2nd Edition Networks and Systems D ROY CHOUDHURY ACANDEEWMIC New Academic Science SCIENCE (cid:78)(cid:69)(cid:84)(cid:87)(cid:79)(cid:82)(cid:75)(cid:83)(cid:32) (cid:65)(cid:78)(cid:68) (cid:83)(cid:89)(cid:83)(cid:84)(cid:69)(cid:77)(cid:83) This page intentionally left blank Networks and Systems Second Edition D ROY CHOUDHURY Professor and Head Electronics and Computer Engineering Department Delhi College of Engineering Delhi, India NEW ACADEMIC SCIENCE New Academic Science Limited The Control Centre, 11 A Little Mount Sion Tunbridge Wells, Kent TN1 1YS, UK www.newacademicscience.co.uk • e-mail: [email protected] Copyright © 2014 by New Academic Science Limited 27 Old Gloucester Street, London, WC1N 3AX, UK www.newacademicscience.co.uk • e-mail: [email protected] ISBN : 978 1 781830 67 3 All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the copyright owner. British Library Cataloguing in Publication Data A Catalogue record for this book is available from the British Library Every effort has been made to make the book error free. However, the author and publisher have no warranty of any kind, expressed or implied, with regard to the documentation contained in this book. Preface In the second edition, I have tried to keep the organization of the first edition with some reorientation. Each chapter opens with historical profile of some electrical engineering pioneers to be mentioned in the chapter or a career discussion on a sub-discipline of electrical engineering. The use of PSPICE is encouraged in a student friendly manner. The chapter ends with a summary (Resume) of the key points and formulas. Thoroughly worked examples are liberally given at every section. Most of the new material are additions aimed at strengthening the weaknesses of the original edition. Some specific changes deserve mention. The most important of these is new chapters on Signals and Systems (Chapter 2), Active Filter Fundamentals (Chapter 14) and Discrete Systems (Chapter 20). The book ends with appendices, one is with the concept of ‘Algebra of Complex Numbers (Phasors) and Matrix Algebra’, the other with Multiple-choice Review Questions and Answers and third is the answers to selected problems. I wish to record my thanks to Niel Roy os M/s Navister, Chicago, USA for his contribution in the second chapter. D. Roy Choudhury This page intentionally left blank Contents Preface...........................................................................................................................................v 1. BASIC CIRCUIT ELEMENTS AND WAVEFORMS 1–48 1.1 Introduction................................................................................................................1 1.2 Circuit Components....................................................................................................3 1.3 Assumptions for Circuit Analysis ............................................................................13 1.4 Definitions ................................................................................................................15 1.5 Conservation of Energy ............................................................................................16 1.6 Source of Electrical Energy......................................................................................21 1.7 Standard Input Signals............................................................................................23 1.8 Sinusoidal Signal......................................................................................................28 1.9 Kirchhoff’s Laws.......................................................................................................34 2. SIGNALS AND SYSTEMS 49–96 2.1 Signals......................................................................................................................49 Conjugate Symmetric...............................................................................................50 The Continuous – Time Unit Step and Unit Impulse Functions............................50 2.1.1 The Discrete – Time Unit Impulse and Unit Step Sequences......................51 Even and Odd Signals....................................................................................52 Periodic Signal...............................................................................................54 Time Scaling..................................................................................................55 Reflection .......................................................................................................57 Time Shifting.................................................................................................59 Types of Sequences ........................................................................................62 Impulse Response ..........................................................................................65 Discrete Convolution......................................................................................66 viii CONTENTS 2.2 New Approach for Solving Problems........................................................................71 2.3 System Properties ....................................................................................................84 System with and without Memory...........................................................................84 Invertibility and Inverse Systems............................................................................84 Causality ..................................................................................................................85 Linearity...................................................................................................................86 3. MESH AND NODE ANALYSIS 97–168 3.1 Introduction..............................................................................................................97 3.2 Kirchhoff’s Laws......................................................................................................97 3.3 Source Transformation...........................................................................................105 3.4 General Network Transformations........................................................................109 3.5 Mesh and Node Analysis........................................................................................119 3.6 Network Equations for RLC Network....................................................................129 3.7 Magnetic Coupling..................................................................................................149 4. FOURIER SERIES 169–224 4.1 Introduction............................................................................................................169 4.2 Trigonometric Fourier Series.................................................................................171 4.3 Evaluation of Fourier Coefficients..........................................................................171 4.4 Waveform Symmetry .............................................................................................179 4.5 Fourier Series in Optimal Sense............................................................................189 4.6 Exponential Form of Fourier Series.......................................................................190 4.7 Effective Value........................................................................................................199 4.8 Fourier Transform .................................................................................................201 4.9 Effective Value of a Non-sinusoidal Wave..............................................................216 5. THE LAPLACE TRANSFORM 225–254 5.1 Introduction............................................................................................................225 5.2 Laplace Transformation.........................................................................................227 5.3 Some Basic Theorems ............................................................................................228 5.4 Gate Function.........................................................................................................239 5.5 Impulse Function...................................................................................................243 5.6 Laplace Transform of Periodic Functions..............................................................244 6. APPLICATION OF LAPLACE TRANSFORM 255–331 6.1 Introduction............................................................................................................255 6.2 Solution of Linear Differential Equation................................................................255 6.3 Heaviside’s Partial Fraction Expansion.................................................................256 6.4 Kirchhoff’s Laws.....................................................................................................262 6.5 Solution of Network Problems ...............................................................................269 CONTENTS ix 6.6 Convolution Integral...............................................................................................294 6.7 Convolution Theorem .............................................................................................295 6.8 Evaluation of the Convolution Integral..................................................................296 6.9 Inverse Transformation by Convolution ................................................................297 6.10 Impulse Response...................................................................................................302 6.11 Graphical Convolution............................................................................................304 7. ANALOGOUS SYSTEM 332–372 7.1 Introduction............................................................................................................332 7.2 Mechanical Elements .............................................................................................333 7.3 D’Alembert’s Principle ...........................................................................................335 7.4 Force-voltage Analogy ............................................................................................336 7.5 Force-current Analogy............................................................................................339 7.6 Mechanical Couplings ............................................................................................340 7.7 Electromechanical System .....................................................................................361 7.8 Liquid Level System...............................................................................................368 8. GRAPH THEORY AND NETWORK EQUATION 373–438 8.1 Introduction............................................................................................................373 8.2 Graph of a Network................................................................................................373 8.3 Trees, Cotrees and Loops........................................................................................376 8.4 Number of Possible Trees of a Graph ....................................................................377 8.5 Incidence Matrix.....................................................................................................378 8.6 Cut-set Matrix........................................................................................................384 8.7 Tie-set Matrix and Loop Currents .........................................................................390 8.8 Inter-relationship Among Various Matrices..........................................................392 8.9 Analysis of Networks..............................................................................................394 8.10 Network Equilibrium Equation .............................................................................400 8.11 Duality....................................................................................................................426 9. NETWORK THEOREMS 439–496 9.1 Introduction............................................................................................................439 9.2 Superposition Theorem...........................................................................................439 9.3 Reciprocity Theorem...............................................................................................449 9.4 Thevenin’s Theorem...............................................................................................450 9.5 Norton’s Theorem...................................................................................................454 9.6 Millman’s Theorem ................................................................................................462 9.7 Maximum Power Transfer Theorem .....................................................................467 9.8 Substitution Theorem ............................................................................................471 9.9 Compensation Theorem..........................................................................................474 9.10 Tellegen’s Theorem.................................................................................................476