Gateway to...PSUs(Synopsis, Multiple Choice Questions and their Explanatory Notes)

For

Electronics & Telecommunication Engineering (ETE), Electronics &

Communication Engineering (ECE), Electrical & Electronics Engineering (EEE),

Electronics & Instrumentation Engineering (EIE)

By

Upkar Prakashan, AGRA-2

Bharat Electronics Limited (BEL).

Bharat Heavy Electronics Limited (BHEL).

Steel Authority India Limited (SAIL).

Hindustan Aeronautics Limited (HAL).

Indian Oil Corporation Limited (IOCL).

National Thermal Power Corporation Limited (NTPC).

National Fertilizers Limited (NFL).

Centre for Development of Telematics (C-DOT).

Power Grid Corporation of India Limited (PGCIL).

Hindustan Petroleum Corporation Limited (HPCL).

Larsen & Tubro Limited (L & T).

Indian Farmers Fertilizers Corporation Limited (IFFCO).

Bhabha Atomic Research Centre (BARC).

Indian Space Research Organization (ISRO).

Defence Research Development Organization (DRDO), etc.

Useful for

Ashish DixitPhd. Persuing

M.Tech. (V.L.S.I. Design) C-DAC, Mohali (Punjab)

B.Tech. (HONs E & T)

Sr. Lecturer (EC Deptt.)

AMITY UNIVERSITY

Lucknow

Seccond Revised Edition

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Heartiest Thanks

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(Ritnand Balved Education Foundation)

AMITY UNIVERSITY

Uttar Pradesh

&

Maj. Gen. K. K. OhriAVSM Retd.

Director General

AMITY UNIVERSITY

Lucknow Campus

Dedicated to

GrandfatherGrandfatherGrandfatherGrandfatherGrandfather

Late Mr. Ram Prasad DixitLate Mr. Ram Prasad DixitLate Mr. Ram Prasad DixitLate Mr. Ram Prasad DixitLate Mr. Ram Prasad Dixit

FatherFatherFatherFatherFather

Mr. Uidya Dhar DixitMr. Uidya Dhar DixitMr. Uidya Dhar DixitMr. Uidya Dhar DixitMr. Uidya Dhar Dixit

&&&&&

MotherMotherMotherMotherMother

Smt. Mithlesh DixitSmt. Mithlesh DixitSmt. Mithlesh DixitSmt. Mithlesh DixitSmt. Mithlesh Dixit

PREFACEPREFACEPREFACEPREFACEPREFACE

It is a matter of great pleasure to present this book ‘‘Gateway to… PSUs’’. Getting a job in the PSUs like BSNL & VSNLand other PSUs like, DRDO, ISRO, NTPC, GAIL, BHEL, BARC, NHPC, SAIL, IOCL etc. has always been an aspiration of engineeringgraduates. Although a number of books are available for entrance examinations for these jobs, the present book makes a significant

departure from its predecessors because of its encyclopaedic subject matter and its innovative and lucid presentation. I am surethat this book will provide complete satisfaction to the readers. Presently the market scenario is confusing and it is hard to find theoptimum quality books. While preparing this manuscript, I continuously kept in mind the requirements of the aspirants regarding

the latest trends of these entrance examinations as well as the most intricate concepts involved.

This book is divided into twenty units. Each unit begins with a synopsis and ends with multiple choice questions and theirexplanatory notes. The variety and quality of problems also makes this book unique and an asset for the aforementioned

examinations.

Salient Features of the Book Synopsis has been given in the beginning of each topic for a quick and easy review just before the examination.

More than 4500 objective solved problems.

Special tricks are given to solve the objective problems in order to save the time during examination.

It covers many objective problems frequently asked in various competitive examinations.

For a better understanding many problems have been solved graphically.

Previous years question papers fully solved.

This work has come into being only because of the unparallel help and support that came from Mr. Ankit Singh, Mr. AnkitJain, Mr. Mayank Srivastava, Mr. Kranti Kumar, Mr. Dheeraj Shukla, Mr. Pawan Kumar Saini. Their contribution to this work is

second to none.

No words are sufficient to express my gratitude to Sri Vidya Dhar Dixit, Avatar Singh, Mrs. Mithlesh Dixit, Mr. RadhesyamDixit, Mr. Prakash Dixit, Paramjit Kaur, Uday Shanker Trivedi, Anita Trivedi, Archana and Amita, Sukhwinder Kaur, Jasvinder Kaur

and elder brother Suresh Kumar Tripathi and wife Annapurna, for their cooperation, patience and support during the preparationof the manuscript.

