Electrical Circuit Analysis and Design Other Macmillan titles of re la ted interest W. A. Atherton, From Compass to Computer B. R. Bannister and D. G. Whitehead, Fundamentals of Modern Digital Systems J. C. Cluley, Transducers for Microprocessor Systems R. F. W. Coates, Modern Communication Systems, second edition Donard de Cogan, Solid State Devices - A Quantum Physics Approach C. W. Davidson, Transmission Lines for Communication, second edition M. E. Goodge, Analog Electronics M. E. Goodge, Semiconductor Device Technology B. A. Gregory, An Introduction to Electrical Instrumentation and Measurement Systems, second edition Paul A. Lynn, An Introduction to the Analysis and Processing of Signals, third edition Paul A. Lynn, Electronic Signals and Systems J. E. Parton, S. J. T. Owen and M. S. Raven, Applied Electromagnet ics, second edition P. Silvester, Electric Circuits Trevor J. Terrell, Introduction to Digital Filters, second edition M. J. Usher, Sensors and Transducers L. A. Warnes, El~ctronic Materials B. W. Williams, Power Electronics - Devices, Drivers, Applications and Passive Components G. Williams, An Introduction to Electrical Circuit Theory Macmillan New Electronics Series Series Editor: Paul A. Lynn Rodney F. W. Coates, Underwater Acoustic Systems W. Forsythe and R. M. Goodall, Digital Control Paul A. Lynn, Digital Signals, Processors and Noise Paul A. Lynn, Radar Systems A. F. Murray and H. M. Reekie, Integrated Circuit Design F. J. Owens, Signal Processing of Speech Dennis N. Pim, Television and Teletext M. J. N. Sibley, Optical Communications Martin S. Smith, Introduction to Antennas P. M. Taylor, Robotic Control G. S. Virk, Digital Computer Control Systems Alan Waters, Active Filter Design Electrical Circuit Analysis and Design N oel M. Morris formerly Principal Lecturer Staffordshire University M MACMILLAN © Noel M. Morris 1993 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1P 9HE. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. First published 1993 by THE MACMILLAN PRESS LTD Houndmills, Basingstoke, Hampshire RG21 2XS and London Companies and representatives throughout the world ISBN 978-0-333-55483-8 ISBN 978-1-349-22560-6 (eBook) DOI 10.1007/978-1-349-22560-6 A catalogue record for this book is available from the British Library Contents Preface xii Acknowledgements xv 1 Elements and Laws 1 1.1 Introduction 1 1.2 Electric current 1 1.3 Voltage, e.m.f. and p.d. 3 1.4 Power in d.c. circuits 4 1.5 Linear passive circuit elements 4 1.6 Resistance, conductance and Ohm's law 5 1.7 Power in a resistive circuit 6 1.8 Energy consumed in a resistive circuit 7 1.9 Independent and dependent supply sources 8 1.9.1 The operational amplifier 9 1.10 Kirchhoff's laws 11 1.11 The double-suffix voltage notation 14 1.12 Practical (non-ideal) sources 16 1.13 Transformation of practical sources 17 1.14 Resistance of aseries circuit 19 1.15 Voltage division in series-connected resistors 20 1.16 Resistance and conductance of a parallel circuit 21 1.17 Current division in a parallel circuit 22 Unworked problems 24 2 Circuit Analysis 28 2.1 Introduction 28 2.2 Definitions and terminology 28 2.3 Mesh analysis 31 v VI Contents 2.4 General mIes for writing mesh equations 34 2.5 Solution of three simultaneous equations 34 2.6 Worked examples using mesh analysis 37 2.7 Nodal analysis 41 2.8 General mIes for writing nodal equations 44 2.9 Worked examples using nodal analysis 44 2.10 Network topology 49 2.11 Loop analysis 52 2.12 Duality 54 Unworked problems 57 3 Circuit Theorems 61 3.1 Introduction 61 3.2 Linearity 61 3.3 Principle of superposition 61 3.4 Thevenin's theorem 63 3.5 Norton's theorem 65 3.6 Relationship between Thevenin's and Norton's circuits 67 3.7 Reciprocity theorem 67 3.8 The maximum power transfer theorem 69 3.9 The parallel-generator (Millman's) theorem 70 3.10 Rosen's theorem or the general star-mesh transformation 72 3.11 The star-delta, tee-wye or tee-pi transformation 73 3.12 The delta-star, wye-tee or pi-tee transformation 74 3.13 Summary of star-delta and delta-star transformations 74 Unworked problems 75 4 Energy Storage Elements 79 4.1 Introduction 79 4.2 Capacitors 79 4.3 Energy stored in a capacitor 81 4.4 Capacitors in parallel 82 4.5 Capacitors in series 82 4.6 Potential division in series-connected capacitors 83 4.7 Inductance 84 4.8 Energy stored in an inductor 87 4.9 Inductors in se ries 87 4.10 Inductors in parallel 88 4.11 Duality between inductors and capacitors 90 4.12 Relationship between inductance, reluctance and the number of turns on a coil 90 Contents vii 4.13 Mutual inductance 91 4.14 Direction of the