Manual for K-Notes

Why K-Notes?

Towards the end of preparation, a student has lost the time to revise all the chapters from his /

her class notes / standard text books. This is the reason why K-Notes is specifically intended for

Quick Revision and should not be considered as comprehensive study material.

What are K-Notes?

A 40 page or less notebook for each subject which contains all concepts covered in GATE

Curriculum in a concise manner to aid a student in final stages of his/her preparation. It is highly

useful for both the students as well as working professionals who are preparing for GATE as it

comes handy while traveling long distances.

When do I start using K-Notes?

It is highly recommended to use K-Notes in the last 2 months before GATE Exam

(November end onwards).

How do I use K-Notes?

Once you finish the entire K-Notes for a particular subject, you should practice the respective

Subject Test / Mixed Question Bag containing questions from all the Chapters to make best use

of it.

Measurement & Instrumentation

State characteristics of measuring system

1) Accuracy

Degree of closeness in which a measured value approaches a true value of a quality under

measurement .

When accuracy is measured in terms of error :

Guaranteed accuracy error (GAE) is measured with respect to full scale deflation.

Limiting error (in terms of measured value)

GAE * FullscaledeflationLE

Measuredvalue

2) Precision

Degree of closeness with which reading in produced again & again for same value of input

quantity.

3) Sensitivity

Change the output quantity per unit change in input quantity.

o

i

qS

q

4) Resolution

Smallest change in input which can be measured by an instrument

5) Threshold

Minimum input required to get measurable output by an instrument

6) Zero Drift

Entire calibration shifts gradually due to permanent set

7) Span Drift

If there is proportional change in indication all along upward scale is called span drift.

8) Dead zone & Dead time

The range of input for which there is no output this portion is called Dead zone.

To respond the pointer takes a minimum time is called dead time.

TYPES OF ERROR

a) Gross Error : Error due to human negligency, i.e. due to loose connection, reading the value

etc.

b) Systematic error : Errors are common for all observers like instrumental errors,

environmental errors and observational errors.

c) Random errors : Error due to unidentified causes & may be positive or negative.

Absolute Errors :

m rA A A

mA Measured value

rA True value

Relative Errors :

r =

A

T

AbsoluteErrors

Truevalue A

mm rT T

r

AA A A 1

1

Composite Error :

i) Sum of quantities

1 2

X X X

x x1 x2

ii) Difference of quantities

1 2

X X X

x x1 x2 So for sum & difference absolute errors are added.

iii) Multiplication of quantities

1 2 3

X X X X

31 2

1 2 3

XX XX

X X X X

iv) Division of quantities

1

2

XX

X

1 2

1 2

X XX

X X X

So, for multiplication & division, fractional or relative errors are added.

If m m

1 2

p

3

X XX

X

31 2

1 2 3

XX XXm n p

X X X X

Precision Index

Indicates the precision for a distribution

1h

2

Probable Error

r = 0.6745

0.4769r

h

Standard deviation of combination of quantities

2 2 2

1 2n1 2

n

2 2 2x x x x

X X X......

X X X

Probable Error

2 2 2

1 2 nn1 2

22 2x x x x

X X Xr r r ...... r

X X X

Electro-Mechanical Instruments

1) Permanent magnet moving Coil (PMMC)

Deflecting Torque

Td = nIAB

Where n = no. of turns

I = current flowing in coil

A = Area of coil

B = magnetic flux density

Deflection G

Ik

G = NBA & K = Spring constant

Eddy current damping & spring control torque in used.

For pure AC signal, the pointer vibrates around zero position.

It is used to measured DC or average quantity.

It can directly read only up to 50mV or 100mA.

Enhancement of PMMC

i) Ammeter

For using PMMC as an ammeter with wide range, we connect a small shunt resistance in

parallel to meter.

m

Im

I multiplication factor

Basically, m is ratio of final range (as an ammeter) to initial range of instrument.

m

sh

RR

m 1

; mR = meter resistance

ii) Voltmeter

A series multiples resistance of high magnitude is connected in series with the meter.