We would like to thank all our friends like Akhilesh Shukla, Satish Shukla, Ankit Kumar Singh, Dheeraj Shukla, Mayank

Srivastava, Pawan Saini, Sarika Mishra, Abhishek Dwivedi, Anand Singh Chauhan, Rohit Singh, Rajeev Singh, Ganga Raju,Talvinder Kaur, Umesh Pandey Mrs. Mehga Singhla and Shivendra Singh Chauhan for their encouragement during the preparationof the book.

We would also like to thank all our parents like Mr. Mohan Chandra Tripathi, Mr. Nageswar Nath Upadhyay (IAS General-Secretary) Mr. Shankari Singh Chauhan, Mr. Surya Kumar Tripathi, Mr. Ram Sevak Singh Chauhan, Mr. Lallan Mishra, Mr. BindadinSingh Chauhan, Mr. Santa Kumar Dixit, Mr. Ram Gopal Singh Chauhan, Mr. Suresh Kumar Tripathi, Mr. Dev Chaitanya Brahmachari,

Mr. Om Prakash Yadav, Dinesh Jaiswal, Sanjay Dixit, Mohd. Irsad Khan (MLA Sarojininagar Lucknow) .

I wish to express my gratitude to Mr. Mahendra Jain, Upkar Prakashan, Agra, for publishing this manuscript.

Suggestions for the improvement of this book will be greatly acknowledged and incorporated in the nextedition.

—Ashish Dixit

ACKNOWLEDGMENTACKNOWLEDGMENTACKNOWLEDGMENTACKNOWLEDGMENTACKNOWLEDGMENT

I am very much grateful to a number of colleagues who have given us enormous contributions and cooperation during the

preparation of the manuscipt of this book. I would like to give my hearty thanks to the following persons for their continuous

support, encouragement throughout this book, like........

Prof. S. K. Jha (CEST, Lucknow), Prof. Anuj Goel (H.O.D. EC B. N. Institute, Lucknow, Miss Shrushut (CEST Lucknow), Mr.

B. K. Chauhan (N.I.E.C. Lucknow), Mrs. Indu Prabha Singh (H.O.D. EC S.I.T.M., Unnao, Mr. Sameer Srivastava (S.I.T.M. Barabanki),

Mr. Rahul Mishra (Dr. M.C. Saxena, Lucknow), Dr. Manish Gupta, AMITY UNIVERSITY, Lucknow, Mr. Marut Tiwari, Mr. Prabhaker

Dubey, N. I. E. C., Lucknow. Mr. Gurumohan Singh & Mrs. Shivani Wadhwa, Mr. H.S. Sagar Director, SBBSIET, Jalandhar, Dr.

Arun Khosla, Dr. B.R. Sood Professor, SBBSIET, Jalandhar, Mr. Kamaljit Singh Professor, NIT, Jalandhar, Dr. Sukhbeer Bindra

Professor GNDN, Amritsar, Mr. Neeraj Sharma Senior Lecturer VIET, Chandigarh, Mr. Mandeep Singh Engineer BEL, Ghaziabad,

Dr. M.L. Singla Scientist F, Deputy Director CSIO, Punjab, Mr. Manish Gupta, Mr. Kidwai Siddique, H.O.D. Electronics AIET,

Lucknow, Mr. Anand Chopra , Mr. Bhabhaya Alankar Lecturer, N.I.T. Jalandhar, Mr. Ankit Kumar Senior Lecturer, AIET Lucknow,

Mr. Sanjay Jain Indian Engineering Services, Mr. Abhishek Agrahari Design Engineer N.I.T. Surat, Mr. Bijendra Saini, (IES), Mrs.

Lalita Sharma Senior Lecturer, I.E.T. Khandari, Agra, Mr. Yogendra Kumar Prajapati Senior Lecturer, V.I.E.T. Bijnour, Mr. Puneet

Chandra Srivastava, Associate Professor R.K.G.I.T. Ghaziabad, Mr. Vivek Srivastava Senior Lecturer, N.I.E.T. Greater Noida, Mr.

