THE COPPERBELT UNIVERSITY
SCHOOL OF ENGINEERING
ELECTRICAL ENGINEERING DEPARTMENT
EE/EG 232 ENGINEERING MEASUREMENT SYSTEMS LECTURE NOTES
CONTACT HOURS:2 hours/ week
The course is divided into two parts, namely the Electrical and the Mechanical. The
electrical part will be taught for two terms by Mr A Tambatamba and the Mechanical part
in the third term by Mr E Langi.
Course content(Electrical)
Module 1
Measurements and errors: definition; accuracy and precision; significant figures; types of errors
statistical analysis, common symbols used in metrology.
Module 2
Measuring instruments: classification; torque generation, controlling and damping; PMMC, MI
and electrodynamometer ammeters and voltmeters; electrodynamometer, induction types in
wattmeters (1 and 3 phase); induction energy meter, compensation, creeping errors testing;
frequency meter: vibration reed type, electrical resonance.
Module 3
Instrument transformers: voltage and current transformers; ration and phase angle errors, phasor
diagram, methods of reducing error; testing and applications
Galvanometers: principles of operation; performance equations of the D’ Arsonval
galvanometer, vibration and ballistic galvanometers
Potentiometer: DC potentiometer, Crompton potentiometer, construction, standardisation,
application, AC potentiometer, Drysdale polar potentiometer,construction, standardisation,
application
Module 4
AC/DC bridges: Genera balance equations of Wheatsone, Kelvin, Maxwell, Hay, Schering, and
Wein bridges.
Transducers:Pizo-Electric, Thermistors, Thermorecouple, Torque meter
Module 5
Cathode ray Oscilloscope: construction, sweep generation, application of CRG in measurements
of frequency, phase, amplitude and rise in time of a pulse.
Digital multi-meter: Block diagram, principle of operation, accuracy of measurements.
Electronic voltmeter: transistor voltmeter, block diagram, principle of operation. Other types of
electronic voltmeter, digital frequency metter : construction and principle of operation.
1
Module 1
Introduction
Metrology (from Greek words metron – to measure and logy- science) is the science of
measurements. Measurements are important to humans as it helps to determine unknown
quantities that they can with their bare hands. A measurement has been developing along side
with advancement in engineering. The first in the world institutions set up to study
measurements was established in German in 1887. It is said to be partly responsible for the
abrupt increase in industrialisation of German, which was the first one in the world. Other
countries which are industrialised today follow the German example.
The main purpose for of metrology is best described by the famous Italian scientist Galileo
Galilei, who said ‘measure everything that can be measured and try to make measurable what is
not yet measurable’.
Measurements are useful in determining the value of a measured quantity. Quantities can include
electric current, distance, mass time and others. A measured value is the reading obtained from a
measuring instruments, with a degree of error. A true value is simply the rated value.
The International System of Units (SI)
SI is the modern and most widely used system of units in measurements. It is made up of seven
base units which include Length measured in meters (m), mass- Kilogram (kg), time- seconds
(s), electric current- Ampere (A), temperature- Kelvin (K), luminous (light) intensity –Candela
(Cd) and amount of substance- mole (mol). SI is the foundation of modern metrology. It was
found in 1960. Apart from the seven base units there are 23 other units which are derived using
formulas. Electric current is the only unit for electrical engineers in the base units. Other
electrical engineering units like voltage, resistance, power and others are derived from formulas
using electric current.
PREFIXES
Since the presentation of the results in the initial units of physical quantities is sometimes
unsuitable, the SI system defines the use of SI prefixes as shown in Table 1 below. The prefixes
hecto, deka, deci, and centi are not used in electrical engineering. The names of SI units form the
name of the starting unit by addingSI prefixes. Thus, it is correct to say 10 teraohms, but not 10
deciohms. Using only a single prefix inthe SI system is also allowed. Therefore, 10 μμW is not
allowed, as it must be 10 pW.
Table 1
Prefix
Symbol Exponent
Number
12
Tera
T
1 000 000 000 000
10
9
Giga
G
1 000 000 000
10
Mega
M
1 000 000
106
3
Kilo
k
1 000
10
0
Base
base
1
10
Deci
d
0.1
10−1
−2
centi
c
0.01
10
−3
Milli
M
0.001
10
Micro
0.000001
μ
10−6
−9
Nano
n
0.000000001
10
2
pico
p
Other prefixes are included in the table 2 below,
10−12
0.000000000001
Table 2
The base is the unit used to measure some quantity, eg amps, watts, meter, second etc. From
table 1 it can be seen that it is convenient to express some numbers in their exponent form i.e10x.
For example, 2.5 Megagram=2.5 Mg=2.5x106g=2500 000g.
Prefix conversion
Example 1, How many dg are there in 2.6 kg?
Solution
Approach (steps)
1. Write the amount and units and given in question
2. Multiply the amount and units given in1 by a fraction having the given units at the
bottom and the require units at the top,
3. Then assign the bigger prefix from the fraction with 1 and the small one with
4. Determine the exponent of the figure 10 assigned to smaller prefix of the fraction in 3 by
subtraction the exponent of the smaller prefix from that of a bigger one.
5. After that cancel the like terms and present the final answer in simple exponent form as
shown below.
104 𝑑𝑔
2.6 𝑘𝑔 ∙
= 2.6 ∙ 104 𝑑𝑔
1 𝑘𝑔
Finding exponent of dg, exponent of kg =3 that of dg—1, therefore 3-(-1)=4, when simplifying
the expression by cancelling the like units and prefixes, we get 𝟐. 𝟔 ∙ 𝟏𝟎𝟒 𝒅𝒈
Example 2, How many cm are there in 18.2 nm?
Solution
The approach is similar to the previous example
18.2 𝑛𝑚 ∙
1 𝑐𝑚
= 18.2 ∙ 10−7 𝑐𝑚
107 𝑛𝑚
3
Finding exponent of nm.since exponent of cm =-2 that of nm=-9, therefore -2-(-9)=7, when
simplifying the expression by cancelling the like units and prefixes, we get 𝟏𝟖. 𝟐 ∙ 𝟏𝟎𝟕 𝒅𝒈
Conversion of quantities that squared and cubed
For quantities like length that get squared and cubed to give area and volume have a slightly
different way of converting their prefixes.
Example 3, How manym2 are there in 1 km2?
Solution
The approach is similar to the previous example; apart from the fact that the exponent will have
to be multiplied by 2 or 3 with we are dealing with squared or cubed quantities respectively.
103(2) 𝑚2
= 1 ∙ 106 𝑚2
1 𝑘𝑚2
Finding exponent of 𝑚2 . Since exponent of km =3 that of m=0, therefore 3-(0)=3, the exponent
is then multiplied by 2 as the units are squared. Then after simplifying the expression by
cancelling the like units and prefixes, we get 1 ∙ 106 𝑚2
1 𝑘𝑚2 ∙
Measurements and errors
Most important terms used in metrology include:
i) Measurement: This is the process of using an instrument to determine the magnitude
or value of a quantity, variable or parameter.
ii) Electrical/ electronic measuring instrument: this is a device based on electrical and
electronic principle for determining the value or magnitude of a quantity, variable or
parameter.
iii) Instrument accuracy: this is the closes of an instrument reading to the true value of
the variable being measured.
iv) Instrument precision: This is the measure of the degree to which successive
measurements taken under the same conditions.differ from each other.
v) Instrument sensitivity: this is the ratio of the output signal or response of the
instrument to change of an input or measured variable.
vi) Instrument resolution: this is the smallest change in measured value the instrument
will respond to.
vii) Instrument Error: this the deviation ( difference) of the measured value from a true
value
viii) Measurement reproducibility: is the closeness of individual measurements of
the same quantity that are measured in changed circumstances (different measurers in
different laboratories, using other measurement methods, instruments, places, and
conditions, etc.).
ix) True value: the arithmetic mean of an infinite set of measurements. Note that its
practically impossible to make infinity measurements.
x) Confidence limits: the range within which its probable to find the true value of a
seasurement.
4
Error Minimisation Techniques
Several techniques may be used to minimize the effects of errors. For example:
1. Making precision measurements,
2. It is advisable to record a series of observations rather than rely on one observation.
3. Alternate methods of measurement, as well as the use of different instruments to perform
the same experiment,
4. Provide a good technique for increasing accuracy.
Although these techniques tend to increase the precision of measurement by reducing
environmental or random error, they cannot account for instrumental error.
Error has been defined above as the deviation of the measured value from a true value.
Mathematically it can be define:
𝐸𝑟𝑟𝑜𝑟 = 𝛿 = 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑣𝑎𝑙𝑢𝑒(𝐴𝑚 ) − 𝑇𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒(𝐴𝑇 )
If 𝐴𝑚 > 𝐴𝑇 𝑡ℎ𝑒𝑛 𝐸𝑟𝑟𝑜𝑟 𝑖𝑠 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒, 𝑤ℎ𝑒𝑟𝑒 𝑎𝑠 𝑖𝑓 𝐴𝑚 < 𝐴𝑇 𝑡ℎ𝑒𝑛 𝐸𝑟𝑟𝑜𝑟 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
Example
A motor is rated 3000 rpm at full speed.When correctly connected to the power supply at full
speed, the reading of the torque meter is used to measure the rpm was 3002 rpm. Determine the
error.
Solution
𝐸𝑟𝑟𝑜𝑟 = 𝛿 = 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑣𝑎𝑙𝑢𝑒(𝐴𝑚 ) − 𝑇𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒(𝐴𝑇 ) = 3002 − 3000 = +𝟐 𝒓𝒑𝒎
Note that 𝛿 is also known as absolute error.
Errors in measurements are usually expressed in percentage (%).
The quality of an instrument in terms of accuracy is determined by its percentage Relative Static
Error (RSE) (also called the limiting Error).
𝐴𝑚 − 𝐴𝑇
𝛿
𝐴𝑚
∙ 100% =
∙ 100% = (
− 1) ∙ 100%
𝐴𝑇
𝐴𝑇
𝐴𝑇
𝐴𝑚 −𝐴𝑇
is also referred to as relative error
𝐴
%𝑅𝑆𝐸 =
𝑇
%𝑅𝑆𝐸is also referred to as Percentage Error
Example
Two instruments A and B had the information as shown below:
𝐼𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡 𝐴: 𝛿𝐴 = 1 𝐴, 𝐴𝑇 = 2 𝐴
𝐼𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡 𝐵 ∶ 𝛿𝐵 = 10 𝐴, 𝐴𝑇 = 1000 𝐴
Determine which instrument was more quality.
Solution
𝛿
1
𝐼𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡 𝐴: %𝑅𝑆𝐸 =
∙ 100% = ∙ 100% = 50%,
𝐴𝑇
2
𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 100% − %𝑅𝑆𝐸 = 100% − 50% = 50%
𝛿
10
𝐼𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡 𝐵: %𝑅𝑆𝐸 =
∙ 100% =
∙ 100% = 1%,
𝐴𝑇
1000
𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 100% − %𝑅𝑆𝐸 = 100% − 1% = 99%
From the results obtained, it can be concluded that instrument B is more quality as it has a higher
accuracy low error.
5
Accuracy and Precision
As mentioned above, Accuracy refers to the degree of closeness or conformity to the true value
of the quantity under measurement. Precision refers to the degree of agreement within a group of
measurements or instruments.
To demonstrate the difference between accuracy and precision, two voltmeters of the same make
and model may be compared. Both meters have knife-edged pointers and mirror-backed scales to
avoid parallax, and they have carefully calibrated scales. They may therefore be read to the same
precision. If the value of the series resistance in one meter changes considerably, its readings
may be in error by a fairly large amount. Therefore the accuracy of the two meters may be quite
different. (To determine which meter is in error, a comparison measurement with a standard
meter should be made).
Precision is composed of two characteristics: conformity (closeness) and the number of
significant figures to which a measurement can be made. Consider, for example, that a resistor,
whose true resistance is 1,273,672 Ω, is measured by an ohmmeter which consistently and
repeatedly indicates 1.3 MΩ. But can the observer “read” the true value from the scale? His
estimates from the scale reading consistently yield a value of 1.3 MΩ. This is as close to the true
value as he can read the scale 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 example
