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Eighth Ed

GATE ELECTRONICS & COMMUNICATIO

Network Analysis Vol 3 of 10

RK Kanodia Ashish Murolia

GAT E Elect ronics & Communicat ion Vol 3, 8e Network Analysis RK Kanodia & Ashish Murolia

Copyright © By NODIA & COMPANY

Information contained in this book has been obtained by aut hor, from sources believes to be r eliabl neither NODIA & COMPANY nor it s aut hor guarantee the accuracy or complet eness of any informa and NODIA & COMPANY nor its aut hor shall be r esponsible for any error, omissions, or damages ar use of this informat ion. This book is published with the underst anding that NODIA & COMPANY and are supplying information but are not att empting to render engineering or other professional services.

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To Our Par ents

Preface to th For almost a decade, we have been receiving tremendous responses from GAT E aspirant s for our earli GAT E Multiple Choice Quest ions, GAT E Guide, and the GAT E Cloud series. Our first book, GATE M Choice Quest ions (MCQ), was a compilat ion of objective questions and solutions for all sub jects of GA Elect ronics & Communicat ion Engineering in one book. The idea behind the book was that Gat e aspir had just completed or about to finish their last semest er to achieve his or her B.E / B.Tech need only to answering questions to crack GATE . The solutions in the book were present ed in such a manner that needs to know fundament al concepts to underst and them. We assumed t hat st udents have learned eno the fundamentals by his or her graduat ion. The book was a great success, but still ther e were a large r aspirant s who needed more preparat ory materials beyond just problems and solutions. This lar ge rat io included average st udents. Lat er, we perceived that many aspirant s couldn’t develop a good problem solving approach in their B Some of them lacked the fundament als of a subject and had difficulty underst anding simple solut ions. we have an idea to enhance our content and present two separat e books for each sub ject: one for t heor contains brief t heory, problem solving methods, fundament al concepts, and point s-t o-remember. The s is about problems, including a vast collection of problems with descript ive and step-by-st ep solut ions t be underst ood by an average student . This was the origin of GAT E Guide (t he theory book) and GAT (the problem bank) series: two books for each subject . GAT E Guide and GAT E Cloud were published subject s only. Thereafter we r eceived an immense number of emails from our readers looking for a complet e study pa for all subject s and a book that combines bot h GAT E Guide and GAT E Cloud. This encouraged us to GAT E Study P ackage (a set of 10 books: one for each subject ) for G ATE Elect ronic and Communicat i Engineering. Each book in this package is adequat e for t he purpose of qualifying GATE for an average Each book contains brief t heory, fundament al concept s, problem solving met hodology, summary of form and a solved question bank. The quest ion bank has three exercises for each chapter: 1) Theoret ical MC Numerical MCQs, and 3) Numerical Type Quest ions (based on the new GATE pat ter n). Solutions are in a descript ive and st ep-by-step manner, which are easy to understand for all aspirant s. We believe that each book of GAT E Study P ackage helps a student learn fundament al concept s and d problem solving skills for a sub ject, which are key essentials t o crack GAT E. Although we have put a v effort in preparing this book, some errors may have crept in. We shall appreciat e and great ly acknowl constr uct ive comment s, criticisms, and suggest ions from t he user s of t his book. You may write to us at [email protected] and [email protected].

Acknowledgements We would like to express our sincere thanks to all the co-authors, editors, and reviewers for t heir effor making this pr oject successful. We would also like t o thank Team NODIA for providing professional su this project through all phases of its development . At last, we express our grat itude to God and our Fa providing moral support and mot ivat ion.

SYLLABUS GATE Electronics & Communications

Networks: Network graphs: matrices associat ed with graphs; incidence, fundamental cut set and fundament al circu Solution methods: nodal and mesh analysis. Network theorems: superposit ion, Thevenin and Norton’s power t ransfer, W ye-Delt a transformation. Steady stat e sinusoidal analysis using phasors. Linear const an differential equations; time domain analysis of simple RLC circuits, Solution of network equat ions us transfor m: frequency domain analysis of RLC circuit s. 2-port network paramet ers: driving point a functions. Stat e equations for networks.

IES Electronics & Telecommunication

Network Theory Network analysis techniques; Network theorems, transient response, st eady stat e sinusoidal respons graphs and t heir applicat ions in network analysis; Tellegen’s theorem. Two port networks; Z, Y, h and t parameters. Combinat ion of two port s, analysis of common two port s. Network functions : part s functions, obtaining a network funct ion fr om a given part. Transmission crit eria : delay and rise time, E other definitions effect of cascading. Element s of network synthesis.