Sanjay Singh GAIL, India, Miss Aplana Sharma, Dr. P.K. Pandey (H.O.D. EC MPGI, Lucknow), Mr. Pawan Yadav, Mr. C.K. Dixit,

Mr. Ravi Kant, Mr. Satendra Yadav HOD EC Deptt., B.B.I.E.T. Bulandsahar, Mr. Rafeek Ahmad, Mr. Vinod Kumar Yadav and

Miss Smriti Agrawal.

I would like to express my special thanks to Mr. Ambika Misra, Secretary Ambalika Institute of Management & Technology

Lucknow, Dr. S. Q. Abbas, Director, Ambalika Institute of Management and Technology, Lucknow, Dr. Alok Kumar Mishra, Additional

Director Ambalika Institute of Management and Technology, Mr. Sunil Kumar, Dean Academic, Ambalika Institute of Management

and Technology, Lucknow, Mr. U. P. Singh, Dr. Mansi Mishra, Mr. Sandeep Dwivedi, Mr. Ravi Kumar Tiwari, Mr. Sonu Kumar Rai,

Miss Anuradha Yadav, Miss Sabika Ahmad, Miss Monika & Prof. Karmakar all from Ambalika Institute of Management and

Technology, Lucknow.

I would also like to express my special thanks Maj Gen. K.K. Ohri, Prof. S.T.H. Abidi, Prof. N.R. Yadav, Dr. O.P. Singh, Mr.

Arun Pathak, Dr. Neetu Agrawal, Mrs. Gitika Srivastawa, Miss Sarika Mishra, Mr. Zia Abbas, Mr. Anupam, Mr. Sachin Kumar, Mr.

Yashu Swami, Dr. Amit kumar Mishra, Mr. Anirudh Banergi, Mr. Sienu Rizavi, Mr. Loknath Mishra, Mr. Rajinder Tiwari, Dr. Nidhi

Mathur, Miss Shilpi Bisth and Mr. Sudhir Pandey AMITY UNIVERSITY Lucknow Campus.

— Ashish Dixit

CONTENTSCONTENTSCONTENTSCONTENTSCONTENTS

1. Network & Systems 1–136 Objective Problems on Switching……………………………………………………………………………… 28 Answers with Explanatory Notes………………………………………………………………………….…… 30 Objective Problems : Level 1…………………………………………………………………………………... 34 Answers with Explanatory Notes………………………………………………………………………….…… 62 Objective Problems : Level 2…………………………………………………………………………………... 105 Answers with Explanatory Notes………………………………………………………………………….…… 116

2. Electronics Devices 1–96 Objective Problems……………………………………………………………………………………………… 28 Answers with Explanatory Notes………………………………………………………………….…………… 51 Miscellaneous Problems…………………………………………………………………………………….….. 74 Answers with Explanatory Notes…………………………………………………………………………….… 88

3. Analog Circuits 1–88 Objective Problems……………………………………………………………………………………………… 39 Answers with Explanatory Notes…………………………………………………………………………….… 64

4. Digital Circuits 1–56 Objective Problems : Level 1…………………………………………………………………………………... 28 Answers………………………………………………………………………………………..…………………. 33 Objective Problems : Level 2

(i) Problems of Boolean Theorems and Logic Gates………………………………………………………… 34 Answers with Explanatory Notes…………………………………………………………………………….… 37

(ii) Problems on Combinational Logic Circuits……………………………………………………………….. 41 Answers with Explanatory Notes…………………………………………………………………………….… 43

(iii) Problems on Sequential Circuits : Flip-Flop……………………………………………………………… 45 Answers with Explanatory Notes…………………………………………………………………………….… 48

(iv) Problems on Counter…………………………………………………………………………………...….. 49 Answers with Explanatory Notes………………………………………………………………………….…… 52

(v) Problems on Register…………………………………………………………………………………...…... 52 Answers ………………………………………………………………………………………..………………... 53

(vi) Problems on Memory Organisation……………………………………………………………………….. 53 Answers with Explanatory Notes…………………………………………………………………………….… 55

5. Microprocessor 1–64 Objective Problems : Level 1…………………………………………………………………………………... 16 Answers with Explanatory Notes………………………………………………………..……………………... 23 Short Answer Type Questions………………………………………………………………………………….. 26 Miscellaneous Problems on Digital Circuits & Microprocessor…………………………………………….. 27 Answers with Explanatory Notes………………………………………………………………………….…… 47

