I.

Measurement System

Measurement System – Functional elements of an instrumentation system - Static and Dynamic

Characteristics - Errors in Measurements and their statistical analysis – Calibration - Primary
and secondary standards. Bridge Circuits: Wheatstone bridge, Maxwell's bridge, Wein's bridge
and Schering bridge.
1.1 Measurements:
Measurement is defined as the action or act or process of measuring something (any
physical quantity). The measurement of a given quantity is essentially an act or the result of
comparison between the input quantity (whose magnitude is unknown or the physical
parameter is being observed or Measurand) and predefined standard, then the result is
expressed in numerical values.
In fact, measurement is the process by which one can convert any physical parameters
to meaningful numbers. The numerical value is meaningless unless followed by a unit used,
since it (unit) identifies the characteristic or property measured. Figure 1.1 shows the basic
process of measurement.

Figure 1.1: Basic Process of Measurement

Examples: To determine the Length of a piece of paper using ruler, Temperature of water
and pressure of air etc.
1.1.1 Methods of Measurements:
Methods of measurements are broadly classified into two basic categories.
(1) Direct method of measurement and
(2) Indirect method of measurement
1) Direct method of measurement.
In Direct method of measurement, Unknown Quantity (measurand) is directly
compared with predefined standard. The result is expressed as a numerical value and a unit.
Direct methods are common for the measurement of physical quantities like length, mass
and time.
For Example, to measure the length of an iron bar, we compare the length of an iron bar with
a standard ruler. The unit length is metre. An iron bar is so many times long because that
many units on our standard having the same length as the bar. Here we have determined the
length of paper by direct comparison with standard ruler.
➢ Direct methods are not always possible, feasible and practicable.
➢ Most of the cases inaccurate because they involve human factors

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 ➢ Direct methods are also less sensitive.
Hence direct methods are not preferred for accurate measurements and are rarely used.
2) Indirect method of measurement.
In indirect method of measurement, the physical parameters to be measured is
compared with the predefined standard through the use of a calibrated system. (Calibration
is the process of checking the accuracy of instrument by comparing the instrument reading
with a standard or against a similar meter of known accuracy)
➢ Indirect methods are used is industries for accurate measurements.
➢ Example, Temperature measurement using thermocouple in industries.

A measurement system is defined as the system (Group of physical components) which

is used for making measurements. It has as its input the quantity being measured and its
output the value of that quantity. Generally, a measurement system is used to know the
unknown value of a quantity or a variable.
➢ Measurement involves the use of instruments as a physical means of determining
quantities or variable.
➢ In simple measurement, a measuring instrument consists of a single unit which
gives an output reading according to the unknown input quantity applied to it.
➢ In complex measurement, a measuring instrument may be consisting of several
separate elements like sensor/ transducer, signal conditioner and display.
➢ Because of this modular nature of the elements within it, it is common to refer the
measuring instrument as a Measurement system.

1.2 Functions of Measurement systems:

The important three functions of measurement system are i) Indicating Function ii)
Recording Function and iii) Controlling Function.
i) Indicating Function (Indicating measurement system)
Instruments and systems use different kind of methods for obtaining information
concerning the input unknown quantity under measurement. Mostly this information is
obtained as a deflection of pointer of a measuring instrument. In this way the instruments
perform a function which known as indicating function.
Example, the speed of automobile is indicated by deflection of pointer of a
speedometer, Ammeter, Voltmeter and Wattmeter.
ii) Recording Function (Recording measurement system)
In many cases the system makes a written record on the paper according to given input
unknown quantity under measurement against time or against some other variable. Thus,
system or instrument performs a recording function.
Example, Monitoring of instantaneous values of temperature records using
potentiometric strip chart recorder with respect to time, monitoring of pressure and
temperature relationship record for boiler and compressor using X-Y recorder.
iii) Controlling Function (Controlling measurement system)
In this case, the information is used by the instrument or the system to control the
original measured input unknown quantity. This controlling function is one of the most

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important functions used in the field of industrial control processes.

1.3 Elements of a generalized measurement system:

A systematic organization and analysis are more important for measurement systems.
The whole operation system can be described in terms of functional elements. The functional
elements of generalized measurement system are shown in figure 1.2.

Figure 1.2: Functional elements of generalized measurement system

Most of the measurement system consists of following functional elements.

