Electrical Circuit Theory and Technology · Electrical Circuit Theory and Technology Revised second...

Electrical Circuit Theory and Technology

John Bird

In Memory of Elizabeth

Electrical Circuit Theory and Technology

Revised second edition

John Bird, BSc(Hons), CEng, MIEE, FIEIE, CMath,FIMA, FCollP

Newnes

OXFORD AMSTERDAM BOSTON LONDON NEW YORK PARISSAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO

NewnesAn imprint of Elsevier ScienceLinacre House, Jordan Hill, Oxford OX2 8DP200 Wheeler Rd, Burlington, MA 01803

First published 1997Second edition 2001Reprinted 2002Revised second edition 2003

Copyright 1997, 2001, John Bird. All rights reserved

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No part of this publication may be reproduced in any materialform (including photocopying or storing in any medium byelectronic means and whether or not transiently or incidentally to someother use of this publication) without the written permission of thecopyright holder except in accordance with the provisions of the Copyright,Designs and Patents Act 1988 or under the terms of a licence issued by theCopyright Licensing Agency Ltd, 90 Tottenham Court Road, London,England W1P 4LP. Applications for the copyright holder’s writtenpermission to reproduce any part of this publication should be addressedto the publisher

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

ISBN 0 7506 5784 7

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Typeset by Laser Words, Madras, IndiaPrinted and bound in Great Britain

Part 1 Basic Electrical Engineering Principles 1....................................1 Units associated with basic electrical quantities 1........................................

SI units 1..........................................................................................................................Charge 4...........................................................................................................................Force 4.............................................................................................................................Work 5..............................................................................................................................Power 5............................................................................................................................Electrical potential and e. m. f. 6......................................................................................Resistance and conductance 6........................................................................................Electrical power and energy 7..........................................................................................Summary of terms, units and their symbols 8..................................................................Further problems on units associated with basic electrical quantities 9...........................

2 An introduction to electric circuits 10..............................................................Standard symbols for electrical components 10.................................................................Electric current and quantity of electricity 11......................................................................Potential difference and resistance 13...............................................................................Basic electrical measuring instruments 13.........................................................................Linear and non- linear devices 13......................................................................................Ohms law 14.....................................................................................................................Multiples and sub- multiples 14..........................................................................................Conductors and insulators 16.............................................................................................Electrical power and energy 16..........................................................................................Main effects of electric current 20......................................................................................Fuses 20.............................................................................................................................Further problems 13...........................................................................................................

3 Resistance variation 23..................................................................................Resistance and resistivity 23..............................................................................................Temperature coefficient of resistance 26...........................................................................Further problems on resistance variation 29......................................................................

4 Chemical effects of electricity 31....................................................................Introduction 31....................................................................................................................Electrolysis 31....................................................................................................................Electroplating 32.................................................................................................................The simple cell 32..............................................................................................................Corrosion 33.......................................................................................................................E.m.f. and internal resistance of a cell 34..........................................................................Primary cells 36..................................................................................................................Secondary cells 37.............................................................................................................Cell capacity 39..................................................................................................................Further problems on the chemical effects of electricity 39.................................................

Assignment 1 41................................................................................................5 Series and parallel networks 42.....................................................................

Series circuits 42................................................................................................................Potential divider 44.............................................................................................................Parallel networks 45...........................................................................................................Current division 48.............................................................................................................Wiring lamps in series and in parallel 52............................................................................Further problems on series and parallel networks 53........................................................

6 Capacitors and capacitance 55......................................................................Electrostatic field 57...........................................................................................................Electric field strength 57.....................................................................................................Capacitance 57..................................................................................................................Capacitors 57.....................................................................................................................Electric flux density 59........................................................................................................Permittivity 59.....................................................................................................................The parallel plate capacitor 61...........................................................................................Capacitors connected in parallel and series 63..................................................................Dielectric strength 67..........................................................................................................Energy stored 68................................................................................................................

Practical types of capacitor 69...........................................................................................Discharging capacitors 70..................................................................................................Further problems on capacitors and capacitance 70.........................................................

7 Magnetic circuits 74........................................................................................Magnetic fields 74..............................................................................................................Magnetic flux and flux density 75.......................................................................................Magnetomotive force and magnetic field strength 76.........................................................Permeability and B Ò H curves 77.......................................................................................Reluctance 80....................................................................................................................Composite series magnetic circuits 81...............................................................................Comparison between electrical and magnetic quantities 84..............................................Hysteresis and hysteresis loss 84......................................................................................Further problems on magnetic circuits 85..........................................................................

Assignment 2 87................................................................................................8 Electromagnetism 89......................................................................................

Magnetic field due to an electric current 89........................................................................Electromagnets 91.............................................................................................................Force on a current- carrying conductor 92.........................................................................Principle of operation of a simple d. c. motor 96................................................................Principle of operation of a moving coil instrument 97.........................................................Force on a charge 98.........................................................................................................Further problems on electromagnetism 98.........................................................................

9 Electromagnetic induction 100.........................................................................Introduction to electromagnetic induction 100......................................................................Laws of electromagnetic induction 101................................................................................Inductance 104.....................................................................................................................Inductors 106........................................................................................................................Energy stored 107................................................................................................................Inductance of a coil 107.......................................................................................................Mutual inductance 108.........................................................................................................Further problems on electromagnetic induction 109............................................................

10 Electrical measuring instruments and measurements 113.............................Introduction 113....................................................................................................................Analogue instruments 113....................................................................................................Moving-iron instrument 113..................................................................................................The moving-coil rectifier instrument 114..............................................................................Comparison of moving- coil, moving- iron and moving- coil rectifier instruments 114.........Shunts and multipliers 115...................................................................................................Electronic instruments 117...................................................................................................The ohmmeter 117...............................................................................................................Multimeters 118....................................................................................................................Wattmeters 118....................................................................................................................Instrument ˛ loadingÌ effect 118............................................................................................The cathode ray oscilloscope 121........................................................................................Waveform harmonics 124....................................................................................................Logarithmic ratios 126..........................................................................................................Null method of measurement 129........................................................................................Wheatstone bridge 129........................................................................................................D.c. potentiometer 130.........................................................................................................A.c. bridges 130...................................................................................................................Measurement errors 131......................................................................................................Further problems on electrical measuring instruments and measurements 133..................