6. Signals & Systems 1–112 Objective Problems……………………………………………………………………………………………… 45 Answers with Explanatory Notes…………………………………………………………………………….… 71

7. Control Systems 1–76 Objective Problems……………………………………………………………………………………………… 24 Answers with Explanatory Notes………………………………………………………………………………. 47

8. Communications 1–56 Objective Problems : Level 1…………………………………………………………………………………... 25 Answers with Explanatory Notes………………………………………………………………………………. 28

( xii )

Objective Problems : Level 2…………………………………………………………………………………… 29 Answers with Explanatory Notes………………………………………………………………………………. 35 Objective Problems : Level 3…………………………………………………………………………………… 36 Answers with Explanatory Notes………………………………………………………………………………. 47

9. Electromagnetics 1–60 Objective Problems : Level 1…………………………………………………………………………………… 23 Answers with Explanatory Notes………………………………………………………………………………. 26 Objective Problems : Level 2…………………………………………………………………………………… 27 Answers with Explanatory Notes………………………………………………………………………………. 42

10.Power Electronics 1–51 Objective Problems : Level 1…………………………………………………………………………………… 35 Answers………………………………………………………………………………………..…………………. 40 Objective Problems : Level 2…………………………………………………………………………………… 41 Answers………………………………………………………………………………………..…………………. 51

11. Microwave Engineering 1–31 Objective Problems : Level 1…………………………………………………………………………………… 19 Answers………………………………………………………………………………………..…………………. 26 Objective Problems : Level 2…………………………………………………………………………………… 27 Answers with Explanatory Notes………………………………………………………………………….…… 31

12.Optical Fibre Communication 1–16 Objective Problems……………………………………………………………………………………………… 5 Answers with Explanatory Notes………………………………………………………………………………. 12

13.Electrical Engineering Materials 1–39 Objective Problems……………………………………………………………………………………………… 15 Answers with Explanatory Notes………………………………………………………………………………. 33

14.Measuring Instruments 1–44 Objective Problems……………………………………………………………………………………………… 23 Answers with Explanatory Notes………………………………………………………………………………. 36

15.Satellite Communication 1–24 Objective Problems……………………………………………………………………………………………… 13 Answers with Explanatory Notes………………………………………………………………………………. 22

16.Radar Engineering 1–11 Objective Problems……………………………………………………………………………………………… 6 Answers with Explanatory Notes………………………………………………………………………………. 9

17.Television 1–16 Objective Problems……………………………………………………………………………………………… 6 Answers with Explanatory Notes………………………………………………………………………………. 16

18.Computer Engineering 1–1619.Dictionary of Electronics 1–1620.General Ability Test 1–31 Model Question Papers 1–92 Practice Paper - 1…………………………………………………………………………………..…………… 3 Answer with Hints…………………………………………………………………………………...…………… 12 Practice Paper - 2…………………………………………………………………………………..……………. 23 Answer with Hints…………………………………………………………………………………..……………. 32 Practice Paper - 3…………………………………………………………………………………...…………… 49 Answer with Hints…………………………………………………………………………………...…………… 57 Practice Paper - 4…………………………………………………………………………………...…………… 67 Answer with Hints…………………………………………………………………………………...…………… 78

Network and Systems

NNNNEEEETTTTWWWWOOOORRRRKKKK AAAANNNNDDDD SSSSYYYYSSSSTTTTEEEEMMMMSSSS Kirchhoff Voltage Law (KVL) is a consequence of the

law of conservation of energy, voltage being the energy(or work) per unit charge.

Kirchhoff Current Law (KCL) is the consequence ofconservation of charge. Since the algebraic summationof the charge must be zero, the time derivative of thissummation must also equal to zero.

Star to Delta Transformation

A

B C

RA

RB RC

A

B C

RAB RCA

RBC

O

RAB

=RA RB + RB RC + RC RA

RC

or RAB

= RA + R

B +

RA RBRC

Similarly,

RBC

=RB RC + RC RA + RA RB

RA

and RCA

=RC RA + RA RB + RB RC

RB

Delta to Star Transformation

A

B C

RA

RB RC

A

B C

RAB RCA

RBC

O

RA

=RAB × RAC

RAB + RBC + RCA

Similarly, RB

=RAB × RBC

RAB + RBC + RCA

and RC

=RAC × RBC

RAB + RBC + RCA

Resistance of a wire depends on its material and itssize. It is given by

R =ρlA

where, ρ = Resistivity of a material in Ω cm

l = Length of the wire, andA = Area of cross-section of wire.

