Network Analysis And Synthesis 9780070144781

Table of contents :
Title
Contents
1 Introduction to Different Types of Systems
2 Introduction to Circuit-Theory Concepts
3 Network Topology (Graph Theory)
4 Network Theorems
5 Laplace Transformand Its Applications
6 Two-Port Network
7 Fourier Series and Fourier Transform
8 Sinusoidal Steady State Analysis
9 Magnetically Coupled Circuits
10 Three Phase Circuits
11 Resonance
12 Network Functions and Their Time-Domain and Frequency-Domain Response
13 Elements of Realizability and Network Synthesis
14 Operational Amplifier and Active Filter
15 Introduction To Software SPICE
16 Indefinite Admittance Matrix (IAM)
17 Symmetrical Components
Appendix A
Appendix B
Appendix C
Index

Citation preview

About the Authors

S P Ghosh obtained his BE (Hons) in Electrical Engineering from National Institute of Technology, Durgapur, and received a Master of Electrical Engineering degree from Jadavpur University with specialization in High Voltage Engineering. He joined College of Engineering and Management, Kolaghat, as a lecturer in 2002. Presently, he is working as an Assistant Professor in the department of Electrical Engineering at College of Engineering and Management, Kolaghat. He has published several papers in national and international conferences. He is also pursuing his PhD at Bengal Engineering and Science University, Shibpur. His areas of interest include Power Systems, Electrical Machines, and Artificial Neural Networks.

A K Chakraborty received his BEE (Hons) from Jadavpur University, MTech in Power System Engineering from IIT Kharagpur and PhD (Engineering) from Jadavpur University. He joined College of Engineering and Management, Kolaghat, in 1998 as Assistant Professor and was elevated to the rank of Professor in the Electrical Engineering Department. He served as HOD from 2002 to 2005 in the same department. Presently, he is working as Professor and Head of the Department of Electrical Engineering. He also worked as a Lecturer in NIT Silchar for five years. He served industries, namely, CESC Ltd. and Tinplate Company of India Ltd (a Tata Enterprise) for over fourteen years before joining this institute. He is a Fellow of Institute of Engineers (India), Chartered Engineer, Member IET (UK) and Life Member of ISTE. He has published several technical papers in national and international conferences and also in reputed journals. He has guided several MTech and PhD scholars. His research interests are in the field of Power System Protection, Economic Operation of Power Systems, Deregulated Power System, and HVDC.

S P Ghosh Assistant Professor Department of Electrical Engineering College of Engineering and Management Kolaghat, West Bengal

A K Chakraborty Professor and Head Department of Electrical Engineering College of Engineering and Management Kolaghat, West Bengal

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To my family My Wife, Lipika Daughter, Adrita

S P Ghosh

To my family Wife, Indira Daughters, Amrita and Ananya

A K Chakraborty

Contents Foreword Preface 1.

Introduction to Different Types of Systems 1.1 1.2 1.3 1.4

2

3

1–22

Introduction 1 Concepts of Signals and Systems 1 Different Types of Signals 2 Different Types of Systems 6 Interconnection of Systems 10 Solved Problems 11 Summary 17 Short-Answer Questions 18 Exercises 20 Questions 20 Multiple-Choice Questions 20 Answers 22

Introduction to Circuit-Theory Concepts 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

xv xvii

23–102

Introduction 23 Some Basic Terminologies of Electric Circuits 23 Different Notations 26 Basic Circuit Elements 27 Passive Circuit Elements 28 Types of Electrical Energy Sources 39 Fundamental Laws 41 Source Transformation 43 Network Analysis Techniques 48 Duality 50 Star-Delta Conversion Technique 52 Solved Problems 55 Summary 79 Short-Answer Questions 81 Exercises 89 Questions 92 Multiple-Choice Questions 92 Answers 102

Network Topology (Graph Theory) Introduction 103 3.1 Graph of a Network 103 3.2 Terminology 104 3.3 Concept of a Tree 105

103–154

viii Contents 3.4 3.5 3.6 3.7 3.8

4

Network Theorems 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

5

Incidence Matrix [Aa] 107 Tie-Set Matrix and Loop Currents 110 Cut-Set Matrix and Node-Pair Potential 112 Formulation of Network Equilibrium Equations 115 Generalized Equations in Matrix Forms for Circuits having Sources 116 Solved Problems 118 Summary 147 Short-Answer Questions 147 Exercises 150 Questions 151 Multiple-Choice Questions 152 Answers 154 Introduction 155 Network Theorems 155 Substitution Theorem 156 Superposition Theorem 156 Reciprocity Theorem 159 Thevenin’s Theorem 160 Norton’s Theorem 161 Maximum Power Transfer Theorem 166 Tellegen’s Theorem 170 Millman’s Theorem 172 Compensation Theorem 175 Solved Problems 177 Summary 217 Short-Answer Questions 218 Exercises 220 Questions 224 Multiple-Choice Questions 225 Answers 230

Laplace Transform and Its Applications 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15

155–230

Introduction 231 Advantages of Laplace-Transform Method 231 Definition of Laplace Transform 232 Concept of Complex Frequency 232 Basic Theorems of Laplace Transform 233 Region of Convergence (ROC) 237 Laplace Transform of some Basic Functions 238 Laplace Transform Table 242 Other Important Laplace Transforms 243 Laplace Transform of Periodic Functions 244 Inverse Laplace Transform 244 Applications of Laplace Transform 248 Transient Analysis of Electric Circuits using Laplace Transform 250 Response with Pulse Input Voltage 268 Steps for Circuit Analysis using Laplace Transform Method 271 Concept of Convolution Theorem 271

231–326

ix Contents Solved Problems 273 Summary 303 Short-Answer Questions 304 Exercises 309 Questions 312 Multiple-Choice Questions 313 Answers 325

6

Two-Port Network 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

7

327–412

Introduction 327 Relationships of Two-Port Variables 327 Conditions for Reciprocity and Symmetry 334 Interrelationships between Two-Port Parameters 338 Interconnection of Two-Port Networks 339 Two-Port Network Functions 344 Transfer Functions of Terminated Two-Port Networks 345 Application of Network Parameters to the Analysis of Typical Two-Port Networks 348 Some Special Two-Port Networks 351 Image Parameters of a Two-Port Network 354 Solved Problems 359 Summary 398 Short-Answer Questions 398 Exercises 402 Questions 405 Multiple-Choice Questions 406 Answers 412

Fourier Series and Fourier Transform Part I: Fourier Series 413 Introduction 413 7.1 Definition of Fourier Series 414 7.2 Dirichlet’s Conditions 414 7.3 Convergence of Fourier Series 414 7.4 Fourier Analysis 415 7.5 Waveform Symmetry 419 7.6 Truncating Fourier Series 422 7.7 Steady-State Response of Network to Periodic Signals 424 7.8 Steps for Application of Fourier Series to Circuit Analysis 424 7.9 Power Spectrum 425 Part II: Fourier Transform 425 Introduction 425 7.10 Definition of Fourier Transform 425 7.11 Convergence of Fourier Transform 426 7.12 Fourier Transform of Some Functions 427 7.13 Properties of Fourier Transforms 429 7.14 Energy Density and Parseval’s Theorem 432 7.15 Comparison between Fourier Transform and Laplace Transform 433 7.16 Steps for Application of Fourier Transform to Circuit Analysis 434 Solved Problems 434

413–472

x Contents Summary 460 Short-Answer Questions 460 Exercises 467 Questions 469 Multiple-Choice Questions 470 Answers 472

8

Sinusoidal Steady State Analysis 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20

9

Magnetically Coupled Circuits 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

473–542

Introduction 473 Advantages of using Alternating Currents in Electrical Engineering 473 Basics of Sinusoids 474 Terminologies 474 Some Values of Alternating Quantities 476 Complex Number Systems 479 Phasor Representation 480 The j Operator 484 Phasor Diagrams 484 Circuit Response to Sinusoids 484 Kirchhoff’s Laws in Phasor Domain 485 Voltage and Current Phasors in Single-Element Circuits 485 Phasor Analysis of R-L Series Circuit 488 Phasor Analysis of RC Series Circuit 490 Phasor Analysis of RLC Series Circuit 492 Steps for Sinusoidal Steady-State Analysis (Phasor Approach to Circuit Analysis) 494 Concept of Reactance, Impedance, Susceptance and Admittance as Phasors 494 AC Power Analysis 496 Power Calculations in Different Electrical Elements 498 Sinusoidal Steady-State Response of Parallel AC Circuits 503 Sinusoidal Steady-State Response of Series–Parallel AC Circuits 507 Solved Problems 507 Summary 527 Short-Answer Questions 528 Questions 534 Exercises 535 Multiple-Choice Questions 537 Answers 542

543–590

Introduction 543 Self-Inductance 543 Coupled Inductor 544 Mutual Inductance 544 Mutual Inductance between Two Coupled Inductors 545 Dot Convention 546 Determination of Coefficient of Coupling from Energy Calculations in Coupled Circuits 548 Inductive Coupling 549 Linear Transformer 551

