Chapter 2
Characteristics of Measurement System Elements
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Objectives
To understand the importance of characteristics of measurement
system: static, dynamic and statistical
Learn about terms used in measurement according to VIM standard
 Graphically, or analytically, determine the static, dynamic and
statistical characteristics of a transducer.
 to understand calibration and standards
 to understand generalized Model of a system element
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Meaning of element characteristics.
The relationships which may occur between the output O and
input I of an element
What do we mean by the word Element?
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Review :
a measurement system consists of different types of element
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The two main characteristics are
steady-state (static )characteristics :
these are the relationships which may occur between the output O and input I of
an element when I is either at a constant value or changing slowly
Dynamic Characteristics
those that appear when an element responds to sudden input changes
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Static characteristics
Systematic
that can be exactly quantified by mathematical or graphical means
Statistical
those that can be quantified by statistical means
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a. Range (VIM : Interval): The limits between which the input (output) can vary.
The range defines the minimum and maximum values of a quantity that the
instrument is designed to measure.
Range is represented as :
from Imin to Imax
from Omin to Omax
Example1(textbook page 9) :
A pressure transducer has an input range of ( 0 to 104 )Pa.
a pressure transducer has an output range of (4 to 20)mA.
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Example 2: What is the range of the following Voltmeter?
Ans: From 0 to 5 Volts
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Example 3: Product Description : Palm-Size Handheld Digital Multimeter - DT830B
Product Description
Model: DT830B
Digital AC/DC multimeter with diode and transistor test function.
Specifications:
DCV:
0-200mV/0.25%
0-2-20-200-1kV/0.5%
ACV:
0-200V-750V/1.2%
DCA:
0-200u-2m-20m/1.0%
0-200mA/1.2%-10A/2.0%
R:0-200-2k-20k-200k/0.8%
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What do we mean by VIM?
VIM :International vocabulary of metrology
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VIM in Jordan
http://www.jism.gov.jo/arabic/index_arab.htm
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‫مؤسسة المواصفات و المقايس االردنية‬
11
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b. Span (VIM : Range of interval) :
it is the maximum variation in input or output.
The span defines the variation of values of a quantity that the instrument is
designed to measure.
Input Span = Imax-Imin
Output Span = Omax-Omin
Example 2 (textbook page 9). in example 1 we can say
a pressure transducer has an input span about 104 Pa.
a pressure transducer has an output span of 16 mA.
What do you know about 4 to16 mA ?
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Example 4: Given the following readings between I(load) and O(deflection):
Load [kg]
0
5
5
8
10
Deflection [mm]
0
30
35
40
80
a. Determine the range of input and output
b. Determine span of input and output
Solution
a.
IRange: form (0 to 10) kg
Orange : form (0 to 80) mm
b.
ISpan= 10 kg
OSpan= 80 mm
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c. The relationship between The input and Output
Case 1.
In many cases O(I ) can be expressed as a polynomial in I:
Example 5 : Thermocouple
For a copper–constantan (Type T) thermocouple junction
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What is a thermocouple ?
If two different metals A and B are joined together, there is a difference
in electrical potential across the junction called the junction potential. This
junction potential depends on
 the metals A and B
the temperature T °C of the junction
Refer to page 172 Textbook: chapter 8
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Case 2: Other expressions
Example of expressions: exponential
Example 6:
Relationship between the variation of resistance in Thermistor and the
temperature change
the resistance R(T)ohms of a thermistor at T °C is given by
Assignment : What is a thermistor ?
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Case 3 :Ideal straight line
An element is said to be linear if corresponding values of I and O lie on
a straight line.
In other words
The relationship between the output reading of an instrument is
assumed to be linearly proportional to the quantity being measured
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Ideal straight line equation
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Example 7: A pressure transducer has an input range of ( 0 to 104 )Pa and
It has an output range of (4 to 20)mA. What is the ideal straight line for the
above pressure transducer ?:
The ideal straight line for the above pressure transducer is:
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Problem 2.8 :
A force sensor has an output range of 1 to 5 V corresponding to an input range of
0 to 2 × 105 N. Find the equation of the ideal straight line.
Problem 2.9
A differential pressure transmitter has an input range of 0 to 2 × 104 Pa and an
output range of 4 to 20 mA. Find the equation to the ideal straight line.
Problem 2.10:
A non-linear pressure sensor has an input range of 0 to 10 bar and an output
range of 0 to 5 V. The output voltage at 4 bar is 2.20 V. Calculate the nonlinearity in volts and as a percentage of span.
Problem 2.11:
A non-linear temperature sensor has an input range of 0 to 400 °C and an
output range of 0 to 20 mV. The output signal at 100 °C is 4.5 mV. Find the
non-linearity at 100 °C in millivolts and as a percentage of span.
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Non-linearity
In many cases the straight-line relationship is not obeyed and the element is
said to be non-linear.
Definition of non-linearity.
The nonlinearity at point
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can be calculated from the following formula
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Quantification of non-linearity
a. maximum non-linearity
Non-linearity is often quantified in terms of the maximum non-linearity ;
expressed as a percentage of full-scale deflection (f.s.d.), i.e. as a
percentage of span. Thus:
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b. Nonlinearity at point I as percentage of f.s.d
Nonlinearity at point I as percentage of f.s.d
N (i )