Self inductance (L) is the property of conductor (orcoil) by virtue of which it opposes any change indirection or magnitude of current flowing through itself.It is given by

L = Nφl henry

where, N = No. of turns in the coil

φ = Flux set by current I.

Also, L = µ0 ur N2A

l henry

Current through inductor is given by

i (t) = 1L ∫

t

–∞ V dt or 1L ∫

t

–∞ V dt + i(0) amp

Voltage across inductor is given by

V = L didt = N

dφdt volt

Power, P = VI = LI didt watt

When n inductors are connected in series the equiva-lent inductance is

Leq

= L1 + L

2 + L

3 + .....+ L

n

When n inductors are connected in parallel the equiva-lent inductance,

1L

eq=

1L

1 +

1L

2 +

1L

3 + ..... +

1L

n

Relation between the mutual inductance and two coilswith inductance L

1 and L

2 is given by relation.

M = k √L1L2 where 0 ≤ k ≤ 1

where k = coefficient of coupling. It is a measure ofamount of linking flux produced by one coil, w.r.t. othercoil.

Capacitance is the property of material by virtue ofwhich it opposes the variation in potential between thetwo sides

C =qV

where, q = charge, V = potential

C =ε0 εrA

d

4 | NETWORK AND SYSTEMS

where C = Capacitance is proportional to the dielectricand area of the plates, and is inversely proportional tothe distance between the plates.

ε0

= 8.86 × 10– 12 F/m

εr

= Relative permittivity

A = Area of the plate

d = Spacing between two plates.

Current through the capacitor is

i = C dvdt Amp.

Voltage across the capacitor is

V = 1C ∫

t

–∞ i (t) dt or 1C ∫

t

–∞ i(t) dt + VC

(0)

where VC

(0) = initial voltage on capacitor.

When n capacitances are connected in series, theequivalent capacitance is given by

1

Ceq

=1C

1 +

1C

2 +

1C

3 + ..... +

1C

n

When n capacitances are connected in parallel, theequivalent capacitance is given by

Ceq

= C1 + C

2 + C

3 + ..... + C

n

Some conservation laws are given below :

(a) The conservation of charge

q1 = q

2 and C

1 V

1 = C

2 V

2 ; i ≠ ∞

(b) The conservation of flux linkage

ψ1 = ψ

2 and L

1 i

1 = L

2 i

2 ; v ≠ ∞

(c) The conservation of momentum

P1 = P

2 and m

1 v

1 = m

2 v

2 ; F ≠ ∞

Mutual Inductance (M)

It is the ability of one coil to produce an inducedvoltage in a nearby coil by electromagnetic induction,when the current being changed in the first coil, theaction is vice-versa. It is measured in terms of thecoefficient of mutual induction.

GRAPH THEORY

Network topology is only concern with the geometricalstructure of the network.

A graph of any Network can be drawn by placing all thenodes which are points of intersection of more than twobranches. Consider a network given below :

R1 R2 R3

C1 C21

23

4

L

12

3

4

Given network Graph of the given network

If there exists a path between every pairs of nodes of agraph, then the graph is said to be a connected graph.

There are at least two branches in a circuit.

Rank of graph = n –1, where n = no. of nodes orvertices.

A subgraph is a subset of the branches and nodes of agraph. The subgraph is said to be proper if it consistsof strictly have branches and nodes less than all thebranches and nodes of the graph.

Any tree contains n – 1 branches where n is thenumber of nodes or vertices. The branches in a treecalled twig the branches other than the twig is calledlink (also called cords).

12

3

4

1

23

4 65

Correspondingtree

1 2 3

5

23

4

Given graph Tree

Fig. : Simple graph and Tree

In the above figure branches 3, 2, 5 are called twigwhile the branches 1, 4, 6 called link.

Properties of Trees

(i) In a tree, there exist one and only one pathbetween any pair of nodes.

(ii) Every connected graph has at least one tree.

(iii) All the links of a tree together constitute thecomplement of the corresponding tree is calledco-tree.

(iv) Each tree has (n – 1) branches.

(v) The rank of tree is (n – 1). This is also the rank ofa graph to which the tree belongs i.e., both treeand graph has some rank. Where n is the numberof nodes in a graph.

