SSC JE Measurements - wifistudy (1).pdf

SSC-JE Crash Course
(1) Which of the following statement is incorrect?
(A) Uranium is a radioactive material.
(B) Anthracite is the best form of coal.
(C) Solar energy is pollution free.
(D) Wind energy is a conventional source of energy.
(2) Match the following:
1- Load curve (a) Everlasting
2- tidal power generation (b) Variation of load on the reference to the time
3- Non- conventional method (c) Photovoltaic effect
Of power generation
4- Solar cell (d) Fuel-cells generation
(A) 1-b,2-a,3-d,4-c (B) 1-a,2-b,3-c,4-d
(C) 1-c,2-d,3-b,4-a (D) 1-a,2-d,3-c,4-d
SSC JE 2019 (Tier-1), UPPSC- AE-2020, DMRC JE
Electrical & Electronic Engineering Measurements
11-02-2020

Error Analysis
Measurement Error (also called Observational Error) is the difference
between a measured quantity and its true value. It includes random error
(naturally occurring errors that are to be expected with any experiment) and
systematic error (caused by a mis-calibrated instrument that affects all
measurements).

Error
Gross error Systematic error Random error
Instrument
al error
Observational
error
Environmental
error

Gross error: -
These error occurs due to human mistakes.
Environmental error: -
These error occurs due to environmental conditions like temperature, pressure.
Instrumental error: -
These error occurs due to defective parts like springs, magnet, due to loading
effect etc.
Observational error: -
As the name suggests, these types of errors occurs due to wrong observations or
reading in the instruments particularly in case of energy meter reading. The wrong
observations may be due to PARALLAX. In order to reduce the PARALLAX error
highly accurate meters are needed: meters provided with mirror scales.

Random error: -
Random errors are caused by the sudden change in experimental conditions
and noise and tiredness in the working persons. These errors are either
positive or negative. An example of the random errors is during changes in
humidity, unexpected change in temperature and fluctuation in voltage.
These errors may be reduced by taking the average of a large number of
readings.
Limiting error: -
The limited deviation of the measured value from the true value is known as
the limiting error or guarantee error. Such type of error is fixed on the
instrument. The magnitude of the limiting error depends on the design,
material and the workmanship used for the construction of the instrument.

• The component like resistor, inductor and capacitor uses in the instrument
has some rated fixed value. The deviation from their rated value causes the
error in the system. The deviation mainly occurs because of the variation in
the environmental condition. The manufacturer already knew about the
limiting error of the instrument.
• The actual value of the instrument along with the limiting error is
expressed as: -

The relative limiting error,
εr =
𝐌𝐞𝐚𝐬𝐮𝐫𝐞𝐝 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲 − 𝐓𝐫𝐮𝐞 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲
𝐓𝐫𝐮𝐞 𝐪𝐮𝐚𝐧𝐭𝐢𝐭𝐲
× 𝟏𝟎𝟎
% εr =
𝐀𝐦−𝐀𝐓
𝐀𝐓
× 𝟏𝟎𝟎
Error at required value: -
% εr =
𝐅𝐮𝐥𝐥 𝐬𝐜𝐚𝐥𝐞 𝐫𝐞𝐚𝐝𝐢𝐧𝐠×𝐅𝐮𝐥𝐥 𝐬𝐜𝐚𝐥𝐞 𝐞𝐫𝐫𝐨𝐫
𝐑𝐞𝐪𝐮𝐢𝐫𝐞𝐝 𝐯𝐚𝐥𝐮𝐞

Composite errors: -
X1 = a ± εr1
X2 = b ± εr2
X3 = c ± εr3
(1) Addition/subtraction terms: -
X = X1 ± X2 ± X3
% εr =±
𝒂
𝒂+𝒃+𝒄
ε𝒓𝟏 +
𝒃
𝒂+𝒃+𝒄
ε𝒓𝟐 +
𝒄
𝒂+𝒃+𝒄
ε𝒓𝟑

(2) Multiplication/division terms: -
X = X1 X2 X3 Or
𝐗𝟏
𝐗𝟐𝐗𝟑
% εr =± ε𝒓𝟏 + ε𝒓𝟐 + ε𝒓𝟑
(3) Power term: -
X = X1
m X2
n X3
p Or
𝐗𝟏
𝐦
𝐗𝟐
𝐧𝐗𝟑
𝐩
% εr =± 𝐦 × 𝛆𝐫𝟏 + 𝐧 × 𝛆𝐫𝟐 + 𝐩 × 𝛆𝐫𝟑

Standard deviation error: -
m = f(x1, x2……..xn)
σ1, σ2,…….. σn = Standard deviation of x1, x2……..xn.
Standard deviation of m: -
σm =
δ𝒎
δ𝒙𝟏
𝟐
σ1
𝟐 +
δ𝒎
δ𝒙𝟐
𝟐
σ2
𝟐 … … … … … .
δ𝒎
δ𝒙𝒏
𝟐
σn
𝟐

Q: - Power is measure across a resistor by connecting an ammeter in series
to the resistance, and voltmeter in parallel to the resistance. This meter
measures 200V & 5A and the standard deviation of this meter are 2V and
0.1A. Calculate the standard deviation of the power.

Accuracy: -
It is the closeness of instrument reading with its true value.
𝑨𝒄𝒄𝒖𝒓𝒂𝒄𝒚 = 𝑬𝒓𝒓𝒐𝒓
Precision: -
• It is a measure of reproducibility of the reading.
• Precision does not guarantee for accuracy.
• If the no. of significant figure is increased then precision is increased.
Not accurate
Not precise
Accurate but
Not precise
Accurate
And precise

Resolution: -
Smallest change in input able to measure by the instrument is called
resolution.
Sensitivity: -
It is the ratio of change in output to change in input.

Q: - The two currents in a parallel branch given as (150± 1) Amp, (250± 2)
Amp. Find the total current in the circuit?

Q: - In the above problem if the errors are in the form of standard deviation
then find the total current in the circuit.

Types of the torque: -
(1) Damping torque
(2) Deflecting torque
(3) Controlling torque
Damping torque: -
A damping torque is produced by a damping or stopping force which acts on
the moving system only when it is moving and always opposes its motion.
Such a torque is necessary to bring the pointer to rest quickly. If there is no
damping torque, then the pointer will keep moving to and fro about its final
deflected position for some time before coming to rest, due to the inertia of
the moving system.

• This damping torque acts only when the pointer is in motion and always
opposes the motion. The position of the pointer when stationary is,
therefore, not affected by damping torque. The degree of damping decides
the behavior of the moving system.
• If the instrument is under-damped, the pointer will oscillate about the
final position for some time before coming to rest. On the other hand, if
the instrument is over damped, the pointer will become slow and
lethargic.

• However, if the degree of damping is adjusted to such a value that the
pointer comes up to the correct reading quickly without oscillating about
it, the instrument is said to be critically damped.
Types of damping torque: -
Air friction damping: -
Used in Moving iron meters and
Electrodynamometer type instrument.

Fluid friction damping: -
• Used in electrostatic meters.
• Jewel bearings are used for reducing wear and tear
of the moving system.
• The friction produced between spindle and
Bearing produces frictional errors so that accuracy of
the instrument will be reduced.
• The instrument which has high torque to weight
ratio has low frictional error so that accuracy is higher.
• Moving system like spindle and pointer are made of aluminium which has
low weight.
• The instrument which has low operating torque uses fluid friction damping
which reduces frictional error between bearing and spindle.

• At the final steady state position damping are minimized due to friction
between fluid and vane(float).
• Used in electrostatic voltmeter type instruments.
Eddy current damping: -
If a core/ former wound with a coil is placed in a magnetic field experiences
a force. Due to interaction of current flowing through the coil and magnetic
field.
An emf in the core which produces circulating eddy currents. Which opposes
the main torque. This is called eddy current damping which depends on
velocity of the moving system if magnetic flux is constant.
Example: - Used in PMMC.

Electro magnetic damping: -
The opposing flux is produced both in the former and coil wound on the former.
The magnitude depends on current flowing through the coil which intern depends
on resistance of the coil and external circuit, By adjusting this external resistance
nearer to critical damped value can be achieved.
It is used in galvanometer.
Deflecting torque: -
By applying different principles like electromagnetic, static, thermal etc, a pointer
is deflected proportional to the quantity to be measured and hence deflecting
torque is used.
Controlling torque or restoring torque: -
Controlling torque is used to keep the pointer at balance position where both
controlling and deflecting torques are equal.
It brings the pointer to the zero initial position if there is no deflecting torque.

Type of controlling torque: -
(1) Gravity control: -
A small weight is kept at the end of the moving system which produces
controlling torque TC.
TC ∝ 𝒘𝒍𝒔𝒊𝒏𝜽
TC ∝ 𝒔𝒊𝒏𝜽
𝒍
𝜽
𝒘
𝒘𝒍𝒔𝒊𝒏𝜽

At balance position: -
Td = TC
Td ∝ I
Td = K1I
K1I = 𝒘𝒍𝒔𝒊𝒏𝜽
𝐬𝐢𝐧𝛉 =
K1I
𝐊𝟐𝐰𝐥
I
𝛉 = 𝐬𝐢𝐧−𝟏 K1I
𝐊𝟐𝐰𝐥
……………..(1)
Equation (1) shows that gravity control has non-linear scale.

• Gravity control is used in vertically mounting instruments.
• Controlling torque is independent of temperature and time.
• Cost is cheaper.
• Disadvantage is not suitable for horizontal mounted instruments.
Spring control instrument: -
• It is most practically used in all the indicating instruments.
• Spring is made of phosphor bronze which is less affected by temperature.
TC ∝
𝑬𝒃𝒕𝟑
𝟏𝟐𝒍
𝛉 = K.𝛉
E = Young modulus k = spring/Tortional/restoring/controlling constant
b = width of spring k =
𝑬𝒃𝒕𝟑
𝟏𝟐𝒍
Nm/rad.
t = Thickness of spring
l = length of spring

At balance position: -
Td = TC
Td ∝ I
Td = K1I
TC = K𝛉
K𝛉= K1I
• Spring has Linear scale.
• Spring tension will be reduced due to ageing and temperature.
𝛉
𝒕
𝟎
𝛉 ∝ 𝐈

PMMC(Permanent magnet moving coil): -
F = n.B.I.𝒍 Sinα
if α = 90°
F = nBI𝒍
Deflecting torque: -
Td = F×b
Td = n.B.I.𝒍 ×b
Td = n.B.I.A
Td = G.I
Where G = nBA
n = No. of turns
B = Maximum flux density
I = Current through the coil
A = Area of cross section of coil

A = 𝒍 ×b
At balance
T = Td
K𝛉= GI
• Deflecting torque is produced due to interaction of permanent magnet flux
and current flows through the coil.
• Spring provides controlling torque.
• If the control spring is failed or snapped then the pointer comes to zero
initial position because the current is passing through spring is zero.
• Eddy current damping is used.
𝛉 ∝ 𝐈
𝛉
𝒕
𝟎
Linear scale

• PMMC measures average quantity of current and voltage.
• For any signal measured average value using : -
Io =
𝟏
𝐓
‫׬‬
𝟎
𝐓
𝐢 𝐭 𝐝𝐭.
Which is read by PMMC.
• If the pure AC signal passing through PMMC then pointer vibrates nearer
to zero because Iavg = 0.
• If a signal i(t) = Io + I1Sinωt + I2Sinωt……is passing through PMMC. It reads
Iavg = Io.
• If a negative value

Enhancement of meters: -
(1) Ammeter: -
For enhancement of the ammeter a shunt resistance is connected in parallel to
the meter. This Rsh is made of manganin.
Which has low value of temperature coefficient of resistance so that the error due
to temperature change can be minimized.
Manganin = α = 0.00015/°C
Copper = α = 0.00395/°C or 0.004/°C
Im = I×
𝐑𝐬𝐡
𝐑𝐬𝐡+𝐑𝐦
𝐑𝐬𝐡+𝐑𝐦
𝐑𝐬𝐡
=
𝐈
𝐈𝐦
= m
jh
Rsh =
𝐑𝐦
𝐦−𝟏

Q: - A PMMC ammeter has meter internal resistance of 50 ohm and
measures current up to 5 mA. Calculate the shunt resistance and voltage
drop across meter at full scale for measuring the currents of: -
(1) 25 mA (2) 250 mA (3) 1 A

Effect of temperature on ammeter: -
Without any compensation: -
Rsh =
𝐑𝐦
𝐦−𝟏
With RSwamp: -
Im = I×
𝐑𝐬𝐡𝟐
𝐑𝐬𝐡𝟐+𝐑𝐦+𝐑𝐬𝐰𝐚𝐦𝐩
𝐑𝐬𝐡𝟐+𝐑𝐦+𝐑𝐬𝐰𝐚𝐦𝐩
𝐑𝐬𝐡𝟐
=
𝐈
𝐈𝐦
= m
1
1
2

𝐑𝐬𝐡𝟐
=
𝐑𝐦
𝐦−𝟏
+
𝐑𝐒𝐰𝐚𝐦𝐩
𝐦−𝟏
𝐑𝐬𝐡𝟐
> 𝐑𝐬𝐡𝟏
By addition of swamp resistance in series to the meter the error due to
change in temperature is compensated.
𝐑𝐬𝐡𝟐
=𝐑𝐬𝐡𝟏
+
𝐑𝐒𝐰𝐚𝐦𝐩
𝐦−𝟏

Effect of frequency on AC ammeters: -
MI and electrodynamometer instrument, are used for measurement of both
AC and DC currents, in case of AC measures rms value of the current.
𝐈𝟏 = 𝟏𝟎 𝟐 𝐬𝐢𝐧(𝟐𝛑 × 𝟓𝟎𝐭) 𝐈𝐫𝐦𝐬𝟏
= 𝟏𝟎𝐀
𝐈𝟐 = 𝟏𝟎 𝟐 𝐬𝐢𝐧(𝟐𝛑 × 𝟏𝟎𝟎𝐭) 𝐈𝐫𝐦𝐬𝟐
= 𝟏𝟎𝐀
𝐙𝐬𝐧 = 𝐑𝐬𝐡 + 𝐣𝛚 𝐋𝐬𝐡
𝐙𝐦 = 𝐑𝐦 + 𝐣𝛚 𝐋𝐦
𝐈𝐦 =
𝐈 .𝐙𝐬𝐡
(𝐙𝐬𝐡+𝐙𝐦)
𝐈𝐬𝐡 =
𝐈.𝐙𝐦
(𝐙𝐬𝐡+𝐙𝐦)
Rsh
Rm
Lsh
Lm

𝐈𝐦
𝐈𝐬𝐡
=
𝐙𝐬𝐡
𝐙𝐦
𝐈𝐦
𝐈𝐬𝐡
=
𝐑𝐬𝐡
𝟐
+𝛚𝟐𝐋𝐬𝐡²
𝐑𝐦
𝟐 +𝛚𝟐𝐋𝐦²
𝐈𝐦
𝐈𝐬𝐡
=
𝐑𝐬𝐡
𝐑𝐦
𝟏+𝛚𝟐 𝐋𝐬𝐡
𝐑𝐬𝐡
𝟐
𝟏+𝛚𝟐 𝐋𝐦
𝐑𝐦
²
If
𝐋𝐬𝐡
𝐑𝐬𝐡
=
𝐋𝐦
𝐑𝐦
Then
𝐈𝐦
𝐈𝐬𝐡
=
𝐑𝐬𝐡
𝐑𝐦
If the time constant of the shunt is equal to the meter time constant then
AC ammeter is independent of frequency.