11 Semiconductor diodes 137.............................................................................Types of materials 137.........................................................................................................Silicon and germanium 138..................................................................................................n-type and p-type materials 138...........................................................................................The p-n junction 139.............................................................................................................Forward and reverse bias 140..............................................................................................Semiconductor diodes 140...................................................................................................Rectification 143...................................................................................................................Further problems on semiconductor diodes 143..................................................................

12 Transistors 145...............................................................................................The bipolar junction transistor 145.......................................................................................Transistor action 147............................................................................................................Transistor symbols 149........................................................................................................Transistor connections 149..................................................................................................Transistor characteristics 150..............................................................................................The transistor as an amplifier 152........................................................................................The load line 154..................................................................................................................Current and voltage gains 155.............................................................................................Thermal runaway 158...........................................................................................................Further problems on transistors 159....................................................................................

Assignment 3 162................................................................................................Main formulae for Part 1 164...............................................................................

General 164..........................................................................................................................Capacitors and capacitance 164..........................................................................................Magnetic circuits 164............................................................................................................Electromagnetism 164..........................................................................................................Electromagnetic induction 164.............................................................................................Measurements 164...............................................................................................................

Part 2 Electrical Principles and Technology 165.......................................13 D.c. circuit theory 167.....................................................................................

Introduction 167....................................................................................................................Kirchhoffs laws 167.............................................................................................................The superposition theorem 171............................................................................................General d.c. circuit theory 174.............................................................................................Th · evenins theorem 176....................................................................................................Constant-current source 181................................................................................................Nortons theorem 181...........................................................................................................Th · evenin and Norton equivalent networks 184.................................................................Maximum power transfer theorem 187.................................................................................Further problems on d. c. circuit theory 189.........................................................................

14 Alternating voltages and currents 193............................................................Introduction 193....................................................................................................................The a.c. generator 194.........................................................................................................Waveforms 194....................................................................................................................A.c. values 195.....................................................................................................................The equation of a sinusoidal waveform 200.........................................................................Combination of waveforms 204............................................................................................Rectification 208...................................................................................................................Further problems on alternating voltages and currents 209.................................................

Assignment 4 212................................................................................................15 Single-phase series a.c. circuits 213..............................................................

Purely resistive a.c. circuit 214.............................................................................................Purely inductive a.c. circuit 214............................................................................................Purely capacitive a. c. circuit 214.........................................................................................R L series a.c. circuit 215...................................................................................................R C series a.c. circuit 220..................................................................................................R L C series a.c. circuit 221.............................................................................................Series resonance 225..........................................................................................................Q-factor 227.........................................................................................................................Bandwidth and selectivity 229..............................................................................................Power in a.c. circuits 230.....................................................................................................Power triangle and power factor 232....................................................................................Further problems on single- phase series a. c. circuits 234.................................................

16 Single-phase parallel a. c. circuits 238...........................................................Introduction 238....................................................................................................................R L parallel a.c. circuit 238.................................................................................................R C parallel a.c. circuit 240................................................................................................

L C parallel a.c. circuit 241.................................................................................................LR C parallel a.c. circuit 243..............................................................................................Parallel resonance and Q- factor 247...................................................................................Power factor improvement 252............................................................................................Further problems on single- phase parallel a. c. circuits 256...............................................

17 D.c. transients 259.........................................................................................Introduction 259....................................................................................................................Charging a capacitor 260.....................................................................................................Time constant for a C R circuit 260....................................................................................Transient curves for a C R circuit 261................................................................................Discharging a capacitor 266.................................................................................................Current growth in an L R circuit 268...................................................................................Time constant for an L R circuit 269...................................................................................Transient curves for an L R circuit 269...............................................................................Current decay in an L R circuit 272....................................................................................Switching inductive circuits 275............................................................................................The effects of time constant on a rectangular waveform 275...............................................Further problems on d. c. transients 276..............................................................................

18 Operational amplifiers 278.............................................................................Introduction to operational amplifiers 278............................................................................Some op amp parameters 280.............................................................................................Op amp inverting amplifier 282............................................................................................Op amp non- inverting amplifier 285....................................................................................Op amp voltage- follower 286..............................................................................................Op amp summing amplifier 286...........................................................................................Op amp voltage comparator 288..........................................................................................Op amp integrator 288.........................................................................................................Op amp differential amplifier 289.........................................................................................Digital to analogue ( D/ A) conversion 291...........................................................................Analogue to digital ( A/ D) conversion 293...........................................................................Further problems on operational amplifiers 294...................................................................

Assignment 5 296................................................................................................19 Three phase systems 297..............................................................................

Introduction 297....................................................................................................................Three-phase supply 298.......................................................................................................Star connection 298.............................................................................................................Delta connection 302............................................................................................................Power in three- phase systems 303.....................................................................................Measurement of power in three- phase systems 306..........................................................Comparison of star and delta connections 312....................................................................Advantages of three- phase systems 312............................................................................Further problems on three- phase systems 312...................................................................

20 Transformers 315...........................................................................................Introduction 315....................................................................................................................Transformer principle of operation 316................................................................................Transformer no- load phasor diagram 319...........................................................................E.m.f. equation of a transformer 320....................................................................................Transformer on- load phasor diagram 324...........................................................................Transformer construction 325..............................................................................................Equivalent circuit of a transformer 326.................................................................................Regulation of a transformer 329...........................................................................................Transformer losses and efficiency 330.................................................................................Resistance matching 334.....................................................................................................Auto transformers 337..........................................................................................................Isolating transformers 340....................................................................................................Three-phase transformers 340.............................................................................................Current transformers 342.....................................................................................................Voltage transformers 343.....................................................................................................Further problems on transformers 344.................................................................................

Assignment 6 349................................................................................................

