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1. ELECTRICAL & ELECTRONICS
MEASUREMENT
1

2. An Overview
2

3. Measurements For Nation’s Progress Is Imperative: Why?
The advancement of Science and Technology is dependent upon
a parallel progress in measurements techniques. The reason is
that the progress in science and technology leads to discovering
new phenomena and relationships, which requires new types of
quantitative measurements.
3

4. Field Of Engineering Applications Of MeasurementSystems
1. Design of equipment and processes.
2. Proper operation and maintenance of equipment and processes.
3. Quality control assurance programs for industrial processes.
4

5. Methods of Measurements
1. Direct Methods: The unknown quantity is directly compared against
a standard, which are common for the measurements of physical
quantities like length, mass and time.
Disadvantages are:
i. Limited accuracy due to human factors.
ii. Less sensitive.
5

6. Methods of Measurements, cont
2. Indirect Methods: The unknown quantity under measurement is
determined via the use of measurement systems as follows:
to be
measured
Quantity
Transducer
For converting
physical quantity
into electric
signal
Signal
Processing
For noise
reduction,
amplification, etc
Measuring
Device
Advantages are:
i. High accuracy and sensitivity can be obtained by using electronic
and digital type instruments.
ii. Availability to use commercial instrument-types at lower costs with
accepted accuracy and sensitivity.
iii. Measurements of non-physical quantities.
6

7. Classification of Instruments
Broadly, instruments are classified into two categories:
1. Absolute instruments: These instruments give the magnitude of the
quantity to be measured in terms of physical constants of the
instruments. For example, Tangent Galvanometer. It is used only in
standard institutions for calibration.
2. Secondary instruments: In these instruments the magnitude of the
quantity to be measured is indicated on graded scale (e.g. analog
instruments) or displayed numerically on screen (e.g. digital
instruments).
7

8. Errors in Measurements
1. TrueValue:
i. It is not possible to determine the “true value” of a quantity by
experimental means. The reason for this is that the positive
deviations from the true value do not equal the negative deviations
and hence do not cancel each other.
ii. In practice the “true value” is measured by a “standard unit”.
8
2. Static (or Absolute) Error A
It is defined as the deviation of the measured value (Am) from its true
one (At), or A  Am At

9. Errors in Measurements
3. Static Correction C
i. It is defined as the correction to be added to the measured value so
as to obtain its true value, or
ii. The true value of an instrument
C  At Am
iii. Generally,
At  A C
9
C  A

10. Errors in Measurements
4. Relative Static Error A
r
i. The percentage static error (%  ) of an instrument=
ii. When At is unknown, the percentage static error (%r) of an
instrument can be expressed as a fraction of the full scale deflection
A
100
At
(f.s.d) as:
iii. The true value At =
Am
1 r
f.s.d
10
r
% 
A
100

11. Example 1:
A voltage has a true value of 1.5 V. An analog indicating instrument with
a scale range of 0-2.5 V shows a voltage of 1.46 V. What are the values of
absolute error and correction. Express the error as a fraction of the true
value and the f.s.d.
Solution 1:
2.5
11
Relative error (expressed as a percentage of f.s.d) 
 0.04
100  1.6%
Absolute error A  Am  At  1.46  1.5  0.04 V
Absolute correction C  A  0.04 V
%Relative error (%r ) 
A

 0.04
100  2.66%
At 1.5

12. Errors in Measurements
5. Limiting Error
The manufacturers have to specify the deviation from the specified value
of a particular quantity in order to enable the purchaser to make proper
selection according to his requirements. The limits of these deviations
from the specified value are defined as limiting errors.
12

13. Example 2:
The inductance of an inductor is specified by a manufacturer as 20 H  5%.
Determine the limits of inductance between which it is guaranteed.
Solution 2:
5
100
13
Limiting value of inductance (A)  Am  A
 Am  r Am  Am 1 r
 201 0.05 20  1 Henery
 0.05
Relative error (r ) 

14. Example 3:
A 10
m
r
A 0-25 A ammeter has a guaranteed accuracy of 1 percent of full
scale reading. The current measured by this instrument is 10A.
Determine the limiting error in percentage.
Solution 3:
The magnitude of limiting error of the instrument,
A  r f .s.d  0.01 25  0.25 amperes
 (of measured value) 
A

0.25
 0.025
Therefore, the current being measured is between the limitsof
10
14
amperes
 %Limiting error 
0.25
100  2.5%
A  Am 1 r  101 0.025 10  0.25

