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1.1
INTRODUCTION
The physical sciences are called exact sciences, for they give us accurate knowledge.
This exactness or accuracy comes from Measurements. The study of physics
involved measurements of various types at every stage. So Physics is often called
the Science of Measurements.
Determining a quantity or variable using a physical means is called the
measurement and the means by which the quantity is determined is called
Measuring Instruments.
Thus, an instrument may be defined as a device for determining the value
or magnitude of a quantity or variable. An instrument enables a person to
determine the value of an unknown quantity.
Some of terms are used in the measurement work, which are defined below.
1. True Value: It is the average of an infinite number of measured values.
2. Error: It is the difference between the measured value and the true
value.
3. Index Scale: It is the set of marks or divisions.
4. Index Number: It is the number of divisions moved.
The essential requirements of a measuring instrument are:
(i) It should not alter the circuit conditions,
(ii) Power consumed by it should be small.
1.2
PERFORMANCE CHARACTERISTICS OF MEASURING
INSTRUMENTS
The characteristics of the instrument and the measurement system play an important
role in determining the performance of the measurement system. The characteristics
of the instruments can be divided into two distinct categories:
1. Static characteristics, and
2. Dynamic characteristics.
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FUNDAMENTALS OF ELECTRICAL AND ELECTRONIC MEASUREMENTS
1. Static Characteristics: The main static characteristics are
(i) Accuracy: It refers to the closeness of an instrument reading to the
true value of the quantity or variable under measurement. This term
describes the algebraic difference between the indicated value and the
true value or theoretical value of the measurand. In practice, accuracy
is usually expressed as a percentage of full scale output or percent of
reading or digits for digital readouts.
(ii) Precision: It refers to the measure of the reproducibility or repeatability
of the measurement. It is a measure of the degree to which successive
measurements differ from one another.
If a number of readings of a voltmeter are taken, then the expected
value of 1.0 volt is not obtained on every occasion. A range of values
such as 0.99, 1.01, 1.00, 1.02, 0.98, etc. are obtained about the expected
value. The effect is termed as a lack of repeatability in the instrument.
(iii) Sensitivity: It is the ratio of the magnitude of the output signal or
response to the input signal. It is expressed in mm/mA, or counts/volt.
In other words, it describes the transfer ratio of the output to the input.
(iv) Resolution: It indicates the smallest change in measurable input to
which the instrument will respond.
(v) Linearity: It is the degree to which variations in the output of an
instrument follow input variations. It is the uniform displacement. It
reflects that the output, in some way, is proportional to the input.
The system is said to be linear when the corresponding values of input
and output lie on a straight line.
(vi) Drift: It indicates a change of baseline, i.e., change of the output
when input is zero, or of sensitivity with time, temperature etc.
(vii) Noise: This is an undesired signal at the output due to either internal
source or to interference.
(viii) Hysteresis: It is the change of output with some value of input with a
different history of input variation.
A hysteresis is observed when the input/output characteristics for an
instrument are different when input increases than when the input
decreases. It results when some of the energy applied to increase inputs
is not recovered when the input is decreased.
The phenomenon, which takes place during magnetizing and
demagnetizing a magnetic material, is called hysteresis.
(ix) Span: It is the total operating range of the instrument.
(x) Threshold or Dead Time: The threshold of the instrument is the
smallest change in the measurand which will result in a measurable
change in output of the instrument.
MEASUREMENT AND ERROR
In other words, Threshold or Dead Time can be defined as the time
required by an instrument or the measurement system to respond to a
change in the measurand.
(xi) Saturation: The output is generally proportional to the input. A point
is reached when the output of the instrument will no longer increase
for the increased input, giving rise to non-linear relationship. The
region in which the output does not change with increase in input is
called Saturation Region.
Saturation Region may also be referred to as Dead Zone. In other
words, the Saturation Region or Dead Zone may be defined as the
largest change in the input for which there is no output of the
instrument.
2. Dynamic Characteristics: It describes the ways in which an instrument or
measurement system responds to sudden changes to the input.
