_1. Types of measurements_37___

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MEASUREMENT THEORY FUNDAMENTALS, 361-1-3151

MEASUREMENT THEORY

FUNDAMENTALS

361-1-3151

Eugene Paperno

http://www.ee.bgu.ac.il/~paperno/

Eugene Paperno, 2006


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MEASUREMENT THEORY FUNDAMENTALS

Measurement theory is a branch of applied mathematics that is useful

in measurement and data analysis. The fundamental idea ofmeasurement theory is that measurements are not the same as the

attribute being measured. Hence, if you want to draw conclusions

about the attribute you must take into account the nature of the

correspondence between the attribute and the measurements.

Measurement theory shows that strong assumptions are required for

certain statistics to provide meaningful information about reality.

Measurement theory encourages people to think about the meaning of

their data. It encourages critical assessment of the assumptions

behind the analysis. It encourages responsible real-world data

analysis.

Mathematical statistics is concerned with the connection betweeninference and data. Measurement theory is concerned with the

connection between data and reality. Both statistical theory and

measurement theory are necessary to make inferences about reality.

Reference: http://www.measurementdevices.com/mtheory.html


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3MEASUREMENT THEORY FUNDAMENTALS. Grading policy

GRADING POLICY

20% participation in lectures

30% home exercises

50% presentation


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4MEASUREMENT THEORY FUNDAMENTALS. Grading policy

HOMEWORKBuild in LabView the following virtual instruments (VI):

1. Lock-in amplifierSR830www.thinksrs.com/mult/SR810830m.htm

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


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MEASUREMENT THEORY FUNDAMENTALS. Recommended literature

Recommended literature

[1] K. B. Klaassen, Electronic measurement and instrumentation,Cambridge University Press, 1996.

[2] H. O. Ott, Noise reduction techniques in electronic systems,

second edition, John Wiley & Sons, 1988.

[3] P. Horowitz and W. Hill, The art of electronics, Second Edition,

Cambridge University Press, 1989.

[4] R. B. Northrop, Introduction to instrumentation and measurements,

second edition, CRC Press, 2005.

[5] D. A. Jones and K. Martin,Analog integrated circuit design,

John Wiley & Sons, 1997.

[6] A. B. Carlson, Communication systems: an introduction to signals and

noise in electrical communication, McGraw-Hill, 2004.

[7] Leon Cohen, The history of noise: on the 100th anniversary of its birth,

IEEE Signal Processing magazine, vol. 20, 2005.

[8] National Instruments, Inc. web site: www.ni.com

[9] IEEE Transactions on Instrumentation and Measurements.


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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 ofmeasurement. (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 measurements1.1. 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

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 ( , )

4.2.1. Systematic errors

4.2.1. Random errors



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5. Sources of errors

5.1. Influencing the measurement object: matching

5.4.1. Anenergetic matching5.4.2. 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/fnoise

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. CalculatingSNR and input noise voltage from NF

5.3.4. Two source noise model

5.4. Low-noise design: noise matching

5.4.1. Maximization ofSNR

5.4.2. 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



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5.5. Interference: environment influence

5.5.1. 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



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7. Measurement devices in electrical engineering

7.1. Input transducers7.1.1. Mechanoelectric transducers

7.1.2. Thermoelectric transducers

7.1.3. Magnetoelectric transducers

7.2. Signal conditioning

7.2.1. Attenuators

7.2.2. Compensator network7.2.3. 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



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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 aspectof 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]


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This aspect too is a necessary but not sufficient aspect of

measurement. Admiring a painting inside an otherwise emptyroom will provide information about onlythe 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.


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]


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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 setS

si

States,phenomena


Source set and image set are isomorphic if the transformation

does copythe source set structure (relationship between the

elements).

Reference: [1]

onto elements of an abstract image set

Image setI

ii

Transformation


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Image space

Example: Measurement as mapping

Empirical space

State (phenomenon):

Static magnetic field

V

R[

Instrumentation

Abstract symbol

Transformation

B=f(R, [ V)

Measurement model



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The field of designing measurement instruments and systems

is called instrumentation.