M = multiplication factor

m

Vm

V

s mR R m 1

Sensitivity of voltmeter

s mv

fsd

R R1S / V

I V

Application of PMMC

1) Half wave rectifier meter

mavg

II I

RMS

s m f

2V

R R R

RMS

avg

s m f

0.45VI

R R R

; For Ac input

For DC input

DCavg

s m f

VI

R R R

avg AC DCavgI 0.45 I (Assuming DC RMSV V )

AC DC(Sensitivity) 0.45(Sensitivity)

2) Full wave rectifier meter

RMS

ACf

avgs m

2 2VI

R R 2R

RMS

s m f

0.9V

R R 2R

DCavg DC

s m f

VI

R R 2R

avg DCACavgI 0.9 I (Assuring RMS DCV V )

AC DCSensitivity 0.9 Sensitivity

2) Moving iron meter

Deflecting torque, 2d1 dL

T I2 d

I = current flowing throw the meter

L = Inductance

= deflection

Under steady state

21 dL

K I2 d

2 I

MI meter measures both ac & dc quantities. In case of AC, It measures RMS value.

1T 2

2

RMS

0

1I i t dt

T

If 0 1 2i t I I sinwt I sin2wt .......

2 2 2RMS 0 1 21

I I I I .......2

Air friction Damping is used

Condition for linearity

dLcons tant

d

MI meter cannot be used beyond 125Hz, as then eddy current error is constant.

3) Elector dynamometer

Deflecting Torque, d 1 2dM

T i id

For DC, 1 2i i I

2ddM

T Id

2 I

For AC, 1 m1i I sin t

2 m2i I sin t

1 2d avg

dMT I I cos

d

Where m11

II

2 & 2

I2I

2

Applications of dynamometer

1) Ammeter

Fixed coils are connected in series.

1 2I I I

0 (Angel between 1 2I & I )

2d

dMT I

d

At balance, c dT T

2dM

K Id

2 I

It reads both AC & DC & for AC it reads RMS.

2) Voltmeter

sR Series multiplier resistance

2 1

s

VI I

R , 0

cos 1

2

d 2

s

V dMT

dR

At balance, cdT T

2

2s

V dMK

dR

2 V

It reads both AC & DC & for AC it reads RMS.

3) Wattmeter

Fixed coils carry same current as load & as called as current coils.

Moving coil is connected across voltage and thus current voltage, a high non-inductive

load is connected in series with MC to limit the current.

d 1 2

dMT I I cos

d

avg

s s

PV dM dMI cosR d R d

At balance, d

k T

avg P

Symbol :

Two wattmeter method

1 RY R RY RW V I cos V & I

L LV I cos 30

2 BY B BY BW V I cos V & I

L LV I cos 30

Where LV is line to line voltage

LI is line current

These expression remain same for -connected load.

1 23P W W

L L3V I cos

2 13Q 3 W W

L L3V I sin

2 1

1 2

3

3

Q 3 W Wtan

P W W

2 11

1 2

3 W Wtan

W W

for lag load

2 11

1 2

3 W Wtan

W W

for lead load

= Remember, In our case 1

W is wattmeter connected to R-phase and 2

W is wattmeter

connected to B-phase.

= If one of the wattmeter indicates negative sign, then pf < 0.5

Errors in wattmeter

a) Due to potential coil connection

2

T

cLr

I r% * 100

P

LI = load current

Cr = CC Resistance

TP = True Power

2

T

rs

V% *100

R P

V = voltage across PC

sR = Series multiplier resistance

TP = True Power

b) Due to self inductance of PC

If PC has finite inductance

p p sZ R R jwLp

p sR R p sZ R jwLp

r% tan tan *100

= load pf angle

p1

s

Ltan

R

4) Energy meter

Energy = Power * Time

T

VIcos tW * kwhr

1000 3600

TW = True energy

It is based on principle of induction.