In general, the dynamic response of the measurement system is expressed in
the form of a differential equation.
For any dynamic system, the order of the differential equation which
describes the system is called the Order of the System.
(i) Zero-order System: It has an ideal dynamic performance, because
the output is proportional to the input for all frequencies and there is
no amplitude or phase distortion. A linear potentiometer is an example
of a zero-order element.
(ii) First-order System: A first-order instrument or system is characterized
by a linear differential equation. The temperature transducer is an
example of first-order measuring devices, since this is characterized
by a single parameter, i.e., time constant, T.
The differential equation for the first-order system is given by
x(t) = y + T ⋅
dy
dx
where,
x = Input
x(t) = Time function of the input
y = Output.
(iii) Second-order System: This type of system is characterized by the
second-order differential equation. The example of the second-order
system is the mass-spring system of the measurement of the force.
The second-order instrument or system is defined by the equation
a2
d2 y
dt
2
+ a1
dy
+ a0 y = b0x
dt
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FUNDAMENTALS OF ELECTRICAL AND ELECTRONIC MEASUREMENTS
The solution of the Equation (1.10) is given as
1 d2 y
ω2n
where,
dt
2
+
2ξ dy
+ y = Kx
ωn dt
K=
wn =
ξ =
b0
= Static frequency
a0
a0
= Natural frequency
a1
a1
= Damping ratio
2 a0 a2
Thus, the second-order system is characterized by the two parameters —
the natural frequency, fn or the angular frequency, wn (= 2p fn), and the
damping ratio, x.
In the second-order system, the natural frequency is the index of speed of
response, whereas the damping ratio is a measure of the system stability.
The second-order instrument is more common than first-order types.
The dynamic characteristics of an instrument or the measurement system
are as follows:
(i) Respond Time,
(ii) Fidelity,
(iii) Measuring lag, and
(iv) Dynamic error.
(i) Respond Time: It is an important parameter to describe the dynamic response
of an instrument. It characterizes the instrument to a step change in the
measurand (input). It includes rise time, delay time and time constant.
(ii) Fidelity: It is defined as the degree of the measurement system. It indicates
changes in the measurand without any dynamic error.
(iii) Measuring Lag: It is the retardation or delay in the response of a
measurement system to changes in the measurand.
(iv) Dynamic Error: It is the difference between the true value of the quantity
under measurement changing with time and the measured value of the
quantity. It also referred to as Measurement error.
Other Characteristics
There are other characteristics which determine the performance characteristics
and the choice of the instruments. These are:
(i) The Input and the output impedance,
(ii) Overload range,
(iii) Recovery time after overload,
(iv) Excitation voltage,
(v) Reliability,
(vi) Size and weight.
MEASUREMENT AND ERROR
The dynamic behaviour of the measurement system is studied in the following
two domains:
(i) Time Domain, and (ii) Frequency Domain.
In the Time Domain analysis time is used in the measurement system as an
independent variable. The dynamic behaviour of the system is studied as a function
of time with the different input signals applied to the measurement system. The
different input signals are step, ramp, parabolic and impulse.
The output of the measurement system is expected to follow the input very
closely as far as possible. Hence, it is necessary to compare the output and the
input variables at all times.
In the Frequency Domain analysis, the frequency is used as an independent
variable. The dynamic behaviour of the system is studied with the sinusoidal
input signal applied to the measurement system. The resulting output of a linear
system is sinusoidal with different from the input waveform in amplitude and
phase in the steady state condition.
1.3
ACCURACY AND PRECISION
Accuracy refers to the degree of closeness to the true value of the quantity under
measurement.
Precision refers to the degree of agreement within a group of measurements
or instruments.
Two identical voltmeters of the same make and same model can give the
reading to the same precision. But, if the value of its component of on voltmeter
changes considerably, then its readings may vary by a large amount. Therefore,
the accuracy of the two meters may be quite different. And to determine which
meter gives the error in reading, a comparison measurement should be made.