1. Primary sensing element

2. Variable conversion element
3. Variable manipulation element
4. Data transmission element
5. Data storage and playback element
6. Data presentation element
1. Primary Sensing Element
An element of an instrument which makes first, the contact with the quantity to be
measured is called primary sensing element. Thus, first detection of measurand is done by
the primary sensing element. In most of the cases, a transducer follows primary sensing
element which converts the measurand into a corresponding electrical signal. Transducer is
defined as a device which converts a physical quantity into an electrical quantity. In many
cases the physical quantity is directly converted into an electrical quantity by a transducer.
So, the first stage of a measurement system is known as a detector transducer stage. Example,
Pressure transducer with pressure sensor, Temperature sensor etc.
2. Variable Conversion Element
The output of primary sensing element is electrical signal of any form like a voltage, a
frequency or some other electrical parameter. Sometime this output not suitable for next level
of system. So, it is necessary to convert the output to some other suitable form while
preserving the information content of the original signal to perform the desired function of
the instrument.
Variable conversion element converts the signal from one physical form to another without
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changing the information content of the signal.
For example, the output of primary sensing element is in analog form of signal and
next stage of system accepts only in digital form of signal. So, we have to convert analog
signal into digital form using an A/D converter. Here A/D converter acts as variable
conversion element.

3. Variable Manipulation Element

The function of variable manipulation element is to manipulate the signal presented to
it preserving the original nature of signal. Here manipulation means only change in the
numerical value of signal.
Examples,
1. Voltage amplifier acts as variable manipulation element. Voltage amplifier accepts a
small voltage signal as input and produces the voltage with greater magnitude. Here
numerical value of voltage magnitude is increased.
2. Attenuator acts as variable manipulation element. It accepts a high voltage signal and
produces the voltage or power with lower magnitude. Here numerical value of voltage
magnitude is decreased.
➢ Linear process manipulation elements: Amplification, attenuation, integration,
differentiation, addition and subtraction etc.,
➢ Nonlinear process manipulation elements: Modulation, detection, sampling,
filtering, chopping and clipping etc.
These are performed on the signal to bring it to desired level to be accepted by the next stage
of measurement system. This process of conversion is called signal conditioning. The
combination of variable conversion and variable manipulation elements are called as Signal
Conditioning Element.

4. Data Transmission Element

The elements of measurement system are actually physically separated; it becomes
necessary to transmit the data from one to another. The element which is performs this
function is called as data transmission element.
Example, Control signals are transmitted from earth station to Space-crafts by a telemetry
system using radio signals.
The combination of Signal conditioning and transmission element is known as
Intermediate Stage of measurement system.
5. Data storage and playback element
Some applications require a separate data storage and playback function for easily
to rebuild the stored data based on the command. The data storage is made in the form of
pen/ink and digital recording. Examples, magnetic tape recorder/ reproducer, X-Y
recorder, X-t recorder, Optical Disc recording etc.
6. Data presentation Element
The information about the measured physical quantity is to be presented to a person
handling the instrument in the proper form for monitoring, control and analysis purposes.
This function is done by data presentation element. If the data is to be monitored then visual

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display devices are used as data presentation element.
These devices may be analog or digital instruments like ammeter, voltmeter, camera,
CRT, printers, analog and digital computers. Computers are used for control and analysis of
measured data of measurement system. This Final stage of measurement system is known as
Terminating stage.

Example of generalized measurement system

Example 1. Bourdon Tube Pressure Gauge:

The simple pressure measurement system using bourdon tube pressure gauge is shown
in figure 1.3. The detail functional elements of this pressure measurement system is given
below.

Figure 1.3: Bourdon tube pressure gauge

In this measurement system, bourdon tube acts as primary sensing and variable
conversion element. Bourdon tube senses the input pressure and on account of input pressure
the closed end of the tube is displaced. Pressure in converted into small displacement. The
closed end of bourdon tube is connected through mechanical linkage to a gearing
arrangement. The gearing arrangement amplifies the small displacement and makes the
pointer to rotate through large angle. The mechanical linkage acts as a data transmission
element while the gearing arrangement acts as a data manipulation element. The final data
presentation stage consists of pointer & dial arrangement which gives an indication of the
pressure signal applied to the bourdon tube. The schematic diagram of this measurement
system is given in Fig:1.4

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 Figure 1.4: Schematic diagram of a Bourdon tube pressure gauge

1.4 Characteristics of Measuring Instruments:

These performance characteristics of an instrument are very important in their selection.

➢ Static Characteristics: Static characteristics of an instrument are considered for
instruments which are used to measure an unvarying process condition.
Performance criteria based upon static relations represent the static
Characteristics. (The static characteristics are the value or performance given
after the steady state condition has reached).
➢ Dynamic Characteristics: Dynamic characteristics of an instrument are
considered for instruments which are used to measure a varying process
condition. Performance criteria based upon dynamic relations represent the
dynamic Characteristics.