illustrates that conformity is a necessary, but not sufficient condition for precision because of the
lack of significant figures obtained. Similarly, precision is a necessary, but not sufficient
condition for accuracy.
Too often the beginning student is inclined to accept instrument readings at face value. He is not
aware that the accuracy of a reading is not guaranteed by its precision. In fact, good
measurement techniques demands continuous scepticism as to the accuracy of the results.
In critical work, good practice dictates that the observer make an independent set of
measurements, using different instruments or different measurement techniques, not subject to
the same systematic errors. He must also make sure that the instruments function properly and
are calibrated against a known standard, and that no outside influence affects the accuracy of his
measurements.
Significant Figures
An indication of precision of the measurement is 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 figures, the greater
the precision of measurement.
For example, if a resistor is specified as having a resistance of 68 Ω, its resistance should be
closer to 68 Ω than to 67 Ω or 69 Ω. If the value of the resistor is described as 68.0 Ω, it means
that its resistance is closer to 68.0 Ω than it is to 67.9 Ω or 68.1 Ω. In 68 Ω there are two
significant figures; in 68.0 Ω there are three. The latter, with more significant figures, expresses a
measurement of greater precision than the former.
Often, however, the total number of digits may not represent measurement precision. Frequently,
large numbers with zeros before a decimal point are used for approximate populations or
amounts of money. For example, the population of a city is reported in six figures as 380,000.
This may imply that the true value of the population lies between 379,999 and 380,001, which is
six significant figures. What is meant, however, is that the population is closer to 380,000 than
6
370,000 or 390,000. Since in this case the population can be reported only to two significant
figures, how can large numbers be expressed?
A more technically correct notation uses powers of ten, 38 × 104 or 3.8 × 105. This indicates that
the population figures is only accurate to two significant figures. Uncertainty caused by zeros to
the left of the decimal point is therefore usually resolved by scientific notation using powers of
ten. Reference to the velocity of light as 186,000 mi/s, for example, would cause no
misunderstanding to anyone with a technical background. But 1.86 × 105 mi/s leave no
confusion.
It is customary to record a measurement with all the digits of which we are sure nearest to the
true value. For example, in reading a voltmeter, the voltage may be read as 117.1 V. This simply
indicates that the voltage, read by the observer to best estimation, is closer to 117.1 V than to
117.0 V or 117.2 V. Another way of expressing this result indicates the range of possible error.
The voltage may be expressed as 117.1 ± 0.05 V, indicating that the value of the voltage lies
between 117.05 V and 117.15 V.
When a number of independent measurements are taken in an effort to obtain the best possible
answer (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 is
illustrated in Example 1.
Example 1
A set of independent voltage measurements taken by four observers was recorded as 117.02 V,
117.11 V, 117.08 V, and 117.03 V. Calculate:
1. The average voltage,
2. The range of error.
Solution
1. Average Voltage
𝑉1 + 𝑉2 + 𝑉3 + 𝑉4 117.02 + 117.11 + 117.08 + 117.03
𝑉𝑎𝑣 =
=
= 117.06 𝑉
𝑛
4
2. Range
Range 𝑉𝑚𝑎𝑥 − 𝑉𝑎𝑣 = 117.11 − 117.06 = 0.05
but also 𝑉𝑎𝑣 − 𝑉𝑚𝑖𝑛 = 117.06 − 117.02 = 0.04
0.05+0.04
The average range of error therefore equals
= ±0.045 = ±0.05
2
When two or more measurements with different degrees of accuracy are added, the result is only
as accurate as the least square measurement. Suppose that two resistances are added in series as
in Example 2.
Example 2
Two resistors R1 and R2 are connected in series. Individual resistance measurements, using a
Wheatstone bridge, give R1 = 18.7Ω and R2 = 3.624Ω. Calculate the total resistance to the
appropriate number of significant figures.
Solution
R1 = 18.7 Ω (three significant figures)
R2 = 3.624 Ω (four significant figures)
RT = R1 + R2 = 22.324 Ω (five significant figures) = 22.3 Ω
7
The doubtful figures are written in italics to indicate that in the addition of R1 and R2 the last
three digits of the sum are doubtful figures. There is no value whatsoever in retaining the last two
digits (the 2 and the 4) because one of the resistances is accurate only to three significant figures
or tenths of an ohm. The result should therefore also be reduced to three significant figures or the
nearest tenth, i.e., 22.3 Ω.
The number of significant figures in multiplication may increase rapidly, but again only the
appropriate figures are retained in the answer, as shown in Example 3.
Example 3
In calculating voltage drop, a current of 3.18 A is recorded in a resistance of 35.68 Ω. Calculate
the voltage drop across the resistor to the appropriate number of significant figures.
Solution
V = IR = 3.18 × 35.68 = 113.4624 = 113V
Since there are three significant figures involved in the multiplication, the answer can be written
only to a maximum of three significant figures.
In Example 3, the current, I, has three significant figures and R has four; and the result of the
multiplication has only three significant figures. This illustrates that the answer cannot be known
to the accuracy greater than the least poorly defined of the factors. Note also that if extra digits
accumulate in the answer, they should be discarded or rounded off. In the usual practice, if the
(least significant) digit in the first place to be discarded is less than five, it and the following
digits are dropped from the answer. This was done in Example 3. If the digit in the first place to
be discarded is five or greater, the previous digit is increased by one. For three-digit precision,
therefore, 113.46 should be rounded off to 113; and 113.74 to 114.
Addition of figures with a range of doubt is illustrated in Example 4.
Example 4
Add 826 ± 5 to 628 ± 3
Solution
N1 = 826 ± 5 (= ±0.605%)
N2 = 628 ± 3 (= ±0.477%)
Sum = 1454 ± 8 (= ±0.55%)
Note in Example 4 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 figures N1 and N2 does not differ greatly from
the percentage doubt in the final result. If the same two numbers are subtracted, as in Example 5,
there is an interesting comparison between addition and subtraction with respect to the range of
doubt.
Example 5
Subtract 628 ± 3 from 826 ± 5 and express the range of doubt in the answer as a percentage.
Solution
N1 = 826 ± 5 (= ±0.605%)
N2 = 628 ± 3 (= ±0.477%)
Difference = 198 ± 8 (= ±4.04%)
8
Again in Example 5, the doubtful parts are added for the same reason as in Example 4.
Comparing the results of addition and subtraction of the same numbers in Example 4 and 5, note
that the precision of the results, when expressed in percentages, differs greatly. The final 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. Consider the case illustrated in Example 6.
Example 6
Subtract 437 ± 4 from 462 ± 4 and express the range of doubt in the answer as a percentage.
Solution
N1 = 462 ± 4 (= ±0.87%)
N2 = 437 ± 4 (= ±0.92%)
Difference = 25 ± 8 (= ±32%)
Example 6 illustrates clearly that one should avoid measurement techniques depending on
subtraction of experimental results because the range of doubt in the final result may be greatly
increased.
Types of Error
No measurement can be made with perfect accuracy, but it is important to find out what the
accuracy actually is and how different errors have entered into the measurement. A study of
errors is a first step in finding ways to reduce them. Such a study also allows us to determine the
accuracy of the final test result.
Errors may come from different sources and are usually classified under three main headings:
Gross Errors -Largely human errors, among them misreading of instruments, incorrect adjustment and improper application of instruments, and computational mistakes.
Systematic Errors -Shortcomings of the instruments, such as defective or worn parts, and
effects of the environment on the equipment or the user.
Random Errors -Those due to causes that cannot be directly established because of random
variations in the parameter or the system of measurement.
Each of these classes of errors will be discussed briefly and some methods will be suggested for
their reduction or elimination.
Gross (grave) Errors
This class of errors mainly covers human mistakes in reading or using instruments and in
recording and calculating measurement results. As long as human beings are involved, some
gross errors will inevitably be committed. Although complete elimination of gross errors is
probably impossible, one should try to anticipate and correct them. Some gross errors are easily
detected; others may be very elusive. Grave errors are avoided with good knowledge and
attention during the measurements, and a proper selection of measurement equipment and the
measurement procedure. It is always useful to evaluate the approximate value of the measurand
before the measurements. One common gross error, frequently committed by beginners in
measurement work, involves the improper use of an instrument. In general, indicating
instruments change conditions to some extent when connected into a complete circuit, so that the
measured quantity is altered by the method employed. For example,
a well-calibrated voltmeter may give a misleading reading when connected across two points in a
high resistance circuit (Example 7). The same voltmeter, when connected in a low-resistance
circuit, may give a more dependable reading (Example 8). These examples illustrate that the
voltmeter has a “loading effect” on the circuit, altering the original situation by the measurement
process.
9
Example 7
A voltmeter, having a sensitivity of 1,000 Ω/V, reads 100 V on its 150-V scale when connected
across an unknown resistor in series with a milliammeter. When the milliammeter reads 5 mA,
calculate:
1. Apparent resistance of the unknown resistor,
2. Actual resistance of the unknown resistor,
3. Error due to the loading effect of the voltmeter.
Solution
1. The total circuit resistance equals
𝑅𝑇 =
𝑉𝑇 100 𝑉
=
= 20 𝑘Ω
𝐼𝑇
5 𝑚𝐴
Neglecting the resistance of the milliammeter, the value of the unknown resistor is RX = 20 kΩ
2. The voltmeter resistance equals
𝑅𝑉 = 1000
Ω
∙ 150 𝑉 = 150 𝑘Ω
𝑉
Since the voltmeter is in parallel with the unknown resistance, we can write
𝑅𝑇 ∙ 𝑅𝑉
20 ∙ 150
𝑅𝑋 =
=
= 23.05 𝑘Ω
𝑅𝑉 − 𝑅𝑇 150 − 20
3. Error
% 𝐸𝑟𝑟𝑜𝑟 =
𝐴𝑐𝑡𝑢𝑎𝑙 − 𝑎𝑝𝑝𝑟𝑒𝑛𝑡
23.05 − 20
∙ 100% =
∙ 100% = 13.23%
𝐴𝑐𝑡𝑢𝑎𝑙
23.05
Example 8
Repeat Example 7 if the milliammeter reads 800 mA and the voltmeter reads 40 V on its 150-V
scale.
Solution
1. The total circuit resistance equals
𝑅𝑇 =
𝑉𝑇 40 𝑉
=
= 50 Ω
𝐼𝑇 0.8 𝐴
2. The voltmeter resistance equal
𝑅𝑋 = 1000
Ω
∙ 150 𝑉 = 150 𝑘Ω
𝑉
Since the voltmeter is in parallel with the unknown resistance, we can write
𝑅𝑇 ∙ 𝑅𝑉
50 ∙ 150
𝑅𝑋 =
=
= 75 Ω
𝑅𝑉 − 𝑅𝑇 150 − 50
3. Error
% 𝐸𝑟𝑟𝑜𝑟 =
𝐴𝑐𝑡𝑢𝑎𝑙 − 𝐴𝑝𝑝𝑎𝑟𝑒𝑛𝑡
75 − 50
∙ 100% =
∙ 100% = 33.33 %
𝐴𝑐𝑡𝑢𝑎𝑙
75
Errors caused by the loading effect of the voltmeter can be avoided by using it intelligently. For
example, a low-resistance voltmeter should not be used to measure voltages in a vacuum tube
10
amplifier. In this particular measurement, a high-input impedance voltmeter (such as VTVM or
TVM) is required.
A large number of gross errors can be attributed to carelessness or bad habits, such as improper
reading of an instrument, recording the result differently from the actual reading taken, or
adjusting the instrument incorrectly. Consider the case in which a multi-range voltmeter uses a
single set of scale markings with different number designations for the various voltage ranges. It
is easy to use a scale which does not correspond to the setting of the range selector of the
voltmeter. A gross error may also occur when the instrument is not set to zero before the
measurement is taken; then all the readings are off.
Errors like these cannot be treated mathematically. They can be avoided only by taking care in
reading and recording the measurement data. Good practice requires making more than one