21 D.c. machines 350..........................................................................................Introduction 350....................................................................................................................The action of a commutator 351...........................................................................................D.c. machine construction 352.............................................................................................Shunt, series and compound windings 353..........................................................................E.m.f. generated in an armature winding 353......................................................................D.c. generators 356..............................................................................................................Types of d.c. generator and their characteristics 356..........................................................D.c. machine losses 362......................................................................................................Efficiency of a d.c. generator 363.........................................................................................D.c. motors 364....................................................................................................................Torque of a d.c. machine 365...............................................................................................Types of d.c. motor and their characteristics 368.................................................................The efficiency of a d. c. motor 373.......................................................................................D.c. motor starter 376...........................................................................................................Speed control of d. c. motors 377........................................................................................Motor cooling 381.................................................................................................................Further problems on d. c. machines 381..............................................................................

22 Three-phase induction motors 386.................................................................Introduction 386....................................................................................................................Production of a rotating magnetic field 387..........................................................................Synchronous speed 388.......................................................................................................Construction of a three- phase induction motor 390............................................................Principle of operation of a three- phase induction motor 390...............................................Slip 391................................................................................................................................Rotor e.m.f. and frequency 393............................................................................................Rotor impedance and current 394........................................................................................Rotor copper loss 395..........................................................................................................Induction motor losses and efficiency 395...........................................................................Torque equation for an induction motor 397........................................................................Induction motor torque - speed characteristics 401.............................................................Starting methods for induction motors 403...........................................................................Advantages of squirrel- cage induction motors 404.............................................................Advantages of wound rotor induction motor 405..................................................................Double cage induction motor 405.........................................................................................Uses of three-phase induction motors 405...........................................................................Further problems on three- phase induction motors 406.....................................................

Assignment 7 408................................................................................................Main formulae for Part 2 409...............................................................................

A.c. theory: 409....................................................................................................................Single-phase circuits: 410....................................................................................................D.c. transients: 410..............................................................................................................Operational amplifiers 411..................................................................................................Three-phase systems: 411...................................................................................................Transformers: 411................................................................................................................D.c. machines: 411...............................................................................................................Three-phase induction motors: 411......................................................................................

Part 3 Advanced Circuit Theory and Technology 413...............................23 Revision of complex numbers 415.................................................................

Introduction 415....................................................................................................................Operations involving Cartesian complex numbers 417........................................................Complex equations 419........................................................................................................The polar form of a complex number 421............................................................................Multiplication and division using complex numbers in polar form 421..................................De Moivres theorem powers and roots of complex numbers 423.....................................Further problems on complex numbers 424.........................................................................

24 Application of complex numbers to series a. c. circuits 429...........................Introduction 429....................................................................................................................Series a.c. circuits 429.........................................................................................................

Further problems on series a. c. circuits 440.......................................................................25 Application of complex numbers to parallel a. c. networks 25......................

Introduction 25....................................................................................................................Admittance, conductance and susceptance 25..................................................................Parallel a.c. networks 448....................................................................................................Further problems on parallel a. c. networks 454..................................................................

26 Power in a.c. circuits 459...............................................................................Introduction 459....................................................................................................................Determination of power in a. c. circuits 459.........................................................................Power triangle and power factor 464....................................................................................Use of complex numbers for determination of power 465....................................................Power factor improvement 470............................................................................................Further problems on power in a. c. circuits 472...................................................................

Assignment 8 475................................................................................................27 A.c. bridges 476.............................................................................................

Introduction 476....................................................................................................................Balance conditions for an a. c. bridge 476...........................................................................Types of a.c. bridge circuit 478............................................................................................Further problems on a. c. bridges 488.................................................................................

28 Series resonance and Q- factor 491..............................................................Introduction 491....................................................................................................................Series resonance 491..........................................................................................................Q-factor 495.........................................................................................................................Voltage magnification 498....................................................................................................Q-factors in series 502.........................................................................................................Bandwidth 504......................................................................................................................Small deviations from the resonant frequency 509..............................................................Further problems on series resonance and Q- factor 512....................................................

29 Parallel resonance and Q- factor 515.............................................................Introduction 516....................................................................................................................The LR C parallel network 516...........................................................................................Dynamic resistance 517.......................................................................................................The LR CR parallel network 517........................................................................................Q-factor in a parallel network 519........................................................................................Further problems on parallel resonance and Q- factor 527..................................................

Assignment 9 530................................................................................................30 Introduction to network analysis 531..............................................................

Introduction 531....................................................................................................................Solution of simultaneous equations using determinants 532...............................................Network analysis using Kirchhoffs laws 535.......................................................................Further problems on Kirchhoffs laws 542............................................................................

31 Mesh-current and nodal analysis 545............................................................Mesh-current analysis 545...................................................................................................Nodal analysis 550...............................................................................................................Further problems on mesh- current and nodal analysis 559................................................

32 The superposition theorem 562......................................................................Introduction 562....................................................................................................................Using the superposition theorem 562...................................................................................Further problems on the superposition theorem 573...........................................................

33 Thevenins and Nortons theorems 5755.........................................................Introduction 575....................................................................................................................Thevenins theorem 575.......................................................................................................Nortons theorem 587...........................................................................................................Thevenin and Norton equivalent networks 593....................................................................Further problems on Thevenins and Nortons theorem 598................................................

Assignment 10 602..............................................................................................34 Delta-star and star-delta transformations 603................................................

Introduction 603....................................................................................................................

Delta and star connections 603............................................................................................Delta-star transformation 603...............................................................................................Star-delta transformation 611...............................................................................................Further problems on delta-star and star-delta transformations 614.....................................

35 Maximum power transfer theorems and impedance matching 617................Maximum power transfer theorems 617...............................................................................Impedance matching 623.....................................................................................................Further problems on maximum power transfer theorems and impedance matching 626....

Assignment 11 629..............................................................................................36 Complex Waveforms 631...............................................................................

Introduction 631....................................................................................................................The general equation for a complex waveform 632.............................................................Harmonic synthesis 633.......................................................................................................Rms value, mean value and the form factor of a complex wave 645...................................Power associated with complex waves 650.........................................................................Harmonics in single- phase circuits 653...............................................................................Resonance due to harmonics 664........................................................................................Sources of harmonics 666....................................................................................................Further problems on complex waveforms 671.....................................................................

37 A numerical method of harmonic analysis 678...............................................Introduction 678....................................................................................................................Harmonic analysis on data given in tabular or graphical form 683.......................................Complex waveform considerations 683...............................................................................Further problems on a numerical method of harmonic analysis 685....................................