(vi) No. of possible trees of a graph = dct [[A] [AT]]where A is the reduced incidence matrix.

If the total number of branches and nodes in a connec-ted graph are b and n respectively, then the number oflinks l is given by

l = b – n + 1

Incidence Matrix

Tells information about the branches connected atwhich nodes and what the orientations relative tonodes. It is denoted by A

a.

The matrix obtained from Incidence Matrix (Aa) by elimi-

nating one of the rows is called the reduced incidencematrix denoted by A. It is of order (n – 1)a × b

where, n = No. of nodesb = Branches

Properties of Complete Incidence Matrix :

(i) Sum of entries in any column is zero.

(ii) The determinant of the complete incidence matrix ofa closed loop is zero.

NETWORK AND SYSTEMS | 5

Fundamental circuit or tie set matrix can be obtainedby taking one link and other twigs at a time. The num-ber of fundamental circuit depends upon the number oflink.

Consider a graph given below and select a tree.

1 23

4

1

24

6

5

12

3

4

3

Given graph Corresponding tree

Fundamental loops = b – n + 1 = No. of link

Number of twig = n – 1

where, n = No. of nodes

b = No. of branches

The fundamental circuit can be drawn as :

1 2 3

4

1

5 4loop1

1 2 3

4

2

34

loop2

loop 31 2

6

1 2 3

4

In matrix form :

fundamental Branches

circuit 1 2 3 4 5 6

5 1 0 0 –1 1 0

3 0 1 1 1 0 0

6 1 1 0 0 0 1

Cut-set matrix is formed by taking one twig and allother link. The set of branches which is cut by theclosed line forms a cut set.

Cut set separates the node of the given graph into twographs.

The no. of cut-set is equal to the no. of nodes.

Cut-set containing only one twig are independent andare known as fundamental cut sets (f-cut sets)

Number of cut-sets = Number of twig = n – 1

where n = No. of nodes in a graph.

Consider a graph having 5 nodes shown below :

1 3

2

4

5

a

b c

d

e

f

g

h

1 3

2

4

5

Given graph Corresponding tree

3

4

1

2

5

C4

C2

C3C1

a

bc

d

e

f

g

h

Fig. : Formation of cut-set

Since in the given graph there are 5 nodes, so 5 – 1= 4 cut-sets are possible. One thing should be kept inmind that the orientation of a cut-set is decided by thedefining branch of the tree.

Line four cut sets for the above choice of the tree are(a, c, b) ; (b, f, c) ; (c, g, d) ; (d, h, a) and each cut setdefines a cut-set row vector. The cut-set (a, e, b), forexample, defines the following cut-set row vector.

Network shown below :

Z1 Z2

Z3

T-Network

Z1 Z2

Z3

π-Network

Z2 Z4

Z1

Ladder Network

Z3 Z5

Z6

Z1

Bridge Network

Z2

Z3

Z4

Note : These network are said to be grounded or un-balanced network.

Planar graph : It is possible to draw a graph on sheetof paper without crossing the lines then such types ofthe graph are said to be planar graph. Some of theplanar graph is shown below :

a

b

c

d

b

c

d

a

Non-planar graph : If graph is drawn with crossinglines then it is called non-planar graph. Fig. shows thenon-planar graph.

(i) Non-planar graph (ii) Non-planar graph

6 | NETWORK AND SYSTEMS

COUPLING

Dot notation : Dot on the coils indicate the polarity ofvoltages induced in the coils. These also indicatewhether the fluxes produced by the coils will be additiveor opposing.

(i) When coils are joined in series with fluxes additive,the equivalent inductance is

Leq

= L1 + L

2 + 2M

M

L1 L2

Leq

= L1 + L

2 – 2M (for opposing fluxes) →

M

L1 L2

Coefficient of coupling : The amount of couplingbetween the inductively coupled coils is expressed interms of the coefficient of coupling, which is defined as

K = M

√L1 L2

where, M = mutual inductance between the coils

L1 = Self inductance of the first coil

L2 = Self inductance of the second coil

Coefficient of coupling, K is always less than unity.

Maximum value of 1 or 100% called perfect coupl-ing, when the entire flux of one coil links the other.

The greater the coefficient of coupling between thetwo coils, the greater the mutual inductance bet-ween them and vice versa.

