MEASUREMENT THEORY FUNDAMENTALS, 361-1-3151

MEASUREMENT THEORY FUNDAMENTALS

361-1-3151

Eugene Papernohttp://www.ee.bgu.ac.il/~paperno/

Eugene Paperno, 2006 ©

3MEASUREMENT THEORY FUNDAMENTALS. Grading policy

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4MEASUREMENT THEORY FUNDAMENTALS. Grading policy

HOMEWORKBuild in LabView the following virtual instruments (VI):

1. Lock-in amplifier SR830www.thinksrs.com/mult/SR810830m.htm

2. Spectrum analyzer SR785http://www.thinksrs.com/mult/SR785m.htm

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

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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, standards2.2.1. 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

MEASUREMENT THEORY FUNDAMENTALS. Contents

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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 (תרגום, העברת שגאיות )

4.2.1. Systematic errors

4.2.1. Random errors

MEASUREMENT THEORY FUNDAMENTALS. Contents

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5. Sources of errors

5.1. Influencing the measurement object: matching

5.4.1. Anenergetic matching

5.4.2. Energic matching

5.4.3. Non-reflective matching

5.4.4. When to match and when not?

5.2. 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 SNR5.4.2. Noise in diodes

5.4.3. Noise in bipolar transistors

5.4.4. Noise in FETs

5.4.5. Noise in differential and feedback amplifiers

5.4.6. Noise measurements

MEASUREMENT THEORY FUNDAMENTALS. Contents

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5.5. Interference: environment influence5.5.1. Thermoelectricity

5.5.2. Piezoelectricity

5.5.3. Leakage currents

5.5.4. Cabling: capacitive injection of interference

5.5.5. Cabling: inductive injection of interference

5.5.6. Grounding: injection of interference by improper grounding

5.5. Observer influence: matching

6. 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

MEASUREMENT THEORY FUNDAMENTALS. Contents

11

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. Attenuators

7.2.2. Compensator network

7.2.3. Measurement bridges

7.2.4. Instrumentation amplifiers

7.2.5. Non-linear signal conditioning

7.2.6. 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

MEASUREMENT THEORY FUNDAMENTALS. Contents

12

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.

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement

Reference: [1]

13

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.

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement

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.

Reference: [1]

14

Image space

Abstract ,well-definedsymbols

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

with the help of a particular transformation (measurement

model).

Empirical space

Source set S

si

States ,phenomena

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement

Source set and image set are isomorphic if the transformation

does copy the source set structure (relationship between the

elements).

Reference: [1]

onto elements of an abstract image set

אבסטרקטי מרחב

Image set I

ii

Transformation

מרחב אמפירי

15

Image space

Example: Measurement as mapping

Empirical space

State (phenomenon):

Static magnetic field

VR

Instrumentation

Abstract symbol

Transformation

B= f (R, V )

Measurement model

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.1. Definition of measurement

מרחב אמפירי אבסטרקטי מרחב

16

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.

1.2. Definition of instrumentation

1 .BASIC PRINCIPLES OF MEASUREMENTS. 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.

Reference: [1]

171 .BASIC PRINCIPLES OF MEASUREMENTS. 1.3. Why measuring?

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]

18

2) perform measurement;

3) alter the pressure if it

was abnormal.

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.

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.3. Why measuring?

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;

Reference: [1]

19

REAL WORLDempirical statesphenomena, etc.

IMAGEabstract numbers

symbols, labels, etc.

SCIENCE

)processing, interpretation(

measurement results

PureApplied

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.3. Why measuring?

Measurement

Verification (measurement)Control/change

Control/change

Hypotheses laws

theories

Illustration: Measurement in pure and applied science

20

These five characteristics are used to determine the five types

(levels) of measurements.

Distinctiveness: A B, A B.

Ordering in magnitude: A B, A B, A B.

Equal/unequal intervals:

ABCD,ABCDABCD.

Ratio: A kB(absolute zero is required).

Absolute magnitude: A ka REF, B kb REF

(absolute reference or unit is required).

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.4. Types of measurements

1.4. Types of measurements

To represent a state, we would like our measurements to have

some of the following characteristics.