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CONTE

CHAPTER 1

BASIC CONCEPTS

1.1

INTRODUCTION TO CIRCUIT ANALYSIS

1.2

BASIC ELECTRIC QUANTITIES OR NETWORK VARIABLES

1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.3

Charge Current Volt age P ower Energy

1

1 1 2 3 4

CIRCUIT ELEMENTS 4

1.3.1 1.3.2 1.3.3 1.3.4 1.4

1

Active and P assive Elements 5 Bilateral and Unilateral Elements Linear and Non-linear Elements Lumped and Distributed Elements

SOURCES

1.4.1 1.4.2

5

Indep endent Sources Dependent Sources

EXERCISE 1.1

8

EXERCISE 1.2

18

SOLUTIONS 1.1

23

SOLUTIONS 1.2

30

CHAPTER 2

5 6

BASIC LAWS

2.1

INTRODUCTION

2.2

OHM’S LAW AND RESISTANCE

37

2.3

BRANCHES, NODES AND LOOPS

39

2.4

KIRCHHOFF’S LAW

2.4.1 2.4.2

5 5 5

37

40

Kirchhoff’s Current Law Kirchoff’s Voltage Law

40 41

2.5

SERIES RESISTANCES AND VOLTAGE DIVISION

2.6

PARALLEL RESISTANCES AND CURRENT DIVISION

27

SOURCES IN SERIES OR PARALLEL

44

41 42

2.8

ANALYSIS OF SIMPLE RESISTIVE CIRCUIT WITH A SINGLE SOURCE

2.9

ANALYSIS OF SIMPLE RESISTIVE CIRCUIT WITH A DEPENDENT SOURCE

2.10

DELTA- TO- WYE(T- Y ) TRANSFORMATION

2.10.1 2.10.2 2.11

Wye To Delta Conversion Delta To Wye Conversion

49

EXERCISE 2.2

67

SOLUTIONS 2.1

78

SOLUTIONS 2.2

101

CHAPTER 3

INTRODUCTION

127

3.2

NETWORK GRAPH

127

Directed and Undirected Graph P lanar and Non-planar Graphs Subgraph 128

3.2.4 3.2.5

Connected Graphs Degree of Vertex 129

3.3.1

130

T wigs and Links

INCIDENCE MATRIX

TIE-SET

CUT-SET

3.6.1 3.6.2 3.6.3 3.6.4

127 128

129 129

TREE AND CO-TREE

3.5.1 3.5.2 3.5.3 3.6

47 47

3.2.1 3.2.2 3.2.3

3.4.1 3.4.2 3.5

46

GRAPH THEORY

3.1

3.4

46

NON-IDEAL SOURCES 48

EXERCISE 2.1

3.3

46

131

P rop erties of Incidence Matrix: Incidence Matrix and KCL

131 132

133

T ie-Set Matrix 134 T ie-Set Matrix and KVL 134 T ie-Set Matrix and Branch Currents

135

136

Fundamental Cut - Set 136 Fundamental Cut-set Matrix 137 Fundamental Cut-set Matrix and KCL 138 Tree Branch Volt ages and Fundamental Cut-set Voltages

EXERCISE 3.1

140

EXERCISE 3.2

149

SOLUTIONS 3.1

151

SOLUTIONS 3.2

156

139

4.4

COMPARISON BETWEEN NODAL ANALYSIS AND MESH ANALYSIS

EXERCISE 4.1

164

EXERCISE 4.2

173

SOLUTIONS 4.1

181

SOLUTIONS 4.2

192

CHAPTER 5

CIRCUIT THEOREMS

5.1

INTRODUCTION

5.2

LINEARITY

5.3

SUPERPOSITION

5.4

SOURCE TRANSFORMATION

5.4.1 5.5

211

211 212 213

Source Transformation For Dependent Source

THEVENIN’S THEOREM

5.5.1 5.5.2 5.5.3 5.6

163

214

T hevenin’s Voltage 215 T hevenin’s Resistance 215 Circuit Analysis Using T hevenin Equivalent

NORTON’S THEOREM

5.6.1 5.6.2 5.6.3

214

216

217

Norton’s Current 217 Norton’s Resistance 218 Circuit Analysis Using Nort on’s Equivalent