(conductively Equivalent Circuit of a Magnetically Coupled Circuit) 552 9.10 Ideal Transformer 553

xi Contents 9.11 Tuned Coupled Circuits 555 Solved Problems 560 Summary 579 Short-Answer Questions 579 Exercises 585 Questions 587 Multiple Choice Questions 588 Answers 590

10 Three Phase Circuits 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

11 Resonance 11.1 11.2 11.3 11.4 11.5 11.6

634–685

Introduction 634 Series Resonance or Voltage Resonance 634 Parallel Resonance or Current Resonance or Anti-Resonance 641 Relation between Damping Ratio and Quality Factor 645 A More Realistic Parallel Resonant Circuit 646 Universal Resonance Curve 652 Applications of Resonance 653 Solved Problems 654 Summary 671 Short-Answer Questions 671 Exercise 678 Questions 679 Multiple-Choice Questions 681 Answers 685

12 Network Functions and Their Time-Domain and Frequency-Domain Response 12.1 12.2 12.3 12.4

591–633

Introduction 591 Advantages of Polyphase Systems 591 Some Terminologies 592 Generation of Balanced Three-Phase Supply 593 Phase Sequence 594 Interconnection of Three-Phase Systems 595 Measurement of Power in Three-Phase Circuits 599 Conversion of Balanced Three-Phase System from Star to Delta 604 Analysis of Balanced Parallel Load 605 Analysis of Unbalanced Load Circuits 606 Solved Problems 610 Summary 625 Short-Answer Questions 626 Exercises 628 Questions 629 Multiple-Choice Questions 630 Answers 633

Introduction 656 Terminal and Terminal Pairs 686 Network Functions for a One-Port Network 687 Network Functions for Two-Port Networks 687 Poles and Zeros of Network Functions 688

686–754

xii Contents 12.5 Pole Zero Diagram 689 12.6 Significance of Poles and Zeros 689 12.7 Natural Response and Natural Frequencies 690 12.8 Relation between Pole Position, Natural Response and Stability 691 12.9 Restriction on the Location of the Poles and Zeros in the s-Plane 692 12.10 Necessary Conditions for Driving Point Functions (Restriction on Pole-Zero Locations in the s-Plane for Driving Point Functions) 693 12.11 Necessary Conditions for Transfer Functions (Restriction on Pole-Zero Locations in the s-Plane for Transfer Functions) 697 12.12 Time Domain Behaviour from Pole–Zero Plot 697 12.13 Frequency Domain Behaviour from Pole–Zero Plot 699 Solved Problems 715 Summary 743 Short-Answer Questions 744 Exercises 747 Questions 749 Multiple-Choice Questions 749 Answers 754

13 Elements of Realizability and Network Synthesis

755–860

Part I: Elements of Realizability 755 Introduction 755 13.1 Elements of Realizability Theory 755 13.2 Hurwitz Polynomial 757 13.3 Positive Real Functions 759 Part II: Synthesis of Driving Point Functions 767 Introduction 767 13.4 Basic Synthesis Procedure 767 13.5 Methods of Synthesis 770 13.6 Driving Point Synthesis of One-Port Networks with Two Types of Elements 771 13.7 Synthesis of RLC Driving point Functions 792 Solved Problems 803 Summary 851 Short-Answer Questions 852 Exercises 855 Questions 857 Multiple-Choice Questions 858 Answers 860

14 Operational Amplifier and Active Filter Introduction 861 14.1 Operational Amplifier (Op-Amp) 861 14.2 Filter 862 14.3 Advantages of Active Filters over Passive Filters 14.4 Application of Active Filters 863 14.5 Types of Active Filters 863 14.6 Low-Pass-Active Filter 864 14.7 High-Pass-Active-Filter 867 14.8 Band-Pass Active Filter 869 14.9 Band-Reject (Notch) Active Filter 876 14.10 Filter Approximation 878 14.11 All-Pass Active Filter 884

861–892

862

xiii Contents Summary 885 Short-Answer Questions 887 Exercises 889 Questions 889 Multiple-Choice Questions 889 Answers 892

15 Introduction To Software SPICE Introduction 893 15.1 Types of Spice 893 15.2 Execution of SPICE (How Spice Works) 894 15.3 Types of Analysis 894 15.4 Model Statements 895 15.5 DC Circuit Analysis 903 15.6 Transient Analysis 903 15.7 AC Circuit Analysis 905 15.8 Fourier Analysis and Harmonic Decomposition using SPICE 15.9 Harmonic Recomposition 906 15.10 DC Sensitivity Analysis 906 Solved Problems 906 Summary 921 Questions 922

895–922

905

16 Indefinite Admittance Matrix (IAM)

923–930

16.1 Definition of Indefinite Admittance Matrix (IAM) 923 16.2 Properties of IAM 924 16.3 Applications of IAM 927 Exercises 931

17 Symmetrical Components Introduction 933 17.1 Advantages of Symmetrical Component Method 933 17.2 a Operator 933 17.3 Symmetrical Components of an Unbalanced Three-Phase System 934 17.4 Component Synthesis (Evaluation of the Components) 935 17.5 Component Analysis 935 17.6 Graphical Method of Determining Sequence Components 935 17.7 Symmetrical Components of Current Phasors 936 17.8 Absence of Zero Sequence Components of Voltage and Current 936 17.9 Three-Phase Power in terms of Symmetrical Components 937 17.10 Sequence Impedances and Sequence Networks 938 17.11 Solution of 3-Phase Unbalanced Loads supplied from Unbalanced Supply 939 17.12 Solution of 3-Phase Unbalanced Loads supplied from Balanced Supply 941 Solved Problems 941 Summary 947 Exercises 948 Questions 949 Multiple-Choice Questions 949 Answers 950 Appendix A, B, C

931–948

Foreword

There is no necessity to emphasize that all engineering systems use electric circuits as components. Again, the knowledge of circuit theory is very much essential to understand the operation of these systems. Circuit Theory and Networks is an important subject which is common to almost all core and modern engineering branches. Having a clear idea of the basic concepts is very much essential to both students, who are pursuing their engineering courses, and the practicing engineers, who run plants and systems in a day-to-day way. The book Network Analysis and Synthesis written by S P Ghosh and A K Chakraborty is the result of their long association with teaching and plant-operation experiences. The book consists of 17 chapters which are nicely written, starting from the fundamentals. Every chapter contains live examples and worked-out problems of standard universities, UPSC, IETE, AMIE and GATE examinations. The book has been written with an up-to-date approach to accommodate the students of present standards and also to overcome the difficulties of teachers. I am sure this book will be an asset not only to the teachers and students but also to practicing engineers and technicians who are engaged in system operations. I wish the publication all success.

S K Sen BE Cal, Ph D (London), FIE, FNAE, DIC FELLOW IMPERIAL COLLEGE (LONDON) Former Minister-in-Charge, Power, Science Technology and Non-Conventional Energy Sources, Govt. of West Bengal Ex-Vice-Chancellor, Jadavpur University (Kolkata) Ex-Prof. and Head, Electrical Engineering, B E College (Shibpur) Hon. Member, Sikkim State Planning Commission Hon. Advisor to the Chief Minister, Govt. of Sikkim, India

Preface

Brief Introduction to the Subject Network analysis and Synthesis is a gateway course to all engineering subjects; Electrical Engineering, Electronics and Communication Engineering, Computer Science and Engineering, Information Technology, Instrumentation Engineering in particular. Almost all engineering systems use electric circuits as components. Knowledge of Network Analysis is very essential to understand the operation of these systems. Also, the subject of Network Analysis provides the background for understanding the behaviour of many other electrical and electronic devices. The present book has been written to provide knowledge of network analysis and synthesis, starting from the fundamentals.

Objectives This book has been written as per the syllabi of Network Analysis and Synthesis, or Circuit Theory and Networks, as it is taught under different universities in India. This text works well in our self-paced course, where students must rely on it as their primary learning resource. Nonetheless, completeness and clarity are equally advantageous when the book is used in a more traditional classroom setting. Cognizance of the present standard of students and the difficulties of the teachers has been given due thought. The conceptual examples and practice problems and a variety of conceptual and multiple choice questions at the end of each chapter give students a chance to check and to enhance their conceptual understanding.

Scope This book is mainly written for the engineering students of different universities all over India for the subject of network analysis and synthesis. However, as this course mitigates a definite percentage in every competitive examination of engineering professionals, viz., IES, UPSC, GATE, etc., we have written this book to help students see that a relatively small number of basic concepts are applied to a wide variety of situations.