Reference: [1]

21

States are only namedNOMINAL

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.4. Types of measurements

States can be orderedORDINAL

Distance is meaningfulINTERVAL

Abs. zeroRATIO

Abs. unitABSOLUTE

Illustration: Levels of measurements (S. S. Stevens, 1946)

22

1. nominal scale,

2. ordinal scale,

3. interval scale,

4. ratio scale,

5. absolute scale.

The types of scales reflect the types of measurements:

1.5. Scaling of measurement results

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

A scale is an organized set of measurements, all of which

measure one property.

231 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

A scale is not always unique; it can be changed without loss

of isomorphism.

Image space

Abstract ,well-definedsymbols

Empirical space

Source set S

si

States ,phenomena

אבסטרקטי מרחב מרחב אמפירי

ii

Transformation

Image set I

241 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

A scale is not always unique; it can be changed without loss

of isomorphism.

Image space

Abstract ,well-definedsymbols

Empirical space

Source set S

si

States ,phenomena

אבסטרקטי מרחב

Image set I

מרחב אמפירי

iiii

Transformation

25

Image1

1 1

0 0

State orthogonality

1. Nominal scale

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

Examples:

numbering of

football

players,

detection

and alarm

systems,

etc.

26

1 1

0 0

Image2=(Image1+1)State orthogonality

1. Nominal scale

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

Examples:

numbering of

football

players,

detection or

alarm

systems,

etc.

27

Image3=Cos(Image2)

1 1

1 1

State orthogonality

1. Nominal scale

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

Examples:

numbering of

football

players,

detection or

alarm

systems,

etc.

28

1 1

1 1

Image4=Image32

2 2

2 2

State orthogonality

1. Nominal scale

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

Examples:

numbering of

football

players,

detection or

alarm

systems,

etc.

29

2 2

2 2

Image5=Cos(Image4)

1 1

1 1

State orthogonality

1. Nominal scale

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

Examples:

numbering of

football

players,

detection or

alarm

systems,

etc.

The structure is lost!

Any one-to-one transformation can be used to

change the scale.

30

A 11

A 21

A 21

A 12

Image1

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State order

2. Ordinal scale

Examples:

IQ test,

etc.

31

A 11

A 21

A 21

A 12

A 11

A 41

A 41

A 14

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State order

2. Ordinal scale

Image2 Image12

Examples:

IQ test,

etc.

32

A 11

A 41

A 41

A 14

A 11

A 41

A 41

A 14

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State order

2. Ordinal scale

Image3 Image2

Examples:

IQ test,

competition

results,

etc.The structure is lost!

Any monotonically increasing transformation, either linear or

nonlinear, can be used to change the scale.

33

Image1

A 44

A0

A 67

A1

A 84

A4

A 54

A1

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State interval

Interval scale

Examples:

time scales,

temperature

scales, etc.,

where the

origin or zero

is not fixed

(floating).

34

A 44

A0

A 67

A1

A 84

A4

A 54

A1

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State interval

Interval scale

Examples:

time scales,

temperature

scales, etc.,

where the

origin or zero

is not fixed

(floating).

Image210Image12

A 4242

A0

A 6272

A10

A 8242

A40

A 5242

A10

Any increasing linear transformation can be used to change

the scale.

35

Image1

A 44

A1

A 67

A6/7

A 84

A2

A 54

A5/4

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State ratio

4. Ratio scale

Examples:

measurement

of any physical

quantities

having fixed

(absolute)

origin.

36

A 44

A1

A 67

A6/7

A 84

A2

A 54

A5/4

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State ratio

4. Ratio scale

Image210Image1

Examples:

measurement

of any physical

quantities

having fixed

(absolute)

origin.

The only transformation that can be used to change the scale

is the multiplication by any positive real number.

A 4040

A1

A 6070

A6/7

A 8040

A2

A 5040

A5/4

37

Image

A 1

A 3/2

A 2

A 5/4

1 .BASIC PRINCIPLES OF MEASUREMENTS. 1.5. Scaling of measurement results

State absolute value

5. Absolute scale

Examples:

measurement

of any physical

quantities by

comparison

against an

absolute unit

(reference).

Ref . Ref .

Ref . Ref .

No transformation can be used to change the scale

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