Sensitivity of Ammeter:
Sensitivity is the ratio of change in o/p to the change in i/p.
• For linear/Uniform meter: -
𝐓𝐜 = 𝐓𝐝 𝐤𝛉 = 𝐆 𝐈
𝛉 =
𝐆
𝐊
𝐈
Static sensitivity S =
θ
𝐈
θ
o/p
ΔI1
I
Δθ1

Non linear Sensitivity for Non-linear Meter: -
𝛉 ∝ 𝐈²
𝐒 =
𝚫 𝛉𝟏
𝚫𝐈𝟏
≠
𝚫 𝛉𝟐
𝚫𝐈𝟐
𝐒 =
𝐝𝛉
𝐝𝐈
=Dynamic Sensitivity
θ
o/p
I

Non linear scale: - Linear scale: -
S1 =
𝛉𝟏
𝐈𝟏
S1 =
𝛉𝟐
𝐈𝟐
• An ammeter which has minimum full scale value current has high
sensitivity.
• With higher sensitivity readability of data is convenient.
0.5
0
0
0.5
1
1.5
1
𝛉𝟏
𝛉𝟐
Cramped
scale

Resolution: -
Smallest change in input able to measure by the instrument is called
resolution.
Enhancement of voltmeter: -
For enhancement of voltmeter series resistance made of mangnin is
connected in series to the meter.
m =
𝐕
𝐕𝐦
Vm = V.
𝐑𝐦
(𝐑𝐦+𝐑𝐒)
RS = Rm (m-1)
RS = Series multiplier resistance
IfS
Rm

Full scale current IfS = Im =
𝐕
(𝐑𝐒+𝐑𝐦)
Voltmeter sensitivity = SV =
𝟏
𝐈𝐟𝐒
=
𝐑𝐬+𝐑𝐦
𝐕
……Ω/m
Loading effect: -
• In case of voltmeter, which is having higher value of voltmeter sensitivity
has more accuracy, because the current passing through the meter is very
small and the meter will not work as a load.
• If the voltmeter having lower sensitivity working similar to the load so that
error in the measurement of voltage will be more.

Effect of frequency on ac voltmeter: -
if,
• By the addition of compensating capacitor in parallel to RS voltmeter is
made independent of frequency.
RS
CC
m
m
CCRS = 0.41
𝑳𝒎
𝑹𝒔

Applications of PMMC: -
Rectifier meters: -
(a) Half wave rectifier meter: -
V = Vm Sinωt
PMMC reads Iavg
I1 =
𝐈𝐦
𝛑
=
𝐕𝐦
𝛑(𝐑𝐬+𝐑𝐦+𝐑𝐟)
Ammeter

Vm = 𝟐Vrms
I1 = 𝟐
𝐕𝐫𝐦𝐬
I1 =
𝟎.𝟒𝟓 𝐕𝐫𝐦𝐬
Ideal diode Rf = 0.
With DC input: - Assume VAC = Vrms
I2 =
𝐕𝐫𝐦𝐬
From (1) and (2)
I1 = 0.45 I2
If with DC input
I2 = 1 A
Vrms

(Si)AC = 0.45 (Si)DC
Full wave rectifier meter: -
(1) AC input: -
I1 =
𝟐𝐕𝐦
𝛑
=
𝟐𝐕𝐦
(𝐑𝐬+𝐑𝐦+𝟐𝐑𝐟)𝛑
Vm = 𝟐Vrms
I1 =
𝟐𝐕𝐦
𝛑
=
𝟐 𝟐𝐕𝐦
I1 =
𝟎.𝟗𝐕𝐦
(𝐑𝐬+𝐑𝐦+𝟐𝐑𝐟)
…………(1)
Ideal diode Rf = 0.

With DC input: -
Assume VDC = Vrms
I1 =
𝐕𝒓𝒎𝒔
…………(2)
From (1) and (2)
I1 = 0.9 I2
(Si)AC = 0.9 (Si)DC
• Rectifier meters used in communication application for calculation of
percentage of dc present in the AC signal.
• For reducing the power loss in place of series resistance RS, capacitance
can be used then the current measured by the PMMC.
I1 = 0.9
𝐕𝐫𝐦𝐬
𝐗𝐂

Calibration errors in PMMC meters: -
Form factor = Ff =
𝐑𝐌𝐒
𝐀𝐯𝐠
For sinusoidal signal F.F. = 1.11 =
𝐑𝐌𝐒
𝐀𝐯𝐠
Rms = 1.11 Avg
For Square Ff = 1 =
𝐑𝐌𝐒
𝐀𝐯𝐠
Rms = avg
% εr =
𝐅.𝐅𝐦𝐞𝐚𝐬𝐮𝐫𝐞𝐝−𝐅.𝐅.𝐓𝐫𝐮𝐞
𝐅.𝐅.𝐓𝐫𝐮𝐞
× 𝟏𝟎𝟎

• If the instrument are calibrated for one particular waveform rms
measurement, then if the other signal are passing through the instrument
producing error in measurement of rms reading. This is called error due to
calibration.
Ratio meter (Ohm-meter): -
For coil (C1)
Td1 = N.B.I1.A.Sinθ
For Coil (C2)
Td2 = N.B.I2.A.Cosθ
At balance condition
Td1 = Td2
N.B.I1.A.Sinθ = N.B.I2.A.Cosθ

tan θ =
𝐈𝟐
𝐈𝟏
if θ is small then tan θ ≈ θ
θ =
𝐈𝟐
𝐈𝟏
For example: -
θ =
𝐈𝟐
𝐈𝟏
=
𝐕/𝐒
𝐕/𝐑
=
𝐑
𝐒
if S is known
then
R(test)
S(known)
C1
V
C2
>
>
I2
I1
θ ∝ R

• Ratio meter is working on the principle of PMMC, used for the
measurement of unknown resistance over wide range. This is called ohm
meter, measure resistance up to 100 kilo ohm.
Megger: -

• θ =
𝐈𝟐
𝐈𝟏
𝐈𝟏=
𝐕
𝐑𝐓+𝐑𝟏
I2 =
𝐕
𝐑𝟐
θ ∝
𝐑𝐓+𝐑𝟏
𝑹𝟐
R1 and R2 = current limiting resistance
RT = test resistance
θ ∝ RT

• Megger works on electromagnetic principle used for measurement of
insulation resistance of electrical motors, generators, transformers etc.
• It is used to measure the wide range of resistance but actually design for
measurement of high resistance in the mega ohm range.
• Hand driven generator is used for supplying the power to the motor.
• It will generate the voltage up to 2000 volts.

Moving iron instrument: -
Attraction type moving iron meter: -

Repulsion type moving iron instrument: -

• Whenever the current is flowing through the fixed coil produces magnetic
flux which is used to attract or repel the iron disc. This principle is used in
moving iron instruments for measurement of both AC and DC quantities of
current and voltages.
• It is most widely used in laboratories because of less cost and more robust.
Deflecting torque(Td): -
At balance,
TC = Td
Kθ =
𝟏
𝟐
𝐈𝟐 𝐝𝐋
𝐝θ
→Scale is non-linear
Td =
𝟏
𝟐
𝐈𝟐 𝐝𝐋
𝐝θ
θ ∝ 𝐈𝟐
θ o/p
I

• Scale is non-uniform because θ ∝ 𝐈𝟐
.
• It is used for measurement of both ac and dc.
• In AC it measures rms quantity of current and voltage.
• For any signal calculate Irms: -
Irms =
𝟏
𝐓
‫׬‬
𝟎
𝐓
𝐢𝟐
𝐭 𝐝𝐭
𝟏/𝟐
• If a current of i(t) = Io + I1Sinωt + I2Sin2ωt………..is passing through rms
reading meter it measures Irms = 𝐈𝟐
+
𝟏
𝟐
(𝐈𝟏
𝟐
+ 𝐈𝟐
𝟐
+ ⋯ )
𝟏/𝟐
.
• Spring provides controlling torque.
• Air friction damping is used.
• The iron disc is made of nickel iron alloy which has low hysteresis area and
hence error due to hysteresis is minimized.
• The eddy current errors becomes constant beyond 125 Hz and hence MI
meters are suitable up to 125 Hz.

• For measurement of high frequency currents in the MHz range
(Communication applications) thermal instruments are used i.e,
thermocouple and hot wire meters are used.
Condition for linearity of scale: -
Td = TC
Kθ =
𝟏
𝟐
𝐈𝟐 𝐝𝐋
𝐝θ
For linear scale: I ∝ θ
𝟏
𝟐
𝐊𝟏
𝟐
θ𝟐 𝐝𝐋
𝐝θ
= Kθ
θ
𝐝𝐋
𝐝θ
=
𝟐𝐊
𝐊𝟏
𝟐 = K2 = constant
θ
𝐝𝐋
𝐝θ
=constant

Note: -
In case of moving iron instrument for AC and DC measurement separate
calibration is required otherwise produces error in the measurement of
voltages and currents.
Transfer instruments: -
If the instrument is calibrated for measurement of AC signal, it can be
directly used without calibration for dc measurement is called transfer
instrument.
Example of transfer instruments is electrodynamometer type instruments.

Electrodynamometer: -
Series aiding
𝐋 = 𝐋𝟏 + 𝐋𝟐 + 𝟐 𝐌
𝐋𝟏, 𝐋𝟐 ⇒ 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭
𝐝𝐋
𝐝 𝛉
= 𝟐
𝐝𝐌
𝐝𝛉
M.I meter: -
𝐓𝐝 =
𝟏
𝟐
𝐈𝟐 𝐝𝐋
𝐝𝛉
Electrodynamometer: -
𝐓𝐝 =
𝟏
𝟐
. 𝐈. 𝐈:
𝐝𝐋
𝐝𝛉
𝐓𝐝 =
𝟏
𝟐
𝐢𝐥. 𝐢𝟐. 𝟐
𝐝𝐌
𝐝𝛉
𝐓𝐝 = 𝐢𝐥𝐢𝟐.
𝐝𝐌
𝐝𝛉
>
>

AC signals:-
𝐢𝟏 = 𝐈𝐦𝟏𝐬𝐢𝐧𝛚𝐭
𝐢𝟐 = 𝐈𝐦𝟐 𝐬𝐢𝐧(𝛚𝐭 − 𝛂)
𝐓𝐝 = 𝐈𝐦, 𝐈𝐦𝟐 𝐬𝐢𝐧 𝛚𝐭 𝐬𝐢𝐧 𝛚𝐭 − 𝛂
𝐝𝐌
𝐝𝛚
𝐓𝐝𝐚𝐯𝐠
=
𝟏
𝟐𝛑
‫׬‬
𝟎
𝟐𝛑
𝐓𝐝. 𝐝(𝛚𝐭)
=
𝐈𝐦𝐢.𝐈𝐦𝟐
𝟐
𝐜𝐨𝐬𝛂
𝐝𝐌
𝐝𝛉
=
𝐈𝐦𝟏
𝟐
.
𝐈𝐦𝟐
𝟐
𝐜𝐨𝐬𝛂
𝐝𝐌
𝐝𝛉
α
i1
i2

= 𝐈𝐫𝐦𝐬𝟏
. 𝐈𝐫𝐦𝐬𝟐
. 𝐜𝐨𝐬𝛂
𝐝𝐌
𝐝𝛉
𝐓𝐝𝐚𝐯𝐠
= 𝐈𝟏. 𝐈𝟐. 𝐜𝐨𝐬𝛂
𝐝𝐌
𝐝𝛉
𝐓𝐜 = 𝐤𝛉
At balance condition,
𝐓𝐝 = 𝐓𝐂
𝐤𝛉 = 𝐈𝟏𝐈𝟐 𝐜𝐨𝐬𝛂. 𝐝𝐌/𝐝𝛉
Applications: -
(1) Ammeter
(2) voltmeter
(3) wattmeter
(4) power factor meter
(5) power frequency meter

(1) Ammeter:-
𝐅𝐜𝟏. 𝐅𝐜𝟐 = 𝐅𝐢𝐱𝐞𝐝 𝐜𝐨𝐢𝐥
MC = Moving coil
𝛂 = 𝟎°, 𝐈𝟏 = 𝐈𝟐 = 𝐈
𝐓𝐝 = 𝐈𝟏𝐈𝟐 𝐜𝐨𝐬𝛂
𝐝𝐌
𝐝𝛉
𝐓𝐝 = 𝐈𝟐 𝐝𝐌
𝐝𝛉
𝐓𝐜 = 𝐊𝛉
At balance: -
𝐓𝐝 = 𝐓𝐜
K𝛉 = 𝐈𝟐
𝐝𝐌/𝐝𝛉
𝛉 ∝ 𝐈²
>
I1
I
>
I2