15. Errors in Measurements: Random errors
i. These errors are of variable magnitude and sign and do not obey any
rule.
ii. The presence of random errors becomes evident when different results
are obtained on repeated measurements of one and the same quantity.
iii. The effect of random errors is minimized by measuring the given
quantity many times under the same conditions and calculating the
arithmetic mean of the values obtained.
iv. The problem of random errors is treated mathematically as one of the
probability and statistics.
15

16. be the number of n-repeated measurements,
then the arithmetic mean equals
Standard deviation is
Errors in Measurements: Random errors
Let x1, x2,L,xn
n
n
xi
x  i1
Probable error =
x
n
2
 
 i
i1
n 1
0.6745
n
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17. Static Characteristics of Instruments
1. Accuracy
It is the closeness with which an instrument reading approaches the
true value of the quantity being measured.
Accuracy (in percent) = 100  %r
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18. Static Characteristics of Instruments
2. Sensitivity
i. The static sensitivity of an instrument is the ratio of the magnitude
of the output signal to the magnitude of the input signal or the
quantity to be measured.
ii. Its units are millimeter per micro-ampere, counts per volt, etc.
depending upon the type of input and output signals.
iii. Deflection factor (or Inverse sensitivity) of an instrument is the
reciprocal of the sensitivity of that instrument.
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19. Example 4:
A Wheatstone bridge requires a change of 7  in the unknown arm
of the bridge to produce a change in deflection of 3 mm of the
galvanometer. Determine the sensitivity. Also determine the
deflection factor.
Solution 4:
1 1
Sensitivity 0.429
19
 /mm
 2.33

mm /
7 
Deflection factor 
Sensitivity 
3 mm
0.429

20. 3. Resolution
It is the smallest increment in input (quantity being measured) which can be
detected with certainty by an instrument.
Static Characteristics of Instruments
Example 5:
A moving coil voltmeter has a uniform scale with 100 divisions, the full-scale
reading is 200 V and 1/10 of a scale division can be estimated with a fair
degree of certainty. Determine the resolution of the instrument in volts.
Solution 5:
100
20
Resolution 
1
scale division 
1
 2  0.2 V
10 10
1 scale division 
200
 2V

21. Example 6:
1
9999
21
1
9999
A digital voltmeter has a read-out range from 0 to 9999 counts.
Determine the resolution of the instrument in volts when the full
scale reading is 9.999 V.
Solution 6:
The resolution of this instrument is1count in 9999
count 
Resolution 9.999 volt 103
V 1mV

22. Accuracy versus Precision
i. The term “Precise” means clearly or sharply defined.
ii. Precision is a measure of the reproducibility ( or consistency) of the
measurements, i.e. given a fixed value of a quantity, precision is a
measure of the degree of agreement within a group of measurements.
iii. Consider the measurement of a known voltage of 100 V with a meter.
Five readings are taken, and the indicated values are 104, 103, 105, 103
and 105 V. From these values it is seen that
The instrument cannot be depended on for an accuracy better than 5%,
While a precision of 1% is indicated since the maximum deviation
from the mean reading of 104 V is only 1 V.
100V
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%Limiting error 
 5 V
 100 5%

23. When a number of independent measurements are taken in order to
obtain the best measured value, the result is usually expressed as
arithmetic mean of all readings. The range of doubt or possible error is
the largest deviation from the mean .
Range of Possible Errors
23

24. Example 7:
2
24
Average range of errors 
0.03  0.05
 0.04 A
A set of independent current measurements were recorded as 10.03,
10.10, 10.11 and 10.08 A. Calculate the range of possibleerrors.
Solution 7:
Average current Iav 
I1  I2  I3  I4 
10.03  10.10  10.11 10.08
 10.08 A
4 4
Maximum value of current Imax  10.11 A
Range  Imax  Iav  0.03A
Minimum value of current Imin  10.03 A
Range  Iav  Imin  0.05A

25. Loading Effects
i. Under practical conditions the introduction of any measuring
instrument in a system results, invariably, in extraction of energy
from the system thereby distorting the original signal under
measurement.
ii. This distortion may take the form of attenuation (reduction in
magnitude), waveform distortion, phase shift and many a time all
these undesirable features put together.
iii. The incapability of the system to faithfully measure, record, or
control the input signal (measurand) in undistorted form is called
loading effect.
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26. Loading Effects
1. Loading Effects due to Shunt Connected Instruments
Thevinin
voltage
source o
E
Zo
Output
impedance
ZL
Load
(instrument)
On connecting shunt connected
the actual voltage decreases to
as follows:
E
Eo EL
instrument, whose impedance is ZL ,
IL o
Zo ZL
EL  Eo  Zo IL
And the %loading error in measurement
100
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EL  Eo
Eo
of Eo equals
%Loading error 