mutually induced e.m.f. - the dot notation 92 4.15 Coefficient of coupling 93 4.16 Mutually coupled coils in se ries and in parallel 94 Unworked problems 96 5 The Sinewave, Phasors and Power 100 5.1 Introduction 100 5.2 Mean or average value of an alternating quantity 100 5.3 The effective value or r.m.s. value of a periodic wave 102 5.4 Phase angle 103 5.5 Phasors and phasor diagrams 105 5.6 Representation of a sinusoidal quantity as a complex value 106 5.7 Impedance of elements 107 5.7.1 Resistance 108 5.7.2 Pure inductance 108 5.7.3 Pure capacitance 109 5.7.4 CIVIL - an a.c. mnemonic 110 5.8 The susceptance of elements 111 5.9 The admittance of elements 111 5.10 The impedance of elements in se ries 112 5.11 The admittance and impedance of elements in parallel 113 5.12 Impedance and admittance of series-parallel circuits 115 5.13 Power and power factor 115 5.14 Power, VA and V Ar absorbed by ideal elements 117 5.15 v-i waveforms 118 5.16 Power consumed in an a.c. circuit 118 5.17 Complex power 121 Unworked problems 122 6 Sinusoidal Steady-state Analysis 126 6.1 Introduction 126 6.2 Nodal, mesh and loop analysis 130 6.3 Principle of superposition 132 6.4 Thevenin's theorem and Norton's theorem 133 6.5 Millman's theorem 135 6.6 Rosen's, star-delta and delta-star theorems 136 6.7 Maximum power transfer theorem 137 6.8 a.c. circuits with dependent sources 138 Unworked problems 140 viii Contents 7 Polyphase Circuits 144 7.1 Introduction 144 7.2 Three-phase generation 144 7.3 Star-connection or Y -connection 145 7.4 Phase sequence 146 7.5 Balanced and unbalanced systems 147 7.6 Phase and line voltages in a star-connected system 147 7.7 Delta-connected or mesh-connected three-phase source 150 7.8 Three-phase, four~wire, star-star system 150 7.8.1 Balanced supply, balanced load 151 7.8.2 Three-phase, four-wire, star-connected system with an unbalanced load 152 7.9 Three-phase, three-wire, star-star system 153 7.10 Delta-connected systems 155 7.10.1 Balanced delta-connected load with a balanced supply 156 7.10.2 Unbalanced delta-connected load with a balanced supply 158 7.10.3 Unbalanced delta-connected load with an unbalanced supply 159 7.11 Delta-connected supply and a star-connected load 159 7.12 Star-connected supply and delta-connected load 159 7.13 Summary of balanced star- and delta-connected loads 160 7.14 Power consumed in a three-phase system 160 7.15 Power measurement in a three-phase system 162 7.16 The two-wattmeter method of power measurement 163 7.17 Introduction to symmetrical components 164 7.18 Analysis of unbalanced conditions 165 7.18.1 Power consumed by symmetrical components 166 Unworked problems 167 8 Two-port Networks 171 8.1 Introduction 171 8.2 Input impedance, output impedance, voltage gain, current gain and power gain 172 8.3 Admittance parameters or y-parameters 176 8.4 Impedance parameters or z-parameters 180 8.5 Hybrid parameters or h-parameters 183 8.6 Transmission parameters 186 8.7 Relationship between the y-, Z-, h- and t-parameters 189 Contents ix 8.8 Interconnection between two-port networks 191 Unworked problems 194 9 The Transformer 198 9.1 Introduction 198 9.2 The ideal transformer 198 9.3 Phasor diagram for an ideal transformer on no-load 201 9.4 Leakage flux in a transformer 203 9.5 Impedance matching with an ideal transformer 204 9.6 The ideal transformer as a two-port network 207 9.7 Thevenin's equivalent circuit of an ideal transformer and voltage regulation 208 9.8 The linear transformer 211 9.9 T- and n-equivalent circuit of a linear transformer 214 Unworked problems 214 10 Transient Solution of Electrical Circuits 218 10.1 Introduction 218 10.2 Classical solution of first-order systems 219 10.2.1 The time constant i of a circuit 223 10.2.2 Circuits reducible to first-order format 223 10.3 Classical solution of second-order systems 224 10.4 The Laplace transform 226 10.5 Step, impulse and ramp functions 227 10.6 Laplace transforms of some useful functions 233 10.7 Properties of the Laplace transform 237 10.8 Representation of circuit elements in the s domain 242 10.9 Introduction to analysis of first-order systems using the Laplace transform 244 10.10 Sinusoidal excitation of first-order systems 249 10.11 Solution of second-order circuits using Laplace transforms 251 10.12 s-Domain transfer functions 258 10.13 Transients in magnetically coupled circuits 259 Unworked problems 262 11 Complex Frequency, the s-Plane and Bode Diagrams 266 11.1 Introduction 266 11.2 The exponential form of a complex number 267 11.3 Complex frequency 267 11.4 The s-plane 268 11.5 Transformation of impedance into the s-domain 269