(2) Voltmeter: -
𝐑𝐬 = Series multiplier resistance
𝛂 = 𝟎, 𝐈𝟏 = 𝐈𝟐 =
𝐕
𝐑𝐬
𝐓𝐝 =
𝐕²
𝐑𝐬²
𝐜𝐨𝐬 𝟎°
𝐝𝐌
𝐝𝛉
=
𝐕²
𝐑𝐬²
𝐝𝐌
𝐝𝛉
TC = Td
𝐊𝛉 =
𝐕²
𝐑𝐬²
𝐝𝐌
𝐝𝛉
𝛉 ∝ V2
M
F.C.1
F.C.2
M.C.
RS
I2
>
>
I1= I2
I
>
Lag load
Cosφ
Pr.
S
V

• Electrodynamometer measures both AC and DC current and voltages. In case of
AC, measures RMS quantity.
• Scale is non-linear.
• Condition for linearity 𝛉
𝐝𝐌
𝐝𝛉
=constant
Measurement of Power: -
DC power: - P=V.I.
A.C. Power :- 𝐩 = 𝐯. 𝐢
𝐯 = 𝐕𝐦 𝐬𝐢𝐧 𝛚𝐭
𝐢 = 𝐈𝐦 𝐬𝐢𝐧(𝛚𝐭 − 𝛟)
𝐩 = 𝐯. 𝐢 = 𝐕𝐦𝐈𝐦 𝐬𝐢𝐧𝛚𝐭 𝐬𝐢𝐧(𝛚𝐭 − 𝛟)
𝐏𝐚𝐯𝐠 =
𝟏
𝟐𝛑
‫׬‬
𝟎
𝟐𝛑
. 𝐩. 𝐝(𝛚𝐭)
=
𝐕𝐦𝐈𝐦
𝟐
𝐂𝐨𝐬𝛟

=
𝐕𝐦
𝟐
.
𝐈𝐦
𝟐
𝐜𝐨𝐬𝛟 = 𝐕𝐫𝐦𝐬. 𝐈𝐫𝐦𝐬. 𝐜𝐨𝐬𝛟
Electrodynamometer wattmeter: -
𝐓𝐝 = 𝐈𝟏 𝐈𝟐 𝐜𝐨𝐬𝛂
𝐝𝐌
𝐝𝛉
𝐈𝐟 𝐈𝟏 = 𝐈 , 𝐈𝟐 =
𝐕
𝐑𝐬
, 𝛂 = 𝛟
𝐓𝐝 =
𝐕.𝐈.𝐜𝐨𝐬 𝛟
𝐑𝐬
.
𝐝𝐌
𝐝𝛉
𝐓𝐝 =
𝐏𝐚𝐯𝐠
𝐑𝐬
𝐝𝐌
𝐝𝛉
• Wattmeter measures avg active power.
𝐏𝐚𝐯𝐠 = 𝐕. 𝐈. 𝐜𝐨𝐬𝛟

• 𝐅𝐂𝟏, 𝐅𝐂𝟐 ⇒ Current coil
• MC ⇒ Potential coil or voltage coil or pressure coil
RS
F.C.1 F.C.2
>
I
I1
>
I2

• Wattmeter reading = P.C. Voltage × C.C. Current × Cos ∠(P.C. voltage & C.C. current)
• Wattmeter works on the principle of electrodynamometer. It consisting of
fixed or current coils which are connecting in series to the load for
measurement of load current.
• For small load currents FC1, and FC2 are connected in series for larger load
current these are connected in parallel.
• M.C. or P.C. used for measurement of voltage which is connected across
the supply.
θ = α
I2 = V/RS
I = I1

• Potential coil of wattmeter must be highly resistive is required so that
error due to potential coil inductance is minimized.
• P.C. of the energy meter must be highly inductive is required so that error
in the measurement of energy is minimized.
Harmonic signals: -
𝐏. 𝐂. = 𝐕 = 𝐕𝐨 + 𝐕𝐦𝟏 𝐬𝐢𝐧 𝛚𝐭 + 𝛉𝟏 + 𝐕𝐦𝟐 𝐬𝐢𝐧 𝟐𝛚𝐭 + 𝛉𝟐 +……………..
𝐂. 𝐂. = 𝐈 = 𝐈𝐨 + 𝐈𝐦𝟏 𝐬𝐢𝐧 𝛚𝐭 + 𝛂𝟏 + 𝐈𝐦𝟐𝐬𝐢𝐧 𝟐𝛚𝐭 + 𝛂𝟐 +………..
𝐏𝐚𝐯𝐠 =
𝟏
𝟐𝛑
‫׬‬
𝟎
𝟐𝛑
𝐯. 𝐢 𝐝𝛚𝐭
𝐏𝐚𝐯𝐠 = 𝐕𝐨𝐈𝐨 +
𝟏
𝟐
[𝐕𝐦𝟏 𝐈𝐦𝟏
𝐜𝐨𝐬 𝛉𝟏 − 𝛂𝟏 + 𝐕𝐦𝟐 𝐈𝐦𝟐 𝐜𝐨𝐬 𝛉𝟐 − 𝛂𝟐 + ⋯ . . ]

Q: - 𝐕 = 𝟓 + 𝟐 𝐬𝐢𝐧(𝛚𝐭 + 𝟒𝟓°)
𝐢 = 𝟒 + 𝟑 𝐬𝐢𝐧 𝛚𝐭 + 𝟏𝟓° + 𝟐 𝐬𝐢𝐧(𝟐𝛚𝐭 + 𝟑𝟎°)
𝐏𝐚𝐯𝐠 = 𝟓 × 𝟒 +
𝟏
𝟐
× 𝟐 × 𝟑 × 𝐜𝐨𝐬 𝟒𝟓° − 𝟏𝟓° = 𝟐𝟐. 𝟔 𝐖
Blondels theorem: -
• For measurement of power In case of n-phase Balanced system the
minimum no of wattmeters required are (𝒏 − 𝟏).
• In case of 𝟑 − 𝛟 power measurement the minimum number of wattmeters
required are 𝟑 − 𝟏 = 𝟐.

(1) Star-load: -
𝐏 = 𝟑 𝐕𝐋 𝐈𝐋 𝐜𝐨𝐬𝛟
W1 = VBY IB ∠𝐕𝐁𝐘 − 𝐈𝐁
W2 = VRY IR ∠𝐕𝐑𝐘 − 𝐈𝐁
W2
W1
Y
IR
B
IY
IB
R

𝐖𝟐 = 𝐕𝟏𝐈𝐋 𝐜𝐨𝐬(𝟑𝟎° + 𝛟)
𝐖𝟏 = 𝐕𝐋𝐈𝐋 𝐜𝐨𝐬(𝟑𝟎° − 𝛟) VR
VRY
VBY
VB
-VY
VY
IR
IB
30°
φ
30-φ°
φ°
30°

Delta load: -
W1 = VBY IB ∠𝐕𝐁𝐘 − 𝐈𝐁
W2 = VRY IR ∠𝐕𝐑𝐘 − 𝐈𝐁
Y
R
B
W2
W1
IR
IB
IY
Z∠ϕ
Z∠ϕ
Z∠ϕ

𝐖𝟐 = 𝐕𝟏𝐈𝐋 𝐜𝐨𝐬(𝟑𝟎° + 𝛟)
𝐖𝟏 = 𝐕𝐋𝐈𝐋 𝐜𝐨𝐬(𝟑𝟎° − 𝛟)
𝐓𝐨𝐭𝐚𝐥 𝟑 − 𝛟 active power: -
W = 𝐖𝟏 + 𝐖𝟐 = 𝐕𝐋𝐈𝐋 𝐜𝐨𝐬 𝟑𝟎 − 𝛟 + 𝐕𝐋 𝐈𝐋 𝐜𝐨𝐬(𝟑𝟎 + 𝛟)
= 𝐕𝐋 𝐈𝐋 . 𝟐 𝐜𝐨𝐬 𝟑𝟎°. 𝐜𝐨𝐬𝛟°
𝐖𝟏 + 𝐖𝟐 = 𝟑 𝐕𝐋𝐈𝐋𝐜𝐨𝐬𝛟 = 𝐏𝟑𝛟

Total 3 –phase reactive power: -
𝐖𝟏 − 𝐖𝟐 = 𝐕𝐋𝐈𝐋[𝐜𝐨𝐬 𝟑𝟎 − 𝛟 − 𝐜𝐨𝐬 𝟑𝟎 + 𝛟 ]
𝐖𝟏 − 𝐖𝟐 = 𝐕𝐋𝐈𝐋𝐬𝐢𝐧𝛟
𝐐𝟑𝛟 = 𝟑 𝐕𝐋 𝐈𝐋𝐬𝐢𝐧𝛟
𝐐𝟑𝛟 = 𝟑(𝐖𝟏 − 𝐖𝟐)
Power factor angle: -
𝐖𝟏−𝐖𝟐
𝐖𝟏+𝐖𝟐
=
𝐕𝐋𝐈𝐋𝐬𝐢𝐧𝛟
𝟑 𝐕𝐋𝐈𝐋 𝐜𝐨𝐬 𝛟
𝛟 = 𝐭𝐚𝐧−𝟏 𝟑(𝐖𝟏−𝐖𝟐)
𝐖𝟏+𝐖𝟐
lag load
𝛟 = − 𝐭𝐚𝐧−𝟏 𝟑(𝐖𝟏−𝐖𝟐)
𝐖𝟏+𝐖𝟐
lead load
𝐩. 𝐟. = 𝐜𝐨𝐬𝛟

Reading of wattmeter’s for different p.f. of load:-
ϕ Cos ϕ 𝐖𝟏 = V𝐋I𝐋 cos(30 − ϕ) 𝐖𝟐 = V𝐋I𝐋 cos(30 + ϕ) W = 𝐖𝟏 + 𝐖𝟐 Remarks
O 1 3VLIL
2
.
3
2
VLIL
3VLIL W1 = W2
30° 0.866 VLIL VLIL
2
3
2
VLIL
W1 = 2W2
60° 0.5 3
2
VLIL
0 3
2
VLIL
W1 = W
W2 = 0
90° 0 VLIL
2
−VL IL
2
0 W1 = +Ve
W2 = −Ve

• If the p.f. is less than 0.5 then one of the wattmeter indicates negative
value. For recording this negative value of the power either potential coil
or current coil terminals are reversed and record the reading with negative
sign.
Measurement of Reactive power: -
• In the measurement of reactive power C.C. is connected to one of the
phase and potential coil is connected between the remaining two phases.
W= 𝐕𝐋𝐈𝐋𝐬𝐢𝐧𝛟
𝐐𝟑𝛟 = 𝟑 𝐖 Vars

Errors in wattmeters:-
Error due to potential coil connection: -
Potential coil on source side: -
𝐏𝐓 = 𝐓𝐫𝐮𝐞 𝐩𝐨𝐰𝐞𝐫 = 𝐕𝐈 𝐜𝐨𝐬𝛟
𝐏𝐦 = 𝐦𝐞𝐚𝐬𝐮𝐫𝐞𝐝 𝐩𝐨𝐰𝐞𝐫 = 𝐏𝐓 + 𝐈𝟐
𝐫𝐜 .

% 𝛜𝐫 =
𝐏𝐦−𝐏𝐓
𝐏𝐓
× 𝟏𝟎𝟎
%𝛜𝐫 =
𝐈𝟐𝐫𝐜
𝐏𝐓
× 𝟏𝟎𝟎
⇒ For small load current error 𝛜𝐫 is small.
Potential coil on load side: -

Power loss 𝐏𝐋𝐢𝐧 𝐩. 𝐜. = 𝐈𝟐
𝟐
𝐑𝐬 =
𝐕²
𝐑𝐬
𝐏𝐦 = 𝐏𝐓 +
𝐕²
𝐑𝐬
%𝛜𝐫 =
𝐏𝐦−𝐏𝐓
𝐏𝐓
× 𝟏𝟎𝟎
% 𝛜𝐫 =
𝐕²
𝐑𝐬.𝐏𝐓
× 𝟏𝟎𝟎
Note:-
P.C. on load side is used in case of large load current to reduce error.

Error due to p.c. current flowing in the C.C.: -
If the p.c. is connected on the load side then the current coil carries both c.c.
and p.c. current, so that power loss occurs in the p.c. coil. For compensating
this power loss compensating coil is connected in series to the p.c.