100%
O max  O min
Why do we need to calculate Nonlinearity at point
I as percentage of f.s.d?
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Q1. How can you define the linearity of the system ?
Q2. What do we mean when we say that the linearity is
2%?
Q2. What do we mean when we say that the non-linearity
is 2%?
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Example 6(problem 2.12)
A thermocouple used to measure temperature between 0C and 500 C has
the following input- output characteristics
T, C
0
E, µV 0
100
200
300
500
5268
10777 16325 27388
a) Find the equation of ideal straight line
b) Find the non linearity at 100 C and as percentage of F.S.D
c) Find the maximum linearity as percentage of F.S.D
d) Find the minimum non linearity as percentage of F.S.D
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How to understand the table ?
T, C
0
E, µV 0
0
100
200
5268
10777 16325 27388
100
200
300
300
500
Measurand
Output
500
Results or measurements
0
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5268
10777 16325 27388
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a) Find the equation of ideal straight line
Oideal  KI  a
Omax  Omin
K
Imax  Imin
27388  0

K
 54.776 V / C
500  0
a  Omin  K  Imin
a  0 K0  0
Oideal  54.776  I
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b) Find the non linearity at 100 C and as percentage of F.S.D
N (i )  O(i )  ( Ki  a)
N (100)  O(100)  (54.766 100  0)
N (100)  5268  (54.776 100  0)  209.6V
No-linearity at 100 C and as percentage of F.S.D

N (100)
 209.6
100% 
100%  0.77%
27388  0
27388  0
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d. Sensitivity (S)
It is the change ΔO in output O for unit change ΔI in input I, i.e. it is the ratio
ΔO/ΔI.
In the limit that ΔI tends to zero, the ratio ΔO/ΔI tends to the derivative
dO/dI, which is the rate of change of O with respect to I
 For a linear element dO/dI is equal to the slope or gradient K of the straight
line;
 For a non-linear element the sensitivity is the slope or gradient of the output
versus input characteristics O(I )
 the unit of sensitivity is
[ Unit of O]/ [Unit of I]
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Sensitivity of linear element
For a linear element dO/dI is equal to the slope or gradient K of the
straight line
Refer to Figure 2.3
O dO
S  lim

dI
I 0 I
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Sensitivity (Static Sensitivity) of non-linear element
For a non-linear element dO/dI = K + dN/dI, i.e. sensitivity is the slope or
gradient of the output versus input characteristics O(I ).
example of nonlinear element characteristic
Page 154 textbook
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Example 8.
A pressure transducer has an input range of ( 0 to 104 )Pa an it has an output
range of (4 to 20) mA. What is the sensitivity of the transducer ?
The sensitivity is 1.6 × 10−3 mA/Pa.
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e. Inverse sensitivity or Threshold (C) {VIM }
largest change in a value of a quantity being measured that causes no
detectable change in the corresponding indication
Simple definition
Threshold is the minimum level of input that produces (causes) a detectable
amount of output
1
C 
S
I
dI
C  lim

dO
O 0 O
Unit of Threshold
[C]=[unit of input/unit of output]
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Example 9: Given the following readings between I(load) and O(deflection):
Load [kg]
0
5
5
8
10
Deflection [mm]
0
30
35
40
80
a. Determine the sensitivity of the instrument
O dO
80  0
S  lim

K
 8mm / kg
dI
10  0
I 0 I
b. Determine threshold of instrument
1 1
C    0.125kg / mm
K 8
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f. Hysteresis
It is a characteristic of measuring system which describes the difference
between two outputs of the same input.
H (i )  O(i )   O(i ) 
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Max. Hysteresis as percentage of f.s.d
Hˆ