Salient Features Some salient features of this book are • Covers both network analysis and synthesis • Rich pedagogy with large number of examples, solved problems, unsolved problems, MCQ's and short-answer type questions and answers • Contains large number of problems and questions from Indian universities, GATE, UPSC, AMIE, IETE and other competitive examinations • Discussion of the software packages PSPICE and MATLAB for solving network analysis problems • Detailed coverage of different types of systems and networks • Simple and student-friendly approach of writing

Organization This book has a total of seventeen chapters. The first chapter provides information about the basic characteristics of different types of systems. The second chapter deals with the basic circuit components, laws and techniques for circuit analysis. Chapter 3 discusses the application of graph-theory concepts in circuit analysis. In this chapter, the application of a

xviii Preface mathematical tool like graph theory has been presented with the help of a large number of practical examples. Chapter 4 is devoted to various network theorems necessary for simplified analysis of electrical problems. For examination purposes, this chapter is very important as several questions frequently are set from this chapter. Chapter 5 introduces a new method of circuit analysis—Laplace Transform method. Starting from the very fundamental concept of Laplace transform, its applications in various complicated circuit problems has been discussed in detail in this chapter. The sixth chapter deals with the concept of two-port network which has a vast application in many fields like transmission lines, filters and attenuators, and so on. Chapter 7 is divided into two parts. Part I presents the fundamentals of Fourier series and its application for circuit analysis. Part II discusses Fourier transform and its applications. Chapter 8 discusses the method of studying steady-state behaviour of electrical networks when sinusoidal excitations are applied. Chapter 9 deals with mutual inductance and magnetically coupled circuits. The tenth chapter explains the different aspects of three-phase circuits. Chapter 11 discusses a very important practical phenomenon of electrical engineering, called resonance. In this chapter, starting from the basic concept, the conditions of different circuits under resonance and its application have been discussed. In Chapter 12, the relations between the various voltages and currents in a circuit have been discussed in terms of network functions and responses. Chapter 13 discusses a new concept in the subject, known as network synthesis which aims at determining a suitable electrical network given some operating characteristics. Chapter 14 is devoted to operational amplifiers and active filters. Chapter 15 deals with a software package for circuit analysis, termed as SPICE. Chapter 16 explains circuit analysis with the help of a tool, called indefinite admittance matrix. However, it should be mentioned that this method was very useful earlier; with the advancement of digital computers, this method is becoming obsolete. The last chapter, Chapter 17 aims at the discussion of symmetrical component method of unbalanced three-phase circuits.

Acknowledgements Authors are indebted to Prof. Nirmal Chatterjee, Prof. Kalyan Dutta, Prof. C K Roy, Prof. A N Sanyal of Jadavpur University for their encouragement. We are grateful to Prof. S N Bhadra, HOD, Department of Electrical Engineering, College of Engineering and Management, Kolaghat, and Ex-Professor of IIT, Kharagpur, Prof. P B Duttagupta, IIT Kharagpur, Prof. H P Bhowmik, Ex-Principal, Institute of Leather Technology and Dr Abhinandan De, Bengal Engineering and Science University, Shibpur, for their constant inspiration and encouragement in the filed of academics. We would also like to thank the reviewers for taking out time to review the book. Their names are given below.

Urmila Kar Tirtha Shankar Das T L Singhal Ashutosh Marathe Arvind Pachorie A A Ansari Shashi Gandhar Vinay Pathak Mahavir Singh

NSEC (Netaji Subhash Engineering College) Garia, West Bengal Guru Nanak Institute of Technology Kolkata, West Bengal Chitkara Institute of Enginering Punjab Pune University Pune, Maharashtra Government Engineering College Jabalpur, Madhya Pradesh Sagar Institute of Research and Technology Bhopal, Madhya Pradesh Bharati Vidyapeeth College of Engineering, GGSIP University, New Delhi Bhopal Institute of Technology Bhopal, Madhya Pradesh Accurate Institute Greater Noida, Uttar Pradesh

xix Preface Pranita Joshi A Subramanian R Joseph Xavier Vishwanath Hegde K Amaresh B Venkata Prasanth

Mumbai University Mumbai, Maharashtra V R S Engineering College Villupuram, Tamil Nadu Ramakrishna Institute of Technology Coimbatore, Tamil Nadu Malnad Engineering College Hasan, Karnataka KSRM Engineering College Kadapa, Andhra Pradesh NBKR Institute of Technology Nellore, Andhra Pradesh

We are also thankful to the editorial and production staff of Tata McGraw Hill Education Private Limited for taking interest in publishing this edition. Last but not the least, we acknowledge the support offered by our respective wives and children without which this work would have not been successful.

Feedback Criticism and suggestions for improvement shall be gratefully acknowledged. Readers may contact S P Ghosh at [email protected] and Dr A K Chakraborty at [email protected]. S P Ghosh A K Chakraborty

Visual Walkthrough

5 Each c that g hapter be g i the c ves an id ins with hapte e a abo an Intr r. o ut th e con duction tents of

Laplace Transform and Its Applications

Introduction Classical methods of solving differential equations become quite cumbersome when used for networks involving higher order differential equations. In such cases, the Laplace transform method is used. The classical methods consist of three steps: (i) determination of complementary function, (ii) determination of particular integral, and iii) determination of arbitrary constants. But, these methods become difficult for the equations containing derivatives; and transform methods prove to be superior. The Laplace transform is an integral that transforms a time function into a new function of a complex variable. The term Laplace comes from the name of the French mathematician Pierre Simon Laplace (1749–1827). The transformation method is a very effective tool for solving integro-differential equations. Laplace transformation is also a very powerful tool for network analysis. Any linear circuit consisting of linear circuit elements can be solved by the knowledge of Laplace transformation. In this chapter, we will first discuss the basics of Laplace transformation and then apply this transform method to study the transient behaviour of electric circuits.

603 Three Phase Circuits

(

W1 = V12 I1 cos 30° +

) =V I

cos 30° +

) =V I )=

cos 30° −

L L

(

)

(

)

1. 2. 3. 4.

For a switch in the position 2, the wattmeter reading,

(

W2 = V13 I1 cos 30° −

)

(

) +V I

(

∴ W1 + W2 = VL I L cos 30 +

L L

(

L L

cos 30 −

W

1

i1

v3 i3

R

v1

i2

3 v2

R 2

Fig. 10.19 One-wattmeter method for a balanced deltaconnected load

Example 10.5 The power input to a three-phase induction motor is read by two wattmeters. The readings are 1000 W and 500 W. Find out the pf of the motor. If the line voltage is 400 V, find the line current. Here, W1

1000 W; W2

500 W, VL

400 V

⬖ power factor of the motor,

(

⎡ 3 W1 − W2 cos = cos ⎢ tan −1 W1 + W2 ⎢ ⎣

)

(

⎤ ⎡ 3 1000 − 500 ⎥ = cos ⎢ tan −1 1000 + 1500 ⎥ ⎢ ⎣ ⎦

) ⎤⎥ = cos ⎡ tan ⎢ ⎣

⎥ ⎦

−1

1 ⎤ ⎥ = 0.5 3⎦

Line current is IL =

P 3VL cos

W1 + W2

=

1000 + 500

=

3VL cos

3 × 400 × 0.5

= 4.33 A

W

10.6.2 Measurement of Reactive Power In case of a balanced three-phase load, the reactive power can be measured using one wattmeter. The connection is shown in Fig. 10.20. Here, the current coil of the wattmeter is connected in one line and the pressure coil is connected across the other two lines. The phasor diagram is shown in Fig. 10.21. The wattmeter reading is

(

W = V32 I1 cos 90° +

) =V I L

p

(

1

i1 N

3 2

V1

i2

Fig. 10.20 Measurement of reactive power for a 3-phase balanced starconnected load

cos 90° +

)

⇒ W = − 3V p I p sin

V2

V32 V2

V3

i3

It gives complete solution. Initial conditions are automatically considered in the transformed equations. Much less time is involved in solving differential equations. It gives systematic and routine solutions for differential equations.

3VL I L cos = total power of the load

Thus, the sum of the wattmeter readings gives the load power, same as in a twowattmeter method. Here, also, if the current coil is to be reversed to obtain one of the wattmeter readings then that reading should be treated as negative. In case of a balanced delta-connected load, for a three-phase power measurement by one-wattmeter method, the resistance (say, R) of value equal to that of the pressure coil of the wattmeter is connected in each of the remaining two phases, as shown in Fig. 10.19. The pressure coil and the resistances form a balanced star-connection.

Solution

5.1 ADVANTAGES OF LAPLACE-TRANSFORM METHOD Laplace-transform methods offer the following advantages over the classical methods:

For a switch in the position 1, the wattmeter reading

I3

(90

␾) V1 I ␾ 1

␾

V3 I2

␾

V2

Fig. 10.21 Phasor diagram for reactive power measurement for balanced 3-phase starconnected load

in ded tter i v o pr be are pic for s o e . rial mpl ch t Exa fter ea xt mate d a e e t r k Wor chapte g of the n h i c d a n e ersta und

560 Network Analysis and Synthesis

When k ⴝ kC

Each c taken hapter co n f all ov rom the tains a lar ge nu quest er Ind m io ia an d oth n papers ber of solv of dif er co ed pr mpet itive ferent uni oblems exam v inatio ersities ns.