Error due potential coil inductance: -
Potential coil is of Highly resistive: -
𝐓𝐝 =
𝐕 𝐈
𝐑𝐬
. 𝐜𝐨𝐬𝛟.
𝐝𝐌
𝐝𝛉
………………….(1)
True power PT = 𝐕𝐈 𝐜𝐨𝐬𝛟

P.C. contains inductance (LP): -
P.C. inductance (Impedance)
𝐙𝐩 = 𝐑𝐩 + 𝐑𝐬 + 𝐣𝛚 𝐋𝐩
Assume 𝐑𝐩 ≪ 𝐑𝐒
𝐙𝐩 = (𝐑𝐬 + 𝐣 𝛚𝐋𝐩)
ZP
RS
𝛚LP
β
V
I2
I1= I
φ- β= α
β
φ

𝐓𝐝 ∝ 𝐈𝟏𝐈𝟐 𝐜𝐨𝐬 𝛟 − 𝛃 .
𝐝𝐌
𝐝𝛉
∝ 𝐈.
𝐕
𝐙𝐩
𝐜𝐨𝐬 𝛟. − 𝛃 .
𝐝𝐌
𝐝𝛉
𝐜𝐨𝐬𝛃 =
𝐑𝐬
𝐙𝐩
⇒
𝟏
𝐙𝐩
=
𝐜𝐨𝐬𝛃
𝐑𝐬
𝛃 = 𝐭𝐚𝐧−𝟏 𝛚 𝐋𝐩
𝐑𝐬
𝐓𝛂 ∝
𝐕𝐈 𝐜𝐨𝐬 𝛃
𝐑𝐬
𝐜𝐨𝐬 𝛟 − 𝛃 .
𝐝𝐌
𝐝𝛉
……………….(2)
Measured power = PM = 𝐕𝐈 𝐜𝐨𝐬𝛃. 𝐜𝐨𝐬(𝛟 − 𝛃)
(1) Correction factor = 𝐂𝐟 =
𝐏𝐓
𝐏𝐦
=
𝐜𝐨𝐬 𝛟
𝐜𝐨𝐬 𝛃.𝐜𝐨𝐬(𝛟−𝛃)

(2) Error: -
𝐏𝐓
𝐏𝐦
=
𝐜𝐨𝐬𝛟
𝐜𝐨𝐬 𝛃[𝐜𝐨𝐬𝛟𝐜𝐨𝐬𝛃+𝐬𝐢𝐧𝐠𝛟𝐬𝐢𝐧𝛃]
=
𝐬𝐞𝐜𝟐𝛃
𝟏+𝐭𝐚𝐧𝛟.𝐭𝐚𝐧 𝛃
=
𝟏+𝐭𝐚𝐧𝟐𝛃
𝟏+𝐭𝐚𝐧𝛟𝐭𝐚𝐧𝛃
For small 𝛃 ⇒ 𝐭𝐚𝐧𝟐
𝛃 ≈ 𝟎
𝐏𝐓
𝐏𝐦
=
𝟏
𝟏+𝐭𝐚𝐧𝛟𝐭𝐚𝐧𝛃
𝐏𝐦
𝐏𝐓
= 𝟏 + 𝐭𝐚𝐧𝛟 𝐭𝐚𝐧𝛃
% 𝛜𝐫 =
𝐏𝐦−𝐏𝐓
𝐏𝐓
× 𝟏𝟎𝟎 = 𝐭𝐚𝐧𝛟 𝐭𝐚𝐧 𝛃 × 𝟏𝟎𝟎

Error in watts: -
𝐏𝐦 − 𝐏𝐓 = 𝐏𝐓. 𝐭𝐚𝐧𝛟. 𝐭𝐚𝐧𝛃
= 𝐕𝐈 𝐜𝐨𝐬 𝛟.
𝐬𝐢𝐧𝛟
𝐜𝐨𝐬𝛟
. 𝐭𝐚𝐧 𝛃
𝐏𝐦 − 𝐏𝐓 = 𝐕𝐈 𝐬𝐢𝐧𝛟𝐭𝐚𝐧𝛃
(a) Lag load : -
𝛟 = positive
𝐏𝐦 = 𝐏𝐓 + (𝐕𝐈 𝐬𝐢𝐧𝛟 𝐭𝐚𝐧𝛃)
𝐏𝐦 > 𝐏𝐓
(b) Lead load : -
𝛟 = Negative
𝐏𝐦 = 𝐏𝐓 − (𝐕𝐈 𝐬𝐢𝐧𝛟 𝐭𝐚𝐧𝛃)
𝐏𝐦 < 𝐏𝐓

• If a U.P.F. wattmeter is used for measurement of power, in case of low pf
load, error becomes maximum. For reducing these errors LPF wattmeter’s
is used.
Example: -
• In the O.C. test of a power transformer, the transformer has p.f. of 0.1 to
0.2. For measuring power accurately, lpf wattmeter is used.
• In case of S.C. test the power factor of transformer is more than 0.9
(nearer to unity) and hence UPF wattmeter is used.

LPF wattmeter: -
• 𝐓𝐝 ↑=
𝐕𝐈 𝐜𝐨𝐬𝛟 ↓
𝐑𝐒↓
.
𝐝𝐌
𝐝𝛉
• ↑ 𝐈 =
𝐏
𝐕 𝐜𝐨𝐬𝛟 ↓
• 𝐏𝐦 − 𝐏𝐓 = 𝐕𝐈 𝐭𝐚𝐧 𝛃 𝐬𝐢𝐧 𝛟
• CC = Compensating capacitor
𝐫. 𝐂𝐜 ≈
𝐋𝐩
𝐫

• Due to low p.f. deflecting torque is small for increasing this deflecting
torque, low value of 𝐑𝐬 is used.
• At low p.f. load current drawing is higher so that potential coil has to be
connected at the load end which produces power loss in the potential coil,
for compensating this effect compensating coil is used.
• For reducing the effect of p.c. inductance compensating compactor is
connected in parallel to some portion of Rs.
Effect of external stray magnetic field: -
• In case of ordinary wattmeter’s iron shielding is used for reducing the
effect of external stray magnetic field.
• If the operating field of the instrument and external stray magnetic field
are parallel to each other then the error in the measurement will be more
because of changes in the operating torque.

• In case of high accuracy wattmeter’s astatic system is used for
compensation of external magnetic field.
• In this method two moving coils are mounted on the spindle which carries
the current in opposite direction so that external magnetic field effect is
nullified.
Iron shielding

Power factor meter: -
At balance
𝛉 = 𝛟
𝛉 = Deflecting angle of pointer
𝛟 = 𝐩. 𝐟. 𝐚𝐧𝐠𝐥𝐞 𝐨𝐟 𝐥𝐨𝐚𝐝.
• Power factor meter works on the principal of electrodynamometer.
• It consisting of two fixed coil (F.C.1 and F.C.2) which are called fixed coils
and two moving coils (M.C.1 and M.C.2). M.C.1 and M.C.2 are kept at 90
degree to each other.
• The magnetic Flux produced in the field coils interact with the crossed coils
so that torque is produced and the deflecting angle is calibrated
proportional to the p.f. of the load.
MC1
MC2

• There is no control spring is used in the p.f. meter for producing controlling
torque.
• Air friction damping is used.
• In case of 3-phase p.f. measurement polarized vane p.f. meter is used,
which consists of a 3 moving coils which are kept at 120 degree with
respect of each other.

Power frequency Meter: -
Power frequency meter working on the principal of electro dynamometer
and resonance.

• LC components connected to FC1 and FC2 are tuned to the frequencies of
45 and 55 Hz.
• Depending on the test frequency magnitude lagging and leading current
flowing in FC1 and FC2 produces deflection in the moving coil which is
calibrated in terms of frequency.
• Control torque is provided by placing small weight on the moving system.
• Air friction dumping is used.
• It is used to measure frequency in the power frequency range.

Energy meter: -
True energy = WT = P×T = V ICosφ×t…………kwhr
Mechanisms: -
(1) Driving mechanism – P.C., C.C.
(2) Braking mechanism - Permanent magnet
(3) Rotating mechanism – R.D.
(4) Recording/registering mechanism – RGT

(V)
IV
I
φ2
φ1
φL
S.L.
S.R.
d
Bearing

S.R. = Shading ring
S.L. = Shading loop
R.D. = Rotating Disc
C.C. = Current coil
P.C. = Potential coil
RGT = Reduced gear train
PM = Permanent Magnet
S = Spindle

Driving Mechanism: -
• Energy Meter consisting of p.c. and c.c. . P.C. is connected across the load
for measurement of vtg and C.C. is connected in series to the load for
measurement of current.
• By using driving Mechanism driving torque is produced on the rotating disc
due to interaction, of fluxes produced due to p.c. and c.c. coil, currents.
Note: -
P.C of the energy meter must be highly inductive is required so that error in
the measurement of energy is minimized.
• Driving torque is proportional to the power consumed by the load.

Driving Torque (𝐓𝐝): -
𝐓𝐝 ∝ [𝛟𝟏 𝐢𝟐 𝐜𝐨𝐬∠𝛟𝟏 𝐢𝟐 − 𝛟𝟐 𝐢𝟏 𝐜𝐨𝐬 ∠𝛟𝟐 𝐢𝟏]
𝐓𝐝 ∝ 𝐕𝐈 {[𝐜𝐨𝐬 𝟗𝟎 + α − 𝚫 − 𝛟 − 𝐜𝐨𝐬 𝟗𝟎° + α + 𝚫 − 𝛟 }
𝐓𝐝 ∝ 𝐕𝐈[𝟐 𝐬𝐢𝐧 𝟗𝟎 + α . 𝐬𝐢𝐧(𝚫 − 𝛟)]
𝐓𝐝 ∝ 𝟐 𝐜𝐨𝐬α. [𝐕𝐈 𝐬𝐢𝐧(𝚫 − 𝛟)]
𝟐 𝐜𝐨𝐬α = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭
𝐓𝐝 ∝ 𝐕𝐈 𝐬𝐢𝐧(𝚫 − 𝛟)
𝐏𝐓 = 𝐕𝐈 𝐜𝐨𝐬. 𝛟
𝐏𝐌 = 𝐕𝐈 𝐬𝐢𝐧(𝚫 − 𝛟)
Error in Power: -
𝐏𝐌 − 𝐏𝐓 = 𝐕𝐈 [𝐬𝐢𝐧 𝚫 − 𝛟 − 𝐜𝐨𝐬𝛟]
If 𝚫 = 𝟗𝟎° ⇒ 𝐏𝐌 − 𝐏𝐓 = 𝟎 ⇒ 𝐏. 𝐂. 𝐢𝐬 𝐨𝐟 𝐇𝐢𝐠𝐡𝐥𝐲 𝐢𝐧𝐝𝐮𝐜𝐭𝐢𝐯𝐞

• As the p.f. is reduced then the error is increased if the potential coil is not
highly inductive.
Lag/Quadrature/p.f. correction coil/shading ring Compensation: -

• Lag coil or shading ring is kept below the potential coil for improving the
p.f. of the potential coil, so that 90° phase displacement can be maintained
between potential coil voltage and flux.
• Variable resistor is used in case of lag coil for adjustment of the angle.
• The position of shading ring is used for adjustment of the angle. Shading
ring is made of copper.
Rotating Mechanism: -
• A low weight aluminum disc is used as a rotating disc for reducing friction.

Braking Mechanism: -
𝐓𝐁 ∝ 𝛟𝐦𝐈𝐦𝐝
𝐈𝐦 ∝ 𝐕𝐦 ∝ 𝐍. 𝛟𝐦
𝐓𝐁 ∝ 𝐍. 𝛟𝐦
𝟐
𝐝
𝐓𝐁 = 𝐊𝟏𝐍 𝛟𝐦
𝟐
𝐝
𝐓𝐁 ∝ 𝐍
𝐍 = speed of disc

At constant speed: -
𝐓𝐁 = 𝐓𝐝 Speed correction
𝐓𝐝 ∝ 𝛒, 𝐓𝐁 ∝ 𝐍 𝐓𝐁 ∝ 𝛟𝐦
𝟐
𝐍. 𝐝
𝐏 = 𝐍 𝐍 =
𝐓𝐁
𝐊𝟏𝛟𝐦
𝟐 𝐝
‫׬‬ 𝐩. 𝐝𝐭 = ‫׬‬ 𝐍. 𝐝𝐭.
• By adjusting the position of P.M. speed of the disc is adjusted.
𝐍 ∝
𝟏
𝐝
Energy=‫׬‬ 𝐍. 𝐝𝐭

Registering or recording mechanism:-
• Reduced gear teeth mechanism is used for registering of energy consumed
by the load during certain period of time. This energy is cumulatively add-
up and hence energy meter is a integrating instrument.
Errors and compensation in energy meter: -
Friction and light load compensation: -
• For compensation of fiction and light load energy measurements, shading
loop is used the position of the shading loop can be adjustable along the
spindle.

Creepy: -
• If the friction is overcompensated by placing the shading loop very closer
to the potential coil then rotating disc rotates very slowly an record the
energy without any load current and with potential coil only excited this is
called creepy this produces error in the measurement of energy.
• For reducing creeping two holes or/slots are made on the rotating disc and
these experiences opposing torque so that disc rotation stops.
• By placing small iron piece on the rotating disc creeping is minimized
because this iron piece is attracted by permanent magnet.

Over voltage & over load
compensations: -
• Saturable shunt magnet is placed
between center and side limbs of
shunt magnet for diverting the
access Flux due to over voltage
• The saturable series magnet is
placed between the limbs of
series magnet for diverting access
Flux due to over local current.

Formulae: -
(1) Energy meter constant = K =
𝐍𝐨 𝐟𝐨 𝐫𝐞𝐯.𝐨𝐟 𝐝𝐢𝐬𝐜
𝐊𝐰𝐡𝐫
K =
𝐍
𝐩.𝐭.
(2) Energy measured = 𝐖𝐦 =
𝐓𝐨𝐭𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐑𝐞𝐯𝐨𝐥𝐮𝐭𝐢𝐨𝐧𝐬 𝐚𝐬 𝐩𝐞𝐫 𝐥𝐨𝐚𝐝
𝐊
(3) True energy = 𝐖𝐓 =
𝐕𝐈𝐜𝐨𝐬𝛟
𝟏𝟎𝟎𝟎
×
𝐭
𝟑𝟔𝟎𝟎
……………..kwhr
Where,
I = current in A
Half load = I/2
Full load =I
V=voltage in Volts
T= time in sec.

% εr =
𝐖𝐦−𝐖𝐓
𝐖𝐓
× 𝟏𝟎𝟎
If εr = Positive ⇒ Disc runs fast
If εr = Negative ⇒ Disc Runs slow
Testing of energy meter and wattmeter: -
Normal loading: -

Testing time = t= 5 hours
Energy consumed = 𝟒. 𝟔 × 𝟓 = 𝟐𝟑 kwhr = 23 units
1 unit = 10 /-
Cost of testing = 𝟐𝟑 × 𝟏𝟎 = 𝟐𝟑𝟎 /-
Phantom loading/fictitious loading: -
𝐏 = 𝐕𝐈 = 𝟐𝟎 × 𝟏𝟎 = 𝟐𝟎𝟎𝐰 = 𝟎. 𝟐 kw
𝐭 = 𝟓 hours
• Energy = 𝟎. 𝟐 × 𝟓 = 𝟏 kwhr
Cost of testing = 𝟏 × 𝟏𝟎 = 𝟏𝟎/−

• Phantom or fictitious loading is used for testing of energy meter and
wattmeter.
• By using this method the cost of calibration is minimized because the
power loss is reduced.
• Normal supply voltage is applied to the p.c. and small voltage source with
variable resistor is connected to the c.c.
• By adjusting this variable resistor rated current is passing through the C.C.
• Energy meter works on the principal of electromagnetic induction.