Precision consists of two characteristics: closeness and the number of
significant figures to which a measurement may be made. For example, a resistor
of the true value of 1584572 W is measured by an ohm-meter which consistently
and repeatedly indicates 1.6 MW form the scale of the meter. This is the estimated
value from the scale reading consistently yield a value of 1.6 MW. This is as close
to the true value as the scale can be read by the observer by estimation. Although
there are no deviations from the observed value, the error created by the limitation
of the scale reading is a ‘precision error’.
The above example illustrates that the accuracy is a necessary condition for
precision, but not sufficient because of the lack of significant figures obtained.
Similarly, precision is a necessary condition for accuracy, but not sufficient.
Often the beginner, as observer, accepts the reading of the instrument at face
value. Good measurement technique demands continuous skepticism as to the
accuracy of the results.
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FUNDAMENTALS OF ELECTRICAL AND ELECTRONIC MEASUREMENTS
1.4
SIGNIFICANT FIGURES
An indication of the precision of the measurement obtained from the number of
significant figures in which the result is expressed.
Significant figures convey actual information regarding the magnitude and
the measurement precision of a quantity. The more significant figure, the greater
is the precision of measurement of quantity.
For example, a resistor is specified as having a resistance of 75 W. It means
its resistance should be closer to 74 W or 76 W. And if its value is described as
75.0, it means that its resistance is closer to 75.0 W than 74.9 W or 75.1 W. In
75 W there are two significant figures and in 75.0 W there are three. The value
75.0 W, with more significant figures, expresses a measurement of greater precision
than 75 W.
It is usual practice to record a measurement with all the digits nearest to the
true value. For example, the voltage may be read as 117.1 V. This indicates that
the voltage, read by the observer to best estimation, is closer to 117.1 V that to
117.0 V or 117.2 V. This result indicates the range of possible error and the
voltage may be expressed as 171.1 ± 0.05 V, indicating that the value of the
voltage lies between 117.05 and 117.15 V.
In a case when a number of independent measurements are taken to obtain
the best possible value closest to the true value, the result is usually expressed as
the arithmetic mean of all the readings, with the range of possible error as the
largest deviation from that mean. This can be illustrated by the following example:
Example 1.1
A set of independent voltage measurements taken by four observers was recorded
as 117.01 V, 117.12 V, 117.07 V and 117.04 V. Calculate (a) the average voltage;
(b) the range of error.
Solution:
(a)
(b)
But also
E1 + E2 + E3 + E4
N
117.01 + 117.12 + 117.07 + 117.04
= 117.06 V.
=
4
Eav =
Range = Emax – Eav = 117.12 – 117.06 = 0.06 V
Range = Eav – Emax = 117.06 – 117.01 = 0.05 V
0.06 + 0.05
= ± 0.055 = ± 0.06 V
\ The average of error =
2
When two or more measurements with different degrees of accuracy are
added, the result is only as accurate as the least accurate measurement. For example,
the two resistances R1 = 21.7 W (three significant figures) and R2 = 8.624 W (four
significant figures) are added in series. The total resistance will be RT = 30.324 W
(five significant figures) = 30.3 W.
MEASUREMENT AND ERROR
In the result, the last three digits of the sum are doubtful figures. There is no
value in retaining the last two digits because one of the resistances is accurate
only to three significant figures or tenths of an ohm. Therefore, the result should
also be reduced to three significant figures or the nearest tenth, i.e. 30.3 W.
The number of significant figures in multiplication may increase rapidly,
but again only the appropriate figures are retained in the result. For example, a
current of 5.28 A is recorded in a resistance of 47.78 W. The voltage drop across
the resistor is
V = I R = (5.28) × (47.78) = 252.2784 V.
The result can be recorded only to a maximum of three significant figures
as 252 V, because three significant figures are involved in the multiplication.