It is an integrating type instrument.

mt

W VIsin * kwhr3600

Where mW = measured Energy

= angle between potential coil voltage & flux produced by it.

= load pf angle

Error = m TW W

Energy constant = No.ofRe voluations N

kwhr P.t

Measured Energy = mTotalno.ofrevolutions

KW

True Energy =

TVIcos t

* kw.hr1000 3600

W

Error = m TrT

W W% *100

W

Creeping Error in energy meter

If friction is over compensated by placing shading loop nearer to PC, then disc starts rotating

slow with only PC excited without connecting any load is creeping.

Otherwise if over voltage is applied on pressure coil then also creeping may happen due to

stray magnetic fields.

To remove creeping holes are kept on either side of disc diametrically opposite & the torque

experienced by both holes is opposite & they stop creeping.

% creeping error = TotalNo.ofRew / kwhr due to creeping

*100TotalNo.ofRew / kwhr due to load

Thermal Instruments

These instruments work on the principle of heating and are called as Thermal Instruments.

These are used for high frequency measurements.

They can measure both AC & DC.

In case of AC, they measure RMS value.

Electrostatic voltmeter

Deflecting torque, 2d

1 dcT V

2 d

At Balance,

cdT T

21 dc

V k2 d

2 V

Condition for linearity

dc

constantd

For increasing the range, we connect another capacitor in series

To increase the range from mV to V

m

s

CC

m 1 ;

m

Vm

V

Potentiometer / Null Detector

wI = working current

B

wh

VI

R l.r _____________(1)

Switch at (A)

If gI 0

s w 1V I l r

sw1

VI

l r _____________(2)

Switch at (B)

2x wV I l r

xw

2

VI

l r ________(3)

s x

1 2

V V

l r l r

2x s1

lV V

l

r = resistance of slide wire (/ m)

l = Total length of slide wire (m)

1l = length at which standard cell ( sV ) is balanced

2l = length at which test voltage ( xV ) is balanced

Measuring a low resistance

R

s

VR S

V

Instrument Transformer

Current transformer

Equivalent circuit

Turns Ratio = Nominal Ratio 2

1

Nn

N

1 l s

l s

X Xtan

R R

R = Actual Ratio

s

I cos I sinn

I

Errors in current transformer

1) Ratio Error :

Current ratio p

s

I

I is not equal to turns ratio due to no-load component of current.

r

K R% *100

R

K = n = Nominal Ratio

R = Actual Ratio

2) Phase Angel Ratio :

Ideally, Phase difference between p sI & I should be 0180 but due to no-load component of

current, it deviates from that value.

Phase angle error =

s

I cos I sin 180*

nI degrees

Phase angle between primary & secondary currents

= 180 degrees

Potential Transformer

Equivalent circuit

Turns Ratio = n = 2

1

N

N

Actual Transformation Ratio = R = P

S

V

V

SP P P P

S

I1R n R cos X sin I R I X

V n , Where

1 XtanR

Phases angle error

SP P P P

s

IX cos R sin I X I R

nnV

AC & DC Bridges

AC Bridges

Balance condition : DI 0

41 2 3Z Z Z Z

41 2 3Z Z Z Z

41 2 3

432 3

1 24

Z ZZ

Z

Quality Factor & dissipation factor

Quality Factor (Q) Dissipation Factor

(D)

1 wLQ

R

RD

wL

2 RQ

wL

wLD

R

3 1Q

wCR

D =wcR

4 Q = wcR 1D

wCR

Measurement of Inductance

(i) Maxwells Inductance Bridge

Here, we try to measure 1

R & 1

L

2 31

4

R RR

R

2 31

4

L LL

R

(ii) Maxwells Inductance Capacitance Bridge

2 31

4

R RR

R

41 2 3L R R C

This bridge is only suitable for coils where 1 < Q < 10

Q = Quality Factor

(iii) Hays Bridge

Used for coils having high Q value

2 24 42 3

1 2

R R R CR

11

Q

42 31 2

R R CL

11

Q

4 4

1Q

R C

(iv) Andersons Bridge

This Bridge is used for low Q coils.