• In this condition, the resistance which the secondary circuit couples into the primary at resonance is equal to the primary resistance. • The secondary current will be maximum. • The curve of the secondary current will be broader and flat-topped. • The curve of the primary current will have two peaks. When k > kC • The double peaks of the primary current become more prominent; the peaks being separated from each other. • The magnitude of the primary current at peaks becomes smaller as the value of k is increased. • The curve of the secondary current will also have two peaks.

Solved Problems Problem 9.1 Find the effective value of the inductance for the following connections: (a)

(b)

2H i

i

i 5H

(c)

i

10 H 1H

2H

4H

1H

i

5H 2H

1H

i

3H 2H

Fig. 9.33

Solution (a) This is a series-aiding connection. The effective inductance is,

)

(

⬗ Leq = L1 + L2 + 2 M = 5 + 10 + 2 × 2 = 19 H (b) This is a series-opposing connection. The effective inductance is,

)

(

⬗ Leq = L1 + L2 − 2 M = 2 + 4 − 2 × 1 = 4 H (c) Since the coils are magnetically coupled in series aiding or they assist each other, therefore,

( ( = (L + M

) ( ) ) ( ) ) = (5 + 2 + 1) = 8 H

effective inductance for the coil 1 is L1eff = L1 + M12 + M13 = 2 + 1 + 2 = 5 H

effective inductance for the coil 2 is L2 eff = L2 + M12 + M 23 = 3 + 1 + 1 = 5 H effective inductance for the coil 3 is L3eff Total effective inductance is

(

3

)

13

+ M 23

(

)

Leff = L1eff + L2 eff + L3eff = L1 + L2 + L3 + 2 M12 + M 23 + M13 = 18 H

527 Sinusoidal Steady State Analysis

Summary 1. For transmission and distribution, alternating current has a number of advantages over direct current. 2. A sinusoid is a signal that has the form of a sine or cosine function and in general can be written as

) )

v (t = V m sin t . A shifted sinusoid can be written as

)

v (t = V m sin( t +

where, Vm is the amplitude, ␻ is

2 , T is the time period 2␲f T of the sinusoid and ␾ is the phase of the sinusoid. Use of sinusoids has several advantages like minimum disturbance in electrical circuits, less interference to nearby communication lines and less iron and copper losses. The value of an alternating quantity at any instant of time is known as the instantaneous value. The maximum value of an alternating quantity attained in each cycle is known as the peak or maximum or crest value. The average value of an alternating quantity over a given time interval is the summation of all instantaneous values divided by the number of values taken T 1 over that interval. Mathematically, V av = ∫vdt , where T 0 T is the time period of the quantity.

the angular frequency

3.

4. 5.

6.

7. The rms or effective value of a continuous periodic t T2 is function f(t) defined over the interval T1 T

f rms =

T 2 2 1 2 ⎡f t ⎤ dt or, f = 1 ⎡f t ⎤ dt ⎦ rms ⎦ T2 −T1 T∫ ⎣ T ∫0 ⎣ 1

()

()

8. Form factor is the ratio of the rms value to the average value for an alternating wave.

( )

∴ form factor K f =

rms Value average value

For a sinusoidal wave, its value is 1.11. 9. Peak factor is the ratio of the peak value to the rms value for an alternating wave.

( )

peak value maximum value = rms value rms value For a sinusoidal wave its value is 1.414. 10. A phasor is a complex quantity that represents both the magnitude and phase angle of a sinusoid. For a sinusoid given as v t = V m cos t + , the corre∴ peak factor K p =

()

( ) sponding phasor is written as, v (t ) = V cos ( m

t+

).

11. The graphical representation of the phasors of sinusoidal quantities taken all at the same frequency and with proper phase relationships with respect to each other is called a phasor diagram. 12. Both KVL and KCL hold good in phasor domain, i.e.,

V1 +V 2 +V 3 + ⋅⋅⋅ +V n = 0 and I 1 + I 2 + I 3 + ⋅⋅⋅ + I n = 0 . 13. The voltage and current in different circuit elements have definite phase relations. For a resistor, the voltage and current are always in phase, i.e., the phase angle is zero. In a pure inductor, the current lags behind the voltage by 900 and in a pure capacitor, the current leads the voltage by 900. 14. Impedance (Z) of any two-terminal network is the ratio of the phasor voltage (V) to the phasor current (I ) V j ∠Z = Z ∠Z . The real part i.e. Z = = R + jX = Z e I of impedance Re[Z] R is called the resistance. The

(

)

imaginary part of impedance Im[Z] X is called the reactance. Impedance, resistance and reactance are all measured by the same unit, ohm ( ).

Z = R ; for aresistor = j L ; for aninductor 1 = ; for a capacitor j C 15. The reciprocal of the impedance Z is called admittance. So, it is the ratio of the phasor current to the phasor voltage, i.e. . The real part of admittance R is called conductance, G = Re ⎡⎣Y ⎤⎦ = 2 . The R +X 2 imaginary part of admittance is called susceptance, X B = Im ⎡⎣Y ⎤⎦ = 2 . Admittance, conductance and R +X 2 susceptance are all measured by the same unit, siemen (S). 16. Instantaneous power absorbed by an element is the product of the instantaneous voltage v(t) and the instantaneous current i(t), i.e., p(t) v(t) i(t) (in watts). 17. Average or real or active power (in watts) is the average of the instantaneous power over a time T 1 interval, i.e., P = ∫ p t dt . For the sinusoidal voltT 0 age and current given as v t = V m cos t + v and

()

()

)

)

(

)

i (t = I m cos ( t + i , the average power is given as 1 P = V m I m cos ( v − i = V rms I rms cos ( v − i . 2

)

)

that ary the m sum d in ns a covere i a t on opics t er c hapt portant c h c m Ea ei s th give ter. chap

744 Network Analysis and Synthesis

()

( ) = K ( s − z )( s − z ) ⋅⋅⋅( s − z ) ( ) ( s − p )( s − p ) ⋅⋅⋅( s − p ) ( j − z )( j − z )( j − z ) ⋅⋅⋅( j − z ) =K ( j − p )( j − p )( j − p ) ⋅⋅⋅( j − p )

F s =

If, j − z i =

Each with chapter c o a writi nswers; w ntains a s ng br ief an hich acts et of shor ta d to-t he-po s a guide answer q uesti to the int an on swer s in e students s for xami natio ns.

N s

1

D s

2

zi

(

)

n

1

2

3

m

(

i

−1

2

pi

)

)

2

zi zi

zi

(

−1

pi pi

of all zeros lines to j ( ) = K Product Productt of all poles lines to j

F j

n

=K

= ri

⎞ ⎟= ⎠

+ j −

⎛ − ∠ j − pi = tan ⎜ ⎝ −

(

s= j

3

⎛ − ∠ j − z i = tan ⎜ ⎝ − and, j − pi =

m

2

+ j −

Hence, the magnitude and phase angle of the complete frequency response may be written as

n

2

2

1

1

∏(j

− zi

)

∏(j

− pi

)

i =1 m

i =1

)

2

⎞ ⎟= ⎠

r1r2 ⋅⋅⋅ rn q1q2 ⋅⋅⋅ qm

and angle

i

pi

K

⬔F( j␻) = qi

(summation of angles of the vectors from zeros to j␻-point) (summation of angles of the vectors from poles to j␻-point) n

)

(

m

(

= ∑ ∠ j − z i − ∑ ∠ j − pi

i

i =1

)=(

then the network function may be written as

1

+

2

+

3

i =1

+ ⋅⋅⋅ +

n

)−(

1

+

) 2

+

3

+ ⋅⋅⋅+

n

)

15. The variation of magnitude and phase of a network function with frequency in logarithmic scale is known as Bode plot.

of all zeros lines to j ( ) = K Product Productt of all poles lines to j

F j

Short-Answer Questions 1. What are the poles and zeros? What information do they provide in respect of the network to which they relate? We consider a network function given by the ratio of two polynomials as

()

F s =

an s n + an−1s n−1 + ⋅⋅⋅+ a2 s 2 + a1s + a0 bm s m + bm−1s m−1 + ⋅⋅⋅+ b2 s 2 + b1s + b0

(1)

It is often convenient to factor the polynomials in the numerator and denominator, and to write the transfer function in terms of those factors:

()

F s =

( ) = K ( s − z )( s − z ) ⋅⋅⋅( s − z ) ( ) ( s − p )( s − p ) ⋅⋅⋅( s − p )

N s

1

2

n

D s

1

2

m

(2)

where, the numerator and denominator polynomials, N(s)and D(s), have real coefficients defined by a the system’s differential equation and K = n bm is a positive constant, known as scale factor. From Eq. (2), we can observe the following: At s zi,i 1,2,3....,n, the numerator polynomial N(s) 0; these complex frequencies are known as the zeros of the network function F(s). At zeros, the value of the network function is zero, i.e., Lim F(s) 0. s →Z i

At s pi, i 1,2,3...., m, the denominator polynomial D(s) 0; these complex frequencies are known as the poles of the network function F(s). At poles, the value of . the network function is infinity, i.e., Lim F(s) s → Pi

• Significance of poles and zeros The values of poles and zeros of F(s) and their locations in the s-plane completely specify a network function. All the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must be either purely real, or appear in complex conjugate pairs. In general for the poles, either pi ␴i , or else pi , pi 1 ␴ j␻i. The existence of a single complex pole without a corresponding conjugate pole would generate complex coefficients in the polynomial D(s). Similarly, the system zeros are either real or appear in complex conjugate pairs. The poles and zeros are properties of the transfer function, and therefore of the differential equation describing the input–output system dynamics. Together with the gain constant K they completely characterize the differential equation, and provide a complete description of the system. 2. What do you understand by driving point impedance of a two-port network? Enumerate important properties of driving point impedance functions of a two-port passive network.