K is a non negative number and is independent ofthe reference directions of the currents in thecoils.

If the two coils are a great distance apart in space,the mutual inductance in very small and k is alsovery small.

K ≈ 0.99 → Iron-core coupled circuits

K varies between 0.4 to 0.8 for air core coupledcircuits.

M ≤ √L1 L2 (GATE 2001)

For better understanding the concept of the dotnotation we must know the types of Coupling.

Types of Coupling :

(i) Electrical Coupling : Two coils are connected by awire, so that physical connection exists.

(ii) Magnetic Coupling : Two coils are magneticallycoupled but physically isolated, so called trans-former coupling.

(i) Magnetic Coupling : Consider two inductors withself inductances L

1 and L

2, mutual inductance M,

and coefficient of coupling K and dots are shown infig. applying KVL to both side we get

L1

i ( t)1

R1V (t)1

+

–

i (t)1

i ( t)1

L2 R2

M

M = K √L1 L2

where, 0 ≤ K ≤ 1

V1 (t) = L

1

di1(t)dt + M

di2(t)dt + R

1 i

1 (t),

and L2

di2(t)dt + M

di1(t)dt + R

2 i

2 (t) = 0

A transformer replaced by its T-equivalent :

Magnetic adding :

L1V1

+

–

i1

M

+

–

i2L2 V2

+

–

i1V1

L1 –M L2 –M

i2

M L2

+

–

Magnetic opposition :

L1V1

+

–

i1

M

+

–

i2L2 V2

+

–

i1V1

L1+M L2+M

i2

–M L2

+

–

(ii) Electrical Coupling :Case 1 : When Coupled inductors in series :

M

L1

L2

LeqM

L1

L2

Leq

= L1 + L

2 + 2M

M

L1

L2

LeqM

L1

L2

Leq

= L1 + L

2 – 2M

Case 2 : Coupled inductors in parallel :M

L2Leq L1

M

L2L1

NETWORK AND SYSTEMS | 7

Leq

= L1 L2 – M2

L1 + L2 – 2M

M

L2Leq L1

M

L2L1

Leq

= L1 L2 – M2

L1 + L2 + 2M

Case 3 :

L1 L2

M

Leq

or

M

Leq

Leq

= L1 – M2

L2

L1 L2

M

Leq

L3

Leq

= L1 – M2

L2 + L3

→Transformer as an impedance matching device :

+

–

V1

1

ZinV2

2

+

–N : N

I : nl1 l2

ZL

Zin

= V1I1

=

1n V2

– I2 n

and V2

V1 =

N2N1

= n = I1I2

and Zin

= ZL

n2

PHASORS AND PHASOR DIAGRAMS : Phasor is a directed line segment, whose magnitude is

the maximum value of the cosine function and angle isthe initial phase of the cosine function.

Phasor can be treated as complex number. When thecircuit is analyzed using phasors, it is said to be in thefrequency domain.

There are three equivalent notations for a phasor.

Polar form V = Vm

∠θRectangular form V = V

m (cos θ + j sin θ)

Exponential form V = Vm

ejθ

The cosine function can be written from the givenPhasor.

V (t) = Re [V ejωt ]

= Re [V

m ejθ . ejωt ]

= Re [V

m ej (ωt + θ)]

V (t) = Vm

cos (ωt + θ).

Both polar and exponential forms are used for the multi-plication or division of two phasors, whereas rectan-gular form is used for addition and subtraction of twoPhasors.

Phasor Diagrams :

RL Series Circuit :

+

–~

+ –VR

RI

L +

–VLV

VR

Vθ

VL

I

VR

V = √V2R + V2

L

θ = tan– 1 (VL/V

R)

Power factor = cos θ (lagging)

RC Series Circuit :

θ+

–~

+ –VR

RI

C +

–VCV

VR

V

VC

I

V = √V2R + V2

C

θ = tan– 1 (VC

/VR

)

Power factor = cos θ (leading)

RLC Series Circuit :

+

–~

+ –VR

RI C

+

–VCV

+ –VL

For VL > V

C :

θ

VL

(V –V )L C

VR

VC

V

I

Gateway to PSUs (Electronics &Telecom, Electronics &

Communication, Electrical, Electronics& Instrumentation)

Publisher : Upkar Prakashan ISBN : 9788174829429 Author : Ashish Dixit

Type the URL : http://www.kopykitab.com/product/10399

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