2 31 1

4

R RR r

R

3 41 2 2 34

CRL R R r R R

R

(v) Owens Bridge

431

2

R CR

C

2 41 3L R R C

Measurement Of Capacitance

De-Sautys Bridge

31 2 2 14

Rr R r R

R

42

31

RC C

R

D = dissipation factor

= 1 1

C r

1r = internal resistance of

1C

Schering Bridge

431

2

R CR

C

4 21

3

R CC

R

dissipation factor = D = 4 4C R

Measurement of frequency

Wien Bridge Oscillator

Balancing Condition

3 1 2

4 2 1

R R C

R R C

Frequency of Osculation

1 2 1 2

1f

2 R R C C

Measurement of Resistance

Classification of Resistance

1) Low Resistance : R 1 , Motor and Generator

2) Medium Resistance : 1 < R < 100k , Electronic equipment

3) High Resistance : R > 100 k, winding insulation of electrical motor

DC Bridges

Medium Resistance Measurement

1. Wheatstone Bridge

Finding Theremin Equivalent

thggth

VI

R R

Th

P RV V

P Q R S

Th

PQ RSR

P Q R S

For Balance Condition

gI 0

Th

V 0

PS = RQ

Sensitivities

1) Current sensitivity , i

g

SI

mm/mA

= deflection of Galvan meter in mm

2) Voltage sensitivity,

Th

SV

mm/V

3) Bridge Sensitivity , B

SR /R

mm

ThB

vV SSR / R

B

vV.SSSR 2

S R

For Maximum Sensitivity

SRS R = 1

B, maxvV.SS

4

2. Carey foster slide wire Bridge

r = slide wire resistance in m

.

for case (1).

At balance

1

1

R rP

Q S L r

.(1)

For case (2)

R & S is reversed

2

2

S rP

Q R L r

..(2)

From (1) & (2)

1 2

1 2

R r S r

S L r R L r

3. Voltmeter Ammeter Method

a) Ammeter near the load

vm X AA

VR R R

I

vV = voltage across voltmeter

A

I = Ammeter current

X

R = Test resistance, A

R = Animator resistance

% error = m T A

xT

R R R100 100%

R R

b) Voltmeter near the load

X

v Xm

vA

VVR

I I I

vXm

X v vX

X X

R R1R

I I R R

V V

% error = m X

X

R R100%

R

If a vXR R R , voltmeter is connected near the load

a vXR R R , ammeter is connected near the load

4. Ohmmeter

a) Series Type

when XR 0

m FSDI I = Full scale deflection

when XR

mI 0 = zero deflection

for Half scale deflection

msh

seX hmsh

R .RR R R

R R

b) Shunt Type

S

R = current limiting resistor

If X

R 0

mI 0 = zero deflection

If xR

m FSDI I = Full scale deflection

For Half scale Deflection

m S

x hm S

R RR R

R R

Measurement of Low Resistance

Kelvins Double Bridge Method

Unknown resistance

qr pP P

R SQ p q r Q q

P, Q = outer ratio arms

p, q = inner ratio arms

S = standard resistance

r = lead resistance

R = Test resistance

High Resistance Measurement

Loss of charge Method

tRc

eCV t V

10

C

0.4343tR

VC log

V

t = time in (seconds)

V = source voltage

CV = Capacitor voltage

Cathode Ray Oscilloscope (CRO)

The velocity of e is charged by charging the pre-accelerating & accelerating anode

potential

KE =PE

2

a1

mv qV2

a2qV

m

Deflection sensitivity

D = deflection height on screen

d = distance between plates

d = length of vertical deflecting plates

L = distance between centre of plate & screen

aV = anode potential

yV = Vertical plate Potential

yd

a

L VVD

mm2dV

deflection sensitivity

d

y a

LD VSmmV 2dV

Lissajous Pattern

If both horizontal & vertical deflection plates of CRT is applied with the sinusoidal signal,

the wave form pattern appearing on screen is called Lissajous Pattern.