Thermal Meters: -
Thermo-couple meter: -
𝐕 ∝ α𝟏 𝐓𝟏 − 𝐓𝟐 + α𝟐 𝐓𝟏 − 𝐓𝟐
𝟐
+ ⋯ α𝟏𝟎 𝐓𝟏 − 𝐓𝟐
𝟏𝟎
(Platinum)
(Rhodium)
T1
T2
PMMC

𝐓𝟏 = Hot junction temp
𝐓𝟐 = cold junction temp.
• If two different metals having different work function, forming a junction
and operating at diff temp, a voltage is induced across the junction which
is proportional to the temp diff, this process is called “seeback effect”.
• Thermocouple meter works on this principal.
• Thermocouple output voltage is of “DC quantity” which is measured which
is measured by PMMC meter.
Advantages: -
• It is used to mess are both ac had dc quantities, in case of AC measures
RMS quantity.
• It is used to measure high-frequency currents up to MHz range.

• It is independent of external electrostatic and magnetic fields.
• By using thermocouple higher temperatures can be measured up to
1500° C.
Disadvantage: -
• Not suitable for overloading.
• The output voltage is very small and sensitivity is very low.

Hot wire meter: -
• The current flowing through the hotwire produces I2R losses, so that temp
increases and hence the hotwire expands which drives the pulley which is
attached to the pointer and hence, the pointer reads the current to be
measured on the scale which is calibrated in terms of the current.

Advantage: -
• Measures Both AC and DC.
• In case of Ac measures RMS quality.
• It is used to measure high frequency currents.
Disadvantage: -
• Not suitable for over loading.
• Sluggish or slow operation.

Electrostatic meter (Electrostatic voltmeter): -
For M.I.: -
𝐓𝐝 =
𝟏
𝟐
𝐈𝟐 𝐝𝐋
𝐝𝛉
Electrostatic: -
𝐓𝐝 =
𝟏
𝟐
𝐕𝟐 𝐝𝐜
𝐝𝛉

At balance: -
𝐓𝐜 = 𝐓𝐝
𝐊𝛉 =
𝟏
𝟐
𝐕𝟐
.
𝐝𝐜
𝐝𝛉
𝛉 ∝ 𝐕²
Enhancement of electrostatic voltmeter: -
𝐕𝐦 =
𝐕×𝐗𝐜𝐦
𝐗𝐜𝐦+𝐗𝐜𝐬
𝐦 = 𝐯/𝐯𝐦
𝐂𝐬 =
𝐂𝐦
(𝐦−𝟏)

• Electrostatic, voltmeters are working on the principal of electrical field.
• These are used to measure higher voltages in the kilovolts range.
• By using series multipliers the range of the meter can be enhanced hence
it is used in substations for measurement of very high voltages.
• CVT works on this principal.

Potentiometer: -
• Potentiometer is working on the principle of comparison between two
electrical quantities
• The accuracy of the instrument is higher because the reading is taken
whenever the galvanometer current is zero and hence it is called null
detector.
• The power consumption is very low.
• It is used for measurement of: -
(i) Unknown battery voltages.
(ii) Low resistance up to μΩ range
(iii) Non-electrical quantities like displacement, force, pressure etc.

• In case of DC potentiometer, galvanometer is used as a balanced detector.
• In case of AC potentiometer high precision electrodynamometer ammeter
is used as a balanced detector.
• In case of AC potentiometers both magnitude and phase angle must be
balanced.
• In the measurement of non-electrical quantities the output voltmeter
must have higher internal resistance so that errors due to loading effect
are minimized.
Standardization: -
• The method of converting cm scale in terms of voltage is called
standardization. This is done by using a known or standard voltage source.

SW = Slide Wire
SC = Sliding Contact
𝐫 = resistance of SW……….Ω /m
𝒍 = length of SW……..m
𝐈𝐰 = working current
𝐈𝐰 =
𝐕𝐁
𝐑𝐡+𝒍𝐫
Switch at standardization: -
𝐕𝐬 = 𝐈𝐰(𝒍𝟏𝐫)
𝐈𝐰 =
𝐕𝐬
𝒍𝟏𝐫
––––––(1)

Switch at calibration: -
𝐕𝐱 = 𝐈𝐰𝒍𝟐 𝐫
𝐈𝐰 =
𝐕𝐱
𝒍𝟐 𝐫
………….(2)
Equation (1) = Equation (2)
𝐕𝐬
𝒍𝟏𝐫
=
𝐕𝐱
𝒍𝟐 𝐫
𝒍𝟏 = length of at which VS is balanced
𝒍𝟐 = 𝐥𝐞𝐧𝐠𝐭𝐡 𝐨𝐟 𝐰𝐡𝐢𝐜𝐡 𝐕𝐱 𝐢𝐬 𝐛𝐚𝐥𝐚𝐧𝐜𝐞𝐝
𝐕𝐱 = 𝐕𝐬 .
𝒍𝟐
𝒍𝟏

Application: -
(1) Measurement of unknown voltage: -
2 V = 100 cm
1 cm = 0.02 V
At 60 cm: -
Vx = 60 × 0.02 = 1.2 V

(2) Measurement of displacement
(3) Measurement of low resistance
(4) Potentiometer is used for calibration of voltmeter, ammeter and
wattmeter’s.
Note: -
Once the potentiometer is balanced it is independent of internal resistance
of the voltage source.

Q: - For the network shown, if the potentiometer is balanced at point A,
calculate VS.
𝐕𝐬 =
𝟔×𝟑
𝟑+𝟏+𝟐
𝐕𝐬 = 𝟑 V……..Ans

Instrument transformers: -
Current transformers: -
Instrument transformers are used for the measurement of: -
(1) Higher voltages and higher currents
(2) The equipments and meters are isolated from the higher voltages and
higher currents.
(3) Power loss is minimum because no need of using RSh and RS.
(4) Used for multiple applications like provide voltage and current signals to
the ammeters, voltmeters, wattmeters and energy meters and
protective relays.
(5) Easy maintenance.
• C.T. has single turn primary with multiple secondary turns with multiple
windings, the nominal secondary current of the C.T. is 1A/5A.

• If the secondary current is of low value then power consumption or
burden on C.T. is minimized so that sensitivity of the C.T. is increased.
Note: -
With primary excited secondary of the C.T. can never be open circuited
because when load is connected in series to the primary winding, if
secondary is opened then opposing secondary flux becomes zero and
primary flux is not affected(not reduced) and hence huge voltage is induced
in the secondary which damages the insulation and dangerous to the person
working nearer to C.T.
Turns ratio = n =
𝐍𝟐
𝐍𝟏
Nominal ratio = K =
𝐈𝐏
𝐈𝐒
Actual (or) transformation ratio R =
𝐈𝐏
𝐈𝐒
= 𝐧 +
𝐈𝐨𝐒𝐢𝐧(𝛂+𝛅)
𝐈𝐒

R = 𝐧 +
𝐈𝐨𝐒𝐢𝐧𝛂𝐂𝐨𝐬𝛅+𝐈𝐨𝐂𝐨𝐬𝛂𝐒𝐢𝐧𝛅
𝐈𝐒
Where 𝛅 = Secondary burden angle = tan-1 𝐗𝐥+𝐗𝐒
𝐑𝐋+𝐑𝐒
Note: -
If purely resistive burden 𝛅 = 0
Ip = 𝐧𝐈𝐬
𝟐 + 𝐈µ
𝟐
R = n +
𝐈𝒘𝐂𝐨𝐬𝛅+𝐈µ𝐒𝐢𝐧𝛅
𝐈𝐒
R =
𝐈𝐏
𝐈𝐒
=
𝐧𝐈𝐬
𝟐+ 𝐈µ
𝟐
𝐈𝐒

Errors in C.T.: -
Due to no-load component of current Io , the ratio and phase angle errors
occur in C.T. and P.T.
Ratio error: -
This is given by σ.
Nominal ratio = K =
𝐈𝐏
𝐈𝐒
Actual (or) transformation ratio R =
𝐈𝐏
𝐈𝐒
= 𝐧 +
𝐈𝐒
R =
𝐧𝐈𝐬
𝟐+ 𝐈µ
𝟐
𝐈𝐒
(if 𝛅 = 0)
% σ =
𝐊−𝐑
𝐑
× 𝟏𝟎𝟎

Phase angle error (θ): -
Due to Io the phase angle difference between IP and IS is less than 180 degree. This
is called phase angle error measured in degrees or radians.
θ =
𝐈𝐨𝐂𝐨𝐬(𝛂+𝛅)
𝒏𝐈𝐒
……rad.
θ =
𝐈𝐨𝐂𝐨𝐬𝛂𝐂𝐨𝐬𝛅−𝐈𝐨𝐒𝐢𝐧𝛂𝐒𝐢𝐧𝛅
𝐧𝐈𝐒
For purely resistive burden 𝛅 = 0o ⇒ θ = ………….degrees
Phase angle between Ip and IS = (180- θ)
θ =
𝐈µ𝐂𝐨𝐬𝛅−𝐈𝒘𝐒𝐢𝐧𝛅
𝐧𝐈𝐒
……….Rad.
θ =
𝐈µ𝐂𝐨𝐬𝛅−𝐈𝒘𝐒𝐢𝐧𝛅
𝐧𝐈𝐒
×
𝟏𝟖𝟎
𝝅
………..degrees
𝐈µ
𝐧𝐈𝐒
×
𝟏𝟖𝟎
𝝅

Potential transformer: -
P.T. has multiple primary and multiple secondary turns used for
measurement of higher voltages.
Practically used in all the substations for measurement of bus-bar voltages..
The normal secondary voltage is 110 V.
Methods of reducing errors: -
Ratio and phase angle errors are occur due to no load component of current
Io. If the magnitude of Io is minimized then these errors are minimized.
(1) By reducing reluctance of magnetic path Io is minimized.
Reluctance S =
𝐥
µ𝐨µ𝐫𝐀
S∝
𝟏
µ𝐫

As µ𝐫 is more reluctance is less. So we use high µ𝐫 materials like silicon steel,
permendur, hypernik etc.
• For reducing the reluctance toroidal core or strip wound core are used so
that no. of joints are minimized and hence reluctance and Io are reduced.
• By reducing the distance between primary and secondary cores air leakage
flux is minimized so that Io is reduced.
o
o
o
ooo ooo
o
o
o
ooo
o
o
o
ooo
o
o
o
Air
Toroidal core
Strip wound core

• By using single turn primary in the C.T. both ratio and phase angle errors
are minimized.
θ =
𝐧𝐈𝐒
=
𝐈𝐩
n =
𝐍𝟐
𝐍𝟏
=
𝐈𝐏
𝐈𝐒
That is why N1 = 1.
• By reducing turns compensation ratio error is minimized. There is no effect
on the phase angle error.
% σ =
𝐊−𝐑
𝐑
× 𝟏𝟎𝟎 =
𝐧−𝐑
𝐑
× 𝟏𝟎𝟎
R = n +
𝐈𝐒

R = 200 + 2
• So we are using 198 turns in the secondary instead of 200.
= (200-2) + 2 = 198 + 2 = 200 = n
% σ = 0 %.
• For reducing the phase angle error an extra auxiliary secondary short
circuit turn can be used, this method is called Wilson compensation
method.
Due to Io

CRO (Cathode Ray Oscilloscope): -
Advantage: -
1. Linear device
2. X-Y Plotter
3. Wave form display.
4. High Resolution
5. Higher Sensitivity
6. Low Power Consumption
7. Higher Accuracy
8. Higher speed
9. Stores wave form
10. Free from external fields
11. Easily Adjustable scale
12. Multiple wave forms can be observed.

CRT (cathode ray tube): -
Acc - Accelerating

1. CRO or CRT working on the principal of thermionic emission i.e. emitting of
electrons from a heated surface (electron gen).
2. The brightness of electron beam on the screen is control by changing the
cathode grid potential.
3. The velocity of the electron is accelerated by changing pre accelerating and
accelerating anode potential.
𝐊. 𝐄. = 𝐏. 𝐄.
𝟏
𝟐
𝐦𝐯² = 𝐪. 𝐕𝐚
𝐯 =
𝟐𝐪𝐕𝐚
𝐦
𝐯 ∝ 𝐕𝐚
Where,
v = velocity of electron m/sec.
𝐕𝐚 = Anode voltage
m = mass of electron
q = charge of electron

• Electric field and potential fields are orthogonal to each other.
*Double concave electron lenses*

• Laboratory CRO uses electrostatic focus control.
• This is working on the principal of double concave electron lenses.
• By adjusting focus anode potential, the focal point can be change.
• For fine focus control small dc potential is applied to the horizontal and
vertical deflecting plates, this is called ‘Astigmatism’.
• Television picture tube works on the electromagnetic focus control.
• Computer CRO works on both electrostatic and electromagnetic focus
control.

0 0
O
O
O
O
O
O
O O O O O O
5
4
3
2
1
0
VDD
HDP
0 0.5 1 1.5
2 2.5 3 3.5 4
0 1 2 3 4 5
1
2
3
4
VX
Vy
Vm
t
tr tf

For square signal: -
⇒
• Horizontal deflecting plates (HDP) are kept in vertical position.
• These are applied with saw tooth waveform which is produced by UJT
relaxation oscillator or operational amplifier.
• The time period of the saw tooth waveform can be adjusted by using
synchronizing input.
• The saw tooth is maintain equal to test signal time period.
This is not visible in CRO

• The saw tooth time period is maintained equal to test signal time period.
• HDP is used for shifting the electron beam Horizontally i.e. Horizontal time
scale is adjusted by changing HDP potential.
• VDP is applied with test signal, whose waveform has to be observed on
the screen VDP is kept in Horizontal position.
• For collection of Secondary electrons aquadag coated with graphite is used
for maintaining electrical neutrality with in the CRT.
• During retrace or fly back time of the saw tooth signal blanking ckt is
initiated which activates the cathode grid so that high negative potential is
applied and hence the electrons are completely stopped entering into the
CRT so that blurred image shown on the screen can be avoided.
• Different types of phosphorous coating is used on the screen depending on
the application this phosphor converts heat into light energy.