reading of the same quantity, preferably by a different observer. Never place complete
dependence on one reading but take at least three separate readings, preferably under conditions
in which instruments are switched off-on.
Systematic Errors
This type of errors is usually divided into two different categories:
1. Instrumental errors, defined as shortcomings of the instrument;
2. Environmental errors, due to external conditions affecting the measurement.
Instrumental errors are errors inherent in measuring instruments because of their mechanical
structure. For example, in the d’Arsonval movement, friction in bearings of various moving
components may cause incorrect readings. Irregular spring tension, stretching of the spring, or
reduction in tension due to improper handling or overloading of the instrument will result in
errors. Other instrumental errors are calibration errors, causing the instrument to read high or low
along its entire scale. (Failure to set the instrument to zero before making a measurement has a
similar effect.)
There are many kinds of instrumental errors, depending on the type of instrument used. The
experimenter should always take precautions to insure that the instrument he is using is operating
properly and does not contribute excessive errors for the purpose at hand. Faults in instruments
may be detected by checking for erratic behaviour, and stability and reproducibility of results. A
quick and easy way to check an instrument is to compare it to another with the same
characteristics or to one that is known to be more accurate.
Instrumental errors may be avoided by:
1. selecting a suitable instrument for a particular measurement application;
2. applying correction factors after determining the amount of instrumental error;
3. calibrating the instrument against a standard.
Environmental errors are due to conditions external to the measuring device, including
conditions in the area surrounding the instrument, such as the effects of 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 affects the reading of the instrument. Corrective measures to
reduce these effects include air conditioning, hermetically sealing certain components in the
instrument, use of magnetic shields, and the like.
Systematic errors can also be subdivided into static or dynamic errors. Static errors are caused by
limitations of the measuring device or the physical laws governing its behavior. A static error is
11
introduced in a micrometer when excessive pressure is applied in torquing the shaft. Dynamic
errors are caused by the instrument’s not responding fast enough to follow the changes in a
measured variable.
Random Errors
Random errors result from small and variant changes that occur in the standards, measures,
measurement laboratory, and environment. These errors can cause a large number of different
errors in each individual measurement and each time have a different size and sign, causing
measurement results to scatter.
If the same quantity is measured several times in a row under the same external conditions and
with the same instrument, each time the results will scatter around some value due to random
errors that change the size and sign in each measurement. These random errors cannot be solved
by making corrections, as for some systematic errors. Since all measurements are carried out
under the same conditions, all results have the same weight. Thus, the most probable value of the
measured quantity is the arithmetic mean of individual results. If the measurements are repeated
n times, and individual results are then the arithmetic mean of the individual results is:?
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. The arithmetic mean is given by
the following expression:
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + ⋯ + 𝑥𝑛 ∑ 𝑥𝑖
ẍ=
=
𝑛
𝑛
Where ẍ=arithmetic mean
x1, x2, x3, x4,…, xn=readings take
n=number of readings
Example 1 showed how the arithmetic mean is used.
Deviation from the Mean
Deviation is the departure of a given reading from the arithmetic mean of the group of readings.
If the deviation of the first reading, x1, is called d1, and that of the second reading, x2, is called
d2, and so on, then the deviations from the mean can be expressed as
d1 = x1 − ẍ
d2 = x2 – ẍ
dn = xn − ẍ
Note that the deviation from the mean may have a positive or a negative value and that the
algebraic sum of all the deviations must be zero.
Example 9 illustrates the computation of deviations.
Example 9
A set of independent current measurements was taken by six observers and recorded as 12.8 mA,
12.2 mA, 12.5 mA, 13.1 mA, 12.9 mA, and 12.4 mA. Calculate
12
1. The arithmetic mean,
2. The deviations from the mean.
Solution
1. Using Eq. (1), we see that the arithmetic mean equals
x=
12.8 + 12.2 + 12.5 + 13.1 + 12.9 + 12.4
6
= 12.65 mA
9
2. Using Eq. (2), we see that the deviations are
d1 = 12.8 − 12.65 = 0.15 mA
d2 = 12.2 − 12.65 = −0.45 mA
d3 = 12.5 − 12.65 = −0.15 mA
d4 = 13.1 − 12.65 = 0.45 mA
d5 = 12.9 − 12.65 = 0.25 mA
d6 = 12.4 − 12.65 = −0.25 mA
Note that the algebraic sum of all the deviations equals zero.
Average Deviation
The average deviation is an indication of the precision of the instruments used in making the
measurements. Highly precise instruments will yield a low average deviation between readings.
By definition, average deviation is the sum of the absolute values of the deviation divided by the
number of readings. The absolute value of the deviation is the value without respect to sign.
Average deviation may be expressed as
|𝑑1 | + |𝑑2 | + |𝑑3 | + |𝑑4 | + ⋯ + |𝑑𝑛 | ∑|𝑑𝑖 |
𝐷=
=
𝑛
𝑛
Example 10 shows how average deviation is calculated.
Example 10
Calculate the average deviation for the data given in Example 9.
Solution
0.15 + 0.45 + 0.15 + 0.45 + 0.25 + 0.25
𝐷=
= 0.283 𝑚𝐴
6
Standard Deviation
In statistical analysis of random errors, the root-mean-square deviation or standard deviation is a
very valuable aid. By definition, the standard deviation _ of an infinite number of data is the
square root of the sum of all the individual deviations squared, divided by the number of
readings. Expressed mathematically:
∑ 𝑑𝑖2
d12 + d22 + d23 + ⋯ + d2n
√
√
σ=
=
n
𝑛
In practice, of course, the possible number of observations is finite. The standard deviation of a
finite number of data is given by
13
∑ 𝑑𝑖2
d12 + d22 + d23 + ⋯ + d2n
√
√
σ=
=
n−1
𝑛−1
Equation (5) will be used in Example 11.
Another expression for essentially the same quantity is the variance or mean square deviation,
which is the same as the standard deviation except that the square root is not extracted. Therefore
variance(V ) = mean square deviation =σ2
The variance is a convenient quantity to use in many computations because variances are
additive. The standard deviation, however, has the advantage of being of the same units as the
variable, making it easy to compare magnitudes. Most scientific results are now stated in terms
of standard deviation.
Probability of Errors
Normal Distribution of Errors
Table 1 shows a tabulation of 50 voltage readings that were taken at small time intervals and
recorded to the nearest 0.1 V. The nominal value of the measured voltage was 100.0 V. The
result of this series of measurements can be presented graphically in the form of a block diagram
or histogram in which the number of observations is plotted against each observed voltage
reading. The histogram of Figure 1 represents the data of Table 1.
14
Figure 1 shows that the largest number of readings (19) occurs at the central value of 100.0 V,
while the other readings are placed more or less symmetrically on either side of the central value.
If more readings were taken at smaller increments, say 200 readings at 0.05-V intervals, the
distribution of observations would remain approximately symmetrical about the central value
and the shape of the histogram would be about the same as before. With more and more data,
taken at smaller and smaller increments, the contour of the histogram would finally become a
smooth curve, known as a Gaussian curve. The sharper and narrower the curve, the more
definitely an observer may state that the most probable value of the true reading is the central
value or mean reading.
The Gaussian or Normal law of error forms the basis of the analytical study of random effects.
Although the mathematical treatment of this subject is beyond the scope of the text, the
following qualitative statements are based on the Normal law:
1. All observations include small disturbing effects, called random errors.
2. Random errors can be positive or negative.
3. There is an equal probability of positive and negative random errors.
We can therefore expect that measurement observations include plus and minus errors in more or
less equal amounts, so that the total error will be small and the mean value will be the true value
of the measured variable.
The possibilities as to the form of the error distribution curve can be stated as follows:
1. Small errors are more probable than large errors.
2. Large errors are very improbable.
3.There is an equal probability of plus and minus errors so that the probability of a given error
will be symmetrical about the zero value.
The error distribution curve of Figure 2 is based on the Normal law and shows a symmetrical
distribution of errors. This normal curve may be regarded as the limiting form of the histogram
of Figure 1 in which the most probable value of the true voltage is the mean value of 100.0 V.
Probable Error
The area under the Gaussian probability curve of Figure 2, between the limits +∞ and −∞,
represents the entire number of observations. The area under the curve between the +𝜎 and
−𝜎limits represents the cases that differ from the mean by no more than the standard deviation.
Integration of the area under the curve within the ±𝜎 limits gives the total number of cases within
these limits. For normally dispersed data, following the Gaussian distribution, approximately
68% of all the cases lie between the limits of +𝜎 and −𝜎 from the mean. Corresponding values of
other deviations, expressed in terms of 𝜎, are given in Table 2.
15
If, for example, a large number of nominally 100 resistors is measured and the mean value is
found to be 100.00 , with a standard deviation of 0.20 , we know that on the average 68% (or
roughly two-thirds) of all the resistors have values which lie between limits of ±0.20 of the
mean. There is then approximately a two to one chance that any resistor, selected from the lot at
random, will lie within these limits. If larger odds are required, the deviation may be extended to
a limit of ±2_, in this case ±0.40 . According to Table 2, this now includes 95% of all the cases,
giving ten to one odds that any resistor selected at random lies within ±0.40 of the mean value
of 100.00 .
Table 2 also shows that half of the cases are included in the deviation limits of ±0.6745_. The
quantity r is called the probable error and is defined as
probable error, r = ±0.6745𝜎 (
The value is probable in the sense that there is an even chance that any one observation will have
a random error no greater than ±r. Probable error has been used in experimental work to some
extent in the past, but standard deviation is more convenient in statistical work and is given
preference.
Example 11
Ten measurements of the resistance of a resistor gave 101.2Ω, 101.7Ω, 101.3Ω, 101.0 Ω, 101.5Ω
, 101.3Ω, 101.2Ω , 101.4 Ω, 101.3Ω , and 101.1 Ω.
Assume that only random errors are present. Calculate
1. the arithmetic mean,
2. the standard deviation of the readings,
3. the probable error.
Solution
With a large number of readings a simple tabulation of data is very convenient and avoids
confusion and
mistakes.
16
Limiting Errors
In most indicating instruments, the accuracy is guaranteed to a certain percentage of full-scale
reading. Circuit components (such as capacitors, resistors, etc.) are guaranteed within a certain
percentage of their rated value. The limits of these deviations from the specified values are
known as limiting errors or guarantee errors. For example, if the resistance of a resistor is given
as 500 ± 10 per cent, the manufacturer guarantees that the resistance falls between the limits 450
and 550 . The maker is not specifying a standard deviation or a probable error, but promises that
