MEASUREMENT THEORY FUNDAMENTALS, 361-1-3151
MEASUREMENT THEORY
FUNDAMENTALS
361-1-3151
Eugene Paperno
http://www.ee.bgu.ac.il/~paperno/
© Eugene Paperno, 2006
MEASUREMENT THEORY FUNDAMENTALS. Grading policy
GRADING POLICY
20% participation in lectures
30% home exercises
50% presentation
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MEASUREMENT THEORY FUNDAMENTALS. Grading policy
HOMEWORK
Build in LabView the following virtual instruments (VI):
1. Lock-in amplifier SR830
www.thinksrs.com/mult/SR810830m.htm
2. Spectrum analyzer SR785
http://www.thinksrs.com/mult/SR785m.htm
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MEASUREMENT THEORY FUNDAMENTALS
The mathematical theory of measurement is elaborated in:
Krantz, D. H., Luce, R. D., Suppes, P., and Tversky, A. (1971). Foundations of
measurement. (Vol. I: Additive and polynomial representations.). New York: Academic
Press.
Suppes, P., Krantz, D. H., Luce, R. D., and Tversky, A. (1989). Foundations of
measurement. (Vol. II: Geometrical, threshold, and probabilistic representations). New
York: Academic Press.
Luce, R. D., Krantz, D. H., Suppes, P., and Tversky, A. (1990). Foundations of
measurement. (Vol. III: Representation, axiomatization, and invariance). New York:
Academic Press.
Measurement theory was popularized in psychology by S. S. Stevens, who originated the
idea of levels of measurement. His relevant articles include:
Stevens, S. S. (1946), On the theory of scales of measurement. Science, 103, 677-680.
Stevens, S. S. (1951), Mathematics, measurement, and psychophysics. In S. S. Stevens
(ed.), Handbook of experimental psychology, pp 1-49). New York: Wiley.
Stevens, S. S. (1959), Measurement. In C. W. Churchman, ed., Measurement: Definitions
and Theories, pp. 18-36. New York: Wiley. Reprinted in G. M. Maranell, ed., (1974) Scaling:
A Sourcebook for Behavioral Scientists, pp. 22-41. Chicago: Aldine.
Stevens, S. S. (1968), Measurement, statistics, and the schemapiric view. Science, 161,
849-856.
Reference: http://www.measurementdevices.com/mtheory.html
MEASUREMENT THEORY FUNDAMENTALS. Contents
CONTENTS
1. Basic principles of measurements
1.1. Definition of measurement
1.2. Definition of instrumentation
1.3. Why measuring?
1.4. Types of measurements
1.5. Scaling of measurement results
2. Measurement of physical quantities
2.1. Acquisition of information
2.2. Units, systems of units, standards
2.2.1. Units
2.2.1. Systems of units
2.2.1. Standards
2.3. Primary standards
2.3.1. Primary voltage standards
2.3.2. Primary current standards
2.3.3. Primary resistance standards
2.3.4. Primary capacitance standards
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MEASUREMENT THEORY FUNDAMENTALS. Contents
2.3.5. Primary inductance standards
2.3.6. Primary frequency standards
2.3.7. Primary temperature standards
3. Measurement methods
3.1. Deflection, difference, and null methods
3.2. Interchange method and substitution method
3.3. Compensation method and bridge method
3.4. Analogy method
3.5. Repetition method
3.6. Enumeration method
4. Measurement errors
4.1. Systematic errors
4.2. Random errors
4.2.1. Uncertainty and inaccuracy
4.2.2. Crest factor
4.3.
Error propagation ( ‫ העברת שגאיות‬,‫)תרגום‬
4.2.1. Systematic errors
4.2.1. Random errors
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MEASUREMENT THEORY FUNDAMENTALS. Contents
5. Sources of errors
5.1. Influencing the measurement object: matching
5.4.1.
5.4.2.
5.4.3.
5.4.4.
5.2.
Anenergetic matching
Energic matching
Non-reflective matching
When to match and when not?
Noise types
5.2.1. Thermal noise
5.2.2. Shot noise
5.2.3. 1/f noise
5.3.
Noise characteristics
5.3.1. Signal-to-noise ratio, SNR
5.3.2. Noise factor, F, and noise figure, NF
5.3.3. Calculating SNR and input noise voltage from NF
5.3.4. Two source noise model
5.4.
Low-noise design: noise matching
5.4.1. Maximization of SNR
5.4.2.
5.4.3.
5.4.4.
5.4.5.
5.4.6.
Noise in diodes
Noise in bipolar transistors
Noise in FETs
Noise in differential and feedback amplifiers
Noise measurements
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MEASUREMENT THEORY FUNDAMENTALS. Contents
5.5.
Interference: environment influence
5.5.1.
5.5.2.
5.5.3.
5.5.4.
5.5.5.
5.5.6.
6.