467 Fourier Series and Fourier Transform

Exercises Fourier Series

1F

1. Find the Fourier series expansion for the following functions and sketch the frequency spectrum. (a) f (t )

t 0

T

2T

T

f (t) T/2

T T/2 0 T/2 T

2T

t

2

2␲

␲

0

␲␻

2

v ␲

1

2␲ ␻

20

3␲

2

f (t)

(c)

2␲

␲

2H

v(t )

t

0

A

(b)

hat ter t ut p a h c ,b each roblems ents, Fig. 7.63 (a) Fig. 7.63 (b) f o end 4. Find the Fourier series expansion for the waveforms ng p ignm shown in Fig. 7.64. t the practici ks, ass a n ⎡ ⎤ 1 1 1 wor for give [(a) v = −2 ⎢ sin x + sin2 x + sin3 x + sin 4 x + ⋅⋅⋅⎥ 2 3 4 ⎣ ⎦ s is udents tutorial e s i c t V 4V 4V 4V cos x + cos 3 x + cos 5 x + ⋅⋅⋅ ] exer ting he s (b) v = 2 + ns. (3 ) (5 ) t of only t s in set questio e s t r A s no (a) tion ache help the te xamina e s help es and z z i u (b) q v (t ) 10

␲␻

0

2␲ ␻

t

Fig. 7.56

4␲

x

3␲

␲

f (x )

()

[Ans: (a) f t =

A ∞ A +∑ sin n t 2 n =1 n

V

⎤ T 2T ⎡ 1 (b) f (t = − 2 ⎢cos t + 2 cos 3 t + ⋅⋅⋅⎥ 4 3 ⎣ ⎦

)

()

(c) f t =

1

1 2 ∞ 1 + sin t − ∑ 2 cos 2 n t ] 2 n =1 4 n − 1

2. A periodic waveform as shown in Fig. 7.62 feeds an RL 1 H. Calculate the power load with R 10 ohm and L 2

␲

[v =

at the fundamental frequency supplied to the load.

x

V m 2V m ⎛ ⎞ 1 − 2 ⎜ cos x + cos 5 x + ⋅⋅⋅⎟ + 4 25 ⎝ ⎠

⎤

Vm ⎛ ⎞ 1 1 1 ⎜⎝ sin x − 2 sin2 x + 3 sin3 x − 4 sin 4 x + ⋅⋅⋅⎟⎠ ] ⎦

f (t ) A

t 0

0 ␲ 2␲ 3␲

Fig. 7.64 5. A triangular wave increases linearly from 0 to Vm during the interval 0 to ␲. The wave has zero value during the interval ␲ to 2␲ and this cycle is repeated. Find the Fourier series representation of the wave.

T

2T

Fig. 7.62 3. A waveform of the shape shown in Fig. 7.63 (a) is applied to the network shown in Fig. 7.63 (b). Calculate the power dissipated in a 20- resistor. Take ␻ 1 rad/s. [1.17 W]

6. A wave has a constant value Im during the interval −

2 3 and Im during the interval to . This cycle 2 2 2 is repeated in the next intervals. Find the Fourier series for the wave. to

⎡ 4I m ⎛ ⎞⎤ 1 1 1 cos − cos 3 + cos 5 − cos 7 + ⋅⋅⋅⎟ ⎥ ⎢i = 3 5 7 ⎝⎜ ⎠⎦ ⎣

405 Two-Port Network I1

40

2

1

Ques t each ions are g i c prepa hapter to ven at th ee re the h topic elp the nd of reade . rs

40

24. A two-port network has

I2

V1

20

1

V2 2

Fig. 6.147

(i) at Port 1, driving point impedances of 60 and 50 with Port 2 open circuited and short circuited respectively. (ii) at Port 2, driving point impedances of 80 and 70 with Port 1 open circuited and short circuited respectively.

⎤ ⎡ z 11 = z 22 = 60 ; z 12 = z 21 = 20 ; A = D = 3 ; ⎥ ⎢ ⎢⎣ B = 160 ; C = 0.05 mho; Z 0 = 56.57 ; = 1.762 ⎥⎦

Find the image parameters of the network. Derive the expressions used. [54.77 , 74.83 ]

1. (a) Consider a linear passive two-port network and explain what are meant by i) open-circuit impedance parameters, and ii) short-circuit admittance parameters.

4. What are transmission parameters? Where are they most effectively used? Establish, for two-port networks, the relationship between the transmission parameters and the open-circuit impedance parameters.

(b) What are the open-circuit impedance parameters of a two-port network? How can the transmission parameters be obtained from open-circuit impedance parameters?

5. (a) Two two-port networks are connected in parallel. Prove that the overall y-parameters are the sum of corresponding individual y-parameters.

Questions

(c) Establish for two-port networks, the relationship between the transmission parameters and the open-circuit parameters.

(b) Two two-port networks are connected in cascade. Prove that the overall transmission parameter matrix is the product of individual transmission parameter matrices.

(d) Define z and y parameters of a typical four-terminal network. Determine the relationship between the z and y parameters.

(c) Two two-port networks are connected in series. Prove that the overall z-parameters are the sum of corresponding individual z-parameters.

(e) Express h-parameters in terms of z-parameters for a two-port network.

6. (a) Define ‘transfer function’ and ‘driving point function’ of a two-port network.

(f ) Derive expressions for the y-parameters in terms of ABCD parameters of a two-port network.

(b) Derive the expression of input impedance of a two-port network terminated with a load-impedance ZL, in terms of its -parameters.

2. (a) What do you understand by a reciprocal network? What is a symmetrical network? (b) Write technical note on derivation of short-circuit admittance parameter y12 of a symmetrical and reciprocal two-port lattice network. (c) How will you find the ␲-equivalent of a given network when its y-parameters are known? 3. (a) Explain what are meant by the transmission (ABCD) parameters of a two-port network. Derive the conditions necessary to be satisfied for the network to be i) reciprocal, and ii) symmetrical. Or, Prove that for a reciprocal two-port network, T (AD BC ) 1 (b) Prove that for a symmetrical two-port network, h (h11h22 h12h21) 1

(c) Derive the expression of output impedance of a two-port network terminated with a load-impedance ZL, in terms of its transmission parameters. 7. What is a gyrator? Mention some properties of an ideal gyrator. Show that a gyrator is a non-reciprocal device. 8. What is negative impedance converter (NIC)? Show that an NIC is a non-reciprocal device. 9. What are image parameters? Derive expression of image parameters in terms of (i) ABCD parameters (ii) open-circuit and short-circuit impedances. 10. What is a symmetrical network? Derive expressions for characteristic impedance and propagation constant of a symmetrical networks in terms of short-circuit and open-circuit impedances.

225 Network Theorems

Prove that the load impedance which absorbs the maximum power from a source is the conjugate of the impedance of the source.

14. Derive the condition for maximum power transfer for (a) Load impedance with variable resistance and variable reactance

11. Prove the condition for maximum power transfer for an ac circuit.

(b) Load impedance with variable resistance and fixed reactance

12. A source with internal impedance RS jXS delivers power to a variable load impedance RL j0. Show that the condition for maximum power in the load is

15. State and clearly prove with the help of a suitable example the maximum power transfer theorem as applicable to RLC circuits excited from the sinusoidal energy source. Hence explain clearly the concept and its significance in impedance matching.

RL 2 = RS 2 + X S 2 . 13. State the maximum power transfer theorem and verify that only 50% of the total power supplied by the source can be transferred to the load.

16. State and prove the following theorems: ( i) Tellegen’s theorem

Or,

(ii) Millman’ theorems

State and explain the maximum power transfer theorem. Derive the expression for efficiency for maximum power transfer.

(iii) Compensation theorem

Multiple-Choice Questions 1. Which one of the following theorems is a manifestation of the law of conservation of energy? (i) Tellegen’s Theorem (ii) Reciprocity Theorem (iii) Thevenin’s Theorem (iv) Norton’s Theorem 2. Tellegen’s theorem is applicable to (i) circuits having passive elements (ii) circuits having time-invariant elements only (iii) circuits with linear elements only (iv) circuits with active or passive, linear or non-linear and time-invariant or time-varying elements

7.

8.

9.