38 Magnetic materials 688..................................................................................Revision of terms and units used with magnetic circuits 688...............................................Magnetic properties of materials 690...................................................................................Hysteresis and hysteresis loss 692......................................................................................Eddy current loss 696...........................................................................................................Separation of hysteresis and eddy current losses 701.........................................................Nonpermanent magnetic materials 704...............................................................................Permanent magnetic materials 706......................................................................................Further problems on magnetic materials 707.......................................................................

Assignment 12 710..............................................................................................39 Dielectrics and dielectric loss 711..................................................................

Electric fields, capacitance and permittivity 711...................................................................Polarization 711....................................................................................................................Dielectric strength 712..........................................................................................................Thermal effects 714..............................................................................................................Mechanical properties 714...................................................................................................Types of practical capacitor 715...........................................................................................Liquid dielectrics and gas insulation 715..............................................................................Dielectric loss and loss angle 715........................................................................................Further problems on dielectric loss and loss angle 719.......................................................

40 Field theory 720..............................................................................................Field plotting by curvilinear squares 720..............................................................................Capacitance between concentric cylinders 725...................................................................Capacitance of an isolated twin line 733..............................................................................Energy stored in an electric field 737...................................................................................Induced e.m.f. and inductance 741......................................................................................Inductance of a concentric cylinder ( or coaxial cable) 741..................................................Inductance of an isolated twin line 746................................................................................Energy stored in an electromagnetic field 750.....................................................................Further problems on field theory 753...................................................................................

41 Attenuators 758..............................................................................................Introduction 758....................................................................................................................Characteristic impedance 759..............................................................................................Logarithmic ratios 761..........................................................................................................Symmetrical T-and p- attenuators 764.................................................................................Insertion loss 772.................................................................................................................

Asymmetrical Tand p- sections 775.....................................................................................The L-section attenuator 779...............................................................................................Two-port networks in cascade 782.......................................................................................Further problems on attenuators 785...................................................................................

Assignment 13 789..............................................................................................42 Filter networks 790.........................................................................................

Introduction 791....................................................................................................................Basic types of filter sections 791..........................................................................................The characteristic impedance and the attenuation of filter sections 792..............................Ladder networks 795............................................................................................................Low-pass filter sections 797.................................................................................................High-pass filter sections 807................................................................................................Propagation coefficient and time delay in filter sections 815................................................˛m-derivedÌ filter sections 825..............................................................................................Practical composite filters 833..............................................................................................Further problems on filter networks 837...............................................................................

43 Magnetically coupled circuits 841...................................................................Introduction 841....................................................................................................................Self-inductance 841..............................................................................................................Mutual inductance 842.........................................................................................................Coupling coefficient 843.......................................................................................................Coils connected in series 845..............................................................................................Coupled circuits 849.............................................................................................................Dot rule for coupled circuits 857...........................................................................................Further problems on magnetically coupled circuits 864.......................................................

44 Transmission lines 869...................................................................................Introduction 869....................................................................................................................Transmission line primary constants 869.............................................................................Phase delay, wavelength and velocity of propagation 871..................................................Current and voltage relationships 873..................................................................................Characteristic impedance and propagation coefficient in terms of the primaryconstants 875.......................................................................................................................Distortion on transmission lines 882.....................................................................................Wave reflection and the reflection coefficient 885................................................................Standing waves and the standing wave ratio 890................................................................Further problems on transmission lines 897........................................................................

45 Transients and Laplace transforms 901.........................................................Introduction 901....................................................................................................................Response of R C series circuit to a step input 901............................................................Response of R L series circuit to a step input 906.............................................................L R C series circuit response 910....................................................................................Introduction to Laplace transforms 914................................................................................Inverse Laplace transforms and the solution of differential equations 921..........................Laplace transform analysis directly from the circuit diagram 930.........................................L R C series circuit using Laplace transforms 944...........................................................Initial conditions 949.............................................................................................................Further problems on transients and Laplace transforms 952...............................................

Assignment 14 958..............................................................................................Main formulae for part 3 advanced circuit theory and technology 960................

Complex numbers: 960........................................................................................................General: 960.........................................................................................................................R L C series circuit: 9600...................................................................................................LR C network: 961..............................................................................................................LR CR network: 961...........................................................................................................Determinants: 961................................................................................................................Delta-star: 961......................................................................................................................Star-delta: 961......................................................................................................................Impedance matching: 961....................................................................................................Complex waveforms: 961.....................................................................................................Harmonic analysis: 961........................................................................................................Hysteresis and Eddy current: 961........................................................................................

Dielectric loss: 962...............................................................................................................Field theory: 962...................................................................................................................Attenuators: 962...................................................................................................................Filter networks 963...............................................................................................................Magnetically coupled circuits 963.........................................................................................Transmission lines: 964........................................................................................................Transients: 964.....................................................................................................................

Part 4 General Reference 966......................................................................Standard electrical quantities their symbols and units 968...............................Greek alphabet 971.............................................................................................Common prefixes 972.........................................................................................Resistor colour coding and ohmic values 973.....................................................

Colour code for fixed resistors 973.......................................................................................Letter and digit code for resistors 973..................................................................................

Index 975.............................................................................................................

Preface

‘Electrical Circuit Theory and Technology, Revised second Edition’provides coverage for a wide range of courses that contain electricalprinciples, circuit theory and technology in their syllabuses, fromintroductory to degree level. The chapter ‘Transients and Laplacetransforms’, which had been removed from the second edition due to pagerestraints, has been included in this edition in response to popular demand.The text is set out in four parts as follows:

PART 1, involving chapters 1 to 12, contains ‘Basic ElectricalEngineering Principles’ which any student wishing to progress inelectrical engineering would need to know. An introduction to electricalcircuits, resistance variation, chemical effects of electricity, seriesand parallel circuits, capacitors and capacitance, magnetic circuits,electromagnetism, electromagnetic induction, electrical measuringinstruments and measurements, semiconductor diodes and transistors areall included in this section.