218

5.7

TRANSFORMATION BETWEEN THEVENIN & NORTON’S EQUIVALENT CIRCUITS

5.8

MAXIMUM POWER TRANSFER THEOREM

5.9

RECIPROCITY THEOREM

5.9.1 5.9.2

221

Circuit W ith a Volt age Source Circuit W ith a Current Source

5.10

SUBSTITUTION THEOREM

222

5.11

MILLMAN’S THEOREM

223

5.12

TELLEGEN’S THEOREM

223

EXERCISE 5.1

224

EXERCISE 5.2

239

SOLUTIONS 5.1

246

SOLUTIONS 5.2

272

CHAPTER 6 6.1

221 221

INDUCTOR AND CAPACITOR

CAPACITOR

6.1.1

219

297

Voltage-Current Relationship of a Capacitor

297

219

6.3

INDUCTOR

6.3.1 6.3.2 6.3.3 6.4

Voltage-Current Relationship of an Inductor Energy St ored in an Inductor 302 Some P rop erties of an Ideal Inductor 303 Inductors in Series Inductors in P arallel

DUALITY

307

EXERCISE 6.2

322

SOLUTIONS 6.1

328

SOLUTIONS 6.2

347

CHAPTER 7

303 304

305

EXERCISE 6.1

FIRST ORDER RL AND RC CIRCUITS

7.1

INTRODUCTION

7.2

SOURCE FREE OR ZERO-INPUT RESPONSE

7.2.1 7.2.2

359

Source-Free RC Circuit Source-Free RL circuit

359

359 362

7.3

THE UNIT STEP FUNCTION

7.4

DC OR STEP RESPONSE OF FIRST ORDER CIRCUIT

7.5

STEP RESPONSE OF AN RC CIRCUIT 365

7.5.1 7.5.2 7.6

7.8

365

368

STEP RESPONSE OF AN RL CIRCUIT 368

Complete Resp onse 369 Complete Resp onse in terms of Initial and Final Conditions

STEP BY STEP APPROACH TO SOLVE RL AND RC CIRCUITS

370

7.7.1 7.7.2

370

Solution Using Capacitor Voltage or Inductor Current General Method 371

STABILITY OF FIRST ORDER CIRCUITS

EXERCISE 7.1

373

EXERCISE 7.2

392

SOLUTIONS 7.1

397

SOLUTIONS 7.2

452

CHAPTER 8 8.1

364

Complete Resp onse : 367 Complete Resp onse in terms of Initial and Final Conditions

7.6.1 7.6.2 7.7

302

SERIES AND PARALLEL INDUCTORS 303

6.4.1 6.4.2 6.5

301

372

SECOND ORDER CIRCUITS

INTRODUCTION

469

370

8.6

STEP RESPONSE OF PARALLEL RLC CIRCUIT

8.7

THE LOSSLESS LC CIRCUIT

EXERCISE 8.1

478

EXERCISE 8.2

491

SOLUTIONS 8.1

495

SOLUTIONS 8.2

527

CHAPTER 9

476

477

SINUSOIDAL STEADY STATE ANALYSIS

9.1

INTRODUCTION

9.2

CHARACTERISTICS OF SINUSOID

9.3

PHASORS

9.4

PHASOR RELATIONSHIP FOR CIRCUIT ELEMENTS 544

9.4.1 9.4.2 9.4.3 9.5

9.8

9.9

544 545 545

IMPEDANCE AND ADMITTANCE

Admittance

546

548

KIRCHHOFF’S LAWS IN THE PHASOR DOMAIN

9.6.1 9.6.2 9.7

541

543

Resistor Inductor Capacitor

9.5.1 9.6

541

Kirchhoff’s Volt age Law(KVL) Kirchhoff’s Current Law(KCL)

548

548 549

IMPEDANCE COMBINATIONS 549

9.7.1 9.7.2

Imp edances in Series and Voltage Division Imp edances in P arallel and Current Division

9.7.3

Delta-to-W ye Transformation

CIRCUIT ANALYSIS IN PHASOR DOMAIN

551 552

9.8.1 9.8.2 9.8.3 9.8.4

Nodal Analysis 552 Mesh Analysis 552 Sup erp osition T heorem Source Transformation