100%
O max  O min
Note: Large inputs may presently damage sensing element and hence
decrease nonlinearity and increase hysteresis
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Example 7 (problem 2.4)
A liquid level sensor has an input range of (0 to 15)cm. use the calibration results
given in the table to estimate the maximum hysteresis as percentage of f.s.d
Level h, [cm]
0.0
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
15.0
Ovolt
0.00
0.35
1.42
2.40
3.13
4.35
5.61
6.50
7.77
8.85
10.20
Ovolt
0.14
1.25
2.32
3.55
4.34
5.70
6.78
7.80
8.87
9.65
10.20
Solution
Hˆ  1.35 volt
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Hˆ
 100%
Max. Hysteresis as percentage of f.s.d 
O max  O min
Max. Hysteresis as percentage of f.s.d
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
1.35
 100%  13.24%
10.2  0
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Problem 2.11:
A level transducer has an output range of 0 to 10 V. For a 3 m level, the output
voltage for a falling level is 3.05 V and for a rising level 2.95 V. Find the hysteresis
as a percentage of span.
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i. Resolution R
- It is a smallest change in a quantity being measured that causes a
perceptible change in the corresponding indication
- The largest change in I that can occur without any corresponding
change in O.
The resolution R expresses the ability of the system to detect small increments
of the measurand
Max. Resolution as percentage of f.s.d
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R
ΔI step
I max  I min
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 100%
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Example : Resolution of wire-wound potentiometer
-resistance R increases in a series of steps
- the size of each step being equal to the resistance of a single turn.
-The resolution of a 100 turn potentiometer is 1%.
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example Resolution of analog to digital converter (ADC)
Vref
bit0
bit1
Vin
ADC
bitn
R ADC
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V max  V min

2n
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Example 8. (problem 2.16)
An analog to digital converter has an input range of 0 to 10 V. Calculate the
resolution when the output digital signal is 8 bit.
10 10
RADC8  8 
 0.04
2 256
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v
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J. Environmental effects
Environmental inputs such as: ambient temperature, atmospheric pressure, and etc.
Two types of environmental inputs affected on the output
The modifying input IM
The interfering input Ii.
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J. Environmental effects
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Types of environmental effect inputs
Modifying input IM (Sensitivity drift)
It cases the linear sensitivity of measuring element to change
The interfering input Ii(bias or drift)
cases the bias (a) of measuring element to change
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Example of modifying input IM change
Standard ECG signal
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Modified ECG signal
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Example of interfering input Ii : Bias or interfering input Ii
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 The modifying input IM
Example : condition of power supply variation
At standard condition, for example Vs must be 5 V, then When IM=0 , the
Sstandard = K.
When the standard condition is changed , for example the Vs has a value
greater or smaller than 5 V.
Then IM=value, Snew=Sstandard+KMIM
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The interfering input Ii.
The standard operation condition is that the device is operated at 20oC,
then When Ii=0 , the astandard = a.
When the standard condition is changed, for example the temperature increases to 30 oC.
Then Ii=value, anew=astandard+KiIi
When the standard condition is changed, for example the temperature decreases to 10 oC.
Then Ii=value, anew=astandard-KiIi
Example of modifying input: temperature
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Example 9. (problem 2.3)
A displacement sensor has an input range of (0.0 to 3.0) cm and standard
supply voltage Vs=0.5 volts. Using the calibration results given in the table,
estimate :
a) the maximum non linearity as percentage of f.s.d.
b) the constant Ki, KM associated with supply voltage variations.
c) The slope K of the ideal straight line
x,[cm]
0.0
0.5
1.0
1.5
Vout:Vs=0.5 volts.
0.0
16.5
32.0
44.0 51.5 55.5 58.0
Vout:Vs=0.6 volts.
0.0
21.0
41.5
56.0 65.0 70.5 74.0
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2.0
2.5
3.0
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Summary
IM causes K(at standard condition) is to change to K + KMIM
Where
IM is the deviation from standard conditions
IM =(new value – standard value).
KM is the change in sensitivity for unit change in IM
Ii causes a (at standard condition) is to change to a+ KiIi
Where
Ii is the deviation from standard conditions
Ii =(new value – standard value).
Ki is the change in zero bias for unit change in Ii
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Solution
Step 1. Calculate K when Vs= 0.5 volts, and let we call it Kold
Omax  Omin
Imax  Imin
K old 
K old 
58.0  0.0
 19.33mv / cm
3.0  0.0
Step 2. Calculate the modifying input (The power supply is changed )
I M  Vsnew  Vsold  0.6  0.5  0.1 volt
Step 3. Calculate K when Vs= 0.6 volts, and let we call it Knew
K new
Omax  Omin
74.0