PART 2, involving chapters 13 to 22, contains ‘Electrical Principlesand Technology’ suitable for Advanced GNVQ, National Certificate,National Diploma and City and Guilds courses in electrical and electronicengineering. D.c. circuit theory, alternating voltages and currents,single-phase series and parallel circuits, d.c. transients, operationalamplifiers, three-phase systems, transformers, d.c. machines and three-phase induction motors are all included in this section.

PART 3, involving chapters 23 to 45, contains ‘Advanced CircuitTheory and Technology’ suitable for Degree, Higher NationalCertificate/Diploma and City and Guilds courses in electrical andelectronic/telecommunications engineering. The two earlier sections of thebook will provide a valuable reference/revision for students at this level.

Complex numbers and their application to series and parallel networks,power in a.c. circuits, a.c. bridges, series and parallel resonance andQ-factor, network analysis involving Kirchhoff’s laws, mesh and nodalanalysis, the superposition theorem, Thévenin’s and Norton’s theorems,delta-star and star-delta transforms, maximum power transfer theoremsand impedance matching, complex waveforms, harmonic analysis,magnetic materials, dielectrics and dielectric loss, field theory, attenuators,filter networks, magnetically coupled circuits, transmission line theory andtransients and Laplace transforms are all included in this section.

PART 4 provides a short, ‘General Reference’ for standard electricalquantities — their symbols and units, the Greek alphabet, commonprefixes and resistor colour coding and ohmic values.

At the beginning of each of the 45 chapters learning objectivesare listed.

At the end of each of the first three parts of the text is a handy referenceof the main formulae used.

xviii Electrical Circuit Theory and Technology

It is not possible to acquire a thorough understanding of electricalprinciples, circuit theory and technology without working through a largenumber of numerical problems. It is for this reason that ‘Electrical CircuitTheory and Technology, Revised second Edition’contains some 740detailed worked problems, together with over 1100 further problems,all with answers in brackets immediately following each question. Over1100 line diagrams further enhance the understanding of the theory.

Fourteen Assignments have been included, interspersed within thetext every few chapters. For example, Assignment 1 tests understandingof chapters 1 to 4, Assignment 2 tests understanding of chapters 5 to 7,Assignment 3 tests understanding of chapters 8 to 12, and so on. TheseAssignments do not have answers given since it is envisaged that lecturerscould set the Assignments for students to attempt as part of their coursestructure. Lecturers’ may obtain a complimentary set of solutions of theAssignments in an Instructor’s Manual available from the publishersvia the internet — see below.

‘Learning by Example’ is at the heart of ‘Electrical Circuit Theoryand Technology, Revised second Edition’.

JOHN BIRDUniversity of Portsmouth

Free web downloads

Instructor’s ManualFull worked solutions and mark scheme for all the Assignments inthis book.This material is available to lecturers only. To obtain a passwordplease e-mail [email protected] with the followingdetails: course title, number of students, your job title and workpostal address.To download the Instructor’s Manual visithttp://www.newnespress.comand enter the book title in the search box, or use the followingdirect URL:http://www.bh.com/manuals/0750657847/

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Part 1 Basic ElectricalEngineeringPrinciples

1 Units associated withbasic electrical quantities

At the end of this chapter you should be able to:

ž state the basic SI unitsž recognize derived SI unitsž understand prefixes denoting multiplication and divisionž state the units of charge, force, work and power and perform

simple calculations involving these units

ž state the units of electrical potential, e.m.f., resistance,conductance, power and energy and perform simplecalculations involving these units

1.1 SI units The system of units used in engineering and science is the Système Inter-nationale d’Unités (International system of units), usually abbreviated toSI units, and is based on the metric system. This was introduced in 1960and is now adopted by the majority of countries as the official system ofmeasurement.

The basic units in the SI system are listed with their symbols, inTable 1.1.

TABLE 1.1 Basic SI Units

Quantity Unit

length metre, mmass kilogram, kgtime second, selectric current ampere, Athermodynamic temperature kelvin, Kluminous intensity candela, cdamount of substance mole, mol

Derived SI units use combinations of basic units and there are many ofthem. Two examples are:

ž Velocity — metres per second (m/s)ž Acceleration — metres per second squared (m/s2)

4 Electrical Circuit Theory and Technology

SI units may be made larger or smaller by using prefixes which denotemultiplication or division by a particular amount. The six most commonmultiples, with their meaning, are listed in Table 1.2.

TABLE 1.2

Prefix Name Meaning

M mega multiply by 1 000 000 (i.e. ð106)k kilo multiply by 1000 (i.e. ð103)m milli divide by 1000 (i.e. ð10�3)� micro divide by 1 000 000 (i.e. ð10�6)n nano divide by 1 000 000 000 (i.e. ð10�9)p pico divide by 1 000 000 000 000 (i.e. ð10�12)

1.2 Charge The unit of charge is the coulomb (C) where one coulomb is one amperesecond. (1 coulomb D 6.24 ð 1018 electrons). The coulomb is defined asthe quantity of electricity which flows past a given point in an electriccircuit when a current of one ampere is maintained for one second. Thus,

charge, in coulombs Q = It

where I is the current in amperes and t is the time in seconds.

Problem 1. If a current of 5 A flows for 2 minutes, find the quan-tity of electricity transferred.

Quantity of electricity Q D It coulombsI D 5 A, t D 2 ð 60 D 120 sHence Q D 5 ð 120 D 600 C

1.3 Force The unit of force is the newton (N) where one newton is one kilogrammetre per second squared. The newton is defined as the force which, whenapplied to a mass of one kilogram, gives it an acceleration of one metreper second squared. Thus,

force, in newtons F = ma

where m is the mass in kilograms and a is the acceleration in metresper second squared. Gravitational force, or weight, is mg, where g D9.81 m/s2

Units associated with basic electrical quantities 5

Problem 2. A mass of 5000 g is accelerated at 2 m/s2 by a force.Determine the force needed.

Force D mass ð accelerationD 5 kg ð 2 m/s2 D 10kg m

s2D 10 N

Problem 3. Find the force acting vertically downwards on a massof 200 g attached to a wire.

