UNIT - II

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 CONTENT
U2.1.
Basics of Measurements
• Accuracy
• Precision
• Resolution
• Reliability
• Repeatability
• Validity
• Errors and their analysis
• Standards of measurement
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 Basics of Measurements
Terms

Accuracy
It is the degree of closeness with which
the instrument reading approaches the
true value of the quantity to be measured.

Precision
It refers to how close together a series of
measurements are to each other.

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DifferenceAccuracy
between Vs. Precision
Accuracy and Precision

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 Basics of Measurements contd.
Terms
Resolution
It defines the smallest change in measured quantity that can be
observed.

Example
A moving coil voltmeter has a uniform scale with 100 divisions,
the full scale reading is 200V and 1/10 of a scale division can be
estimated with a fair degree of certainty. Determine the
resolution of the instrument in volt.
Solution: 1 scale division = 200/100 = 2V
Resolution = 1/10 *2V = 0.2V

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 Basics of Measurements contd.
Terms
Reliability
The degree to which an instrument produces consistent results.
Validity
It defines as how accurate an instrument is at measuring what it
is trying to measure.
Repeatability
It is a measure of closeness with which a given input may be
measured over again and again.
How close together repeated values in the experiment.
If the repeats are close together then the measurement has high
repeatability.

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 Basics of Measurements contd.
Errors and their analysis
Errors are classified as:
1) Gross Errors
2) Systematic Errors
3) Random Errors

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 Basics of Measurements contd.
Gross Errors
This type of errors arises due to human mistakes in reading
instruments and recording and calculating measurement
results.
Example: - misreading - 32.5mA as 32.5A
25.8oC as 28.5oC
These errors can be avoided by adopting two means such as:-
(i) Great care should be taken in reading and recording the
data.
(ii) Two, three or even more readings should be taken for the
quantity under measurement.

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 Basics of Measurements contd.
Systematic Errors
These type of errors are divided into three categories: -
(i) Instrumental errors
(ii) Environmental errors
(iii) Observational errors

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 Basics of Measurements contd.
(i) Instrumental Errors
These errors arises due to three main reasons: -
 Due to inherent shortcomings in the instrument,
 Due to misuse of the instruments,
 Due to loading effects of instruments.
Inherent shortcomings
 Due to constructions, calibration, etc.
Example: - spring in permanent magnet instrument has
become weak.
Eliminate: -
 Procedure of measurement must be carefully planned.
 Correction factors should be applied.
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 Basics of Measurements contd.
(i) Instrumental Errors
Misuse of instruments
 Errors caused are due to fault of the operator.
Example: - failure to adjust the zero of instruments,
poor initial adjustments.
Loading effects
 Errors arises due to improper use of an instrument for
measurement.
Example: - a well calibrated voltmeter may give a misleading
voltage reading when connected across high
resistance circuit.

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 Basics of Measurements contd.
(ii) Environmental Errors
 Errors arises due to conditions external to the measuring
devices including conditions in the area surrounding the
instrument.
Example: - temperature, pressure, humidity, dust, vibrations,
etc.
(iii) Observational Errors
 These are the errors introduced by the observer.
Example: - parallax errors, etc.

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 Basics of Measurements contd.
Random Errors
Causes of such errors are unknown and hence the errors are
called random errors.
Example:- Variations of readings from one to another.

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 Statistical Analysis.
Mean
Deviation
Standard Deviation
Variance
Probable Error

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 Problem.
(1)

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 Problem-2
(2) A set of independent current measurement was taken by
six observers and recorded as 12.8mA, 12.2mA, 12.5mA,
13.1mA, 12.9mA and 12.4mA. Calculate (a) the arithmetic
mean (b) the deviations from the mean (c) the average
deviation

Ans: - (a) 12.65mA

(b) 0.15mA, -0.45mA, -0.15mA, 0.45mA, 0.25mA, -0.25mA
(c) 0.283mA

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 Problem-3
(3) Ten measurements of the resistance of a resistor gave
101.2Ω, 101.7Ω, 101.3Ω, 101.0Ω, 101.5Ω, 101.3Ω, 101.2Ω,
101.4Ω, 101.3Ω and 101.1Ω. Assume that only random errors
are present. Calculate (a) the arithmetic mean, (b) the standard
deviation of the readings (c) the probable error.

Ans: - (a) 101.3Ω

(b) 0.2Ω
(c) 0.1349Ω

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 Absolute Error and Relative Error
If a resistor is known to have a resistance of 500 Ω with a
possible error of ±50 Ω, the ±50 Ω is an absolute error.
When the error is expressed as a percentage, it becomes
relative error.
Example: - Relative error
Resistance = 500 Ω ± 10%

Relative error is defined as the

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 Problem – 4
(4) The expected value of the voltage across a resistor is 80V.
However, the measurement gives a value of 79V. Calculate (i)
absolute error (ii) % relative error.
Solution:
Expected Value, (Y) = 80V
Measured Value, (X) = 79V,
(i) Absolute Error = Y – X = 1V
(ii) % relative error = = 1.25%

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 Problem – 5
(5) A batch of resistors that each have a nominal resistance of
330 Ω are to be tested and classified as ±5% and ±10%
components. Calculate the maximum and minimum absolute
resistance for each case.

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 Standards of Measurement
A standard is a physical representation of unit of
measurement.
Example: - The fundamental unit of length is metre in SI
system.