Case 1: Both signals have same frequency

x m xV V sin w t

y m yV V sin w t

x y mV V V

x yw w w

= variable

S.No Lissayous Pattern

1

0 or 360

2

0 90

Or

270 360

3

90 or 270

4

90 270

Or

180 270

5

180

Finding

1) Lissajous Pattern in Ist & IIIrd Quadrant

1 11 1

2 2

X Ysin sin

X Y

for clockwise orientation phase difference = (180 )

for anti-clockwise orientation

2) Lissahous Pattern in IInd & IVth Quadrant

2

1 1X

180 sinX

2

1 1Y

180 sinY

for clockwise orientation

for anti-clockwise orientation = 360

Case 2

x yw w

x m xV V sinw t

y m yV V sinw t

y y

x x

w f Number of horizental tangencies

w f Number of vertical tangencies

y

x

f 42

f 2

Digital Meters

Type of converter Maximum Conversion Time

1) Dual slope ADC n 12 Clocks

2) Successive Approximation Register (ADC) n Clocks

3) Counter ADC n2 Clocks

4) Flash ADC 1 Clock

Dual slope A/D Converter

aV = analog input

R

V = Reference input

Ra 2 11

VV T T

T

1n

CLKT 2 T

Maximum conversion time = n 1

CLK2 T

Successive Approximation Register

Suppose = REF aV 1 V

and aV = 12V

3D 2D 1D 0D

10 5 2.5 1.25

1

T 1 0 0 0 10V < 12V

2T 1 1 0 0 15V > 12 V

3T 1 0 1 0 12.5 > 12 V

4T 1 0 0 1 11.25 < 12 V

In first clock cycle, MSB is set to get voltage corresponding to the digital o/p

If 0

V < aV , then in next cycle next bit is set else,

If 0

V > aV , MSB is reset & next bit is set

We continue the same process till we reach LSB.

Specifications of Digital Voltmeter

1) Resolution

The smallest value of input that can be measured by digital meter is called resolution.

n

1R

10

n = No. of full Digits (0, 1,.., 9)

2) Sensitivity

S = Resolution x Range

3) Over Ranging

The extra 12

digit is called over-ranging

If n = 3, we can measure from 0 999

Resolution , 3

1R 0.001

10

if 1n 32

digit, 12

digit can be 0 & 1.

we can measure from 0 1999

Resolution, 1

R 0.0052000

if 34

digit is there than MSB can be 0 3.

4) Total Error

Error = (% error in reading) x reading + (NO. of counts) Full Scale

Range of meter

Q meter / Voltage Magnifier

If works on the principal of series resonance.

At series resonance

L CX X

V

IR

CCV IX

LCX X

V VR R

C

V = V. Q

CV Q

Practical Q-meter

Also includes series resistance of source (oscillator)

True T

wLQ

R

Measured Q,

Tm

sh shsh

QwL wLQ

R RR RR 1 1

R R

msh

T

RQ Q 1

R

Measurement of unknown capacitance

Test capacitance is connected at 43T & T .

Circuit is resonated at C = 1

C

fr= T1

1

2 2 C C (1)

TC = Test Capacitance

TC is removed & circuit is resonated at C = 2C

fr =

2

1

2 LC (2)

from (1) & (2)

T 2 1C C C

Measurement OF self Capacitance

Resonance is achieved at C = 1

C

1

1 d

1fr

2 L C C

At C = 2

C , resonance is achieved at 2

fr

2

2 d

1fr

2 L C C

= n fr1,

21 2

d 2

C n CC

n 1

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