1.4.1 Static Characteristics:

(i)Range and Span

• Range : The region between which the instrument operate is called range. Example:
An ammeter whose scale reads from 0 to 1 mA is said to have a range from 0 to 1m A.
• Span is the algebraic difference between the upper and lower limits of the instrument
range. Example: span :1mA
• For a thermometer calibrated between 200degree centigrade to 500 degree centigrade,
the span is 300 degree centigrade .
(ii) Accuracy
• It is the degree of closeness with which the reading approaches the true value of the
quantity to be measured.
• It determines the closeness to true value of instrument reading. The accuracy can be
expressed in following ways:
• a) Point accuracy: This is the accuracy of the instrument only at one point on its scale.
• b)Accuracy as percentage of scale range: When an instrument as uniform scale, its
accuracy may be expressed in terms of scale range.
• Eg: Accuracy of an instrument is specified by ±5% for the range of 0 to 200ºC in the
temperature scale means the reading might be within + or -10ºC of the true reading.
• C) Accuracy as percentage of true value: The best way to conceive the idea of accuracy

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 is to specify it in terms of the true value of the quantity being measured. Eg 5% of true
value
(iii) Precision
Precision is the degree of repeatability of a series of the measurement. Precision
is measures of the degree of closeness of agreement within a group of measurements
are repeatedly made under the prescribed condition.
Precision is used in measurements to describe the stability or reliability or the
reproducibility of results.
• The precision is composed of two characteristics:
• a) Conformity: Consider a resistor having true value as 2385692ohm , which is being
measured by an ohmmeter. But the reader can read consistently, a value as 2.4 M ohm
due to the nonavailability of proper scale.
• b) Number of significant figures: The precision of the measurement is obtained from
the number of significant figures, in which the reading is expressed. The significant
figures convey the actual information about the magnitude & the measurement precision
of the quantity. More significant figures greater is the precision of an instrument.Eg:
210 V, 210.1V,210.04V
Comparison between accuracy and precision.
S.No Accuracy Precision
1. It refers to degree of closeness of the It refers to the degree of agreement
measured value to the true value. among group of readings
2. Accuracy gives the maximum error that Precision of a measuring system
is maximum departure of the final result gives its capability to reproduce a
from its true value. certain reading with a given accuracy

iv) Static Error

It is defined as the difference between the measured value and the true value of the
quantity.
True value:It is the error free value of the measured variable.
True value= Measured value – Static error
v) Sensitivity:
Sensitivity is defined as the ratio of change in output signal (response) to the
change in input signal (measurand). It is the relationship indicating how much output
changes when input changes.
change in output
Sensitivity =
change in input

∆qo
Sensitivity =
∆qi
If the sensitivity is constant then the system is said to be linear system. If the
sensitivity is variable then the system is said to be non linear system.

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 Figure 1.5: Definition of sensitivity for (a) Linear and (b) Non linear instrument

When the calibration curve is linear as in figure 1.5a the sensitivity of the
instrument can be defined as in slope of the calibration curve. In this case sensitivity is
constant over the entire range of instrument. If the curve is not normally straight line
or nonlinear instrument sensitivity varies with the input or varies from on range to
another as in figure 1.5b.
(vi) Linearity
• Linearity is the best characteristics of an instrument or measurement system.
• Linearity of the instrument refers to the output is linearly or directly proportional
to input over the entire range of instrument.
• So the degree of linear (straight line) relationship between the output to input is
called as linearity of an instrument.

Nonlinearity: The maximum difference or deviation of output curve from the Specified
idealized straight line.

(vi) Repeatability
• Repeatability is defined as the ability of an instrument to give the same output for
repeated applications of same input value under same environmental condition.

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 • It is the closeness between successive measurements of the output quantity for the same
value of input under the same operating conditions.
(vii) Reproducibility
• Reproducibility is defined as the ability of an instrument to reproduce the same output
for repeated applications of same input value under different environment condition.
• In case of perfect reproducibility, the instrument satisfies no drift condition
(viii) Drift
• Drift is an undesirable change in output over a period of time that is unrelated to change
in input, operating conditions. (value of input variable, operating conditions does not
change)
• Drift is occurred in instruments due to internal temperature variations, ageing effects
and high stress etc.
• Drift may be classified into three categories:
• A)zero drift
• B)Span drift
• C)Zonal drift

A)zero drift
If the whole calibration gradually shifts due to slippage, permanent set, or due to
undue warming up of electronic tube circuits, zero drift sets in.

B)Span drift
If there is proportional change in the indication all along the upward scale, the drifts is
called span drift or sensitivity drift.
C)Zonal drift
In case the drift occurs only a portion of span of an instrument, it is called zonal
drift.