the error is no greater than the limits set.
Example 12
A 0-150-V voltmeter has a guaranteed accuracy of 1 per cent full-scale reading. The voltage
measured by this instrument is 83 V. Calculate the limiting error in per cent.
Solution
The magnitude of the limiting error is
0.01 × 150 V = 1.5 V
The percentage error at a meter indication of 83 V is
1.5
∙ 100% = 1.81%
83
It is important to note in Example 12 that a meter is guaranteed to have an accuracy of better
than 1 per cent of the full-scale reading, but when the meter reads 83 V, the limiting error
increases to 1.81 per cent. Correspondingly, when a smaller voltage is measured, the limiting
error will increase further. If the meter reads 60 V, the per cent limiting error is 1.5/60 ×100 =
2.5 per cent; if the meter reads 30 V, the limiting error is 1.5/30 × 100 = 5 per cent. The increase
in per cent limiting error, as smaller voltages are measured, occurs because the magnitude of the
limiting error is a fixed quantity based on the full-scale reading of the meter. Example 12 shows
the importance of taking measurements as close to full scale as possible.
Measurements or computations, combining guarantee errors, are often made. Example 13
illustrates such a computation.
17
Example 13
Three decade boxes, each guaranteed to ±0.1 per cent, are used in a Wheatstone bridge to
measure the resistance of an unknown resistor Rx. Calculate the limits on Rx imposed by the
decade boxes.
Solution
The equation for bridge balance shows that Rx can be determined in terms of the resistance of
the three decade boxes and Rx = R1xR2/R3, where R1, R2, and R3 are the resistances of the
decade boxes, guaranteed to ±0.1 per cent. One must recognize that the two terms in the
numerator may both be positive to the full limit of 0.1 per cent and the denominator may be
negative to the full 0.1 per cent, giving a resultant error of 0.3 per cent. The guarantee error is
thus obtained by taking the direct sum of all the possible errors, adopting the algebraic signs
which give the worst possible combination.
As a further example, using the relationship P = I2R, as shown in Example 14, consider
computing the power dissipation in a resistor.
Example 14
The current passing through a resistor of 100 ± 0.2 is 2.00 ± 0.01 A. Using the relationship P =
I2R, calculate the limiting error in the computed value of power dissipation.
Solution
Expressing the guaranteed limits of both current and resistance in percentages instead of units,
we obtain
I = 2.00 ± 0.01 A = 2.00 A ± 0.5%
R = 100 ± 0.2 = 100 ± 0.2%
If the worst possible combination of errors is used, the limiting error in the power dissipation is
(P = I2R):
(2 × 0.5%) + 0.2% = 1.2%
Power dissipation should then be written as follows:
P = I2R = (2.00)2 × 100 = 400 W± 1.2% = 400 ± 4.8 W
18
Common symbols used in metrology.
Continuation
19
Module 2
Measuring instruments: classification; torque generation, controlling and damping; PMMC, MI
and electrodynamometer ammeters and voltmeters; electrodynamometer, induction types in
wattmeters (1 and 3 phase); induction energy meter, compensation, creeping errors testing;
frequency meter: vibration reed type, electrical resonance.
MEASURING INSTRUMENTS
Definition of instruments
A Measuring instrument is a device in which we can determine the magnitude or value of the
quantity to be measured. The measuring quantity can be voltage, current, power and energy etc.
Generally instruments are classified in to two categories.
Electrical Measuring Instruments
Basically there are three types of measuring instruments and these are:
a. Electrical measuring instruments
b. Mechanical measuring instruments.
c. Electronic measuring instruments
In this section we are interested in electrical measuring instruments that measure electrical
quantities like power factor, power, voltage, current, frequency etc.
Broadly speaking electrical measuring instruments are divided into Absolute and secondary
instruments
Absolute and secondary instrument
Absolute instruments are those that give the output of the measurements in terms of parameters
of the instruments that have to be inserted in formulas to obtain the value of the quantity that
was been measured whereas the secondary instruments are those that give out the measured
parameter directly. Tangent galvanometer give the current measurements in terms of tangent of
the angles of the pointers deflections, is the most common type of an absolute instrument while
ammeter, voltmeter and ohmmeter are examples of secondary instruments. Practically secondary
instruments are suitable for measurement while absolute instrument are mostly used for lab
experiments and calibrating other instruments. Absolute instrument have no history of been
calibrated but secondary instruments are calibrated by using absolute instruments. Secondary
instrument deflection is meaningless if the instrument is not calibrated. Secondary measuring
instruments can either be analog of digital.
Analog Instruments
These instruments can be classifies according to the following factors:
1. Quantity to be measured
I.
Current, Amperes, Instrument –Ammeter
II.
Resistance, Ohms, instrument- Ohmmeter or Megga
III.
Voltage, Volts, Instrument-voltmeter
IV.
Active power, Watts, Instrument Wattmeter
V. Reactive power, Vars, Instrument, Varmeter
VI.
Energy, Watt-hour, instrument, Energymeter
2. Working principle: to operate, all instruments depend on many physical effect of
electrical current and magnetism.
I.
Magnetic effect ( ammeters and voltmeters)
II.
Eletromagmetic field of attraction or repulsion (ammeters, voltmeters, wattmeters,
watt-hour meters
20
III.
Electrostatic effect ( voltmeters only
IV.
Induction effect (ammeter, voltmeter, wattmeter, watt-hour meter)
V. Heating effect (ammeters, voltmeters)
3. Representation
I.
Indicating instrument
II.
Recording instrument
III.
Integrating instrument
IV.
Electromechanical indicating instrument
V. Null detector
Indicating instrument
This instrument uses a dial and pointer to determine the value of measuring quantity. The pointer
Indication gives the magnitude of measuring quantity. Its give reading only when connected to
the circuit with power supplied. All it does is to display the value using a point. It has no data
storage capacity.
The Permanent Magnet Moving Coil (PMMC) technology can be used to make indicating analog
Ammeters and Voltmeters. The Electro-Magnetic Moving Coil set-up is used also for ammeter,
voltmeter and also for Wattmeters.
Moving Iron phenomenon is used for indicating ammeters and voltmeters
Ohmmeters and megga can also be indicating instruments.
Recording instrument
This type of instruments records the magnitude of the quantity to be measured continuously over
a specified period of time. It also indicates the magnitude on its scale. Such instruments may find
their applications in the medical field to monitoring patients (eg electro-cardiograph), in aviation
as black boxes (flight and voice recorders) for recording various quantities. monitoring of
earthquakes (seismograph), Cathode Ray Oscilloscope (CRO).
Integrating instrument
This type of instrument gives the total amount of the quantity to be measured over a specified
period of time. It has display and recording and other capacities depend of the requirements.
Examples of such meters include Energy-meters, which can record and indicate how power is
been consumed.
Null Detector Type of Instruments
In opposite to deflection type of instruments, the null or zero type electrical measuring
instruments tend to maintain the position of pointer stationary. They maintain the position of the
pointer stationary by producing opposing effect. Thus for the operation of null type instruments
following steps are required:
1. Value of opposite effect should be known in order to calculate the value of unknown
quantity.
2. Detector shows accurately the balance and the unbalance condition accurately.
The detector should also have the means for restoring force.
Let us look at the advantages and disadvantages of deflection and null type of measuring
instruments:
1. Deflection type of instruments is less accurate than the null type of instruments. It is
because, in the null deflecting instruments the opposing effect is calibrated with the high
degree of accuracy while the calibration of the deflection type instruments depends on the
value of instrument constant hence usually not having high degree of accuracy.
2. Null point type instruments are more sensitive than the Deflection type instruments.
21
3. Deflection type instruments are more suitable under dynamic conditions than null type of
instruments as the intrinsic responses of the null type instruments are slower than
deflection type instruments.
Electromechanical indicating instrument
The main difference between indicating and electrochemically (and electronic, electrochemical
and electronic) indicating instrument is that to the EMMC is added an electronic circuit which
increases the instrument sensitivity and output impedance. For satisfactory operation of
electromechanical indicating and indicating instruments, three forces are necessary i.e.
Deflecting, Controlling and Damping forces.
1. Deflecting force (torque)(Td): this is some time called the operation torque. It
can be generated by magnetic, electro-magnetic, electrodynamic (dynamometer) ,
electrostatic, hot wire and other effects of using current and voltage. The process
of producing the deflecting torque depends of the type of instrument.
2. Controlling force (TC): This is the opposing force to the deflection force and
increases with an increase in the deflection force. The pointer comes to rest at a
position where the deflection and controlling forces are equal. The controlling
torque ensures that current applied to the instrument produces the deflection
proportional to its size. Without the controlling torque, the pointer would swing to
the maximum regardless of the magnitude of the current applied. Furthermore
without the controlling torque, one the pointer swings to the maximum when
some current is applied, it would remain there even after the applied current is
disconnected, as there is no force to push the pointer back to zero. Controlling
force can be produced using spring and gravitational forces.
Spring controlled
In these instruments the spring made of phosphor-bronze is attached to the
moving system (pointer). When the pointer moves the then spring twist in the
opposite direction, thus producing a force opposing the deflection force that is
causing the pointer to move
In PMMC instruments, the deflection torque produced is proportional to the
current passing through them i.e 𝑇𝑑 ∝ 𝐼, and for spring control 𝑇𝑐 ∝ 𝜃 𝑎𝑠 𝑇𝑐 =
𝑇𝑑 , ∴ 𝜃 ∝ 𝐼
Spring controlled instruments have a uniform or equally spaced scale over the entire
range because the deflection angle is proportional to the current passing in the
instrument i.e 𝜃 ∝ 𝐼.
22
Fig shows deflection type permanent magnet moving coil ammeter.
To ensure that the controlling torque is promotional to the angle of deflection, the
spring should have a fairly large number of turns so that angular deformation per unit
length , of a full scale deflection is small.
A good spring should have the fooling characteristic
 Non magnetic
 Not prone to fatigue
 Have low resistance
 Low temperature coefficient
Gravity controlled
Gravity controlled is obtained by attaching small adjustable weights to the
moving system such that the weights exert torque in opposite directions; such an
arrangement is shown in the figure above. In this type, the controlling torque is
proportional to the sine of the angle of deflection .i.e 𝑇𝑐 ∝ sin 𝜃. The deflection
torque is proportional to the current in its .i.e 𝑇𝑑 ∝ 𝐼. The pointer come to rest
where 𝑇𝑑 = 𝑇𝑐 , ∴ 𝐼 ∝ sin 𝜃
Gravity controlled instruments do have a uniform or equally spaced scale over the
entire range because the sine of deflection angle is proportional to the current passing
in the instrument i.e sin 𝜃 ∝ 𝐼. Meaning that when the deflection and approaches 90o
the graduations of the scale increases while at the beging the scale was cramped up or
crowded
Comparisons of spring and gravity control
 Gravity control scale not uniform
 Gravity control instrument to be kept vertical
23