 24.7 mv/cm
Imax  Imin
3.0
Step 3. Calculate KM
KM 
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Knew= Kold+KMIM
K new - K old 24.7 - 19.6

 53.3mv/cm
IM
0.1
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K. Error band
The term of error band is used to define the performance of the element.
Any input value I, the output O will be within ±h of ideal straight line value
OIDEAL.
Quantification of Performance
- Systematic statement of performance is replaced by a statistical statement in
terms of a probability density function p(O).
- a probability density function p(x) is defined so that the integral p(x) dx
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Error bands and rectangular probability density function
Probability density function
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Generalized Model of a system element
Direct Model
O  KI  a  N (i)  KM I M I  K I I I
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Examples of element characteristics
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Strain gauge
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Copper–constantan thermocouple
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Accelerometer
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Identification of static characteristics – calibration
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Assignment
Refer to VIM and discuss the term “calibration”
Calibration of an Element
Calibration is a type of experiment in which the measurement variable I,
O, environmental effects and static characteristics.
What do we need for calibration ?
Standard Instrument with known accuracy
Determining the true value with known accuracy
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Measurement standards
primary measurement standards
Centers of Primary standard
-National Physical Laboratory (NPL).
Examples: time, length, mass, current and temperature
-National Engineering Laboratory (NEL)
Examples : density and flow rate of gases and liquids
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Transfer standards
Transfer standards held at accredited centres are calibrated against national
primary and secondary standards
Centers
UKAS (United Kingdom Accreditation Service)
Purpose of Transfer standard:
a manufacturer can calibrate his products against the transfer standard at a
local centre.
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Traceability ladder
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Traceability ladders
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Case study: defining the International Temperature Scale 1
From thermodynamic we know the following relationship
PV = Rθ
The relationship between the pressure P and temperature θ of a fixed
volume V of an ideal gas.
The relationship between the Kelvin and Celsius scales is
International Practical Temperature Scale (IPTS)
Because of the limited reproducibility of real gas thermometers the
International Practical Temperature Scale (IPTS) was devised.
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(a)several highly reproducible fixed points corresponding to the freezing, or
triple points of pure substances under specified conditions;
(b)standard instruments with a known output versus temperature
relationship obtained by calibration at fixed points.
What do we mean by ITS90
It is an International Temperature Scale developed in 1990
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The primary fixed points defining the International Temperature
Scale 1990 – ITS90.
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Solved problem in calibration
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Important Characteristics
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1. Accuracy
The accuracy is the closeness of agreement between the measurement result
and the true value.
2. Precision
The degree of agreement within a group of measurements or instruments
Refer to : Measurement and Instrumentation Principles, Third Edition
Link: www.avaxhome.ws
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Relationship between accuracy and precision
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3. Repeatability ( it will be disused later in more details )
Repeatability is the ability of an element to give the same output for
the same input, when repeatedly applied to it.
4. Reproducibility
It describes the closeness of output readings for the same input when
there are changes in the method of measurement,
5. Tolerance
Tolerance is a term that is closely related to accuracy and defines the
maximum error that is to be expected in some value.
Refer to page 17
Measurement and Instrumentation Principles, Third Edition
Link: www.avaxhome.ws
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6. Dead space
Refer to page 24
Measurement and Instrumentation Principles, Third Edition
Link: www.avaxhome.ws
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7. Saturation
Every sensor has its operating limits. Even if it is considered linear, at some
levels of the input stimuli, its output signal no longer will be responsive. A
further increase in stimulus does not produce a desirable output. It is said
that the sensor exhibits a span-end saturation
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Dynamic characteristic of measurement system
The dynamic characteristics of a measuring instrument describe its behavior
between the time a measured quantity changes value and the time when the
instrument output attains a steady value in response
In any linear, time-invariant measuring system, the following general relation can
be written between input and output for time (t) > 0:
d n f o (t )
d n1 f o (t )
df o (t )
d n f i (t )
d m1 f i (t )
df i (t )
an

a

...