27. Example 8:
An oscilloscope (CRO) having an output resistance of 1 M shunted
by 50pF capacitance is connected across a circuit having an output
resistance of 10 k . If the open circuit voltage has 1.0 V-peak for a
sinusoidal AC-source, calculate %loading effect error of the voltage
measured when frequency is: (i) 100kHz , (ii)1MHz
Solution 8:
1.0
27
%Loading error
Z
at 100kHz
o
o
ZL
Eo
3

0.954 1.0
100  4.6%
 0.954 17.4 V peak
 j32103
11010
1.00o
3

1
EL 
ZL at 100 kHz
 R // jXc  j32 10 

28. Continue Solution 8:
%Loading error
o

0.3031.0
100  70%
 0.303  72 V peak
 j3183
Zo 10103
ZL
1
Eo 1.00o
1
EL  
ZL at 1MHz
 R // jXc  j3183
1.0
28
at 100kHz

29. Loading Effects
2. Loading Effects due to Series Connected Instruments
Thevinin
voltage
source
Io
Zo
Output
impedance
On connecting series connected
the actual current changes to
as follows:
Io IL
instrument, whose impedance is ZL ,
And the %loading error in measurement
 Io

Zo
Zo  ZL Zo  ZL
IL 
Eo
100
IL  Io
Io
of Io equals
%Loading error 
Thevinin
voltage
source
IL
Zo
Output
impedance
ZL
Load
(instrument)
29

30. Uncertainty Analysis and Propagation of Errors
1. Errors vs. Uncertainty
i. Uncertainty is the range that is likely to contain the deviation of the
measured value from the true value based on random-type errors.
ii. Uncertainty can be expressed in absolute terms or relative terms, just
as error can.
For example, consider a meter stick that indicate centimeter and
millimeter divisions. Thus
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1) The smallest value you can read on this meter stick is 1 mm.
2) Any measurement you make will be quoted to the nearest millimeter.
3) So your answer will take the form 55 mm  1 mm (55 mm is the
measured value & 1 mm is the range of uncertainty).

31. Uncertainty Analysis and Propagation of Errors
2. Propagation of Uncertainties
i. Any calculations done using a measurement will have a degree of
uncertainty. This uncertainty is a measure of how confident you are in
the result of your calculation.
ii. The propagation of the uncertainties through various calculations has
to be carefully considered.
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32. Uncertainty Analysis and Propagation of Errors
3. Propagation of Uncertainties: Sum of Two or More Quantities
Let the final result w be the sum of the measured quantities (x  x) & (y  y)
w  x y
w  x y
or
w

x

x

y

y
w w x w y
Finally, the maximum calculation of w is
w
32
y
x y
w
x
w
      

33. Uncertainty Analysis and Propagation of Errors
4. Propagation of Uncertainties: Difference of Two Quantities
Let the final result w be the difference of the measured quantities
(x  x) & (y  y)
w  x y
w  x y
or
w

x

x

y

y
w w x w y
Finally, the maximum calculation of w is
 
33
y
w
y
x
w
x
w   
  

34. Uncertainty Analysis and Propagation of Errors
5. Propagation of Uncertainties: Product of Two or More Quantities
quantities
Let the final result w be the product of the measured
(x  x) & (y  y)
w  x y
ln w  ln x  ln y
Differentiating w.r.t w yields
1

1

x

1

y
w x w y w
w

x

y
w x y
Finally, the maximum calculation of w is
w  x  y 
34

35. Uncertainty Analysis and Propagation of Errors
6. Propagation of Uncertainties: Division of Two Quantities
w 
x
Let the final result w be the division of the measured quantities
(x  x) & (y  y)
y
ln w  ln x  ln y
Differentiating w.r.t w yields
35
1

1

x

1

y
w x w y w
w

x

y
w x y
Finally, the maximum calculation of w is
w  x  y 

36. Uncertainty Analysis and Propagation of Errors
7. Propagation of Uncertainties: w = f (x , y)
Using Root Sum Square (RSS) method, the error in w is defined by
2


  y
y






x
w 
2
 w 
2
w  x
36

37. 1
2
y
x




2 
w  x y
y
x
Let w 
2
 3
2 
2
 1
2 
&
w
  1 x y
3
2
w
 y 1
2
Example 9:
2
37
y
2
1
2
x
w
2
2
2
 y  2 xy
dividing both sides by w2
yields
2
2
     
2



 1 y
  2 y

 
  
x


  x





x y
2 xy
 1
 y




 x y
 xy
 
 
w
 
 w
 

 
1
2
 3
2 
1
2
 1
2 