Thermoelectricity
Piezoelectricity
Leakage currents
Cabling: capacitive injection of interference
Cabling: inductive injection of interference
Grounding: injection of interference by improper grounding
5.5. Observer influence: matching
Measurement system characteristics
6.1. Sensitivity
6.2. Sensitivity threshold
6.3. Signal shape sensitivity
6.4. Resolution
6.5. Non-linearity
6.6. System response
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MEASUREMENT THEORY FUNDAMENTALS. Contents
7. Measurement devices in electrical engineering
7.1. Input transducers
7.1.1. Mechanoelectric transducers
7.1.2. Thermoelectric transducers
7.1.3. Magnetoelectric transducers
7.2.
Signal conditioning
7.2.1.
7.2.2.
7.2.3.
7.2.4.
7.2.5.
7.2.6.
Attenuators
Compensator network
Measurement bridges
Instrumentation amplifiers
Non-linear signal conditioning
Digital-to-analog conversion
8. Electronic measurement systems
8.1. Frequency measurement
8.2. Phase meters
8.3. Digital voltmeters
8.4. Oscilloscopes
8.5. Data acquisition systems
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1. BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement
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1. BASIC PRINCIPLES OF MEASUREMENTS
1.1. Definition of measurement
Measurement is the acquisition of information about
a state or phenomenon (object of measurement)
in the world around us.
This means that a measurement must be descriptive
with regard to that state or object we are measuring: there
must be a relationship between the object of measurement
and the measurement result.
The descriptiveness is necessary but not sufficient aspect
of measurement: when one reads a book, one gathers
information, but does not perform a measurement.
Reference: [1]
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement
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A second aspect of measurement is that it must be selective:
it may only provide information about what we wish to measure
(the measurand) and not about any other of the many states or
phenomena around us.
This aspect too is a necessary but not sufficient aspect of
measurement. Admiring a painting inside an otherwise empty
room will provide information about only the painting, but does
not constitute a measurement.
A third and sufficient aspect of measurement is that it must be
objective. The outcome of measurement must be independent
of an arbitrary observer.
Reference: [1]
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement
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In accordance with the three above aspects: descriptiveness,
selectivity, and objectiveness, a measurement can be described
as the mapping of elements from an empirical source set
onto elements of an abstract image set
with the help of a particular transformation (measurement
model).
Image space
Empirical space
Transformation
si
States,
phenomena
Source set S
‫מרחב אמפירי‬
Abstract,
well-defined
symbols
ii
Image set I
‫מרחב אבסטרקטי‬
Source set and image set are isomorphic if the transformation
does copy the source set structure (relationship between the
elements).
Reference: [1]
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement
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Example: Measurement as mapping
Empirical space
Image space
Transformation
State (phenomenon):
Abstract symbol
Static magnetic field
B= f (R, w, V )
R
w
Measurement model
V
Instrumentation
‫מרחב אמפירי‬
‫מרחב אבסטרקטי‬
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.2. Definition of instrumentation
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1.2. Definition of instrumentation
In order to guarantee the objectivity of a measurement, we
must use artifacts (tools or instruments). The task of these
instruments is to convert the state or phenomenon into a
different state or phenomenon that cannot be misinterpreted by
an observer.
The field of designing measurement instruments and systems
is called instrumentation.
Instrumentation systems must guarantee the required
descriptiveness, the selectivity, and the objectivity of the
measurement.
Reference: [1]
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.3. Why measuring?
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1.3. Why measuring?
Let us define ‘pure’ science as science that has sole purpose
of describing the world around us and therefore is responsible
for our perception of the world.
In ‘pure’ science, we can form a better, more coherent, and
objective picture of the world, based on the information
measurement provides. In other words, the information allows
us to create models of (parts of) the world and formulate laws
and theorems.
We must then determine (again) by measuring whether this
models, hypotheses, theorems, and laws are a valid
representation of the world. This is done by performing
tests (measurements) to compare the theory with reality.
Reference: [1]
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.3. Why measuring?
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We consider ‘applied’ science as science intended to change
the world: it uses the methods, laws, and theorems of ‘pure’
science to modify the world around us.
In this context, the purpose of measurements is to regulate,
control, or alter the surrounding world, directly or indirectly.
The results of this regulating control can then be tested and
compared to the desired results and any further corrections
can be made.
Even a relatively simple measurement such as checking the
tire pressure can be described in the above terms:
1) a hypothesis: we fear that the tire pressure is abnormal;
2) perform measurement;
3) alter the pressure if it
was abnormal.
Reference: [1]
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.3. Why measuring?
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Illustration: Measurement in pure and applied science
REAL WORLD
IMAGE
empirical states
phenomena, etc.
abstract numbers
symbols, labels, etc.
Measurement
SCIENCE
Applied
Pure
(processing, interpretation)
measurement results
Control/change
Verification (measurement)
Control/change
Hypotheses
laws
theories
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.4. Types of measurements
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1.4. Types of measurements
To represent a state, we would like our measurements to have
some of the following characteristics.