Mass D 200 g D 0.2 kg and acceleration due to gravity, g D 9.81 m/s2Force acting downwards D weight D mass ð acceleration

D 0.2 kg ð 9.81 m/s2D 1.962 N

1.4 Work The unit of work or energy is the joule (J) where one joule is one newtonmetre. The joule is defined as the work done or energy transferred whena force of one newton is exerted through a distance of one metre in thedirection of the force. Thus

work done on a body, in joules W = Fs

where F is the force in newtons and s is the distance in metres movedby the body in the direction of the force. Energy is the capacity fordoing work.

1.5 Power The unit of power is the watt (W) where one watt is one joule per second.Power is defined as the rate of doing work or transferring energy. Thus,

power in watts, P =Wt

where W is the work done or energy transferred in joules and t is thetime in seconds. Thus

energy, in joules, W = Pt


Problem 4. A portable machine requires a force of 200 N to moveit. How much work is done if the machine is moved 20 m and whataverage power is utilized if the movement takes 25 s?

Work done D force ð distance D 200 N ð 20 m D 4000 Nm or 4 kJ

Power D work donetime taken

D 4000 J25 s

D 160 J=s = 160 W

Problem 5. A mass of 1000 kg is raised through a height of 10 min 20 s. What is (a) the work done and (b) the power developed?

(a) Work done D force ð distance and force D mass ð accelerationHence, work done D 1000 kg ð 9.81 m/s2� ð 10 m�

D 98 100 Nm D 98.1 kNm or 98.1 kJ

(b) Power D work donetime taken

D 98100 J20 s

D 4905 J/sD 4905 W or 4.905 kW

1.6 Electrical potentialand e.m.f.

The unit of electric potential is the volt (V) where one volt is one jouleper coulomb. One volt is defined as the difference in potential betweentwo points in a conductor which, when carrying a current of one ampere,dissipates a power of one watt, i.e.

volts D wattsamperes

D joules/secondamperes

D joulesampere seconds

D joulescoulombs

A change in electric potential between two points in an electric circuit iscalled a potential difference. The electromotive force (e.m.f.)providedby a source of energy such as a battery or a generator is measured in volts.

1.7 Resistance andconductance

The unit of electric resistance is the ohm (Z) where one ohm is onevolt per ampere. It is defined as the resistance between two points in aconductor when a constant electric potential of one volt applied at thetwo points produces a current flow of one ampere in the conductor. Thus,

resistance, in ohms R =VI


where V is the potential difference across the two points in volts and I isthe current flowing between the two points in amperes.

The reciprocal of resistance is called conductanceand is measured insiemens (S). Thus,

conductance, in siemens G =1R

where R is the resistance in ohms.

Problem 6. Find the conductance of a conductor of resistance(a) 10 �, (b) 5 k� and (c) 100 m�

(a) Conductance G D 1R

D 110

siemen D 0.1 s

(b) G D 1R

D 15 ð 103 S D 0.2 ð 10

�3 S D 0.2 mS

(c) G D 1R

D 1100 ð 10�3 S D

103

100S D 10 S

1.8 Electrical power andenergy

When a direct current of I amperes is flowing in an electric circuit andthe voltage across the circuit is V volts, then

power, in watts P = VI

Electrical energy D Power ð timeD VIt Joules

Although the unit of energy is the joule, when dealing with large amountsof energy, the unit used is the kilowatt hour (kWh) where

1 kWh D 1000 watt hourD 1000 ð 3600 watt seconds or joulesD 3 600 000 J

Problem 7. A source e.m.f. of 5 V supplies a current of 3 A for10 minutes. How much energy is provided in this time?

Energy D power ð time and power D voltage ð current. Hence


Energy D VIt D 5 ð 3 ð 10 ð 60� D 9000 Ws or JD 9 kJ

Problem 8. An electric heater consumes 1.8 MJ when connectedto a 250 V supply for 30 minutes. Find the power rating of theheater and the current taken from the supply.

i.e. Power rating of heater= 1 kW

Power P D VI, thus I D PV

D 1000250

D 4 A

Hence the current taken from the supply is 4 A

1.9 Summary of terms,units and their symbols Quantity Quantity Unit Unit symbol

Symbol

Length l metre mMass m kilogram kgTime t second sVelocity v metres per second m/s or m s�1

Acceleration a metres persecond squared m/s2 or m s�2

Force F newton NElectrical charge coulomb C

or quantity QElectric current I ampere AResistance R ohm �Conductance G siemen SElectromotive volt V

force EPotential volt V

difference VWork W joule JEnergy E (or W) joule JPower P watt W

As progress is made through Electrical Circuit Theory and Technologymany more terms will be met. A full list of electrical quantities, togetherwith their symbols and units are given in Part 4, page 968.


1.10 Further problemson units associated with

basic electrical quantities(Take g = 9.81 m/s2 where

appropriate)

1 What force is required to give a mass of 20 kg an acceleration of30 m/s2? [600 N]

2 Find the accelerating force when a car having a mass of 1.7 Mgincreases its speed with a constant acceleration of 3 m/s2 [5.1 kN]

3 A force of 40 N accelerates a mass at 5 m/s2. Determine the mass.[8 kg]

4 Determine the force acting downwards on a mass of 1500 gsuspended on a string. [14.72 N]

5 A force of 4 N moves an object 200 cm in the direction of the force.What amount of work is done? [8 J]

6 A force of 2.5 kN is required to lift a load. How much work is doneif the load is lifted through 500 cm? [12.5 kJ]

7 An electromagnet exerts a force of 12 N and moves a soft ironarmature through a distance of 1.5 cm in 40 ms. Find the powerconsumed. [4.5 W]

8 A mass of 500 kg is raised to a height of 6 m in 30 s. Find (a) thework done and (b) the power developed.

[(a) 29.43 kNm (b) 981 W]

9 What quantity of electricity is carried by 6.24 ð 1021 electrons?[1000 C]

10 In what time would a current of 1 A transfer a charge of 30 C? [30 s]

11 A current o

f 3 A flows for 5 minutes. What charge is transferred?

[900 C]

12 How long must a current of 0.1 A flow so as to transfer a charge of30 C? [5 minutes]

13 Find the conductance of a resistor of resistance (a) 10Z (b) 2 kZ(c) 2 mZ [(a) 0.1 S (b) 0.5 mS (c) 500 S]

14 A conductor has a conductance of 50 µS. What is its resistance?[20 k�]

15 An e.m.f. of 250 V is connected across a resistance and the currentflowing through the resistance is 4 A. What is the power developed?