Gravity control cheaper
Gravity control not affected by temperature
Gravity not affected by fatigue or deterioration with time
3. Damping force: This is the force that acts on the moving system of the
instrument on when it is moving and opposes its motion regardless of the
direction, thus bring the pointer to rest quickly. Without the damping force the
pointer would oscillate about its final deflection position for longer time before
coming to rest. The degree of damping set to a value than enable the pointer to
rise quickly from the initial position and the deflected position without
overshooting. Over-damping makes the instrument slow and be out of calibration.
Damping can be produced by air friction, eddy currents and fluid friction.
Air friction damping
The piston is mechanically connected to a spindle through the connecting rod (see immediate
Fig. below). The pointer is fixed to the spindle and moves over a calibrated dial. When the
pointer oscillates about on the dial, the piston also moves about in the cylinder and air in it gets
compressed and reduced, thus oppose the motion of the pointer
Fluid friction damping is similar to air friction excerpt that in this type of damping oil is used
in place of air. This is because oil has a greater viscosity and provides a more effective damping
than air. However oil damping is not much used because such instruments may be associated
with oil creeping, such instruments are supposed to be used in the vertical position ans are not
portable.
1.6.2 Eddy current damping
Eddy current damping is the most effective of the three methods. Two types of eddy current
damping are in figure 10.6 and 10.7, in 10.6 (a) is a disk type, in which a thing aluminum
(conducting but non magnetic material) circular disc is fixed to the spindle . This disc is made to
move in the magnetic field produced by a permanent magnet as the pointer moves. The disk is
positioned such that when it is rotating it cuts the magnetic flux of the magnet; hence eddy
currents are produced in the disk flows and produces the damping force in accordance with
Lenz’s law.
24
Figure 10.7 shows the second type of eddy current damping . this kind is mostly used in PMMC
instruments. The coil is wound on a thing light aluminium former in which the sddy currents are
produced when the coil moves in the field magnet. The direction of the induced current and the
damping force produced are shown in the figure.
When the
Example
The torque of an ammeter varies as a square of the current through it. If a current of 5
A produces a deflection of about 90°, what deflection will occur for a current of 3 A
when the instrument is (a) spring controlled (b) gravity controlled.
Solution
Since the defletion torque varies as a (current)2 ,we have 𝑇𝑑 ∝ 𝐼 2
For spring control, 𝑇𝑐 ∝ 𝜃, ∴ 𝜃 ∝ 𝐼 2
For gravity control 𝑇𝑐 ∝ sin 𝜃 , ∴ sin 𝜃 ∝ 𝐼 2
I.
II.
𝟑𝟐
For gravity control: 𝟗𝟎° ∝ 𝟓𝟐 𝒂𝒏𝒅 𝜽 ∝ 𝟑𝟐 , 𝜽 = 𝟗𝟎° ∙ 𝟓𝟐 = 𝟑𝟐. 𝟒°
For gravity control:𝒔𝒊𝒏 𝟗𝟎° ∝ 𝟓𝟐 𝒂𝒏𝒅 𝒔𝒊𝒏 𝜽 ∝ 𝟑𝟐 , 𝒔𝒊𝒏 𝜽 =
𝟗
𝟐𝟓
= 𝟐𝟏. 𝟏
1.6 Damping force
The deflection torque and controlling torque produced by systems are electro mechanical.
Due to inertia produced by this system, the pointer oscillates about it final steady position before
coming to rest. The time required to take the measurement is more. To damp out the oscillation
is quickly, a damping force is necessary. This force is produced by different systems.
(a) Air friction damping
(b) Fluid friction damping
(c) Eddy current damping
1.6.1 Air friction damping
25
The piston is mechanically connected to a spindle through the connecting rod (Fig. 1.6). The
pointer is fixed to the spindle moves over a calibrated dial. When the pointer oscillates in
clockwise direction, the piston goes inside and the cylinder gets compressed. The air pushes the
piston upwards and the pointer tends to move in anticlockwise direction.
Fig. 1.6
If the pointer oscillates in anticlockwise direction the piston moves away and the pressure of the
air inside cylinder gets reduced. The external pressure is more than that of the internal pressure.
Therefore the piston moves down wards. The pointer tends to move in clock wise direction.
1.6.2 Eddy current damping
Fig. 1.6 Disc type
An aluminum circular disc is fixed to the spindle (Fig. 1.6). This disc is made to move in the
magnetic field produced by a permanent magnet.
When the disc oscillates it cuts the magnetic flux produced by damping magnet. An emf is
induced in the circular disc by faradays law. Eddy currents are established in the disc since it has
several closed paths. By Lenz’s law, the current carrying disc produced a force in a direction
opposite to oscillating force. The damping force can be varied by varying the projection of the
magnet over the circular disc.
Fig. 1.6 Rectangular type
26
1.7 Permanent Magnet Moving Coil (PMMC) instrument
One of the most accurate type of instrument used for D.C. measurements is PMMC instrument.
Construction: A permanent magnet is used in this type instrument. Aluminum former is
provided in the cylindrical in between two poles of the permanent magnet (Fig. 1.7). Coils are
wound on the aluminum former which is connected with the spindle. This spindle is supported
with jeweled bearing. Two springs are attached on either end of the spindle. The terminals of the
moving coils are connected to the spring. Therefore the current flows through spring 1, moving
coil and spring 2.
Damping: Eddy current damping is used. This is produced by aluminum former.
Control: Spring control is used.
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
16
Fig. 1.7
Principle of operation
When D.C. supply is given to the moving coil, D.C. current flows through it. When the current
carrying coil is kept in the magnetic field, it experiences a force. This force produces a torque
and the former rotates. The pointer is attached with the spindle. When the former rotates, the
pointer moves over the calibrated scale. When the polarity is reversed a torque is produced in the
opposite direction. The mechanical stopper does not allow the deflection in the opposite
direction. Therefore the polarity should be maintained with PMMC instrument.
If A.C. is supplied, a reversing torque is produced. This cannot produce a continuous deflection.
Therefore this instrument cannot be used in A.C.
Torque developed by PMMC
Let Td =deflecting torque
TC = controlling torque
= angle of deflection
K=spring constant
b=width of the coil
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l=height of the coil or length of coil
N=No. of turns
I=current
B=Flux density
A=area of the coil
The force produced in the coil is given by
F BIL sin(1.4)
When
90
For N turns, F NBIL (1.5)
Torque produced Td Fr distance (1.6)
Td NBIL b BINA 
(1.7)
Td BANI 
(1.8)
Td I 
(1.9)
Advantages
Torque/weight is high
Power consumption is less
27
Scale is uniform
Damping is very effective
Since operating field is very strong, the effect of stray field is negligible
Range of instrument can be extended
Disadvantages
Use only for D.C.
Cost is high
Error is produced due to ageing effect of PMMC
Friction and temperature error are present
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1.7.1 Extension of range of PMMC instrument
Case-I: Shunt
A low shunt resistance connected in parrel with the ammeter to extent the range of current. Large
current can be measured using low current rated ammeter by using a shunt.
Fig. 1.8
Let Rm =Resistance of meter
Rsh=Resistance of shunt
Im = Current through meter
Ish =current through shunt
I= current to be measure
Vm Vsh
(1.10)
ImRm I shRsh
m
sh
sh
m
R
R
I
I (1.11)
Apply KCL at ‘P’ I Im Ish
(1.12)
Eqn (1.12) ÷ by Im
m
sh
mI
I
I
I 1
(1.13)
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sh
m
R
R
I
I 1
(1.14)