Instrumentation systems must guarantee the required

descriptiveness, the selectivity, and the objectivityof 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]


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

1.3. Why measuring?

Let us define pure science as science that has sole purpose

ofdescribingthe 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]


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2) perform measurement;

3) alter the pressure if it

was abnormal.

We considerapplied 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 correctionscan 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]


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REAL WORLDempirical states

phenomena, etc.

IMAGEabstract numbers

symbols, labels, etc.

SCIENCE

(processing, interpretation)

measurement results

PureApplied


Measurement

Verification (measurement)Control/change

Control/change

Hypotheseslawstheories

Illustration: Measurement in pure and applied science


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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:AB CD,AB !CD

AB "CD .

Ratio:A ! kB (absolute zero is required).

Absolute magnitude:A !kaREF, B!kbREF(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]


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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)


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1. nominalscale,

2. ordinalscale,

3. intervalscale,

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.


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A scale is notalways unique; it can be changed without loss

of isomorphism.

Image space


Empirical space

Source setS

si

States,phenomena

ii

Transformation

Image setI


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A scale is notalways unique; it can be changed without loss

of isomorphism.

Image space


Empirical space

Source setS

si

States,phenomena

Image setI

ii

ii

Transformation


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Image1

1 1

0 0

State | orthogonality

1. Nominal scale


Examples:

numbering of

football

players,

detection

and alarm

systems,

etc.


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1 1

0 0

T T

T T

Image2=(Image1+1)vTState | orthogonality

1. Nominal scale


Examples:

numbering of

football

players,

detection or

alarm

systems,

etc.


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T T

T T

Image3=Cos(Image

2)

1 1

1 1


1. Nominal scale


Examples:

numbering of

football

players,

detection or

alarm

systems,

etc.


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1 1

1 1

Image4=Image

3v2T

2T 2T

2T 2T


1. Nominal scale


Examples:

numbering of

football

players,

detection or

alarm

systems,

etc.


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2T 2T

2T 2T

Image5=Cos(Image

4)

1 1

1 1


1. Nominal scale


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.


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A !1&!1

A !2&!1

A !2&!1

A !1&!2

Image1


State | order

2. Ordinal scale

Examples:

IQ test,

etc.


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A !1&!1

A !2&!1

A !2&!1

A !1&!2

A !1&!1

A !4&!1

A !4&!1

A !1&!4


State | order

2. Ordinal scale

Image2! Image1

2

Examples:

IQ test,

etc.


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A !1&!1

A !4&!1

A !4&!1

A !1&!4

A !1&!1

A !4&!1

A !4&!1

A !1&!4


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.


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Image1

A ! 4&! 4

A&!0

A ! 6&! 7

A&!1

A ! 8&! 4

A&!4

A ! 5&! 4

A&!1


State | interval

Interval scale

Examples:

time scales,

temperature

scales, etc.,

where the

origin or zero

is not fixed

(floating).


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A ! 4&! 4

A&!0

A ! 6&! 7

A&!1

A ! 8&! 4

A&!4

A ! 5&! 4

A&!1


State | interval

Interval scale

Examples:

time scales,

temperature

scales, etc.,

where the

origin or zero

is not fixed

(floating).

Image2! 10vImage12

A ! 42&! 42

A&!0

A ! 62&! 72

A&!10

A ! 82&! 42

A&!40

A ! 52&! 42

A&!10

Any increasing linear transformation can be used to

change the scale.


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Image1

A ! 4&! 4

A&! 1

A ! 6&! 7

A&! 6/7

A ! 8&! 4

A&! 2

A ! 5&! 4

A&! 5/4


State | ratio

4. Ratio scale

Examples:

measurement

of any physical

quantities

having fixed

(absolute)

origin.


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A ! 4&! 4

A&! 1

A ! 6&! 7

A&! 6/7

A ! 8&! 4

A&! 2

A ! 5&! 4

A&! 5/4


State | ratio

4. Ratio scale

Image2! 10vImage1

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 ! 40&! 40

A&! 1

A ! 60&! 70

A&! 6/7

A ! 80&! 40

A&! 2

A ! 50&! 40

A&! 5/4


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Image

A ! 1

A ! 3/2A ! 2

A ! 5/4


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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_1. Types of measurements_37___