[1 kW]

16 450 J of energy are converted into heat in 1 minute. What power isdissipated? [7.5 W]

17 A current of 10 A flows through a conductor and 10 W is dissipated.What p.d. exists across the ends of the conductor? [1 V]

18 A battery of e.m.f. 12 V supplies a current of 5 A for 2 minutes.How much energy is supplied in this time? [7.2 kJ]

19 A dc electric motor consumes 36 MJ when connected to a 250 Vsupply for 1 hour. Find the power rating of the motor and the currenttaken from the supply. [10 kW, 40 A]

2 An introduction toelectric circuits

At the end of this chapter you should be able to:

ž recognize common electrical circuit diagram symbolsž understand that electric current is the rate of movement of

charge and is measured in amperes

ž appreciate that the unit of charge is the coulombž calculate charge or quantity of electricity Q from Q D Itž understand that a potential difference between two points in a

circuit is required for current to flow

ž appreciate that the unit of p.d. is the voltž understand that resistance opposes current flow and is

measured in ohms

ž appreciate what an ammeter, a voltmeter, an ohmmeter, amultimeter and a C.R.O. measure

ž distinguish between linear and non-linear devices

ž state Ohm’s law as V D IR or I D VR

or R D VI

ž use Ohm’s law in calculations, including multiples andsub-multiples of units

ž describe a conductor and an insulator, giving examples of eachž appreciate that electrical power P is given by

P D VI D I2R D V2

Rwatts

ž calculate electrical powerž define electrical energy and state its unitž calculate electrical energyž state the three main effects of an electric current, giving

practical examples of each

ž explain the importance of fuses in electrical circuits

2.1 Standard symbols forelectrical components

Symbols are used for components in electrical circuit diagrams and someof the more common ones are shown in Figure 2.1.

An introduction to electric circuits 11

Figure 2.1

2.2 Electric current andquantity of electricity

All atomsconsist of protons, neutronsand electrons. The protons, whichhave positive electrical charges, and the neutrons, which have no electricalcharge, are contained within the nucleus. Removed from the nucleus areminute negatively charged particles called electrons. Atoms of differentmaterials differ from one another by having different numbers of protons,neutrons and electrons. An equal number of protons and electrons existwithin an atom and it is said to be electrically balanced, as the positiveand negative charges cancel each other out. When there are more thantwo electrons in an atom the electrons are arranged into shellsat variousdistances from the nucleus.

All atoms are bound together by powerful forces of attraction existingbetween the nucleus and its electrons. Electrons in the outer shell of anatom, however, are attracted to their nucleus less powerfully than areelectrons whose shells are nearer the nucleus.


It is possible for an atom to lose an electron; the atom, which is nowcalled an ion, is not now electrically balanced, but is positively chargedand is thus able to attract an electron to itself from another atom. Electronsthat move from one atom to another are called free electrons and suchrandom motion can continue indefinitely. However, if an electric pressureor voltage is applied across any material there is a tendency for electronsto move in a particular direction. This movement of free electrons, knownas drift , constitutes an electric current flow. Thus current is the rate ofmovement of charge.

Conductors are materials that contain electrons that are looselyconnected to the nucleus and can easily move through the material fromone atom to another.

Insulators are materials whose electrons are held firmly to theirnucleus.

The unit used to measure the quantity of electrical charge Q is calledthe coulomb C �where 1 coulomb D 6.24 ð 1018 electrons

If the drift of electrons in a conductor takes place at the rate of onecoulomb per second the resulting current is said to be a current of oneampere.

Thus, 1 ampere D 1 coulomb per second or 1 A D 1 C/sHence, 1 coulomb D 1 ampere second or 1 C D 1 AsGenerally, if I is the current in amperes and t the time in seconds during

which the current flows, then I ð t represents the quantity of electricalcharge in coulombs, i.e.

quantity of electrical charge transferred, Q = I × t coulombs

Problem 1. What current must flow if 0.24 coulombs is to betransferred in 15 ms?

Since the quantity of electricity, Q D It, then

I D Qt

D 0.2415 ð 10�3 D

0.24 ð 10315

D 24015

D 16 A

Problem 2. If a current of 10 A flows for four minutes, find thequantity of electricity transferred.

Quantity of electricity, Q D It coulombsI D 10 A; t D 4 ð 60 D 240 sHence Q D 10 ð 240 D 2400 C

Further problems on Q D I ð t may be found in Section 2.12, problems 1to 3, page 21.


2.3 Potential differenceand resistance

For a continuous current to flow between two points in a circuit a poten-tial difference (p.d.) or voltage, V, is required between them; a completeconducting path is necessary to and from the source of electrical energy.The unit of p.d. is the volt, V

Figure 2.2 shows a cell connected across a filament lamp. Current flow,by convention, is considered as flowing from the positive terminal of thecell, around the circuit to the negative terminal.

The flow of electric current is subject to friction. This friction, or oppo-sition, is called resistanceR and is the property of a conductor that limitscurrent. The unit of resistance is the ohm; 1 ohm is defined as the resis-tance which will have a current of 1 ampere flowing through it when1 volt is connected across it, i.e.

resistanceR =potential difference

currentFigure 2.2

2.4 Basic electricalmeasuring instruments

An ammeter is an instrument used to measure current and must beconnected in series with the circuit. Figure 2.2 shows an ammeterconnected in series with the lamp to measure the current flowing throughit. Since all the current in the circuit passes through the ammeter it musthave a very low resistance.

A voltmeter is an instrument used to measure p.d. and must beconnected in parallel with the part of the circuit whose p.d. is required. InFigure 2.2, a voltmeter is connected in parallel with the lamp to measurethe p.d. across it. To avoid a significant current flowing through it avoltmeter must have a very high resistance.