m
28




sh
m
m
R
R
II1
(1.15)







sh
m
R
R
1 is called multiplication factor
Shunt resistance is made of manganin. This has least thermoelectric emf. The change is
resistance, due to change in temperature is negligible.
Case (II): Multiplier
A large resistance is connected in series with voltmeter is called multiplier (Fig. 1.9). A large
voltage can be measured using a voltmeter of small rating with a multiplier.
Fig. 1.9
Let Rm =resistance of meter
Rse =resistance of multiplier
Vm =Voltage across meter
Vse= Voltage across series resistance
V= voltage to be measured
Im Ise
(1.16)
se
se
m
m
R
V
R
V 
(1.17)
m
se
m
se
R
R
V
V 
29
(1.18)
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Apply KVL, V Vm Vse
(1.19)
Eqn (1.19) ÷Vm







m
se
m
se
mR
R
V
V
V
V
11
(1.20)







m
se
m
R
R
VV1
(1.21)







m
se
R
R
1 Multiplication factor
1.8 Moving Iron (MI) instruments
30
One of the most accurate instrument used for both AC and DC measurement is moving iron
instrument. There are two types of moving iron instrument.
Attraction type
Repulsion type
1.8.1 Attraction type M.I. instrument
Construction: The moving iron fixed to the spindle is kept near the hollow fixed coil (Fig.
1.10).
The pointer and balance weight are attached to the spindle, which is supported with jeweled
bearing. Here air friction damping is used.
Principle of operation
The current to be measured is passed through the fixed coil. As the current is flow through the
fixed coil, a magnetic field is produced. By magnetic induction the moving iron gets magnetized.
The north pole of moving coil is attracted by the south pole of fixed coil. Thus the deflecting
force is produced due to force of attraction. Since the moving iron is attached with the spindle,
the spindle rotates and the pointer moves over the calibrated scale. But the force of attraction
depends on the current flowing through the coil.
Torque developed by M.I
Let ‘’ be the deflection corresponding to a current of ‘i’ amp
Let the current increases by di, the corresponding deflection is ‘d’
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Fig. 1.10
There is change in inductance since the position of moving iron change w.r.t the fixed
electromagnets.
Let the new inductance value be ‘L+dL’. The current change by ‘di’ is dt seconds.
Let the emf induced in the coil be ‘e’ volt.
dt
dL
i
dt
di
Li L
dt
d
e ( ) 
(1.22)
Multiplying by ‘idt’ in equation (1.22)
idt
dt
dL
idt i
dt
di
eidt L 
(1.23)
e idt Lidi i dL 2 (1.24)
Eqn (1.24) gives the energy is used in to two forms. Part of energy is stored in the inductance.
Remaining energy is converted in to mechanical energy which produces deflection.
Fig. 1.11
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Change in energy stored=Final energy-initial energy stored
31
22
2
1
( )( )
2
1
L dL i di  Li
{( )( 2 ) }
2
1 2 2 2 L dL i di idi  Li
{( )( 2 ) }
2
1 2 2 L dL i idi  Li
{22}
2
1 2 2 2 Li Lidi i dL ididL  Li
{2 }
2
1 2Lidi i dL
Lidi i dL 2
2
1 
(1.25)
Mechanical work to move the pointer by d
Td d
(1.26)
By law of conservation of energy,
Electrical energy supplied=Increase in stored energy+ mechanical work done.
Input energy= Energy stored + Mechanical energy
Lidi i dL Lidi i dL Td d2 2 
2
1
(1.27)
i dL Td d2 
2
1
(1.28)
d
dL
Tid
2
2
1 
(1.29)
At steady state condition Td TC