a

a
f
(
t
)

b

a

...

b
 b0 f i (t )
n 1
1
0 o
m
m 1
1
n
n 1
m
m 1
dt
dt
dt
dt
dt
dt
Where fi(t)is the measured quantity, fo(t) is the output reading and a0 . . . an, b0 . . .
bm are constants.
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an
d n f o (t )
d n 1 f o (t )
df o (t )

a

...

a
 a0 f o (t )
n 1
1
n
n 1
dt
dt
dt
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In this course we limit consideration to that of step changes in the measured
quantity only, then equation above reduces to:
d n f o (t )
d n1 f o (t )
df o (t )
an
 an1
 ...  a1
 a0 f o (t )  b0 f i (t )
n
n 1
dt
dt
dt
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1. Zero order instrument
If all the coefficients a1 . . . an other than a0 in equation 1 are assumed zero, then:
a0 f o (t )  b0 f i (t )
or
b0
f o (t ) 
f i (t )
a0
and
RAB
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b0
K 
a0
d
 RP
 RP xd
dT
84
First order instrument
If all the coefficients a2 . . . an except for a0 and a1 are assumed zero in
equation ) then:
df o (t )
a1
 a0  b0 f i (t )
dt
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second-order element
d 2 f o (t )
df o (t )
a2
 a1
 a0 f o (t )  b0 f i (t )
2
dt
dt
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Definition of element transfer function
 d2

d
 a2 2  a1  a0  f o (t )  b0 f i (t )
dt
 dt

a s
2
2
d
 s    j
dt

 a1s  a0 Fo ( s)  b0 Fi ( s)
b0
Fo ( s )
b0
a0


2
Fi ( s) a2 s  a1s  a0  a2 2 a1

 s  s  1
a0
 a0


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
Instruments and transducers
87
Transfer function for a first-order element
K
G (s) 
1  s
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O
K
 steady - state sensitivit y
I
Instruments and transducers
88
Transfer function for a second -order element
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Instruments and transducers
89
Obtaining Transfer function for a second -order element
 d2

d
 a2 2  a1  a0  f o (t )  b0 f i (t )
dt
 dt

a s
2
2
d
 s    j
dt

 a1s  a0 Fo ( s)  b0 Fi ( s)
Fo ( s )
b0

Fi ( s )
a2 s 2  a1s  a0

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
b0
a0

 a2 2 a1


s 
s  1
a0
 a0

Instruments and transducers
90
Laplace transforms of common time functions f(t).
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Instruments and transducers
91
response of first- and second-order elements
Response of a first order element
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Response of a second-order element
Instruments and transducers
92
“simple TF of sensor”
Simple transfer function with delay
K
 st
G(s) 
e
1  s
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Instruments and transducers
93
Temperature, Co
Dynamic characteristics of temperature sensor
thermocouple
t=t0
time, s
t=t0
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Instruments and transducers
t=t0
94
Temperature, Co
thermocouple
T=T1
T=T0
time, s
T=T1
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Instruments and transducers
T=T0
95
Temperature, Co
V,mv

T=T1
V=V1
T
V
T=T0
V=V0
time, s
V
K 
T
time, s
t
K
st
G (s) 
e
1  s
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Instruments and transducers
96
Statistical characteristics
1. Statistical variations in the output of a single element with time –
repeatability
What do we mean by a lack of repeatability in the element?
Example 5: The following measurements are for pressure sensor for the same value ,
for several days
P [bar]
1
1
1
1
1
V [Volts]
1
0.99
1.01
0.98
1.02
The most common cause of lack of repeatability in the output O is random
fluctuations with time in the environmental inputs IM, II: if the coupling constants
KM, KI are non-zero, then there will be corresponding time variations in O.
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Instruments and transducers
97
probability density function of the element output O
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Instruments and transducers
98
How to expresses the independent variable O in terms of the independent
variables I, IM and II?
if ΔO is a small deviation in O from the mean value O caused by deviations ΔI,
ΔIM and ΔII from respective mean values I , I and I , then:
i
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Instruments and transducers
M
99
if x1, x2 and x3 have normal distributions with standard deviations σ1, σ2 and σ3
respectively, then the probability distribution of y is also normal with standard
deviation σ given by:
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Instruments and transducers
100
Standard deviation of output for a single element
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Instruments and transducers
101
probability density function of Output of element
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Instruments and transducers
102
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Instruments and transducers
103