3. In any lumped network with elements in b branches, b

∑

k =1

k

(t ) ⋅ i k (t ) = 0 , for all t, holds good according to

(i) Norton’s theorem (ii) Thevenin’s theorem (iii) Millman’s theorem (iv) Tellegen’s theorem 4. Millman’s theorem yields (i) equivalent voltage source (ii) equivalent voltage or current source (iii) equivalent resistance (iv) equivalent impedance 5. The superposition theorem is applicable to (i) current only (ii) voltage only (iii) both current and voltage (iv) current, voltage and power 6. Superposition theorem is not applicable for

10.

11.

(i) voltage calculations (ii) bilateral elements (iii) power calculations (iv) passive elements Thevenin’s theorem can be applied to calculate the current in (i) any load (ii) a passive load only (iii) a linear load only (iv) a bilateral load only Norton’s equivalent circuit consists of a (i) voltage source in parallel with impedance (ii) voltage source in series with impedance (iii) current source in parallel with impedance (iv) current source in series with impedance The superposition theorem is applicable to (i) linear responses only (ii) linear and non-linear responses (iii) linear, non-linear and time-variant responses When a source is delivering maximum power to a load, the efficiency of the circuit (i) is always 50% (ii) depends on the circuit parameters (iii) is always 75% (iv) none of these. Maximum power transfer occurs at a (i) 100% efficiency (ii) 50% efficiency (iii) 25% efficiency (iv) 75% efficiency

12. Which of the following statements is true? (i) A Norton’s equivalent is a series circuit. (ii) A Thevenin’s equivalent circuit is a parallel circuit. (iii) R-L circuit is a dual pair. (iv) L-C circuit is a dual pair.

ns estio re u q a e hoic ions le c e quest d help p i t l n s u The ons a ject r, m apte ovided. minati the sub h c exa ach n of e pr of e wers ar petitive hensio d n e ans com he e mpr At t Q) with ifferent lear co c (MC from d have a n take eader to r the atter. m

1

Introduction to Different Types of Systems

Introduction An electrical network is one of the many important physical systems. In order to understand the basic characteristics of an electric network, we must first know the different concepts of systems. In this chapter, different types of systems have been discussed.

1.1

CONCEPTS OF SIGNALS AND SYSTEMS

1.1.1 Signal A signal is defined as a function of one or more variables, which provides information on the nature of a physical phenomenon. When the function depends on a single variable, the signal is said to be one-dimensional. Example A speech signal whose amplitude varies with time, depending on the spoken word and who speaks it. When the function depends on two or more variables, the signal is said to be multidimensional. Example An image (2-D signal).

1.1.2 Systems A system is an entity that takes an input signal and produces an output signal. It is a combination and interconnection of several components to perform a desired task. Input signals x1(t ) x2(t)

Output signals y1(t ) System

xn(t)

Fig. 1.1

y2(t ) yn(t )

Block-diagram representation of a system

2 Network Analysis and Synthesis

The system responds to one or more input quantities, called input signals or excitation, to produce one or more output quantities, called output signals or response.

1.2

DIFFERENT TYPES OF SIGNALS

Signals can be classified into different categories, as given below. 1. Continuous-time and discrete-time signals 2. Periodic and non-periodic signals 3. Odd and even signals

1.2.1 Continuous-Time and Discrete-Time Signals

x(t )

Signals are represented mathematically as functions of one or more independent variables. We classify signals as being either continuous-time (functions of a real-valued variable) or discrete-time (functions of an integer-valued variable). In other words, a continuous-time signal has a value defined for each point in time and a discrete-time signal is defined only at discrete points in time. To signify the difference, we (usually) use round parenthesis around the argument for continuous time signals, e.g., x(t) and square brackets for discrete-time signals, e.g., x[n]. We will also use the notation xn for discretetime signals. The sequences of values of the discrete-time signal shown in Fig. 1.2 (b) defined at discrete points in time are called samples, and the spacing between them is called the sample spacing. For equal sample spacing, the sequences of values are expressed as a function of the signed integer n as x[n], where n X [ 4] is termed as a sequence of samples or sequence, in short.

time (t)

Fig. 1.2(a) Continuous-time signal X [n] X [3] X [ 2]

1

4

A signal f (t) is said to be periodic if f (t ) = f (t ± nT )

X [ 3]

(1.1)

X [2]

X [0] 3

1.2.2 Periodic and Non-Periodic Signals

X [1]

2 0 X [ 1]

1

2

3

time [n]

Fig. 1.2(b) Discrete-time signal

where n is a positive integer and ‘T’ is the period. Thus, a periodic signal repeats itself every T seconds. Some periodic signals are shown in Fig. 1.3. v(t ) V

2T (a)

Fig. 1.3 Periodic signals

T

0

T (b)

2T

3T

4T

t

3 Introduction to Different Types of Systems

A signal not satisfying the above condition of Eq. (1.1) is called a non-periodic signal. Examples of some non-periodic signals are et, t, etc.

1.2.3 Odd and Even Signals A signal f (t) is said to be odd if

()

( )

f t = − f −t

(1.2)

Some examples of odd signals are sine functions, triangular functions and square function, as shown in Fig. 1.4. v(t) f (t )

V T/2 0 T/4 −V

− T/2

−T

Fig. 1.4

t

T

0

ωt

Odd signals

A signal f (t) is said to be even if

() ( )

f t = f −t

(1.3)

Some examples of even signals are shown in Fig. 1.5. f(t ) f (t) V

− T/2

0

T/2

t

0

ωt

−V Fig. 1.5 Even signals

Decomposition of a signal into odd and even components For any function f(t), let the odd component be denoted by f0(t) and the even component by fe(t), so that,

() () () ∴ f ( −t ) = f ( −t ) + f ( −t ) = − f ( t ) + f ( t ) f t = f0 t + f e t

0

e

0

(1.4) (1.5)

e

[by Eq. (1.2) and (1.3)] By addition and subtraction of Eqs (1.4) and (1.5), we get, 1 f e t = ⎡⎣ f t + f −t ⎤⎦ 2

()

() ( )

(1.6)

()

() ( )

(1.7)

1 f 0 t = ⎡⎣ f t − f −t ⎤⎦ 2 By these two equations, we can decompose a signal into its odd and even components.

4 Network Analysis and Synthesis

Example 1.1 Decompose the following signal into its odd and even components. Solution To find the even and odd components we need the folded signal, i.e. f ( t), as shown in Fig. 1.6 (b). By point-by-point addition and subtraction, we get the even and odd components as shown in Fig. 1.6 (c) and Fig. 1.6 (d). f0(t ) f (t ) 1

1/2

fe(t )

f( t ) 1

1

1/2

1

0 0

1

Fig. 1.6 (a) Signal of Ex .1.1

1/2

−1

t

t

0

t

1

0

1

t

Fig. 1.6 (c) Even component of signal of Fig. 1.6 (a)

Fig. 1.6 (b) Folded signal of Fig. 1.6 (a)

1.2.4 Some Standard Signals

Fig. 1.6 (d) Odd component of signal of Fig 1.6 (a)

f (t)

There are some standard signals which can be generated easily in the laboratory. Some of these standard signals are discussed below. Sinusoidal Signal A sinusoid is a signal that has the form of a sine or cosine function.

t

()

We consider a sinusoidal voltage, v t = Vm sin t where Vm is the amplitude, t is the argument of the sinusoid, is the angular frequency of the sinusoid in rad/s 2 f and T is the time period of the sinusoid. As the sinusoid is periodic, it repeats itself; such that ⎛ 2 ⎞ v t = v t + T = Vm sin ⎜ t + ⎟⎠ = Vm sin t + = Vm sin t ⎝

() (

)

(

()

A shifted sinusoid can be written as, v t = Vm sin where is the phase of the sinusoid. Thus, we see that,

( t = cos( t = sin ( t = cos(

− sin t = sin − cos ± cos sin

) t ± 180 ) t ± 90 ) t ± 90 ) t ± 180

Fig. 1.7(a) sin t

2 , T f (t )

)

(

t+

)

t

Fig. 1.7 (b) cos t

±

Exponential Signal An exponential signal is a function of time defined as

()

f t = 0,

t 0 = 0 for t < 0 and is undefined at t 0. A step function of magnitude K is defined as

u(t − T ) 1

f (t ) = Ku(t ) = K for t > 0 = 0 for t < 0 and is undefined at t 0. A shifted or delayed unit step function is defined as

0

Ramp Signal A unit ramp function is defined as

T

t

Fig. 1.9 (c) Shifted unit step function

f (t ) = u(t − T ) = 1 for t > T = 0 for t < T and is undefined at t T. Another function, called gate function, can be obtained from step function as follows. Therefore, g(t) Ku(t − a) Ku(t − b)

t

Fig. 1.9 (b) Step function of magnitude K

K

0

a

b

Fig. 1.9 (d) Gate function

f (t ) = r (t ) = t for t ≥ 0 = 0 for t < 0 A ramp function of any slope K is defined as f (t ) = Kr (t ) = Kt for t ≥ 0 = 0 for t < 0 A shifted unit ramp function is defined as f (t ) = r (t − T ) = t for t ≥ T = 0 for t < T Impulse Signal This function is also known as Dirac Delta function, denoted by d(t). This is a function of a real variable t, such that the function is zero everywhere except at the instant t 0. Physically, it is a very sharp pulse of infinitesimally small width and very large magnitude, the area under the curve being unity.