K
d
dL
i2 
32
2
1
(1.30)


d
dL
i
K
2
2
1 
(1.31)
2 i (1.32)
When the instruments measure AC, i rms 2
Scale of the instrument is non uniform.
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Advantages
MI can be used in AC and DC
It is cheap
Supply is given to a fixed coil, not in moving coil.
Simple construction
Less friction error.
Disadvantages
It suffers from eddy current and hysteresis error
Scale is not uniform
It consumed more power
Calibration is different for AC and DC operation
1.8.2 Repulsion type moving iron instrument
Construction: The repulsion type instrument has a hollow fixed iron attached to it (Fig. 1.12).
The moving iron is connected to the spindle. The pointer is also attached to the spindle in
supported with jeweled bearing.
Principle of operation: When the current flows through the coil, a magnetic field is produced
by
it. So both fixed iron and moving iron are magnetized with the same polarity, since they are kept
in the same magnetic field. Similar poles of fixed and moving iron get repelled. Thus the
deflecting torque is produced due to magnetic repulsion. Since moving iron is attached to
spindle, the spindle will move. So that pointer moves over the calibrated scale.
Damping: Air friction damping is used to reduce the oscillation.
Control: Spring control is used.
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Fig. 1.12
1.9 Dynamometer (or) Electromagnetic moving coil instrument (EMMC)
Fig. 1.13
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This instrument can be used for the measurement of voltage, current and power. The difference
between the PMMC and dynamometer type instrument is that the permanent magnet is replaced
by an electromagnet.
33
Construction: A fixed coil is divided in to two equal half. The moving coil is placed between
the
two half of the fixed coil. Both the fixed and moving coils are air cored. So that the hysteresis
effect will be zero. The pointer is attached with the spindle. In a non metallic former the moving
coil is wounded.
Control: Spring control is used.
Damping: Air friction damping is used.
Principle of operation:
When the current flows through the fixed coil, it produced a magnetic field, whose flux density
is
proportional to the current through the fixed coil. The moving coil is kept in between the fixed
coil. When the current passes through the moving coil, a magnetic field is produced by this coil.
The magnetic poles are produced in such a way that the torque produced on the moving coil
deflects the pointer over the calibrated scale. This instrument works on AC and DC. When AC
voltage is applied, alternating current flows through the fixed coil and moving coil. When the
current in the fixed coil reverses, the current in the moving coil also reverses. Torque remains in
the same direction. Since the current i1 and i2 reverse simultaneously. This is because the fixed
and moving coils are either connected in series or parallel.
Torque developed by EMMC
Fig. 1.14
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Let
L1=Self inductance of fixed coil
L2= Self inductance of moving coil
M=mutual inductance between fixed coil and moving coil
i1=current through fixed coil
i2=current through moving coil
Total inductance of system,
Ltotal L1 L2 2M 
(1.33)
But we know that in case of M.I
d
dL
Tid
()
2
1 2
(1.34)
(2)
2
1
12
LLM
d
d
T i d 

(1.35)
The value of L1 and L2 are independent of ‘’ but ‘M’ varies with 
d
dM
2
34
T i d2
2
1 2 
(1.36)
d
dM
Tid
2
(1.37)
If the coils are not connected in series i1 i2
d
dM
T i i d 1 2 
(1.38)
TC Td
(1.39)
(1.40)
Hence the deflection of pointer is proportional to the current passing through fixed coil and
moving coil.


d
dM
K
i i1 2
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1.9.1 Extension of EMMC instrument
Case-I Ammeter connection
Fixed coil and moving coil are connected in parallel for ammeter connection. The coils are
designed such that the resistance of each branch is same.
Therefore
I I I 1 2
Fig. 1.15
To extend the range of current a shunt may be connected in parallel with the meter. The value
Rsh is designed such that equal current flows through moving coil and fixed coil.
d
dM
Td I1I2 
(1.41)
Or
d
dM
Td I
2
(1.42)
TC K
(1.43)


35
d
dM
K
I2

(1.44)
2 I (Scale is not uniform) (1.45)
Case-II Voltmeter connection
Fixed coil and moving coil are connected in series for voltmeter connection. A multiplier may be
connected in series to extent the range of voltmeter.
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Fig. 1.16
2
2
2
1
1
1,
Z
V
I
Z
V
I 
(1.46)
d
dM
Z
V
Z
V
Td 
2
2
1
1
(1.47)
d
dM
Z
KV
Z
KV
Td 
2
2
1
1
(1.48)
d
dM
ZZ
KV
36
Td 
12
2
(1.49)
2 Td V 
(1.50)
2 V (Scale in not uniform) (1.51)
Case-III As wattmeter
When the two coils are connected to parallel, the instrument can be used as a wattmeter. Fixed
coil is connected in series with the load. Moving coil is connected in parallel with the load. The
moving coil is known as voltage coil or pressure coil and fixed coil is known as current coil.
Fig. 1.17
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Assume that the supply voltage is sinusoidal. If the impedance of the coil is neglected in
comparison with the resistance ‘R’. The current,
R
v wt
I m sin
2 
(1.52)
Let the phase difference between the currents I1 and I2 is 
1 sin( ) I Im wt  
(1.53)
d
dM
Td I1I2 
(1.54)


d
dM
R
V wt
T I wt m
dm
sin
sin(  ) 
(1.55)


d
dM
I V wt wt
R
T d m m ( sin sin( ))
1  
(1.56)


d
37
dM
I V wt wt
R
T d m m sin .sin( )
1  
(1.57)
The average deflecting torque
Td avg Td dwt 


2
20
1
()
(1.58)
dwt 
d
dM
I V wt wt
R
T d avg m m  


2
0
sin .sin( )
1
2
1
()


(1.59)






  

wt dwt
d
dM
R
VI
T mm
d avg {cos cos(2 )}
1
22
38
( ) 

(1.60)




  



dwt wt dwt
d
dM
R
VI
T mm
d avg cos . cos(2 ).
4
()
2
0
2
0


(1.61)
 


2
0 cos
4
( ) wt
d
dM
R
VI
T mm
d avg 

(1.62)
cos (2 0)
4
( )  


d
dM
R
VI
39
T mm
d avg
(1.63)


cos
1
2
( ) 
d
dM
R
VI
T mm
d avg
(1.64)


d
dM
R
Td avg Vrms Irms 1
( ) cos
(1.65)
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(Td )avg KVI cos
(1.66)
TC
(1.67)
KVI cos(1.68)
VI cos(1.69)
Advantages
It can be used for voltmeter, ammeter and wattmeter
Hysteresis error is nill
Eddy current error is nill
Damping is effective
It can be measure correctively and accurately the rms value of the voltage
Disadvantages
Scale is not uniform
Power consumption is high(because of high resistance )
Cost is more
Error is produced due to frequency, temperature and stray field.
Torque/weight is low.(Because field strength is very low)
Errors in PMMC
The permanent magnet produced error due to ageing effect. By heat treatment, this error
can be eliminated.
The spring produces error due to ageing effect. By heat treating the spring the error can
be eliminated.
When the temperature changes, the resistance of the coil vary and the spring also
produces error in deflection. This error can be minimized by using a spring whose
temperature co-efficient is very low.
40
1.10 Difference between attraction and repulsion type instrument
An attraction type instrument will usually have a lower inductance, compare to repulsion type
instrument. But in other hand, repulsion type instruments are more suitable for economical
production in manufacture and nearly uniform scale is more easily obtained. They are therefore
much more common than attraction type.
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1.11 Characteristics of meter
1.11.1 Full scale deflection current( IFSD )
The current required to bring the pointer to full-scale or extreme right side of the
instrument is called full scale deflection current. It must be as small as possible. Typical value is
between 2A to 30mA.
1.11.2 Resistance of the coil( Rm )
This is ohmic resistance of the moving coil. It is due to , L and A. For an ammeter this should
be as small as possible.
1.11.3 Sensitivity of the meter(S)
V
Z
volt S
I
S
FSD

(/ ),
1
It is also called ohms/volt rating of the instrument. Larger the sensitivity of an instrument, more
accurate is the instrument. It is measured in Ω/volt. When the sensitivity is high, the impedance
of meter is high. Hence it draws less current and loading affect is negligible. It is also defend as
one over full scale deflection current.
1.12 Error in M.I instrument
1.12.1 Temperature error
Due to temperature variation, the resistance of the coil varies. This affects the deflection of the
instrument. The coil should be made of manganin, so that the resistance is almost constant.
1.12.2 Hysteresis error
Due to hysteresis affect the reading of the instrument will not be correct. When the current is
decreasing, the flux produced will not decrease suddenly. Due to this the meter reads a higher
value of current. Similarly when the current increases the meter reads a lower value of current.
This produces error in deflection. This error can be eliminated using small iron parts with narrow
hysteresis loop so that the demagnetization takes place very quickly.
1.12.3 Eddy current error
The eddy currents induced in the moving iron affect the deflection. This error can be reduced by
increasing the resistance of the iron.
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1.12.4 Stray field error
Since the operating field is weak, the effect of stray field is more. Due to this, error is produced
in deflection. This can be eliminated by shielding the parts of the instrument.
1.12.5 Frequency error
When the frequency changes the reactance of the coil changes.
2 2 Z (Rm RS ) XL
(1.70)
2 2 (Rm RS ) XL
41
V
Z
V
I


