. If the system is subjected to the tem­
44, we see that the probe will not have sufficient time to
produce a readout approximating
If, however, at the end of the 1 8-s pulse, the
system indicates 135°C, what must be the value of Tm.,.'!
perature input shown in Fig.
t
Tmax ·
1s•c
- - ­
ao•c
-, - - - - - - - - - -·
l!
t
�
I
I
Os
._
___
25 s
rune -
FIGURE 44: Temperature-time relationship for Problem 37.
557
Temperature Measurements
38.
1be behavior of a iemperature-measuring system approximates two lime-coosiani theory
with fp = 6 s and fj = 1 4 s. ffthe systemexperiences a penurbation as sbown in Flg. 4S
�
what will be the indicaledlemperarure al / = 1 5 s? Al l = 25 s? At 25 s the iem
the same; however, on the assu mption that both
of the probe and the jacket will not be
are at probe iemperarurc, estimale the indicated temperature at 60 s. Will the calculated
value be too high or too low? State your reasoning in answering the last question.
t
j
t
F
Tmp
- - -
�-------.
75°C - 1 - - - - - - - - - - - �
1 ----1
1
1
1
1
1
18s
Os
Time -
FIGURE 45: Temperature-time relationship for Problem 38.
39.
We wish to have both a continuous record and an instantaneous readout of energy flow
rate from
each
healed water passing through a pipe. Temperature, prcssuie. and rale of flow
vary over a range of values.
{ a ) Analyze the problem and prepare a block diagram
40.
( b)
of the
various measurements
and functional problems that must be solved.
Insofar as you can, detai I the sleps to a solution.
A thin-film layered heat-flux gage consists of a Kapton film with a single thin- foil ther­
mocouple junction coaled on either side (Fig. 38). The fi l m has a thickness of 25 µm
{± I µm at 95% confidence) and a conductivity of 0.20 W/m
·
confidence).
·
K (±0.01 W/m · K at 95%
{ a ) If the thennocouples operate in a range where their output is 40 µV f°C, and if the
tlicnnocouplc voltage can be measured to ± I µV {at 95% confidence), what is the
smallest heat flux that can be measured to an uncenainty of no more than 10% {at
95% confidence)?
( b ) Describe how the hcat-llu x gage could be modified to measure a heat fl11x one-tenth
as large as that found in pan {a) with the same percentage uncenainty.
( c ) Thermocouple junctions may have systematic error of up to 2°C. What should be
done to ensure accurate heat flux readings with this gage?
558
Temperature Measurements
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