6 Network Analysis and Synthesis Kr(t)

r(t)

r(t − T ) K

1

1

1

1 t

0

Fig. 1.10 (a) Unit ramp function

0

∞

∫

t

Fig. 1.10 (b) Ramp function

Consider a gate function as shown in Fig. 1.11. The function is compressed along the time-axis and stretched along the y-axis, keeping the area under 1 the pulse as unity. As a 0, the value of a and the resulting function is known as impulse. It is defined as (t ) = 0 for t ≠ 0 and

1

(t )dt = 1

−∞

0

T

t

Fig. 1.10 (c) Shifted unit ramp function

f (t ) δ (t )

3/a

∞

2/a 1/a 0

a/3

a/2

a

Fig. 1.11(a) Generation of impulse function from gate function

t

0

t

Fig. 1.11(b) Impulse signal

1 d Also, (t ) = Lim ⎡⎣ u(t ) − u(t − a ) ⎤⎦ = ⎡⎣ u(t ) ⎤⎦ dt a ⎯⎯ →0 a

1.3

DIFFERENT TYPES OF SYSTEMS

Systems can be classified from different points of view as given below. 1. 2. 3. 4. 5. 6. 7. 8. 9.

Continuous and discrete-time systems Fixed and time-varying systems Linear and non-linear systems Lumped and distributed systems Instantaneous and dynamic systems Active and passive systems Causal and non-causal systems Stable and unstable systems Invertible and non-invertible systems

1.3.1 Continuous- and Discrete-Time Systems A continuous-time system is a system which accepts only continuous-time signals to produce continuoustime internal and output signals. On the other hand, a discrete-time system is a system that transforms discrete-time input(s) into discrete-time output(s). Examples Continuous-Time Systems (i) Atmospheric pressure as a function of altitude (ii) Electric circuits composed of resistors, inductors, capacitors driven by continuous-time sources

7 Introduction to Different Types of Systems

Discrete-Time Systems (i) Weekly stock market index (ii) Balance in a bank account from month to month

1.3.2 Time-Invariant (Fixed) and Time-Varying Systems A system is time-invariant or fixed if the behavInput x(t) Input x(t T) iour and characteristics of the system do not change with time. Otherwise, the system is time-varying. time time 0 0 T Mathematically, if the input x(t) gives the output Output y(t) Output y(t T ) y(t) then the system is time-invariant if the input x(t − T ) gives the output y(t − T ) for any delay T. Hence, a time-shift of the input gives the same time time T 0 0 time-shift of the output. Fig. 1.12 Time-invariant system Whether a system is time-invariant or timevarying can be seen in the differential equation (or difference equation) describing it. Time-invariant systems are modeled with constant-coefficient equations. A constant-coefficient differential (or difference) equation means that the parameters of the system are not changing over time and an input now will give the same result as the input later. Example 1.2 A continuous-time system is modeled by the equation y(t) ⴝ tx(t) ⴙ 4, and a discrete-time system is modeled by y(n) ⴝ x 2[n]. Are these systems time-invariant? Solution For continuous-time system For input x(t) x1(t), output y1(t) tx1(t) 4 For input x(t) x1(t − T ), output, y2(t) tx1(t − T )

(i) (ii)

4

From the condition of time-invariance, the output should be y1(t − T ) (t − T) x1 (t − T) From equations (ii) and (iii), y2(t) y1(t − T) Hence, the system is not time-invariant.

4

For discrete-time system For input x1[n ], output y1[n] x12[n] For input x1[n − n0], output x12[n − n0] From the condition of time-invariance, the shifted output y1[n − n0] Hence, the system is time-invariant.

(iii)

x12[n − n0]

1.3.3 Linear and Non-Linear Systems A system, in continuous-time or discrete-time, is said to be linear, if it obeys the properties of superposition, i.e. additivity and homogeneity (or scaling); while a system is non-linear that does not obey at least any one of these properties.

8 Network Analysis and Synthesis

The superposition principle says that the output to a linear combination of input signals is the same linear combination of the corresponding output signals. Mathematically, the linearity condition is based on two properties: Additivity If the input signals x1(t) and x2(t) correspond to the output signals y1(t) and y2(t), respectively then the input signal {x1(t) x2(t)} should correspond to the output signal {y1(t) y2(t)}. Homogeneity If the input signal x1(t) corresponds to the output signal y1(t), then the input signal a1x1(t) should correspond to the output signal a1y1(t) for any constants a1. Combining these two properties, the condition for a linear system can be written as, if the input signals x1(t) and x2(t) correspond to the output signals y1(t) and y2(t), respectively then the input signal a1x1(t) a2x2(t) should correspond to the output signal a1y1(t) a2y2(t) for any constants a1 and a2. Example 1.3 Check whether the systems with the input–output relationship given below are linear: (a) y(t) ⴝ mx(t) ⴙ c, (b) y(t) ⴝ tx(t) Solution (a) For an input x1(t), output, y1(t) mx1(t) c For an input x2(t), output, y2(t) mx2(t) c For an input {x1(t) x2(t)}, output, y3(t) m{x1(t) x2(t)} c From the condition of linearity, the output should be { y1(t) y2(t)} m{x1(t) x2(t)} 2c From equations (i) and (ii), we conclude that the system is non-linear. (b) For an input x1(t), output, y1(t) tx1(t) For an input x2(t), output, y2(t) tx2(t) For an input {k1x1(t) k2x2(t)}, output, y3(t) t{k1x1(t) k2x2(t)} where k1 and k2 are any arbitrary constants. From the condition of linearity, the output should be {k1y1(t) k2y2(t)} k1tx1(t) k2tx2(t) t{k1x1(t) From equations (i) and (ii), we conclude that the system is linear.

(i) (ii)

(i)

k2x2(t)}

(ii)

1.3.4 Lumped and Distributed Systems All physical systems contain distributed parameters because of the physical size of the system components. For example, the resistance of a resistor is distributed throughout its volume. However, if the size of the system components is very small with respect to the wavelength of the highest frequency present in the signals associated with it then the system components behave as if it all were occurring at a point. This system is said to be a lumped-parameter system. Distributed parameter systems are modeled • •

by partial differential equations if they are continuous-time systems, and by partial difference equations if they are discrete-time systems.

Lumped parameter systems are modeled with ordinary differential or difference equations.

9 Introduction to Different Types of Systems

Example Consider an electric power system of frequency 50 Hz. The wavelength of the signal is obtained as, C 3 × 105 n =C ⇒ = = km = 6000 km n 50 Thus, the electrical system inside a room can be treated as a lumped-parameter system, but will be treated as distributed system for long-distance transmission lines.

1.3.5 Instantaneous (Static or Memoryless) and Dynamic Systems An instantaneous or static or memoryless system is a system where the output at any specific time depends on the input at that time only. On the other hand, a dynamic system is one whose output depends on the past or future values of the input in addition to the present time. A static system has no memory. Physically, it contains no energy-storage elements; while a dynamic system has one or more energy-storage element(s). Example An electrical circuit containing resistance R, has the v i relationship as, v(t)

Ri(t), and so the t 1 system is static. But an electrical circuit containing the capacitor C has the v i relationship as v (t ) = ∫ i(t )dt , C0 and so, the system is a dynamic system.

1.3.6 Active and Passive Systems A system having no source of energy is known as a passive system. Examples of passive systems are electric circuits containing resistance, capacitance, inductance, diodes, etc. A system having a source of energy together with other passive elements is known as an active system. Examples of active systems are electric circuits containing voltage sources or current sources or op-amps.

1.3.7 Causal and Non-Causal Systems A system is said to be causal if the output of the system depends only on the input at the present time and/or in the past, but not the future value of the input. Thus, a causal system is non-anticipative, i.e. output cannot come before the input. On the other hand, the output of a non-causal system depends on the future values of the input.

Input x(t)

0

time

Output y (t)

Example The moving-average system described by 1 y[ n] = {x[ n] + x[ n − 1] + x[ n − 2 ]} 3 is causal; but the moving-average system described by 1 y[ n] = {x[ n + 1] + x[ n] + x[ n − 1]} 3 is non-causal since the output depends on the future value of the input x[n 1]. It is obvious that the idea of future inputs does not have any physical meaning if we take time as our independent variable and for that reason all

0

time

Output y (t)

0

time

Fig. 1.13 (a) Causal systems

10 Network Analysis and Synthesis

real-time systems are causal. However, for the case of image processing, the dependent variable may by the pixels to the left and right (the ‘future’) of the current position on the image, and thus, we can have a non-causal system.