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 (ix) Hysteresis

• Hysteresis is Non-coincidence of loading and unloading curves on output.

• When input of an instrument is slowly varied from zero to full scale and then back to
zero, its output varies as shown in fig.

• If input is decreases from maximum value and output also decreases but does not follow
the same curve, then there is a residual output when input is zero. This phenomenon is
called Hysteresis. The difference between increasing change and decreasing change of
output values is known as hysteresis error.

(x) Threshold

• The minimum value of input which is necessary to activate an instrument to produce

an output is termed its threshold
• Threshold is the minimum value of the input required to cause the pointer to move
from zero position
• If the input is increased very gradually from zero, there will be some minimum value
below which no output change can be detected.
• This minimum value defines the threshold of an instrument.

(xi) Dead time

• It is the time required by a measurement system to begin to respond to a change in the

measurand.
(xii) Dead zone
• Dead zone or dead band is defined as the largest change of input quantity for which
there is no output the instrument.

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 • The region upto which the instrument does not respond for an input change is called
dead zone.
(xiii) Resolution
Resolution or Discrimination is the smallest change in the input value that is required
to cause an appreciable change in the output. (The smallest increment in input or input
change which can be detected by an instrument is called as resolution or
discrimination). So, if a non-zero quantity is slowly increased, output reading will not
increase until some minimum change in the input takes place. The minimum change
which causes the change in the output is called discrimination.
(xiii) Loading Effect

• Loading effect is the incapability of the system to faith fully measure, record or control
the input signal in accurate form
1.4.2 Dynamic characteristics

• Dynamic characteristics of an instrument are considered for instruments which are used
to measure a varying process condition.

• The set of criteria defined for the instruments, which are changes rapidly with time, is
called ‘dynamic characteristics.

• As the input varies from instant to instant, output also varies from instant to instant.
• The dynamic behaviour of an instrument is determined by applying some standard form
of known and predetermined input to its primary element (sensing element) and then
studies the output.

• Generally dynamic behaviour is determined by applying following three types of

inputs.
1. Step Input.
2.Linear Input
3.Sinusoidal input
• Step change in which the primary element is subjected to an instantaneous and finite
change in measured variable and then remains constant

• Linear Input: Linear change, in which the primary element is, follows a measured
variable, changing linearly with time.
• Sinusoidal input: Sinusoidal change, in which the primary element follows a measured
variable, the magnitude of which changes in accordance with a sinusoidal function of
constant amplitude
The dynamic characteristics of an instrument are
• (i) Speed of response

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 • (ii) Lag
• (iii) Fidelity

• (iv) Dynamic error

(i) Speed of response

• It is the rapidity with which an instrument responds to changes in the measured quantity.
• It gives information about how fast the system reacts to the changes in the input.
(ii) Lag

• It is the retardation or delay in the response of an instrument to changes in the measured

variable. The measuring lags are two types:
➢ Retardation type: In this case the response of an instrument begins immediately
after a change in measured variable has occurred.
➢ Time delay type: In this case the response of an instrument begins after a dead
time after the application of the input quantity.
(iii) Fidelity

• It is the degree to which an instrument indicates the changes in the measured variable
without dynamic error.

• (faithful reproduction or fidelity of an instrument is the ability of reproducing an input

signal faithfully (truly)).
(iv) Dynamic error

• It is the difference between the true values of a quantity changing with time and the
value indicated by the instrument, if no static error is assumed.

• It is also called as Measurement Error.

1.5 Errors in Measurements and their statistical analysis
Errors:
The difference between the measured value of quantity and true value (Reference
Value) of quantity is called as Error.
Error = Measured value - True
Value δA= Am - At

1.5.1 Classification of Errors:

All measurement can be made without perfect accuracy (degree of error must always
be assumed). In reality, no measurement can ever made with 100% accuracy. It is important
to find that actual accuracy and different types of errors can be occurred in measuring
instruments. Errors may arise from different sources and usually classified as follows,
Classification of Error
➢ Gross Errors
➢ Systematic Errors