Distinctiveness: A  B, A  B.
Ordering in magnitude: A < B, A  B, A > B.
Equal/unequal intervals: A-B < C-D, A-B  C-D,
A-B > C-D .
Ratio: A  k B (absolute zero is required).
Absolute magnitude: A  ka REF, B  kb REF
(absolute reference or unit is required).
These five characteristics are used to determine the five types
(levels) of measurements.
Reference: [1]
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.4. Types of measurements
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Illustration: Levels of measurements (S. S. Stevens, 1946)
ABSOLUTE Abs. unit
RATIO
Abs. zero
INTERVAL Distance is meaningful
ORDINAL States can be ordered
NOMINAL States are only named
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
1.5. Scaling of measurement results
A scale is an organized set of measurements, all of which
measure one property.
The types of scales reflect the types of measurements:
1. nominal scale,
2. ordinal scale,
3. interval scale,
4. ratio scale,
5. absolute scale.
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1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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A scale is not always unique; it can be changed without loss
of isomorphism.
Image space
Empirical space
Transformation
si
States,
phenomena
Source set S
‫מרחב אמפירי‬
Abstract,
well-defined
symbols
ii
Image set I
‫מרחב אבסטרקטי‬
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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A scale is not always unique; it can be changed without loss
of isomorphism.
Image space
Empirical space
Transformation
si
States,
phenomena
Source set S
‫מרחב אמפירי‬
Abstract,
well-defined
symbols
iii i
Image set I
‫מרחב אבסטרקטי‬
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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1. Nominal scale
State  orthogonality
Image1
Examples:
numbering of
football
1
1
0
0
players,
detection
and alarm
systems,
etc.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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1. Nominal scale
State  orthogonality
Image2=(Image1+1)p
Examples:
numbering of
football
1
2p
1
2p
0
p
0
p
players,
detection or
alarm
systems,
etc.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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1. Nominal scale
State  orthogonality
Image3=Cos(Image2)
Examples:
numbering of
football
1
2p
1
2p
-p1
-p1
players,
detection or
alarm
systems,
etc.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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1. Nominal scale
State  orthogonality
Image4=Image32p
Examples:
numbering of
football
2p
1
2p
1
-2p
-1
-2p
-1
players,
detection or
alarm
systems,
etc.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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1. Nominal scale
State  orthogonality
Image5=Cos(Image4)
Examples:
numbering of
football
players,
2p
1
2p
1
The structure is lost!
detection or
alarm
-2p
1
systems,
etc.
Any one-to-one transformation can be used to
change the scale.
-2p
1
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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2. Ordinal scale
State  order
Examples:
IQ test,
etc.
Image1
A 1
B 1
A 2
B 1
A 2
B 1
A 1
B 2
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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2. Ordinal scale
State  order
Examples:
IQ test,
etc.
Image2  Image12
A 1
B 1
A 4
2
B 1
A 4
2
B 1
A 1
B 4
2
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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2. Ordinal scale
State  order
Examples:
IQ test,
competition
results,
Image3  -Image2
AA-1
1
BB-1
1
AA-4
4
BB-1
1
The structure is lost!
etc.
AA-4
4
BB-1
1
AA-1
1
BB-4
4
Any monotonically increasing transformation, either linear or
nonlinear, can be used to change the scale.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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Interval scale
State  interval
Examples:
time scales,
temperature
scales, etc.,
Image1
A 4
B4
A-B 0
A 5
B4
A-B 1
A 8
B4
A-B 4
A 6
B7
A-B 1
where the
origin or zero
is not fixed
(floating).
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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Interval scale
State  interval
Examples:
time scales,
temperature
scales, etc.,
Image2 10Image1+2
A  442
B  442
A-B 0
A  552
B  442
A-B 110
A  882
B  442
A-B 440
A  662
B  772
A-B 110
where the
origin or zero
is not fixed
(floating).
Any increasing linear transformation can be used to
change the scale.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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4. Ratio scale
State  ratio
Examples:
measurement
of any physical
quantities
Image1
A 4
B4
A/B  1
A 5
B4
A/B  5/4
A 8
B4
A/B  2
A 6
B7
A/B  6/7
having fixed
(absolute)
origin.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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4. Ratio scale
State  ratio
Examples:
measurement
of any physical
quantities
Image2 10Image1
A  440
B  440
A/B  1
A  550
B  440
A/B  5/4
A  880
B  440
A/B  2
A  660
B  770
A/B  6/7
having fixed
(absolute)
origin.
The only transformation that can be used to change the
scale is the multiplication by any positive real number.
1. BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results
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5. Absolute scale
State  absolute value
Examples:
measurement
Image
Ref.
Ref.
A 1
A  5/4
Ref.
Ref.
A 2
A  3/2
of any physical
quantities by
comparison
against an
absolute unit
(reference).
No transformation can be used to change the scale
Next lecture
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