Addition of figures with a range of doubt is illustrated in the following
example:
Example 1.2
Add 736 ± 5 to 538 ± 3
Solution:
N1 = 736 ± 5 (= ± 0.679%)
N2 = 538 ± 3 (= ± 0.557%)
Sum = 1274 ± 8 (= ± 0.627%)
It is to be noted in the above example that the doubtful parts are added,
since the ± sign means that one number may be high and the other low. The worst
possible combination of range of doubt should be taken in the answer. The
percentage doubt in the original figure N1 and N2 does not differ greatly from the
percentage doubt in the final result.
If the same two numbers are subtracted, as in the following example, there
is an interesting comparison between addition and subtraction with respect to the
range of doubt.
Example 1.3
Subtract 538 ± 3 from 736 ± 5 and express the range of doubt in the answer as a
percentage.
Solution:
N1 = 736 ± 5 (= ± 0.679%)
N2 = 538 ± 3 (= ± 0.557%)
Difference = 198 ± 8 (= ± 4.04%)
In the above example 1.3, the doubtful parts are added for the same reason
as in the example 1.2. Comparing the results of addition and subtraction of the
same numbers in both the two examples, precision of the results expressed in
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FUNDAMENTALS OF ELECTRICAL AND ELECTRONIC MEASUREMENTS
percentages, differs greatly. The result after subtraction shows a large increase in
percentage doubt compared to the percentage doubt after addition.
The percentage doubt increases even more when the difference between the
numbers is relatively small, as shown in the following example 1.3.
Example 1.4
Subtract 437 ± 4 from 462 ± 4 and express the range of doubt in the answer as a
percentage.
Solution:
N1 = 462 ± 4 (= ± 0.679%)
N2 = 437 ± 4 (= ± 0.557%)
Difference = 25 ± 8 (= ± 32%)
1.5
ERROR AND TYPES OF ERROR
Error is the difference between the true value and the measured value of a quantity
such as displacement, pressure, temperature, and the like.
No electronic component of instrument is perfectly accurate. All have some
error or inaccuracy. The measurements cannot be made with perfect accuracy. It
is important to find out how different errors have entered into the measurement
and what the accuracy is. A study of error is a first step in finding ways to reduce
them. Such study also determines the accuracy of the final result.
Error is inevitable in any measurement. Well-designed electronic
instrumentation systems limit the error to a value that is acceptable in terms of the
accuracies required in an engineering analysis or the control of a process.
1.5.1 Limiting or Guarantee Errors
The design, the materials used and the workmanship are the important factors for
the accuracy and precision of an instrument. In most of the instruments the accuracy
is guaranteed by the manufacturer for the quality of the instrument to be within a
certain percentage of full scale reading. The values of the circuit components,
like resistors, capacitors and inductors, are mentioned within a certain percentage
of the rated values specified by the manufacturer. The limits of these deviations
from the specified value are defined as Limiting errors.
The magnitude of a quantity Q having a specified value Q1 and a limiting
error ± dQ must have a value between the limits (Q1 – dQ) and (Q1 + dQ) or
Q = (Q1 ± dQ).
For example, the specified value of a resistor is 560 W with a limiting error
of ±10 W. So, the value of the resistor will be between the limits (100 ± 10) ohm.
An error of ± 2 ampere is negligible for the 1000 ampere of current measured,
but the same error is not tolerable for the measurement of current of 10 ampere.
MEASUREMENT AND ERROR
Thus, the quality of measurement is obtained by the relative error, i.e. the
ratio of limiting error dQ to the true value Q of the quantity under measurement.
The relative error, er is given by
er =
Absolute error δ Q ε0
=
=
True value
Q
Q
1.5.2 Known Errors
When the error of an instrument is known, the effect of this error can be computed
when combined with other errors. The signs of relative errors are given and must
be preserved in the calculation.
Types of Errors
There is no measurement with perfect accuracy, but it is important to find out
what accuracy actually is and how the different errors are present into the
measurement.
The aim of study of errors is to find out the ways to minimize them. Errors
may be introduced from different sources.
Errors are usually classified as follows:
1. Gross Errors,
2. Systematic Errors, and
3. Random Errors.
1. Gross Errors: These errors are largely due to human errors in reading of
instruments, incorrect adjustment and improper application of instruments,
and computational mistakes.