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 1. Instrumental errors
i. Inherent shortcomings of instruments
ii. Misuse of instruments
iii. Loading effects
2. Environmental errors
3. Observational errors
➢ Random Errors
1.Gross Errors
• These gross errors mainly occur due to carelessness or lack of experience of a human
being.
• The main source of Gross errors is human mistakes in reading or using instruments and
in recording and calculating measured quantity.
• As long as human beings are involved and they may grossly misread the scale reading,
then definitely some gross errors will be occurred in measured value.
• Example, (i)Due to an oversight, Experimenter read the voltage as 31.5 V, While the
actual reading is 21.5 V
• (ii) The reading may be transposed while recording. For example, 25.8 V actual reading
may be recorded as 28.5 V.
• The complete elimination of gross errors is maybe impossible, one should try to predict
and correct them.
• Some gross errors are easily identified while others may be very difficult to detect.
• The complete elimination of gross errors is not possible but one can minimize by the
following ways.
• Great care should be taken in reading and recording the data.
• Two, three or even more readings should be taken for the quantity being measured by
using different experimenters
• So, it is suitable to take a large number of readings as a close agreement between
readings assures that no gross error has been occurred in measured values.
2.Systematic Errors
Systematic errors are divided into following three categories.
i. Instrumental Errors
ii. Environmental Errors
iii. Observational Errors

i) Instrumental Errors
These errors are arises due to following three reasons (sources of error).
a) Due to inherent shortcoming of instrument
b) Due to misuse of the instruments, and
c) Due to loading effects of instruments
a) Inherent Shortcomings of instruments
➢ These errors are inherent in instruments because of their mechanical
structure due to construction, calibration or operation of the instruments or

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 measuring devices.
➢ These errors may cause the instrument to read too low or too high.
Example, if the spring (used for producing controlling torque) of a permanent
magnet instrument has become weak, so the instrument will always read
high. Errors may be caused because of friction, hysteresis .
Elimination or reduction methods of these errors,
➢ The instrument may be re-calibrated carefully.
➢ The procedure of measurement must be carefully planned. Substitution methods or
calibration against standards may be used for the purpose.
➢ Correction factors should be applied after determining the instrumental errors.
(b) Misuse of the Instruments
➢ In some cases the errors are occurred in measurement due to the fault of the operator
than that of the instrument.
➢ A good instrument used in an unintelligent way may give wrong results.
➢ Examples, Misuse of instruments may be failure to do zero adjustment of instrument,
poor initial adjustments, using leads of high resistance and ill practices of instrument
beyond the manufacturer’s instruction and specifications etc.
➢ Such things do not cause the permanent damage to the instruments but definitely cause
errors.

(C) Loading effects of Instruments

• The errors committed by loading effects due to improper use of an instrument for
measurement work.
• In measurement system, loading effects are identified and corrections should be made
or more suitable instruments can be used.

• Example, a well calibrated voltmeter may give a misleading (may be false) voltage
reading when connected across a high resistance circuit. The same voltmeter, when
connected across a low resistance circuit may give a more reliable reading (dependable
or steady or true value). In this example, voltmeter has a loading effect on the circuit,
altering the actual circuit conditions by measurement process.

• So errors caused by loading effect of the meters can be avoided by using them
intelligently.

(ii) Environmental Errors

Environmental error occurs due to external environmental conditions of the
instrument, such as effects of temperature, pressure, humidity, dust, vibration or external
magnetic or electrostatic fields.
Elimination or reduction methods of these undesirable errors are
➢ Arrangements should be made to keep the conditions as nearly as constant as
possible. Example, temperature can be kept constant by keeping the instrument in the

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 temperature-controlled region.
➢ The device which is used against these environmental effects.
Example, variations in resistance with temperature can be minimized by using very
low resistance temperature co-efficient of resistive material.
➢ Employing techniques which eliminate the effects of these disturbances. For
example, the effect of humidity dust etc., can be entirely eliminated by tightly sealing
the equipment.
➢ The external or electrostatic effects can be eliminated by using magnetic or
electrostatic shield on the instrument.
➢ Applying computed corrections: Efforts are normally made to avoid the use of
application of computed corrections, but where these corrections are needed and are
necessary, they are incorporated for the computations of the results.

(iii) Observational Errors

There are many sources of observational errors. As an example, the pointer of a
voltmeter rests slightly above the surface of the scale. Thus an error on account of
PARALLAX will be acquired unless the line of vision of the observer is exactly above the
pointer. To minimize parallax errors, highly accurate meters are provided with mirrored
scales as shown in figure 1.6.

Correct reading 250V

Figure 1.6: Errors due to parallax

When the pointer’s image appears hidden by the pointer, observer’s eye is directly in
line with the pointer. Although a mirrored scale minimizes parallax error, an error is
necessarily presented through it may be very small.