1.3.8 Stable and Unstable Systems A stable system is one where the output does not diverge as long as the input does not diverge. A bounded input produces a bounded output. For this reason, this type of system is known as a bounded input–bounded output (BIBO) stable system. Mathematically, a stable system must have the following property: If x(t) be the input and y(t) be the output then the output must satisfy the condition y (t ) ≤ M y < ∝;

Input x(t )

Output y (t)

for all t

whenever the input satisfies the condition x (t ) ≤ M x < ∝;

time

0

time

0

for all t

Fig. 1.13 (b) Non-causal system

where Mx and My both represent a set of finite positive numbers. If these conditions are not met, i.e. the output of the system grows without limit (diverges) from a bounded input then the system is unstable.

1.3.9 Invertible and Non-Invertible Systems A system is referred as an invertible system if

x(t )

y(t ) System

(i) distinct inputs lead to distinct outputs, and (ii) the input can be recovered from the output.

w (t) = x (t) Inverse system

Fig. 1.14 Invertible system

The property of invertibility is important in the design of communication systems. When a transmitted signal propagates through a communication channel, it becomes distorted due to the physical characteristics of the channel. An equalizer is connected in cascade with the channel in the receiver to compensate this distortion. By designing the equalizer to be inverse of the channel, the transmitted signal is restored.

1.4

INTERCONNECTION OF SYSTEMS

Most of the physical systems are built as interconnections of several subsystems. Different types of interconnections are shown below. Series or Cascade Interconnection The output of the system 1 is the input to the system 2. Parallel Interconnection The same input signal is applied to systems 1 and 2.

Input

System 1

Fig. 1.15

System 1 Input

+ System 2

Fig. 1.16

Output

System 2

Output

11 Introduction to Different Types of Systems

Combination of Both Cascade and Parallel Interconnections System 1

System 2

Input

System 4

Output

System 3

Fig. 1.17

Feedback Interconnection The output of the system 2 is fed back and added to the external input to produce the actual input to the system 1. Input

System 1

Output

System 2

Fig. 1.18

Solved Problems Problem 1.1 Check whether the system defined by y(t) ⴝ sin[x( t)] is time-invariant. Solution For input x(t) x1(t), output y1(t) sin[x1(t)] For input x(t) x1(t − T), output, y 2 (t) sin[x1(t − T )] From the condition of time-invariance, the output should be y1(t − T ) sin[x1(t − T )] From equations (ii) and (iii), y2(t) y1(t − T ) Hence, the system is time-invariant.

(i) (ii) (iii)

Problem 1.2 Consider a system S with input x[n] and output y[n] related by, y [n] ⴝ x[n]{g[n] ⴙ g[n − 1]} (a) If g[n] 1, for all n, show that S is time-invariant. (b) If g[n] n, show that S is not time-invariant. (c) If g[n] 1 (−1)n, show that S is time-invariant. Solution (a) If g[n] 1, for all n then y[n] x[n]{1 1 1} 2x[n] For input x[n] x1[n], output y1[n] 2x1[n] For input x[n] x1[n − n0], output, y2[n] 2x1[n − n0] From the condition of time-invariance, the output should be y1[n − n0] 2x1[n − n0] From equations (ii) and (iii), y2[n] y1[n − n0] Hence, the system is time-invariant.

(i) (ii) (iii)

12 Network Analysis and Synthesis

(b) If g[n] n, then y[n] x[n]{n n − 1} (2n − 1) x[n] For input x[n] x1[n], output y1[n] (2n − 1)x1[n] For input x[n] x1[n − n0], output, y2[n] (2n − 1)x1[n − n0] From the condition of time-invariance, the output should be y1[n − n0] {2(n − n0) − 1}x1[n − n0] From equations (ii) and (iii), y2[n] y1[n − n0] Hence, the system is not time-invariant.

(i) (ii) (iii)

(c) If g[n] 1 (−1)n, then y[n] x[n]{1 (−1)n 1 (−1)n−1} 2x[n] This relation is same as that of Part (a). Hence the system is time-invariant. Problem 1.3 Consider the systems S whose input and output are related by y(t) ⴝ x 2(t) Check whether S is linear. Solution For an input x1(t), output, y1(t) x12(t) For an input x2(t), output, y2(t) x22(t) For an input {k1x1(t) k2x2(t)}, output, y3(t) [k1x1(t) k2 x2(t)]2 where, k1 and k2 are any arbitrary constants. From the condition of linearity, the output should be {k1y1(t) k2y2(t)} k1x12(t) k2x22(t) From equations (i) and (ii), we conclude that the system is not linear.

(i)

(ii)

Problem 1.4 Consider the following discrete-time systems with input-output relationships as given, y[n] ⴝ Re{x[n]} Check whether the system is linear.

Solution Let, the input be, x1[n] r[n] js[n] Therefore, the output is, y1[n] Re{x1[n]} Re {r[n] js[n]} r[n] Now we consider scaling of the input x1[n] by a complex number, say, (a x2[n]

(a

jb)x1[n]

Corresponding output is, y2[n]

(a

jb){r[n]

Re{x2[n]}

js[n]}

jb) , i.e. the input is,

{ar[n] − bs[n]}

Re{ar[n] − bs[n]}

But the scaled output for linear system is, (a jb)y1[n] ar[n] As the two outputs are not the same, the system is not linear.

j{br[n]

j{br[n] as[n]} as[n]}

ar[n] − bs[n]

jbr[n]

Problem 1.5 Consider a discrete-time system whose output y[n] is the average of the three most recent values of the input signal, x[n], given as 1 y [n] = x [n] + x [n −1] + x [n − 2 ] 2 Show that the system is BIBO stable.

{

Solution Let us assume that, x[n] < Mx
0 1 for t > 0 and is undefined at t 0. u(t)

Ku(t )

1

K

f (t) T/2

V T

T/2

0 T/4 V

Fig. 1.32 Odd functions

T

t 0

ωt

0

t

Fig. 1.33 (a) Unit step function

0

t

Fig. 1.33 (b) Step function of magnitude K

19 Introduction to Different Types of Systems

A step function of magnitude K is defined as f (t)

Ku(t)

and is undefined at t

f (t )

K for t > 0 1 for t > 0

3/a

0.

(t )

2/a 1/a

K

0 0

a

a/3

a/2

a

0

t

Fig. 1.34 (b) Impulse Signal

Fig. 1.34 (a) Generation of impulse function from gate function

b

Fig. 1.33 (c) Gate function

t

It is defined as, ∞

(b) Gate function A gate function can be obtained from a step function as shown in Fig. 1.33 (c). Therefore, g(t)

Ku(t−a)−Ku(t−b).

r(t)

t for t 0 for t

Kr(t)

Kt for t 0 0 for t < 0

K

1 1

1 0

t

Fig. 1.33 (d) Unit ramp function

0

0 and

∫

(t )dt = 1

−∞

Also, (t ) = Lim 1 [ u(t ) − u(t − a )] = d [ u(t )] dt a ⎯→ ⎯oa

The superposition principle says that the output to a linear combination of input signals is the same linear combination of the corresponding output signals. Mathematically, the linearity condition is based on two properties:

Kr (t)

r (t )

for t

A system in continuous-time or discrete-time, is said to be linear if it obeys the properties of superposition, i.e. additivity and homogeneity (or scaling); while a system is non-linear that does not obey at least any one of these properties.

0 0

A ramp function of any slope K is defined as f(t)

0

4. What are the conditions for a system to be a linear system?

(c) Ramp function A unit ramp function is defined as f(t)

(t)

t

Fig. 1.33 (e) Ramp funtion

(d) Impulse function This function is also known as Dirac Delta function, denoted by (t). This is a function of a real variable t, such that the function is zero everywhere except at the instant t 0. Physically, it is a very sharp pulse of infinitesimally small width and very large magnitude, the area under the curve being unity. Consider a gate function as shown in Fig. 1.34 (a). The function is compressed along the timeaxis and stretched along the y-axis, keeping 0, the area under the pulse unity. As a value of [1/a] and the resulting function is known as impulse.

1. Additivity If the input signals x1(t) and x2(t) correspond to the output signals y1(t) and y2(t), respectively then the input signal {x1(t) x2(t)} should correspond to the output signal {y1(t) y2(t)}. 2. Homogeneity If the input signal x1(t) corresponds to the output signal y1(t) then the input signal a1x1(t) should correspond to the output signal a1y1(t) for any constants a1. Combining these two properties, the condition for a linear system can be written as if the input signals x1(t) and x2(t) correspond to the output signals y1(t) and y2(t), respectively then the input signal a1x1(t) a2x2(t) should correspond to the output signal a1y1(t) a2y2(t) for any constants a1 and a2. 5. Give the conditions for a BIBO stability of a system. A stable system is one where the output does not diverge as long as the input does not diverge. A bounded input produces a bounded output. For this

20 Network Analysis and Synthesis

reason, this type of system is known as bounded inputbounded output (BIBO) stable system. Mathematically, a stable system must have the following property: If x(t) be the input and y(t) be the output then the output must satisfy the condition y (t ) ≤ M y