(1.71)
Fig. 1.18
Deflection of moving iron voltmeter depends upon the current through the coil. Therefore,
deflection for a given voltage will be less at higher frequency than at low frequency. A capacitor
is connected in parallel with multiplier resistance. The net reactance, ( X L  XC ) is very small,
when compared to the series resistance. Thus the circuit impedance is made independent of
frequency. This is because of the circuit is almost resistive.
2()
0.41
RS
L
C 
(1.72)
1.13 Electrostatic instrument
In multi cellular construction several vans and quadrants are provided. The voltage is to be
measured is applied between the vanes and quadrant. The force of attraction between the vanes
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and quadrant produces a deflecting torque. Controlling torque is produced by spring control. Air
friction damping is used.
The instrument is generally used for measuring medium and high voltage. The voltage is reduced
to low value by using capacitor potential divider. The force of attraction is proportional to the
square of the voltage.
Fig. 1.19
Torque develop by electrostatic instrument
V=Voltage applied between vane and quadrant
C=capacitance between vane and quadrant
Energy stored= 2
2
1
CV
(1.73)
Let ‘’ be the deflection corresponding to a voltage V.
Let the voltage increases by dv, the corresponding deflection is’d’
When the voltage is being increased, a capacitive current flows
dt
dV
VC
dt
dC
dt
d CV
dt
dq
i ( )
(1.74)
42
V dt multiply on both side of equation (1.74)
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
34
Fig. 1.20
dt
dt
dV
V dt CV
dt
dC
Vidt 2 
(1.75)
Vidt V 2dC CVdV
(1.76)
Change in stored energy= 2 2
2
1
( )( )
2
1
C dC V dV  CV
(1.77)


V dC CVdV
CV CdV CVdV V dC dCdV VdVdC CV
C dC V dV VdV CV

 
 
2
22222
222
2
1
2
1
22
2
1
2
1
()2
2
1
V 2dC CVdV V 2dC CVdV F rd
2
1
(1.78)
Td d V dC
43
2
2
1 
(1.79)





d
dC
Td V
2
2
1
(1.80)
At steady state condition,Td TC






d
dC
KV2
2
1
(1.81)







d
dC
V
K
2
2
1
(1.82)
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
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Advantages
It is used in both AC and DC.
There is no frequency error.
There is no hysteresis error.
There is no stray magnetic field error. Because the instrument works on electrostatic
44
principle.
It is used for high voltage
Power consumption is negligible.
Disadvantages
Scale is not uniform
Large in size
Cost is more
1.14 Multi range Ammeter
When the switch is connected to position (1), the supplied current I1
Fig. 1.21
Ish Rsh ImRm 1 1
(1.83)
m
mm
sh
mm
sh
II
IR
I
IR
R


11
1
(1.84)
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
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




m
m
sh
m
m
sh
I
I
m
m
R
R
I
I
R
R1
1
1
1
1
45
1,
1
,
1
Multiplying power of shunt
21
2 

m
R
Rm
sh ,
Im
I
m2
2 
(1.85)
31
3 

m
R
Rm
sh ,
Im
I
m3
3 
(1.86)
41
4 

m
R
Rm
sh ,
Im
I
m4
4 
(1.87)
1.15 Ayrton shunt
R1 Rsh1  Rsh2
(1.88)
R2 Rsh2  Rsh3
(1.89)
R3 Rsh3  Rsh4
(1.90)
R4 Rsh4
(1.91)
Fig. 1.22
46
Ayrton shunt is also called universal shunt. Ayrton shunt has more sections of resistance. Taps
are brought out from various points of the resistor. The variable points in the o/p can be
connected to any position. Various meters require different types of shunts. The Aryton shunt is
used in the lab, so that any value of resistance between minimum and maximum specified can be
used. It eliminates the possibility of having the meter in the circuit without a shunt.
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
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1.16 Multi range D.C. voltmeter
Fig. 1.23
( 1)
( 1)
( 1)
33
22
11
 
 
 
RRm
RRm
RRm
sm
sm
sm
(1.92)
m m Vm
V
m
V
V
m
V
V
m3
3
2
2
1
1 ,
, 
(1.93)
We can obtain different Voltage ranges by connecting different value of multiplier resistor in
series with the meter. The number of these resistors is equal to the number of ranges required.
1.17 Potential divider arrangement
The resistance R1,R2,R3 and R4 is connected in series to obtained the ranges V1,V2,V3 and V4
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
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Fig. 1.24
Consider for voltage V1, (R1 Rm)Im V1 
mm
m
m
m
m
m
m
47
RR
V
V
R
R
V
V
R
I
V
R  





1  1  1
1
()
(1.94)
R1 (m1 1)Rm  
(1.95)
For V2 , m
m
mmR
R
I
V
R R R I V R   1
2
(21)22
(1.96)
mm
m
m
mRR
R
V
V
R    






( 1 1)
2
2
(1.97)
R2 m2Rm Rm (m1 1)Rm    
( 2 1 1 1) Rm m   m 
(1.98)
48
R2 (m2 m1)Rm  
(1.99)
For V3 R3 R2 R1 Rm Im V3 
m
m
RRR
I
V
R    2 1
3
3
mmmm
mmmm
m
mRmmRmRR
RmmRmRR
V
V
     
     
( ) ( 1)
( ) ( 1)
3211
211
3
R3 (m3 m2)Rm  
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
39
For V4 R4 R3 R2 R1 RmIm V4 
m
m
RRRR
I
V
R     3 2 1
4
4
mmmmm
m
RmmRmmRmRR
V
V        





( 3 2) ( 2 1) ( 1 1)
4

m
m
49
RmmR
RRmmmmmm
443
44322111
1
 
    
Example: 1.1
A PMMC ammeter has the following specification
Coil dimension are 1cm1cm. Spring constant is 0.15 10 N m/ rad  6  
, Flux density is
3 2 1.5
10 wb / m
 .Determine the
no. of turns required to produce a deflection of 900 when a current
2mA flows through the coil.
Solution:
At steady state condition Td TC
BANI K
BAI
K
N


A= 4 2 1 10 m
 
K=
rad
N  m
 6 0.15 10
B= 3 2 1.5 10 wb / m
 
I= A 3 2 10
 
rad
2
90


N=785 ans.
Example: 1.2
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
40
The pointer of a moving coil instrument gives full scale deflection of 20mA. The potential
difference across the meter when carrying 20mA is 400mV.The instrument to be used is 200A
for full scale deflection. Find the shunt resistance required to achieve this, if the instrument to be
used as a voltmeter for full scale reading with 1000V. Find the series resistance to be connected
it?
Solution:
Case-1
Vm =400mV
Im 20mA
I=200A
20
50
20
400
m
m
m
I
V
R







sh
m
m
R
R
II1




 
Rsh
20
200 20 10 1 3
 3Rsh 2 10
Case-II
V=1000V







m
se
m
R
R
VV1




 
20
4000 400 10 1 3 Rse
51
Rse 49.98k
Example: 1.3
A 150 v moving iron voltmeter is intended for 50HZ, has a resistance of 3kΩ. Find the series
resistance required to extent the range of instrument to 300v. If the 300V instrument is used to
measure a d.c. voltage of 200V. Find the voltage across the meter?
Solution:
Rm 3k,Vm 150V,V 300V
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
41







m
se
m
R
R
VV1





R k
R
se
se 3
3
300 150 1
Case-II 






m
se
m
R
R
VV1





52
3
3
200 Vm 1
Vm 100V
Ans
Example: 1.4
What is the value of series resistance to be used to extent ‘0’to 200V range of 20,000Ω/volt
voltmeter to 0 to 2000 volt?
Solution:
Vse V  V 1800
Sensitivity
IFSD
1
20000
1 
Vse Rse iFSD Rse 36M
ans.
Example: 1.5
A moving coil instrument whose resistance is 25Ω gives a full scale deflection with a current of
1mA. This instrument is to be used with a manganin shunt, to extent its range to 100mA.
Calculate the error caused by a 100C rise in temperature when:
(a) Copper moving coil is connected directly across the manganin shunt.
(b) A 75 ohm manganin resistance is used in series with the instrument moving coil.
The temperature co-efficient of copper is 0.004/0C and that of manganin is 0.000150/C.
Solution:
Case-1
Im 1mA
Rm 25
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
42
I=100mA







sh
m
m
R
R
II1







53

0.2525
99
25
99
25 25
100 1 1
sh
sh sh
R
RR
Instrument resistance for 100C rise in temperature, 25(10.00410) Rmt
Rt Ro(1 t t) 
26
m/ t 10
R
Shunt resistance for 100C, rise in temperature
0.2525(10.0001510) 0.2529
sh / t 10
R
Current through the meter for 100mA in the main circuit for 100C rise in temperature
tC
sh
m
m
R
R
I I 






10
1





0.2529
26
100 Imt 1
I mA
m t 0.963
10


But normal meter current=1mA
Error due to rise in temperature=(0.963-1)*100=-3.7%
Case-b As voltmeter
Total resistance in the meter circuit= Rm Rsh 2575 100
54







sh
m
m
R
R
II1







Rsh
100
100 1 1


1.01
100 1
100
Rsh
CLASS NOTES ON ELECTRICAL MEASUREMENTS & INSTRUMENTATION 2015
43
Resistance of the instrument circuit for 100C rise in temperature


25(1 0.004 10) 75(1 0.00015 10) 101.11
m t 10 R
Shunt resistance for 100C rise in temperature


1.01(1 0.00015 10) 1.0115
sh t 10 R







sh
m
m
R
55
R
II1





1.0115
101.11
100 Im 1
Im t 10 0.9905mA 
Error =(0.9905-1)*100=-0.95%
Example: 1.6
The coil of a 600V M.I meter has an inductance of 1 henery. It gives correct reading at 50HZ and
requires 100mA. For its full scale deflection, what is % error in the meter when connected to
200V D.C. by comparing with 200V A.C?
Solution:
Vm 600V, Im 100mA
Case-I A.C.
6000
0.1
600
m
m
m
I
V
Z
XL 2fL 314
 (6000)  (314) 59902 2 2 2
Rm Zm XL
mA
Z
V
I AC
AC 33.33
6000
200 
Case-II D.C
mA
R
V
I
m
DC
DC 33.39
5990
200 
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