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 So we can eliminate this parallax error by having the pointer and scale in the same
plane as shown in figure 1.7

Figure 1.7: Arrangements showing scale and pointer in the same plane

The observational errors are also occurring due to involvement of human factors. For
example, there are observational errors in measurements involving timing of an event
Different observer may produce different results, especially when sound and light
measurement are involved.
The complete elimination of this error can be achieved by using digital display of
output.
3. Random Errors
• These errors are occurred due to unknown causes and are observed when the
magnitude and polarity of a measurement fluctuate in changeable (random) manner.
• The quantity being measure is affected by many happenings or disturbances and
ambient influence about which we are unaware are lumped together and called as
Random or Residual. The errors caused by these disturbances are called Random
Errors. Since the errors remain even after the systematic errors have been taken care,
those errors are called as Residual (Random) Errors
• Random errors cannot normally be predicted or corrected, but they can be minimized
by skilled observer and using a well-maintained quality instrument.
• These errors may be reduced by taking the average of a large number of readings.
1.6 Statistical Analysis:
• Statistical Evaluation of measured data is obtained in two methods of tests as shown
in below.
o Multi Sample Test: In multi sample test, repeated measured data have been
acquired by different instruments, different methods of measurement and
different observers.
o Single Sample Test: measured data have been acquired by identical conditions
(same instrument, methods and observer) excepting time
o Inorder to get the exact value of the quantity under measurement, tests should
be done using many different procedures, techniques and experimenters.
• Statistical Evaluation methods will give the most probable true value of measured
quantity. The mathematical background statistical evaluation methods are Arithmetic
Mean, Deviation Average Deviation, Standard Deviation and variance.

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 1.6.1 Arithmetic Mean
• The most probable value of measured reading is the arithmetic mean of the number
of reading taken.
• The best approximation is made when the number of readings of the same quantity is
very large.
• Arithmetic mean or average of measured variables X is calculated by taking the sum
of all readings and dividing by the number of reading.
• The Average is given by,
X = (𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑛)/ n = Σx /n
Where, X= Arithmetic mean,
x1, x2....... xn = Readings or variable or samples and
n= number of readings

1.6.2 Deviation (Deviation from the Average value)

• The Deviation is departure of the observed reading from the arithmetic mean of the
group of reading.
• Let the deviation of reading x1 be d1 and that of x2 be d2 etc.,
d1= x1- X
d2= x2- X
.. .. ..
dn= xn- X
• The algebraic sum deviation is Zero (d1+ d2+....+ dn= 0)
Algebraic sum of deviations= d1+d2+d3+… +dn
= (x1- X)+ (x2- X )+ ……..+ (xn- X)
= (x1+x2+..+xn)-nX=0
Where (x1+x2+..+xn)= nX

1.6.3 Average Deviation: Average deviation defined as the average of the modulus
(without respect to its sign) of the individual deviations and is given by,
• D = |𝑑1 | + |𝑑2 | + |𝑑3 | + ⋯ + |𝑑𝑛 | n = Σ|d| n
• Where, D= Average Deviation.
• The average deviation is used to identify precision of the instruments which is used
in making measurements. Highly precise instruments will give a low average
deviation between readings.

1.6.4 Standard Deviation :

• Standard deviation is used to analysis random errors occurred in measurement.
• The standard Deviation of an infinite number of data is defined as the square root of
the sum of individual deviations squared, divided by the number of readings (n).

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 1.6.5 Variance
The variance is the mean square deviation, which is the same as S.D except Square
root. Variance is Just the squared standard deviation.

1.6.6 Histogram (Frequency distribution Curve)

• When a number of Multisample observations are taken experimentally there is a
scatter of the data about some central value.
• One graphical method to present test results is called histogram. A histogram is also
called a frequency distribution curve.
• This histogram indicates the number of occurrences of particular value.
• The ordinate indicates the number of observed readings (frequency of occurrence) of
a particular value.
• The steps have smaller increments and we get a smoother curve.

Length (mm) Number of observed readings

(frequency or occurrence)
99.7 1
99.8 4
99.9 12
100.0 19
100.1 10
100.2 3
100.3 1

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 Example: Table shows a set of 50 readings of length measurement. The most
probable or central value of length is 100mm represented as shown in figure 1.8
Histogram.

figure 1.8 Histogram.

1.6.7 Measure of Dispersion from the Mean

The property which denotes the extent to which the values are dispersed about the
central value is termed as dispersion. The other name of dispersion is spread or scatter.
Measure of dispersion from central value is an indication of the degree of consistency
(precision) and regularity of the data.
Example: Figure 1.9 shows the two sets of data and curve 1 vary from x1 to x2 and curve 2
vary from x3 to x4. Curve 1 is having smaller dispersion from central value than the curve
2. Therefore curve 1 is having greater precision than the curve 2.

Figure 1.9: Curves showing different ranges and precision index

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1.6.8 Probable error of finite number of readings

1.6.9 Limiting error

• In most of the instruments the accuracy is guaranteed to be within a certain percentage
of full scale reading.

• The manufacturer has to specify the deviations from the nominal value of a particular
quantity.