An ohmmeter is an instrument for measuring resistance.A multimeter , or universal instrument, may be used to measure

voltage, current and resistance. An ‘Avometer’ is a typical example.The cathode ray oscilloscope (CRO)may be used to observe wave-

forms and to measure voltages and currents. The display of a CROinvolves a spot of light moving across a screen. The amount by whichthe spot is deflected from its initial position depends on the p.d. appliedto the terminals of the CRO and the range selected. The displacement iscalibrated in ‘volts per cm’. For example, if the spot is deflected 3 cmand the volts/cm switch is on 10 V/cm then the magnitude of the p.d. is3 cm ð 10 V/cm, i.e. 30 V (See Chapter 10 for more detail about elec-trical measuring instruments and measurements.)

Figure 2.3

2.5 Linear andnon-linear devices

Figure 2.3 shows a circuit in which current I can be varied by the variableresistor R2. For various settings of R2, the current flowing in resistorR1, displayed on the ammeter, and the p.d. across R1, displayed on thevoltmeter, are noted and a graph is plotted of p.d. against current. Theresult is shown in Figure 2.4(a) where the straight line graph passingthrough the origin indicates that current is directly proportional to the p.d.Since the gradient i.e. (p.d./current) is constant, resistance R1 is constant.A resistor is thus an example of a linear device.


Figure 2.4

If the resistor R1 in Figure 2.3 is replaced by a component such as alamp then the graph shown in Figure 2.4(b) results when values of p.d.are noted for various current readings. Since the gradient is changing, thelamp is an example of a non-linear device.

2.6 Ohm’s law Ohm’s law states that the current I flowing in a circuit is directly propor-tional to the applied voltage V and inversely proportional to the resistanceR, provided the temperature remains constant. Thus,

I =VR

or V = IR or R =VI

Problem 3. The current flowing through a resistor is 0.8 A whena p.d. of 20 V is applied. Determine the value of the resistance.

From Ohm’s law, resistance R D VI

D 200.8

D 2008

D 25Z

2.7 Multiples andsub-multiples

Currents, voltages and resistances can often be very large or very small.Thus multiples and sub-multiples of units are often used, as stated inchapter 1. The most common ones, with an example of each, are listedin Table 2.1

TABLE 2.1

Prefix Name Meaning Example

M mega multiply by 1 000 000(i.e., ð 106)

2 M D 2 000 000 ohms

k kilo multiply by 1000(i.e., ð 103)

10 kV D 10 000 volts

m milli divide by 1000(i.e., ð 10�3)

25 mA D 251000

A

D 0.025 amperes

µ micro divide by 1 000 000(i.e., ð 10�6)

50 µV D 501 000 000

V

D 0.000 05 volts


A more extensive list of common prefixes are given on page 972.

Problem 4. Determine the p.d. which must be applied to a 2 k

resistor in order that a current of 10 mA may flow.

Resistance R D 2 k D 2 ð 103 D 2000

Current I D 10 mA D 10 ð 10�3 A or 10

103or

10

1000A D 0.01 A

From Ohm’s law, potential difference, V D IR D �0.01�2000 D 20 V

Problem 5. A coil has a current of 50 mA flowing through it whenthe applied voltage is 12 V. What is the resistance of the coil?

Resistance, R D VI

D 1250 ð 10�3 D

12 ð 10350

D 12 00050

D 240Z

Problem 6. A 100 V battery is connected across a resistor andcauses a current of 5 mA to flow. Determine the resistance of theresistor. If the voltage is now reduced to 25 V, what will be thenew value of the current flowing?

Resistance R D VI

D 1005 ð 10�3 D

100 ð 1035

D 20 ð 103 D 20 kZ

Current when voltage is reduced to 25 V,

I D VR

D 2520 ð 103 D

25

20ð 10�3 D 1.25 mA

Problem 7. What is the resistance of a coil which draws a currentof (a) 50 mA and (b) 200 µA from a 120 V supply?

(a) Resistance R D VI

D 12050 ð 10�3

D 1200.05

D 12 0005

D 2 400Z or 2.4 kZ

(b) Resistance R D 120200 ð 10�6 D

120

0.0002

D 1200 0002

D 600 000Z or 600 kZ or 0.6 MZ

Further problems on Ohm’s law may be found in Section 2.12, problems 4to 7, page 21.


2.8 Conductors andinsulators

A conductor is a material having a low resistance which allows electriccurrent to flow in it. All metals are conductors and some examples includecopper, aluminium, brass, platinum, silver, gold and carbon.

An insulator is a material having a high resistance which does notallow electric current to flow in it. Some examples of insulators includeplastic, rubber, glass, porcelain, air, paper, cork, mica, ceramics andcertain oils.

2.9 Electrical power andenergy

Electrical power

Power P in an electrical circuit is given by the product of potentialdifference V and current I, as stated in Chapter 1. The unit of power isthe watt, W. Hence

P = V × I watts �2.1

From Ohm’s law, V D IRSubstituting for V in equation (2.1) gives:

P D �IR ð Ii.e. P = I 2R watts

Also, from Ohm’s law, I D VR

Substituting for I in equation (2.1) gives:

P D V ð VR

i.e. P =V 2

Rwatts

There are thus three possible formulae which may be used for calculatingpower.

Problem 8. A 100 W electric light bulb is connected to a 250 Vsupply. Determine (a) the current flowing in the bulb, and (b) theresistance of the bulb.

Power P D V ð I, from which, current I D PV

(a) Current I D 100250

D 1025

D 25

D 0.4 A

(b) Resistance R D VI

D 2500.4

D 25004

D 625Z


Problem 9. Calculate the power dissipated when a current of4 mA flows through a resistance of 5 k

Power P D I2R D �4 ð 10�32�5 ð 103D 16 ð 10�6 ð 5 ð 103 D 80 ð 10�3D 0.08 W or 80 mW

Alternatively, since I D 4 ð 10�3 and R D 5 ð 103 then from Ohm’s law,voltage V D IR D 4 ð 10�3 ð 5 ð 10�3 D 20 VHence, power P D V ð I D 20 ð 4 ð 10�3 D 80 mW

Problem 10. An electric kettle has a resistance of 30 . Whatcurrent will flow when it is connected to a 240 V supply? Findalso the power rating of the kettle.

Current, I D VR

D 24030

D 8 A

Power, P D VI D 240 ð 8 D 1920 W D 1.92 kWD power rating of kettle

Problem 11. A current of 5 A flows in the winding of an electricmotor, t

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