Complete elimination of such errors is probably impossible. The common
gross error is the improper use of an instrument for measurement. For
example, a well calibrated voltmeter can give an error in reading when
connected across a high resistance circuit. The same voltmeter will give
more accurate reading when connected in a low resistance circuit. It means
the voltmeter has a “loading effect” on the circuit, altering the characteristics
by the measurement process.
2. Systematic Errors: These errors are shortcomings of instruments, such as
defective or worn parts, and effects of the environment on the equipment or
the user.
This type of error is usually divided into two different categories:
(i) Instrumental Errors,
(ii) Environmental Errors.
(i) Instrumental Errors: These errors are defined as shortcomings of
the instrument.
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FUNDAMENTALS OF ELECTRICAL AND ELECTRONIC MEASUREMENTS
Instrumental errors are inherent in measuring instruments due to their
mechanical structure. For example, in the deflection type instrument
friction in bearings of various moving components, irregular spring
tension, stretching of the spring or reduction in tension due to improper
handling or overloading of the instrument may cause incorrect readings,
which will result in errors. Other instrumental errors may be due to
calibration; improper zero setting, variation in the air gap, etc.
Instrumental errors may be avoided by following methods:
(a) Selecting a suitable instrument for the particular measurement;
(b) Applying correction factors after determining the amount of
instrumental error;
(c) Calibrating the instrument against a standard instrument.
(ii) Environmental Errors: These errors are due to external conditions
surrounding the instruments which affect the measurements. The
surrounding conditions may be the changes in temperature, humidity,
barometric pressure, or of magnetic or electrostatic fields. Thus, a
change in ambient temperature at which the instrument is used causes
a change in the elastic properties of the spring in a moving-coil
mechanism and so causes an error in the reading of the instrument.
To reduce the effects of external conditions surrounding the instruments
the corrective measures are to be taken as follows:
(a) To provide air-conditioning,
(b) Certain components in the instrument should be completely closed
i.e., hermetically sealed, and
(c) To provide magnetic and electrostatic shields.
Systematic errors can also be subdivided into:
(a) Static errors, and
(b) Dynamic errors.
Static errors are caused by limitations of the measuring device or the
physical laws governing its behaviour. A static error is introduced in a
micrometer when excessive pressure is applied in twisting or rotating
the shaft.
Dynamic errors caused by the instrument do not respond fast enough
to follow the changes in a measured variable.
3. Random Errors: These errors are those errors which are due to unknown
causes and they occur even when all systematic errors have been taken care
of. This error cannot be corrected by any method of calibration or other
known methods of control. Few random errors usually occur in well-designed
experiments, but they become important in high-accuracy work.
MEASUREMENT AND ERROR
For example, a voltmeter with accurately calibrated is being used in ideal
environmental conditions to read voltage of an electric circuitry system. It
will be found that the readings vary slightly over the period of observation.
This variation cannot be corrected by any method of calibration or other
known method of control and it cannot be explained without minute
investigation.
The only way to offset these errors is by increasing the number of readings
and using statistical methods in order to obtain the best approximation of
the true value of the quantity under measurement.
1.5.3 Summary of Static Errors
1.6
STATISTICAL ANALYSIS
A statistical analysis of measurement data allows an analytical determination of
the uncertainty of the final test result. The result of a certain measurement method
may be predicted on the basis of sample data without having detailed information
on all the disturbing factors. To make statistical methods and interpretation
meaningful, a large number of measurements are usually required. And also,
systematic errors should be small compared with residual or random errors, because
statistical treatment of data cannot remove a fixed bias contained in all the
measurements.
1.6.1 Arithmetic Mean
The most probable value of a measured variable is the arithmetic mean of the
number of readings taken. The best approximation will be made when the number
of readings of the same quantity is very large. Theoretically, an infinite number
of readings would give the best result, although in practice, only a finite number
of measurements can be made.
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