• The limits of these deviations from the specified value are called as Limiting Errors
or Guarantee Errors.

1.6.10 Relative limiting error

• The relative limiting error is defined as the ratio of the error to the specified (nominal)
magnitude of the quantity. This is also called fractional error.

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Example.1: The set of voltage measurement that were recorded by eight different students in
the laboratory as follows: 532V, 548V, 543V, 535V, 546V, 531V, 543V and 536. Calculate
the Arithmetic mean, Deviations from mean, average deviation, the standard deviation,
variance and probable error on recorded voltage data.

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Example 2

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1.7 Calibration

Calibration is the process of checking the accuracy of instrument by comparing the

instrument reading with a standard or against a similar meter of known accuracy. So using
calibration is used to find the errors and accuracy of the measurement system or an
instrument.
Calibration is an essential process to be undertaken for each instrument and
measuring system regularly. The instruments which are actually used for measurement work
must be calibrated against some reference instruments in which is having higher accuracy.

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Reference instruments must be calibrated against instrument of still higher accuracy or
against primary standard or against other standards of known accuracy.
The calibration is better carried out under the predetermined environmental
conditions. All industrial grade instruments can be checked for accuracy in the laboratory by
using the working standard.
Certification of an instrument manufactured by an industry is undertaken by National
Physical Laboratory and other authorizes laboratories where the secondary standards and
working standards are kept.

1.7.1 Process of Calibration

The procedure involved in calibration is called as process of calibration. Calibration
procedure involves the comparison of particular instrument with either
➢ A primary standard,
➢ A secondary standard with higher accuracy than the instrument to be calibrated
➢ An instrument of known accuracy.
Procedure of calibration as follows.
➢ Study the construction of the instrument and identify and list all the possible inputs.
➢ Choose, as best as one can, which of the inputs will be significant in the application
for which the instrument is to be calibrated.
➢ Standard and secure apparatus that will allow all significant inputs to vary over the
ranges considered necessary.
➢ By holding some input constant, varying others and recording the output, develop
the desired static input-output relations.
Theory and Principles of Calibration Methods
Calibration methods are classified into following two types,
1) Primary or Absolute method of calibration
2) Secondary or Comparison method of calibration
i. Direct comparison method of calibration
ii. Indirect comparison method of calibration
1) Primary or Absolute method of calibration
If the particular test instrument (the instrument to be calibrated) is calibrated against
primary standard, then the calibration is called as primary or absolute calibration. After the
primary calibration, the instrument can be used as a secondary calibration instrument.

Test Meter (The instrument Primary

to be calibrated) standar
d meter or
Figure 1.10: Representation of Primary Calibration

2) Secondary or Comparison calibration method

If the instrument is calibrated against secondary standard instrument, then the
calibration is called as secondary calibration. This method is used for further calibration of
other devices of lesser accuracy. Secondary calibration instruments are used in laboratory
practice and also in the industries because they are practical calibration sources.

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 Test Meter (The instrument Secondary standard
to be calibrated) meter or
instrument
Figure 1.11: Representation of Secondary Calibration

Secondary calibration can be classified further two types,

i) Direct comparison method of Calibration
Direct comparison method of calibration with a known input source with same order
of accuracy as primary calibration. So the instrument which is calibrated directly is also used
as secondary calibration instruments.
Standard instrument Test Instrument (The voltmeter to be
calibrated
Figure 1.12: Representation of Direct method of Calibration

ii) Indirect comparison method of Calibration

The procedure of indirect method of calibration is based on the equivalence of two
different devices with same comparison concept.

Source Reading Test Instrument (Device 1)

Standard instrument (Device 2)

Figure 1.13: Representation of indirect method of Calibration

1.8 Standards of measurement:

• A standard is a physical representation of a unit of measurement. A known accurate
measure of physical quantity is termed as standard. These standards are used to determine the
accuracy of other physical quantities by the comparison method.
• Example, the fundamental unit of mass in the International System is the Kilogram and
defined as the mass of a cubic decimetre of water at its temperature of maximum of density
of 4oC.
• Different standards are developed for checking the other units of measurements and all
these standards are preserved at the International Bureau of Weight and Measures at Serves,
Paris.
1.8.1 Classification of Standards
• Based on the functions and applications, standards are classified into four categories
1)International standards
2) Primary standards
3) Secondary standards