𝐏𝟐 − Black/White CRO
𝐏𝟒 − Colour CRO
𝐏𝟔 − General purpose CRO
𝐏𝟏𝟏 − High frequency/sampling CRO
𝐏𝟏𝟓 − storage CRO
Deflection sensitivity (S): -
d
Vy
𝒍 L
D
Screen
D =
𝐋𝒍𝐕𝐲
𝟐𝐝𝐕𝐚
S =
𝐃
𝐕𝐘
=
𝐋𝒍
𝟐𝐝𝐕𝐚
…………mm/V
Deflection factor (G)
G =
𝟏
𝑺
=
𝟐𝐝𝐕𝐚
𝐋 𝒍
……….V/mm

Va = Anode voltage
Vy = VDP voltage = test signal
𝒍 = length of VDP
d = distance between VDP plates
L = Distance between center of VDP to screen
D = height at which beam is hitting on the screen (Deflection height)
Special CRO’S: -
(1) Dual Trace CRO: -
This CRO consisting of due electron beam and two vertical deflecting plates
with a selection switch. By changing the selection switch one of the vertical
plate waveform can be observed on the screen.
i.e. waveforms can be alternatively observed. Simultaneous visibility is not
possible.

• By using multiplexer the two waveforms can be observed one after the
other. This is called alternate mode.
• The control signal of the multiplexer frequency can be maintained = twice
the frequency of saw tooth or sweep oscillator.

Dual beam CRO: -
It consisting of two Horizontal, two vertical deflecting plates with two
electron guns, So that two different waveforms can be observed
simultaneously on the screen.
• Cost is higher, It is normally used in research laboratories.
High frequency or Sampling CRO: -
• This consisting of an extra post accelerating anode which is used for
controlling the brightness of beam on the screen. Short persisting
phosphorous coating is used in the High frequency CRO’s.
Storage CRO: -
Memory storing elements are used with in the CRO for storage of waveform
pattern in the CRO’s.

Application of CRO: -
Lissajous pattern: -
If both horizontal and vertical deflecting plates are applied with sinusoidal
signals then waveform pattern appearing on the screen is called lissajous
pattern.
By using this pattern,
• Phase angle difference between two signals can be measured.
• The unknown frequency of the signal from the known frequency of signal
and from the known lissajous pattern can be measured.
Note: -
At any point of time the electron beam hitting on the screen is the vector
sum of voltages applied to the both horizontal and vertical deflecting plates.
The vector sum consisting of both magnitude and phase angle.

Point 𝐕𝐱 𝐕𝐲 |𝐌| 𝛉 = 𝐭𝐚𝐧−𝟏 𝐕𝐲
𝐕𝐱
0 0 0 0 0o
A Vm Vm 𝟐 𝐕𝐦 45°
B 0 0 0 0o
C −𝐕𝐦 −𝐕𝐦 𝟐 𝐕𝐦 225°
D 0 0 0 0o
𝐕𝐱 = 𝐕𝐦 𝐬𝐢𝐧 𝛚𝐱𝐭
𝐕𝐲 = 𝐕𝐦𝐬𝐢𝐧(𝛚𝐲𝐭 + 𝛟)

Case1: -
𝐕𝐱 = 𝐕𝐲 = 𝐕𝐦
𝛚𝐱 = 𝛚𝐲 = 𝛚 ; 𝛟 → charging
𝐕𝐱 = 𝐕𝐦𝐬𝐢𝐧𝛚𝐭
𝐕𝐲 = 𝐕𝐦 𝐬𝐢𝐧(𝛚𝐭 + 𝛟)
(1) 0 to 45o (2) 0o < 𝛟 < 90o
or
270o < 𝛟 < 360o
Straight line

(3) 𝛟 = 90o or 270o (4) 90o < 𝛟 < 180o
or
180o < 𝛟 < 270o

Calculation of 𝛟 from the given Lissajous pattern: -
𝛟 = 180o

(1) LP in 1 and 3rd quardrants: -
𝛟 = Sin-1 𝐗𝟏
𝐗𝟐
= Sin-1 𝐘𝟏
𝐘𝟐
Second possibility
= (360o - 𝛟)
(2) LP in 2nd and 4th quardrants: -
𝛟 = 180o - Sin-1 𝐗𝟏
𝐗𝟐
= 180o - Sin-1 𝐘𝟏
𝐘𝟐
Second possibility
= (360o - 𝛟)
X1
X2
Y2
Y1
Y1
Y2
X1
X2

Case (2): -
ωx = ωy
𝛚𝐲
𝛚𝐱
=
𝐟𝐲
𝐟𝐱
=
𝐍𝐨.𝐨𝐟 𝐡𝐨𝐫𝐢𝐳𝐨𝐧𝐭𝐚𝐥 𝐭𝐚𝐧𝐠𝐞𝐧𝐜𝐢𝐞𝐬
𝐍𝐨.𝐨𝐟 𝐯𝐞𝐫𝐭𝐢𝐜𝐚𝐥 𝐭𝐚𝐧𝐠𝐞𝐧𝐜𝐢𝐞𝐬
Example: -
VX = 𝐕𝐦𝐬𝐢𝐧 (𝟑𝟏𝟒𝐭)
Find fy= ? Or 𝛚y = ?
LP: -
𝛚y
ω𝑿
=
𝟐
𝟒
𝛚y
𝟑𝟏𝟒
=
𝟏
𝟐
ωy = 157

Bandwidth of CRO: -
The maximum undistorted frequency observed on the CRO screen is called
Bandwidth of the CRO, it is related with the rise time of the signal.
𝐁𝐖 × 𝐭𝐫 = 𝟎. 𝟑𝟓
𝐁𝐖 = Bandwidth in Hz
𝐭𝐫 = rise time in sec.
BW = Bandwidth in Hz.
Calibration of CRO: -
CRO is calibrated by applying square or pulse signal of known frequency and
magnitude, by using these parameters the horizontal and vertical scales of
the CRO are calculate.

• X-scale : ms/cm or ms/Division
• Y-scale: mv/cm or mv/Division
T = 5×8=40 msec.
𝐕 = 𝐕𝐦 𝐬𝐢𝐧 𝟑𝟏𝟒𝐭
Given No. Of cycles visible on the CRO screen=?
Let cycles = n
n =
𝐓𝐦𝐚𝐱
𝐓

If 𝐕𝐩𝐩 > 𝐕𝐦𝐚𝐱 then Signal is clipped.
Example: -
X-scale : 5 ms/cm
T= 5×4=20ms
𝐟 =
𝟏
𝐓
=
𝟏
𝟐𝟎𝐦𝐬
= 𝟓𝟎 𝐇𝐳
Frequency can be calculated by observing this CRO.

Probes in CRO: -
• Probe is a interconnecting channel between function generator and CRO.
• Probe must have high input resistance and low resistance, so that the
signal taken from the function generator should not be attenuated.
• The probe must be independent of frequency and hence the time constant
of probe and CRO is maintained equal.
• Laboratory CRO’s uses 10X or high impedance probe.
• In the computers active probes are used between internal drives to the
CPU.
• These active probes consisting of BJT emitter followers, MOSFETs and op-
amp’s which has Ri = ∞ and Ro is very low.

Q-meter or quality factor meter (voltage magnifier): -
Q – meter works on the principle of series resonance, this is used for: -
(1) Measurement of Q meter of the coil.
(2) Measurement of unknown capacitance.
(3) Measurement of unknown inductance and resistance.
(4) Measurement of self or distributed capacitance

Practical Q-meter: -
Measurement of Q: -
QT = Qm 𝟏 +
𝐑𝐒
𝐑
QT = True Quality factor
Qm = Measured quality factor
For maintaining QT = Qm and for reducing error, R is used in mΩ range.
Measurement of unknown capacitance: -
CT = C2 – C1
Measurement of self or distributed capacitance of coil: -
Cd =
𝐂𝟏−𝐧𝟐𝐂𝟐
𝐧𝟐−𝟏

Digital voltmeter: -
Advantage: -
(1) Low power consumption
(2) Easily readable scale
(3) No parallax error
(4) High speed
(5) High accuracy
(6) High sensitivity
(7) Low cost
(8) Compact size
(9) No maintenance
(10) Free from external fields

(11) Easily adjustable scale
(12) Multiple applications
Disadvantage: -
Requires battery supply.
BCD converter: -
(1) Flash – 1 clock
(2) Successive approximation register (SAR) – n clock
(3) Counter – 2n clocks
(4) Dual slope – 2n+1 clocks (Most accurate)
n = no. of bits

Performance parameters of DVM: -
(1) Resolution (R): -
Resolution is the smallest change in input that can be able to detect by the
digital meter is called resolution.
R =
𝟏
𝟏𝟎𝐍
N – No. of full digits (0 to 9)
(2) Sensitivity (S): -
The smallest value of the input that can be able to measure by the digital
meter in a given range (scale) is called sensitivity.
S = Full scale Range × resolution

Over-ranging: -
Low High R =
𝟏
𝟏𝟎𝐍
000 999
𝟏
𝟏𝟎𝐍 = 0.001
Low High Range
0000 1999
𝟏
𝟏𝟎𝐍 = 0.001
0
1
½ digit
3½ digit display

Measurement of R, L,C : -
AC bridges: -
These are used for measurement of
inductance capacitance, quality factor,
frequency, µr and εr and dissipation
factor.
At balance: -
Ig = 0
V1 = V3
I1 Z1∠𝛉𝟏 = I3 Z3∠𝛉𝟑
𝐈𝟏
𝐈𝟑
=
Z3∠𝛉𝟑
Z1∠𝛉𝟏
………….(1)
I3
I2
I4
V1
V3 V4
V2
VAC , f
Ig
>

V2 = V4
I2 Z2∠𝛉𝟐 = I4 Z4∠𝛉𝟒
𝐈𝟐
𝐈𝟒
=
Z4∠𝛉𝟒
Z2∠𝛉𝟐
…………..(2)
If Ig = 0 , I1 = I2, I3 = I4
Z3∠𝛉𝟑
Z1∠𝛉𝟏
=
Z4∠𝛉𝟒
Z2∠𝛉𝟐
Z1∠𝛉𝟏 =
Z2Z𝟑
Z4
∠𝛉𝟐 + 𝛉𝟑 − 𝛉𝟒
Z1 =
Z2Z𝟑
Z4
; ∠𝛉𝟏 = ∠𝛉𝟐 + 𝛉𝟑 − 𝛉𝟒

Measurement of Q: -
(1) Maxwell L-C bridge = Medium Q- coil – 1 < Q < 10
(2) Hay’s bridge = High Q coil – Q > 10
(3) Anderson bridge = Low Q coil – Q < 10
(4) Owns bridge – Incremental Inductance µr

Maxwell L-C bridge: -
At balance: -
Z1Z4 = Z2Z3
(R1 + jωL1)
𝐑𝟒
𝟏+𝐣ω𝐂𝟒𝐑𝟒
= R2R3
R4 R1 + jωL1R4 = R2R3 +𝐣ω𝐂𝟒𝐑𝟒R2R3
Separate real and imaginary part
R4 R1 = R2R3
R1 =
R2R3
R4
L1 = R2R3C4
Q =
ω𝐋𝟏
𝐑𝟏
= ωR4C4

• If R2 and R3 are selected as a balance variables then more number of
operations are required for bridge balance, this is called sliding balance.
• For faster bridge balance and to avoid sliding balance, the uncommon
parameters i.e; R4 and C4 are selected as variables.
Convergence to balance point: -
If the variables are lying in the same arm of bridge then time for balancing is
minimized.

Wagner earth device: -
• Wagner earth device is used for eliminating stray capacitance (CS)
occurring between bridge nodes and nodes to the earth.

Procedure: -
• Initially switch is connected at B, by varying R4 & C4, Bridge is balanced
now the switch is moved to A. By varying Rw , the voltage across the Cs is
discharged to the ground.
• Now change the switch to B, vary R4, C4 until the bridge is balanced. This
process of balancing with a variation of Rw & R4 , C4 is repeated until Ig
becoming zero without varying R4C4 and Rw , this indicates the voltage
across the stray capacitance is at ground potential.
• Now take the readings of R4, C4 and R2 ,R3 and measure the accurate values
of R1 and L1.
• At very high Q coil higher value of C4 and R4 has to be maintained, so that
which is not practical and difficult for bridge balance.
• At very low Q coils sliding balance occurs if R4 and C4 of very low value is
selected.