9.8.5

T hevenin and Norton Equivalent Circuits

PHASOR DIAGRAMS

EXERCISE 9.1

556

EXERCISE 9.2

579

SOLUTIONS 9.1

583

SOLUTIONS 9.2

618

CHAPTER 10

549 550

553 553

554

AC POWER ANALYSIS

553

10.5

COMPLEX POWER

10.5.1

630

Alt ernat ive Forms For Complex P ower

10.6

POWER FACTOR

10.7

MAXIMUM AVERAGE POWER TRANSFER THEOREM

10.7.1

632

AC POWER CONSERVATION

636

10.9

POWER FACTOR CORRECTION

636

EXERCISE 10.1

638

EXERCISE 10.2

648

SOLUTIONS 10.1

653

SOLUTIONS 10.2

669

THREE PHASE CIRCUITS

11.1

INTRODUCTION

11.2

BALANCED THREE PHASE VOLTAGE SOURCES

11.2.1 11.2.2 11.3

11.4

634

Maximum Average P ower Transfer, when Z is Restrict ed 635

10.8

CHAPTER 11

631

683 683

Y-connected T hree-P hase Voltage Source T-connect ed T hree-P hase Volt age Source

BALANCED THREE-PHASE LOADS

688

11.3.1 11.3.2

688 689

Y -connected Load T-connect ed Load

683 686

ANALYSIS OF BALANCED THREE-PHASE CIRCUITS 689

11.4.1 11.4.2

Balanced Y -Y Connection Balanced Y -T Connection

689 691

11.4.3 11.4.4

Balanced T-T Connection Balanced T -Y connection

692 693

11.5

POWER IN A BALANCED THREE-PHASE SYSTEM

11.6

TWO-WATTMETER POWER MEASUREMENT 695

EXERCISE 11.1

697

EXERCISE 11.2

706

SOLUTIONS 11.1

709

SOLUTIONS 11.2

722

CHAPTER 12

694

MAGNETICALLY COUPLED CIRCUITS

12.1

INTRODUCTION

729

12.2

MUTUAL INDUCTANCE

12.3

DOT CONVENTION

730

729

12.7

ENERGY STORED IN A COUPLED CIRCUIT

12.7.1 12.8

Coefficient of Coupling

THE LINEAR TRANSFORMER

12.8.1 12.8.2 12.9

737

THE IDEAL TRANSFORMER

739

12.9.1

740

Reflected Imp edance 742

EXERCISE 12.2

751

SOLUTIONS 12.1

755

SOLUTIONS 12.2

768

CHAPTER 13

INTRODUCTION

13.2

TRANSFER FUNCTIONS

777

13.2.1

778

13.4

737 738

FREQUENCY RESPONSE

13.1

13.3

736

T -equivalent of a Linear Transformer p -equivalent of a Linear Transformer

EXERCISE 12.1

735

777

P oles and Zeros

RESONANT CIRCUIT 778

13.3.1

Series Resonance

778

13.3.2

P arallel Resonance

784

PASSIVE FILTERS

13.4.1 13.4.2 13.4.3 13.4.4

788

Low P ass Filter High P ass Filter Band P ass Filter Band Stop Filter

788 789 790 791

13.5

EQUIVALENT SERIES AND PARALLEL COMBINATION

13.6

SCALING

13.6.1 13.6.2 13.6.3

793

Magnitude Scaling 793 Frequency Scaling 793 Magnitude and Frequency Scaling

EXERCISE 13.1

795

EXERCISE 13.2

804

SOLUTIONS 13.1

807

SOLUTIONS 13.2

821

CHAPTER 14

792

794

CIRCUIT ANALYSIS USING LAPLACE TRANSFORM

14.1

INTRODUCTION

827

14.2

DEFINITION OF THE LAPLACE TRANSFORM

827

14.4

PROPERTIES OF THE LAPLACE TRANSFORM

14.4.1 14.5

Initial Value and Final Value T heorem

CIRCUIT ELEMENTS IN THE S -DOMAIN

Resistor in the s -domain Inductor in the s -domain Capacitor in the s -domain

14.5.1 14.5.2 14.5.3 14.6

CIRCUIT ANALYSIS IN THE

14.7

THE TRANSFER FUNCTION

14.7.1

836

EXERCISE 14.2

850

SOLUTIONS 14.1

853

SOLUTIONS 14.2

880

INTRODUCTION

15.2

IMPEDANCE PARAMETERS

15.2.1 15.2.2 15.2.3

834

834

835

887 887

Some Equivalent Networks 889 Input Imp edance of a Terminated T wo-port Network in Terms of Imp edance Paramete T hevenin Equivalent Across Output Port in Terms of Impedance Parameters 890 891

Some Equivalent Networks 892 Input Admittance of a Terminated T wo-p ort Networks in Terms of Admitt ance P arame

HYBRID PARAMETERS

15.4.1 15.4.2 15.4.3 15.5

831 832 833

s -DOMAIN

ADMITTANCE PARAMETERS

15.3.1 15.3.2 15.4

831

TWO PORT NETWORK

15.1