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4) Working standards

1)International standards
• International standards are defined and established upon internationally.
• They are maintained at the International Bureau of Weights and measures and are not
accessible to ordinary users for measurements and calibration.
• They are periodically evaluated and checked by absolute measurements in terms of
fundamental units of physics.
• International Ohms: It is defined as the resistance offered by a column of mercury
having a mass of 14.4521gms, uniform cross sectional area and length of 106.300cm, to the
flow of constant current at the melting point of ice.
2)Primary Standards
• Primary standards are maintained by the National Standards Laboratories (NSL) in
different parts of the world.
• The principle function of primary standards is the calibration and verification of
secondary standards.
• They are not available outside the National Laboratory for calibration.
• These primary standards are absolute standards of high accuracy that can be used as
ultimate reference standards.
3)Secondary Standards
• These standards are basic reference standards used by measurement and calibration
laboratories in industries.
• These secondary standards are maintained by the particular industry to which they
belong.
• Each industry has its own secondary standard.
• Each laboratory periodically sends its secondary standard to the National Standards
Laboratory for calibration and comparison against the primary standards.
• After comparison and calibration , the National Standards Laboratory returns the
secondary standards to the particular industrial laboratory with a certification of measuring
accuracy in terms of primary standards.
4)Working Standards
• The working standards are used for day-to-day use in measurement laboratories. So this
standard is the principle tools of a measurement laboratory.
• It is used to check and calibrate laboratory instruments for accuracy and performance.

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• Example, manufacturers of electronic components such as capacitors, resistors etc , use
a standard called working standard for checking the component values being manufactured, a
standard resistor for checking of resistance value manufactured.
1.9 Bridge Circuits
• Bridge circuits are used for measurement of resistance, inductance and capacitance.
• It consists of a network of 4 resistance arms forming a closed circuit.
• A source of current is applied to opposite junctions and the current detector is connected
to other two junctions.
• It uses the comparison measurement methods and operate on null- indication principle.
• Bridge circuit compares the value of unknown component with that of an accurately
known component, so its measurement accuracy is high.
• At balance condition, no current flows through the galvanometer.
• The components to be measured in one branch of the network and the network is
adjusted until the detector indicates no output. At this condition, the bridge is said to be
balanced.
• Then the unknown value can be found from the known values of the circuit.
1.9.1 Types of bridges
• Two types of bridges
• (i) D.C bridges
• (ii) A.C bridges
• (i) D.C bridges- used for the measurement of resistance, use the d.c voltage as the
excitation voltage.
• (ii) A.C bridges- used for the measurement of impedances consisting of inductance
and capacitances, use the alternating voltage as the excitation voltage.
1.9.2 Wheatstone bridge (D.C bridge)

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1.9.3 A.C bridges
• Bridge arms are replaced by impedances
• Bridge is excited by an AC source.
• Galvanometer is replaced by detectors (Head phones, vibration galvanometers,
tuneable amplifier detectors)

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1.9.3.1 Maxwells inductance bridge
• The bridge used for the measurement of self-inductance of the circuit is known as the
Maxwell bridge.
• It is the advanced form of the Wheatstone bridge.
• The Maxwell bridge works on the principle of the comparison, i.e., the value of
unknown inductance is determined by comparing it with the known value or standard value.
• Two methods are used for determining the self-inductance of the circuit. They are
• Maxwell’s Inductance Bridge
• Maxwell’s inductance Capacitance Bridge

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Maxwell’s inductance bridge

Figure 1.15 Maxwell’s inductance Bridge

• the value of unknown inductance is determined by comparing it with the known value
of the standard self-inductance.
• Let, L1– unknown inductance of resistance R1.
• L2– Variable inductance of fixed resistance r2 .
• R2 – variable resistance connected in series with inductor L2.
• R3, R4 – known non-inductance resistance

• Impedance of arm ab, Z1 = (R1+jωL1)

• Impedance of arm ad, Z2 = (R2+r2+jωL2)

• Impedance of arm bc, Z3 = R3

• Impedance of arm cd, Z4 = R4

• Hence for balanced bridge,

Z1Z4 =Z2 Z3

• (R1+jωL1) R4=(R2+r2+jωL2) R3

• R1 R4 + jωL1 R4 = R2 R3+r2 R3 +jωL2R3

• R1 R4 + jωL1 R4 = ( R2 +r2 ) R3 +jωL2R3

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1.9.3.2 Maxwell’s Inductance Capacitance Bridge

Figure 1.16 Maxwell’s inductance Capacitance Bridge

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1.9.3.3 Schering Bridge

Figure 1.17 Schering Bridge

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1.9.3.4 Wien’s bridge

Figure 1.18 Wien’s Bridge

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TEXT / REFERENCE BOOKS

1. A. K. Sawhney, A course in electronic Measurements and Instruments, Dhanpat Rai

Sons, 2014
2. H.S. Kalsi, Electronic Instrumentation & Measurement, Tata McGraw Hill, 2011.

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