Hay’s bridge: -
At balance: -
R1 =
𝛚𝟐𝐑𝟒
𝟐
𝐂𝟒
𝟐
𝐑𝟐𝐑𝟑
𝟏+𝛚𝟐𝐑𝟒
𝟐𝐂𝟒
𝟐
L1 =
𝐑𝟐𝐑𝟑𝐂𝟒
𝟏+𝛚𝟐𝐑𝟒
𝟐𝐂𝟒
𝟐
Q =
𝟏
𝛚𝐑𝟒𝐂𝟒

Convert delta to star
x =
𝐑𝟒/𝐣𝛚𝐂
𝐑𝟒+𝐫+
𝟏
𝐣𝛚𝐂
z =
𝐑𝟒𝐫
𝐑𝟒+𝐫+
𝟏
𝐣𝛚𝐂

Under bridge balance: -
L 1 =
𝐑𝟒𝐑𝟑𝐂𝟒(𝐑𝟒+𝐫)
𝐑𝟒
+ 𝐑𝟑𝐂𝟒𝐫
R1 =
𝐑𝟑𝐑𝟐
𝐑𝟒
− 𝐫𝟏
Q =
𝛚𝐋𝟏
𝐑𝟏
• For balancing of Hay’s bridge initially vary R4 and the either R2 or R3.
• Hay’s bridge is used for the measurement of high Q coils (Q>10).
• Anderson bridge is most complicated bridge, balance is obtained by
varying r and r1.
• It is used for measurement of low Q coils.
•

Owns bridge: -
At balance: -
R1 =
𝐂𝟒𝐑𝟑
𝐂𝟐
L1 = R2R3C4
` Q =
ω𝐋𝟏
𝐑𝟏

Modified Owns bridge: -
L1 =
𝐍𝟐µ𝐨µ𝐫𝐀
𝐥
µr =
𝐑𝟐𝐑𝟑𝐂𝟒𝐥
𝐍𝟐µ𝐨𝐀
• By using owns bridge unknown coil
resistance and inductance and Q factor can
be measured from the known value of
capacitance.
• By using modified owns bridge incremental
inductance of any core material can be
calculated.
• Due to hysteresis effect, some residual
current is present in the core material even
though the bridge is balanced.

• Because of this accurate value of µr can not be able to calculate.
• For obtaining accurate value a DC voltage with variable resistor is
connected in the modified owns bridge. Which will inject external dc
current which is used to nullify the effect. Now by taking the readings of
known parameters and physical dimensions of the core material µr is
calculated.
µr =
𝐑𝟐𝐑𝟑𝐂𝟒𝐥
𝐍𝟐µ𝐨𝐀
• LB and CB are used for blocking the AC and DC respectively.
Measurement of mutual inductance: -
(1) Heavy side bridge = Measures M from known L
(2) Heavy side campbell bridge = Meausres L from known M

Measurement of capacitance (C): -
(1) Desauty bridge: - Measures C, D = tanδ
At balance: -
r1 = (R2+r2)
𝐑𝟑
𝐑𝟒
- R1
C1 =
𝐑𝟒
𝐑𝟑
𝐂𝟐
Dissipation factor, D = tanδ = ωC1R1

Schering bridge: -
At balance: -
r1 =
𝐑𝟑
𝐂𝟐
𝐂𝟒
C1 =
𝐑𝟒
𝐑𝟑
𝐂𝟐
Q =
𝟏
𝐃
=
𝟏
ωC4R4

Measurement of εr using Schering bridge : -
εr =
𝐭
𝐭−𝐱
t = thickness of dielectric

Measurement of frequency: -
Weins bridge: -
At balance: -
f =
𝟏
𝟐𝛑 𝐑𝟏𝐑𝟐𝐂𝟏𝐂𝟐
If R1 = R2 = R
C1 = C2 = C
Then;
f =
𝟏
𝟐𝛑 𝐑𝐂

• Weins bridge is used for measurement of frequency up to MHz range.
• If the test signal containing harmonics then bridge balancing is difficult.
Measurement of Resistance: -
Properties of resistance: -
• Resistance should not change with time.
• It should be independent of frequency.
• Resistivity should be higher.
• Temperature coefficient must be lower (α).
• For reducing the effect of inductance in resistive material bifilar winding is
used.
• Thermo-electric effect must be low.

Classification: -
(1) Low resistance
(2) High resistance
(3) Medium resistance
(1) Low resistance – R < 1 Ω - Motor/ generator/ transformer winding.
(2) Medium Resistance- 1 Ω < R < 100 kΩ - Electronic equipments
(3) High Resistance – R > 100 kΩ - Cable, insulation Resistance of electrical
motor, generator, transformer etc.

Low resistance method: -
(1) Kelvin’s double bridge
(2) Potentiometer method
(3) Voltmeter-ammeter (V-I) method
Medium Resistance method: -
(1) Voltmeter ammeter method ( V-I method)
(2) Substitution method
(3) Ohm –meter
(4) Wheat stone bridge
(5) Carry foster slide wire bridge

High resistance method: -
(1) Megger
(2) Loss of charge method
(3) Direct deflection method
Low resistance method: -
Kelvin’s double bridge: -
• It is used for measurement of low resistance up to micro ohms range.
• The effect of lead resistance is eliminated using kelvin’s double bridge by
maintaining inner and outer ratio arms, resistance ratio are equal.
• It is practically used in the measurement of winding resistance of electric
motors, generators, transformer and earth conductor resistance.
• Reversible switch is used for reducing thermo-electric emf at the junctions of the
resistance.

• Convert delta into star x =
𝐩𝐫
𝐩+𝐪+𝐫
y =
𝐪𝐫
𝐩+𝐪+𝐫

• Under balance bridge balance
𝐏
𝐐
=
𝐑+𝐱
𝐒+𝐲
………….(1)
Substitute value of x,y in (1)
If
𝐏
𝐐
=
𝐩
𝐪
R =
𝐏
𝐐
S
R =
𝐏
𝐐
𝐒 +
𝐪𝐫
𝐩+𝐪+𝐫
𝐏
𝐐
−
𝐩
𝐪
Including lead resistance
No lead resistance

Medium resistance method: -
(1) V-I method: -
True resistance – RT
Measured resistance – Rm =
𝐕
𝐈
Ammeter neared to RT : -
Assume RV is very high
IV = 0
I = Ia
V = I (Ra+ RT)
𝐕
𝐈
= Rm = Ra + RT
% εr =
𝐑𝐦−𝐑𝐓
𝐑𝐓
× 100 =
𝐑𝐚
𝐑𝐓
× 𝟏𝟎𝟎

εr is low if RT >> Ra
• Suitable for measurement high resistance in medium range.
Voltmeter nearer to RT: -
I = IR + IV
I =
𝐕
𝐑𝐓
+
𝐕
𝐑𝐕
𝐈
𝐕
=
𝟏
𝐑𝐓
+
𝟏
𝐑𝐕
=
𝟏
𝐑𝐦
𝟏
𝐑𝐕
=
𝐑𝐓−𝐑𝐦
𝐑𝐦𝐑𝐓
% εr =
𝐑𝐦−𝐑𝐓
𝐑𝐓
× 100 = -
𝐑𝐦
𝐑𝐕
× 𝟏𝟎𝟎
% εr = -
𝐕
𝐈𝐑𝐕
× 𝟏𝟎𝟎

Approximate: -
Assume Rm = RT
εr is low if RT << RV
⇒ Suitable for measurement of low R.
Substitution method: -
% εr = -
𝐑𝐓
𝐑𝐕
× 𝟏𝟎𝟎

Switch at A: -
I1 =
𝐕
𝐑𝐡+𝐑𝐓+𝐑𝐦
V = I1(𝐑𝐡 + 𝐑𝐓 + 𝐑𝐦)…………(1)
Switch at B: -
V = I2(𝐑𝐡 + 𝐒 + 𝐑𝐦)…………(2)
I1(𝐑𝐡 + 𝐑𝐓 + 𝐑𝐦) = I2(𝐑𝐡 + 𝐒 + 𝐑𝐦)
• Switch is initially kept at position A, by changing the rheostat, readable
amount of current is passing through the circuit and the current is
measured in the ammeter.
RT = S (if I1 = I2)

• Now the switch is moved to position B, Now by changing standard
resistance box until same current is flowing through the ammeter and the
value of standard resistance is recorded which is equal to the test
resistance RT, i.e. standard resistance is substituted in place of test
resistance.
Wheat stone bridge: -
𝐏
𝐐
=
𝐑
𝐒

• Sensitivity is a important parameter in bridge circuit compare to linearity,
accuracy etc.
• Small change in the measured has to be detected and displayed by the
bridge.
Applications: -
• Used for measurement of medium resistance.
• Used for measurement of non-electrical quantities like, temperature,
vacuum pressure, liquid level, liquid flow rate etc.
• In communication applications used for measurement of distance of earth
fault occur in a underground cable, by using Murray-Varley loop test.
Limitations: -
• Not suitable for measurement of low resistance.
• Because it does not eliminate the effect of lead resistance.

Carry foster slide wire: -
SW- Slide wire
𝐏
𝐐
=
𝐑+𝐥𝟏𝐑
𝐒+ 𝐋−𝐥𝟏 𝐫
………..(1)

By changing R,S
𝐏
𝐐
=
𝑺+𝒍𝟐𝑹
𝑹+ 𝑳−𝒍𝟐 𝒓
………..(2)
• It is working on the principle of Wheatstone bridge used for measurement
of medium resistance by comparing with standard known value of the
resistance.
R – S = (𝒍𝟐 − 𝒍𝟏)𝒓

Measurement of high resistance: -
Loss of charge method: -
VC = Ve-t/RC
R =
𝐭
𝐂𝐥𝐧 𝐕/𝐕𝐂
or
𝟎.𝟒𝟑𝟒𝟑𝐭
𝐂𝐥𝐨𝐠𝟏𝟎 𝐕/𝐕𝐂
Where t is in seconds.
VC
V
0 t
Switch open

Direct deflection method: -
R =
𝐕
𝐈
V = voltmeter reading
I = micro- ammeter reading

Transducer
F (or) P, T D Vo (or) Io
Example: -
Helical spring
Primary transducer or
mechanical transducer
Secondary transducer
or electrical transducer

Primary transducer: -
If the output of transducer displacement or angular displacement, known as
primary transducer. These are also called as mechanical transducer.
Secondary transducer: -
If the output of transducer is either voltage or resistance i.e. electrical
quantity known as secondary transducer.
Under secondary transducer, inductive transducer etc are comes.
Usually in the resistive type, potentiometer is one of the resistive transducer
and also strain gauge.
In the inductive type variable reluctance type RVDT, LVDT.
In the capacitive type only capacitive type transducer comes.

• All the temperature measuring devices are come in mechanical transducer
category except thermos couple and thermopile.
Active transducer: -
These are the transducer which do not require any external energy source
i.e. which will consist of self generating type source.
Passive transducer: -
These are the transducer which require any external energy source, i.e.
which do not have any self generating type source.

Analog transducer: -
These are the transducers whose output signal is in the analog form i.e.
which is continuous in nature.
Digital transducer: -
These are the transducer whose output is in the digital form i.e. which are in
discrete nature.
Example: -
Optical shaft encoder.
Inverse transducer: -
These are the transducer which can convert electrical energy into non-
electrical energy.

Example: -
Motor
Electrical Mechanical
Piezo electric
Transducer
Piezo-electric
transducer
Ultrasound
vibration

Displacement measuring devices: -
(1) Potentiometer – used in cm range – 10-2 m
(2) Inductive transducer – Used in mm range - 10-3 m
(3) Capacitive transducer – Used in mm range – 10-3 m
(4) Hall effect transducer – Used in µm range – 10-6 m
Accuracy: -
LVDT > Capacitive Transducer > Potentiometer
(Most accurate) (More Accurate) (Accurate)

Advantages: -
• Potentiometer is used to measure large displacement over a wide range in
order of cm.
• Potentiometer has higher sensitivity.
• It has linear characteristics.
• Easy to construct
• Simple in operation
• Lesser cost
• Electrical efficiency is high.
• No need of amplification of output voltage.

Disadvantages: -
• Large amount of force is required in order to move sliding contact.
• There is a friction developed between the slide wire and sliding contact.
• To measure the output voltage across potentiometer a voltmeter is
connected which has finite resistance which causes loading effect, to
reduce the loading effect, a high sensitive voltmeter is preferred.
• Materials used in potentiometer are platinum, nickel- chromium.
Power rating of potentiometer: -
Power rating P =
𝒆𝟐
𝑹𝒕
= e2 = P.Rt
emax ∝ 𝐑𝐭
S ∝ Vo ∝ emax ∝ 𝐑𝐭

• In order to design higher sensitivity of potentiometer Vo should be high
which in turn required high input voltage so that the total resistance of
potentiometer should be high.
Requirements of transducer: -
• It has higher sensitivity.
• It has linear characteristics.
• It should be reliable and highly stable.
• It should be maintenance free.
• It can be used for wide range of applications.
• Less affected by surroundings atmosphere conditions.

Angular potentiometer: -
Vo ∝ θ
S =
𝑽𝒐
θ
…….volt/degree
or
volt/radian

Helipot: -
Helipot is a combination of linear and angular potentiometer. These are
highly stable at very high temperature.
The material used to prepare heliport usually is platinum ( Because platinum
is highly stable at high temperature).
Strain gauge: -
It works on the principle of piezoelectric effect.
Tensile
𝒍
𝒍 + ∆𝒍

Tensile increase, so that length 𝒍 increases, and area A decreases.
Volume will be constant V = A × 𝒍
𝒍′
= 𝒍 + ∆𝒍 , R’ = R + ∆𝐑 , R’ > R
Longitudinal strain = L =
∆𝒍
𝒍
= positive
Lateral strain = -
∆θ
θ
= negative
According to piezo resistive effect, the resistance of strained wire always
greater than that of unstrained wire with same material and same
dimensions.
This phenomenon is known as piezo resistive effect.