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 Classification of Standards
Standards are classified as of the following types:

Standards
International Primary Secondary Working
Standards Standards Standards Standards

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 Standards of Measurement
(1) International Standards:
The international standard represent certain unit of
measurement.
The different types of international standards are:
(a) International Ohms
(b) International Amperes
(c) Absolute units

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 Standards of Measurement
(a) International Ohms
At the melting point of ice, when a constant current flows
through a column of mercury having a mass of 14.4521 grams,
with uniform cross-sectional area and length of 106.300 cm,
the resistance offered by it is called international ohms.
(b) International amperes
It is a constant current, which when passed through a solution
of silver nitrate in water, deposits silver at the rate of
0.001118000 gm/sec.

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 Standards of Measurement
(c) Absolute units
International units were replaced in 1948 by absolute units as
these units are more accurate than international units.
But it differs slightly from the international units as for
example:
1 international ohm = 1.00049 absolute ohm
1 international ampere = 0.99985 absolute ohm

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 Standards of Measurement
(2) Primary Standards:
The primary standards are mainly used for calibration and
verification of secondary standards.
It is maintained at the National Standards Laboratories in
different countries.

(3) Secondary Standards:

Secondary standards are basic reference standards used by
measurement and calibration laboratories in industries.

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 Standards of Measurement
(4) Working Standards:
Working standards are considered as the principal tools of a
measurement laboratory. These standards are used to check
and calibrate laboratory instrument for accuracy and
performance.

Example: - Manufacturers of capacitor and resistors etc, use

working standard for checking the component values after
manufacturing.

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 WHEATSTONE’S BRIDGE

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 WHEATSTONE’S BRIDGE
Problem
A Wheatstone’s Bridge as shown
in Fig. has P =3.5 kΩ, R = 7 kΩ
and galvanometer null is
obtained when S=5.51 kΩ.
(a) Calculate the value of Q.
(b) Determine the resistance
measurement range for the
bridge if S is adjustable from
1 kΩ to 8 kΩ.

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 WHEATSTONE’S BRIDGE
Problem
Fig shows the schematic
diagram of a Wheatstone bridge
with values of the bridge
elements. The battery voltage is
5V and its internal resistance is
negligible. The galvanometer has
a current sensitivity of 10mm/µA
and internal resistance of 100Ω.
Calculate the deflection of the
galvanometer caused by the 5Ω
unbalance in arm BC.

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 WHEATSTONE’S BRIDGE

Limitations
• It cannot measure resistance of low range (< 1Ω).
• For high resistance measurement, the galvanometer is
insensitive to imbalance.

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 KELVIN’S BRIDGE
Introduction
• It is the modification of
Wheatstone’s bridge.
• This bridge is used to measure low
value resistance (<1Ω).
• The limitation of Wheatstone’s
bridge is removed by connecting
additional resistors.
• The circuit diagram shows the
limitation of Wheatstone’s bridge.
• The voltage drop across the
connecting leads m and n
introduces errors when measuring
low value resistances.
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 KELVIN’S BRIDGE
Circuit Diagram
• The circuit diagram of
Kelvin’s bridge is as shown
in Fig.
• It consists of two additional
resistors p and r.
• If the ratio , the error
due to voltage drop across Y
is eliminated.

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 KELVIN’S BRIDGE
Derivation of Balance Equation
• When the bridge is balanced, the potential difference between the
galvanometer terminals is zero.
• The voltage across R is equal to the sum of the voltage drops
across r and S:
+
from which we have

Or
……………….. (1)

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 KELVIN’S BRIDGE
Derivation of Balance Equation
• Also the voltage drop across P is equal to the sum of the voltage
drops across p and Q:
+
from which we have

Or
……………….. (2)

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 KELVIN’S BRIDGE
Derivation of Balance Equation
• Dividing equation (2) by (1), we get:

and we have or

Therefore giving
In this case Q is the unknown resistance, S is a standard low value
resistor, and P, R, p, and r are precision adjustable resistors.

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 KELVIN’S BRIDGE
Problem
A four terminal resistor with an approximate value of 0.15 Ω is to be
measured by use of a Kelvin bridge. A standard resistor of 0.1 Ω is
available. Determine the required ratio of R/P and r/p.
Solution:
S = 0.1 Ω
Q ≈ 0.15 Ω

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 AC BRIDGES

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 AC BRIDGES
The general form of ac
bridge is as shown:

Z1, Z2, Z3 and Z4 are the

Z2
Impedances.
Z1

D is the detector. AC D
Balance is obtained by
varying one or more bridge Z3
arms. Z4

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 AC BRIDGES
The general equation for
balance of the ac bridge is
given by:
Z2
Z1

or
In terms of admittance, the
AC D
balanced equation can be
written as
Z3 Z4

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 MAXWELL’S BRIDGE
The schematicdiagram of
Maxwell’s bridge is as shown:

Maxwell’s bridge measures

an unknown inductance in
terms of a known
capacitance.

The balanced equation is

given by:

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 MAXWELL’S BRIDGE
We have,

For balanced,

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 MAXWELL’S BRIDGE
Separating real and imaginary terms,

and

The quality factor of the coil is given by

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 PROBLEM

A Maxwell inductance bridge uses a standard capacitor

C1=0.1µF and operates at a supply frequency of 100Hz.
Balance is achieved when R1=470Ω, R2=1.26kΩ, R3=500Ω.
Calculate
(i) the inductance and resistance of the measured inductor,
(ii) Its Q – factor.

Ans: - Lx = 63mH
Rx = 1.34kΩ
Q = 0.03

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 HAY’S BRIDGE

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 SCHERING’S BRIDGE

The schematic diagram of Schering Bridge is as shown:

This bridge is used for measuring

unknown capacitance.

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 SCHERING’S BRIDGE
We have,

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 SCHERING’S BRIDGE
The balanced equation is given by,

or

or

Equating real and imaginary parts we have,

And

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