Compressive increase, so that length 𝒍 decreases, and area A increases.
Volume will be constant V = A × 𝒍
𝒍′
= 𝒍 − ∆𝒍 , R’ = R - ∆𝐑 , R’ < R
Longitudinal strain = L = -
∆𝒍
𝒍
= Negative
Lateral strain = -
∆θ
θ
= Positive
Compressive
𝒍
𝒍 − ∆𝒍

Poission’s ratio: -
Poission’s ratio (n) =
𝐋𝐚𝐭𝐞𝐫𝐚𝐥 𝐬𝐭𝐫𝐚𝐢𝐧
𝐋𝐨𝐧𝐠𝐢𝐭𝐮𝐝𝐢𝐧𝐚𝐥 𝐬𝐭𝐫𝐚𝐢𝐧
n = 0 to 0.5 for metals.
Strain =
∆𝐨𝐮𝐭𝐩𝐮𝐭
∆ 𝐢𝐧𝐩𝐮𝐭
= gauge factor (Gf)
Gf =
∆𝐑/𝐑
∆𝐥/𝐥
Gf is = 2 to 3 for metals
= 3000 to 4000 for semiconductors
∆𝑹 = Gf R ε

Gauge factor in terms of poission’s ratio: -
Consider a circular wire of length 𝒍 and diameter D. The area of cross section
=
𝝅𝑫𝟐
𝟒
Let S = Stress developed in the wire =
𝑭
𝑨
N/m2
𝝆 = resistivity of wire
R = 𝝆
𝒍
𝑨

Partial differentiation of R with respect to S,
Valid for semiconductors
Rosettes gauge: -
A simple strain gauge can measure the stress only one direction.
Rosette is a combination of two (or) more strain gauge which are used to
measure strain in any direction.
Gf = 1 + 2n+
∆𝛒/𝛒
ε

Load cell: -
Load cell is used to measure mechanical load. Load cell is a combination of
elastic member acting like primary transducer and strain gauge acting like
secondary transducer so that by change in pressure we can measure the
mechanical load.
Elastic devices: -
These are used to measure pressure.
Bourden tube: -
𝒙 ∝ 𝑷
This is used to measure high pressure greater than atmospheric pressure.
1 atm = 760 mm of Hg
1 atm = 760 Torr
1 Torr = 1 mm of Hg

SB.T. =
𝐨𝐮𝐭𝐩𝐮𝐭
𝐢𝐧𝐩𝐮𝐭
mm/N/m2
Types of Bourden type: -
(1) C- type
(2) Spiral type
(3) Helical type
Sensitivity: -
Spiral type > Helical type > C- type

Diaphragm: -
For low pressure less than atmospheric pressure.
Types of diaphragm: -
(1) Flat type
(2) Corrugated type
Scorrugated-type > Sflat
Capsules: -
When two corrugated types diaphragm are joined together capsule is
formed.
Scapsule > SDiaphram

Diaphragm: -
A thin metal whose ends are fixed between two parallel plates knows as
diaphragm. It is used to measure the low pressure. The operating principle is
the applied pressure is converted into displacement.
Materials used to prepare the diaphragm are phosphor bronze, beryllium,
copper, stainless steel, Nickel.
The materials used for preparation of diaphragm depends upon
temperature range.
It also depends upon the chemical nature of the fluid coming in contact with
diaphragm during pressure measurement of fluid.
Advantage: -
• High accuracy
• Good dynamic response.

Disadvantage: -
They are prone to shock vibrations.
Applications: -
Used to measure, displacement, force and pressure.
Note: -
All pressure measurement and flow measurement devices comes under the
category of mechanical transducers.

(1) Bourden tube – High pressure measurement device
(2) Diaphragm
(3) Capsule Low pressure
(4) Bellows
(5) Pirani gauge - Low pressure (10-3 mm of Hg to 10-1 mm of Hg)
Flow measurement: -
• Orifice plate
• Rotameter
• Turbine flowmeter
• Wirles
• Pitot tube
• Flow nozzle
• Elbow
• Vanes – Velocity of fluids

Temperature measuring devices: -
RTD(resistance temperature detector): -
RTD is a temperature measuring device, which works on the principle of
change in resistance of a metallic conductor due to heat produced by heater
element which has positive temperature coefficient. Most commonly used
material to prepare RTD is platinum because it is highly stable at very high
temperature, Copper is not commonly used material around upto 120oC we
can use it. Gold are not commonly used material because of low resistivity
and high conductivity.
Applications: -
• It is used as temperature sensing element.
• It is also used in stator winding protection of an alternator.
• It is also used in small capacity boilers.
• It is also used in turbines and electric oven.

Thermistors: -
• Materials used is sintered mixture of metallic oxides (Fe+Ni+Co).
Working principle: -
Thermistor is a temperature measuring device which works on the principle of
change in resistance of a semiconducting material due to heat produced by heater
element. Which has negative temperature coefficient.
To improve the linearity of thermistor a fixed value of resistor is connected in
parallel with thermistor.
Applications: -
• It is used as temperature sensing element.
• It is also used as temperature control unit in electronic circuit.
• It is used as Q-point stabilization in BJT and electronic circuit.
• Used as time delay unit.
• For measurement of non-electrical quantities like flow rate, pressure, liquid level
and thermal conductivity.

Thermocouple: -
Works on seeback effect. Inverse of seeback effect is ‘Peltier effect.’
Note: -
The produced thermal emf is DC in nature detected by PMMC (avg value)
but the scale is calibrated to read rms value of AC current flowing through
heater element.
Note: -
RTD and thermistor are passive transducers whereas thermocouple is an
active transducer.
(Sensitivity)thermistor > (Sensitivity)RTD> (Sensitivity)thermocouple

Non-Linearity: -
Thermistor > Thermocouple > RTD
Pyrometer: -
Works on the principle of black body radiation. Pyrometer used to measure
temperature for greater than 2500o C.
Material used is Pyrex(Powdered silica).
Bimetallic strip: -
Works on the principle of linear expansion of length of the metal due to
heat produced by the metal.

Applications: -
(1) In AC refrigerator tube light starter.
(2) In oil burners, in thermal relays.
• Bimetallic strip is prepared by manganese and chromium, nickel-iron alloy with
combination of manganese and chromium.
• Temperature range for use is around up to 400o C.
• To measure low temperature thermometers is used, mercury glass
thermometers upto 300o C.
• Platinum resistance thermometers used upto 600o C to 900o C.
• RTD is used upto 183o C.
• Thermistor is used up to -55o C to 15o C.
• Thermocouple upto 1100o C/2000o C
• Pyrometer upto > 2500o C.

Thermopile: -
Thermopile is a combination of two or more thermocouples.
Inductive type transducers: -
Hall effect transducer: -
Which are works on the principle of hall effect, which are used to measure
the displacement in the order of micrometer.
KH =
𝟏
Ƞ𝒆
KH = Hall coefficient
Ƞ = density of charge carriers
e = charge of electron
e = positive = P-type = KH = Positive
e = negative = N-type = KH = Negative

Application: -
(1) VH ∝ B = Used as magnetometer
(2) Mainly used to measure small displacement in order of micrometer.
According to hall effect whenever a current carrying semiconducting plate
having thickness t placed in a magnetic field between two opposite edges of
a semiconducting plate an emf is induced, this phenomenon is known as
Hall effect.
This induced voltage perpendicular to magnetic flux density and current
flowing through semiconductor. It is given by:
VH ∝ B.I. ∝
𝟏
𝒕

Application: -
• It is used to find density or charge carrier.
• Used to estimate the type of semiconductor i.e. whether it is P type or N
type semiconductor by knowing hall coefficient.
• It is used to measure conductivity of charge carrier.
• VH ∝ B, implies, hence it is used as magnetometer like gauss meter.
• VH ∝ B.I, hence it is used as multiplication of two signals.
LVDT (Linear variable differential transformer or Linear variable
displacement transformer): -
Works on the principle of change in reluctance.

Disadvantage: -
It is more sensitive to stray magnetic field.
Advantage: -
(1) High accuracy.
(2) It is used to measure wide range of displacements in the order of 1.25
mm to 250mm.
(3) High sensitivity in the order of 40 V/mm
(4) Power loss is less, low power consumption also.
(5) There are no rotating parts so that no frictional loss.
(6) High reliability.

Disadvantage: -
• It is more sensitive to stray magnetic field.
• It is sensitive to mechanical vibrations.
• Its dynamic response is limited due to the mass of the core.
Applications: -
(1) LVDT is used as primary T.D. for measurement of force and pressure.
(2) LVDT is used as secondary T.D. for measurement of displacement.
LVDT is an inductive T.D. works on the principle of change in reluctance of
path of flux so that flux linkage are changed and the output induced voltage
is changed.

Construction: -
• LVDT consist of high permeability low reluctance core prepared by Nickel
iron, allow which is hydrogen annealed.
• LVDT consist of two windings one of them is known as primary winding
energized by supply voltage, another winding known as secondary winding
usually split up into two parts and connected in series opposition in order
to obtain differential output.
• In normal transformer, if output voltage is zero secondary side gets short
circuited but in LVDT when the core is at null position the output voltage is
zero.

Capacitive T.D.: -
(1) Capacitive T.D. are good accurate.
(2) Capacitive T.D. are extremely sensitive.
(3) Capacitive T.D. have high resolution in the order of 2.5 mm = 2.5×10-3 .
(4) They have good frequency response.
(5) Their loading effect is negligible because of high input impedance.
(6) The output is not affected by stray magnetic field.
Disadvantage: -
(1) There output affected by stray capacitance.
(2) There behavior becomes non-linear due to fringing effect or edge effect.
(3) This edge effect can be reduced by providing guard ring.

Flow measuring devices: -
(1) Electromagnetic flowmeter: -
In all mechanical flow measurement devices put some obstruction on the
flow of the fluid whereas electrical flow measurement devices do not put
any obstruction on the flow of the fluid.
So, mechanical flow measurement devices are called as obstructive type,
whereas electrical flow measurement devices are called as non-linear type.
(1) Electro magnetic flow meter
(2) Turbine flow meter
(3) Hot wire anemometer

• Electromagnetic flow meter works on the principle of faraday law of
electromagnetic induction.
• This type of meters do not put any obstruction in the flow of the fluid.
Through them so these are also called non-obstructive type flow meters.
• These are used for flow measurement of electrically conducting fluid.
• An electromagnetic flow meter consists of a non-conducting and non-
magnetic pipe carrying the liquid whose flow rate is to be measured.
• The pipe is surrounded by an electromagnet which produces magnetic
field, the fluid flowing through the pipe is in conducting nature.
When the conducting fluid, whose flow rate is to be measured is made to
flow through a pipe, so that it cuts the magnetic field causing some emf to
be induced across the electrodes given by:
e = B𝒍V

Advantages: -
(1) Accuracy is good.
(2) It can be used with pipes of any size.
(3) The relation between output voltage and flow rate is linear.
Disadvantage: -
(1) It is costly
(2) The fluid must be conductive nature i.e. it can not be employed for non-
conducting fluids.
Turbine flow meter: -
• Consist of a multiplate turbine.
• Turbine flowmeter is used to measure the flow rate of the fluid.
• Turbine flowmeter consists of a multiblade turbine wheel mounted along the
axis parallel to the direction of fluid flow through the pipe. The speed of rotation
is proportional to the volume flow rate,

Principle: -
Whenever the fluid is flowing through the pipe, due to this the turbine
starts rotating, the blades of the turbine rotating through the flux produced
by magnetic pickup which will (blade) cut flux produce by magnetic pickup.
So that an emf is induced in the form of pulses as the turbine rotates
continuously a series voltage pulses will be generator, these pulses can be
counted by the counter.

• Hot wire anemometer is used to measure the flow rate.
• Hot wire anemometer is placed inside the pipe to measure the flow rate of
a gas flowing through it.
• The leads of anemometer connected to one of the Wheatstone bridge and
constant magnitude of current is made to flow through anemometer.
• When gas is flowing through pipe, it absorbs heat from sensing element
(anemometer).
• Due to this the temperature of hot wire anemometer decreases which
intern changes its resistance.
• Change in resistances causes the bridge becomes unbalanced and
galvanometer deflects, indicating the value of the flow rate.

Measurement of pressure: -
Pressure is defined as the force acting over a unit surface area. Pressure is
one kind of stress.
1 Pascal = 106 N/m2
Largest unit or pressure is Pascal. Smallest unit of pressure is Torr.
1 Torr = 1 mm of Hg
High pressure means greater than atmospheric pressure > 760 mm of Hg.
Low pressure < 1 atm
1 atmosphere pressure = 760 mm of Hg
= 760 Torr

Low pressure measuring devices: -
(1) Pirani gauge
(2) Vacuum gauge
(3) Ionization gauge
(4) Thermo couple gauge Works on the principle of thermal conductivity
(5) Knudsen gauge
(6) MC- lead gauge
Bridge man gauge > 70000 Mega pascal

Bourden gauge < 70000 Mega Pascal & strain gauge cells T.D.
Strain pressure: -
It is the pressure which is exerted when moving along stream.
Absolute pressure: -
It is the pressure exhibited by the fluid on the watts of the container.

• Pressure gauge is used to measure low pressure in the order of 10-3 mm of
Hg to 10-1 mm of Hg.
• Below the atmospheric pressure the thermal conductivity of a gas
molecules is a linear function of pressure.
Working pressure: -
Pirani gauge works on the principle of change is conductivity of the gas
molecules with pressure, which changes the temperature of platinum wire.
Change in resistance causes bridge becomes unbalanced, the output voltage
of the bridge is directly proportional to the pressure.

Bridge man gauge: -
Bridgeman gauge works on the principle of change in resistance with change in
pressure.
Bellows is enclosed
with coil – Gold chrome
R = R1 {1+b.∆𝐏}
∆𝐏 = gauge pressure
b – pressure coefficient
R1 – Resistance of coil milliohm at 1 atm.

• The coil whose resistance changes with the applied pressure is placed at
the bellows containing kerosene coil.
• When the pressure is applied to the gauge the compressed, the resistance
of coil is changed according to linear relationship given by: -
R = R1 {1+b.∆𝐏}
• Change in resistance is a measure of applied pressure.
• The Bridgeman coil mounted on one of the Wheatstone bridge. As the
pressure changes, resistance changes, so that bridge become unbalance
and the output of the bridge is directly calibrated in terms of pressure.

(28) In a steam power plant…………heats the feed water on its way to the
boiler by delivering heat from the flue gases: -
(A) Superheater (B) Economizer
(B) Preheater (D) Turbine
(29) Power generation of thermal power plants is based on: -
(A) Rankine cycle
(B) Otto cycle
(C) Diesel cycle
(D) Carnot cycle

SSC JE Measurements - wifistudy (1).pdf

Recommended

Recommended

More Related Content

Similar to SSC JE Measurements - wifistudy (1).pdf

Similar to SSC JE Measurements - wifistudy (1).pdf (20)

Recently uploaded

Recently uploaded (20)

SSC JE Measurements - wifistudy (1).pdf