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Power and Energy Series 62 Lightning Electromagnetics L Lightning i g h t n in Electromagnetics g Lightning research is an interdisciplinary subject where several branches of engineering and Vernon Cooray is a leading physics converge. Lightning Electromagnetics is a book that caters for the needs of both professor of lightning research E physicists and engineers. It provides: at Uppsala University. He has le published more than 270 scientific • The physicist with information on how to simulate: the charge generation in articles on lightning and electrical c thunderclouds, different discharge processes in air that ultimately lead to a lightning discharges. Two other books tr flash, and the mechanism through which energetic radiation in the form of X-rays and edited by him on this subject, o Gamma rays are produced by lightning flashes. The Lightning Flash and Lightning m Protection, were published in • The power engineer with several numerical tools to study the interaction of lightning 2003 and 2009, respectively, a flashes with power transmission and distribution systems. by the IET, London. g n • The telecommunication engineer with numerical procedures with which to calculate e the electromagnetic fields generated by lightning flashes and their interactions with t overhead and underground telecommunication systems. i c • The electromagnetic specialist with the basic theory necessary to simulate the s propagation of lightning electromagnetic fields over the surface of the Earth. • The atmospheric scientist with numerical procedures to quantify interactions between lightning flashes and the Earth’s atmosphere, including the production of NOx by lightning flashes occurring in the atmosphere. This book also contains a chapter on the stimulation of visual phenomena in humans by electromagnetic fields of lightning flashes, which is essential reading for those who are interested in ball lightning. Edited by Vernon Cooray E d ite d b y The Institution of Engineering and Technology C o www.theiet.org o ra 978-1-84919-215-6 y Lightning Electromagnetics.indd 1 21/05/2012 13:34:50 IET POWER AND ENERGY SERIES 62 Lightning Electromagnetics Other volumes in this series: Volume 1 Power circuit breaker theory and design C.H. Flurscheim (Editor) Volume 4 Industrial microwave heating A.C. Metaxas and R.J. Meredith Volume 7 Insulators for high voltages J.S.T. Looms Volume 8 Variable frequency AC motor drive systems D. Finney Volume 10 SF6 switchgear H.M. Ryan and G.R. Jones Volume 11 Conduction and induction heating E.J. Davies Volume 13 Statistical techniques for high voltage engineering W. Hauschild and W. Mosch Volume 14 Uninterruptible power supplies J. Platts and J.D. St Aubyn (Editors) Volume 15 Digital protection for power systems A.T. Johns and S.K. Salman Volume 16 Electricity economics and planning T.W. Berrie Volume 18 Vacuum switchgear A. Greenwood Volume 19 Electrical safety: a guide to causes and prevention of hazards J. Maxwell Adams Volume 21 Electricity distribution network design, 2nd edition E. Lakervi and E.J. Holmes Volume 22 Artifi cial intelligence techniques in power systems K. Warwick, A.O. Ekwue and R. Aggarwal (Editors) Volume 24 Power system commissioning and maintenance practice K. Harker Volume 25 Engineers’ handbook of industrial microwave heating R.J. Meredith Volume 26 Small electric motors H. Moczala et al. Volume 27 AC-DC power system analysis J. Arrillaga and B.C. Smith Volume 29 High voltage direct current transmission, 2nd edition J. Arrillaga Volume 30 Flexible AC transmission systems (facts) Y-H. Song (Editor) Volume 31 Embedded generation N. Jenkins et al. Volume 32 High voltage engineering and testing, 2nd edition H.M. Ryan (Editor) Volume 33 Overvoltage protection of low-voltage systems, revised edition P. Hasse Volume 34 The lightning fl ash V. Cooray Volume 36 Voltage quality in electrical power systems J. Schlabbach et al. Volume 37 Electrical steels for rotating machines P. Beckley Volume 38 The electric car: development and future of battery, hybrid and fuel-cell cars M. Westbrook Volume 39 Power systems electromagnetic transients simulation J. Arrillaga and N. Watson Volume 40 Advances in high voltage engineering M. Haddad and D. Warne Volume 41 Electrical operation of electrostatic precipitators K. Parker Volume 43 Thermal power plant simulation and control D. Flynn Volume 44 Economic evaluation of projects in the electricity supply industry H. Khatib Volume 45 Propulsion systems for hybrid vehicles J. Miller Volume 46 Distribution switchgear S. Stewart Volume 47 Protection of electricity distribution networks, 2nd edition J. Gers and E. Holmes Volume 48 Wood pole overhead lines B. Wareing Volume 49 Electric fuses, 3rd edition A. Wright and G. Newbery Volume 50 Wind power integration: connection and system operational aspects B. Fox et al. Volume 51 Short circuit currents J. Schlabbach Volume 52 Nuclear power J. Wood Volume 53 Condition assessment of high voltage insulation in power system equipment R.E. James and Q. Su Volume 55 Local energy: distributed generation of heat and power J. Wood Volume 56 Condition monitoring of rotating electrical machines P. Tavner, L. Ran, J. Penman and H. Sedding Volume 57 The control techniques drives and controls handbook, 2nd edition B. Drury Volume 58 Lightning protection V. Cooray (Editor) Volume 59 Ultracapacitor applications J.M. Miller Volume 63 Energy storage for power systems, 2nd edition A. Ter-Gazarian Volume 65 Protection of electrical distribution networks, 3rd edition J. Gers Volume 905 Power system protection, 4 volumes Lightning Electromagnetics Edited by Vernon Cooray The Institution of Engineering and Technology Published by The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). © 2012 The Institution of Engineering and Technology First published 2012 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the author and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the author nor the publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the author to be identifi ed as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library ISBN 978-1-84919-215-6 (hardback) ISBN 978-1-84919-216-3 (PDF) Typeset in India by MPS Limited Printed and bound in the UK by CPI Group (UK) Ltd, Croydon, CR40 4YY ‘In appreciation of the work of a merciful God’ Acknowledgements First of all I wish to thank all my colleagues who have spent a good deal of their free time to write the chapters of this book. My special thanks to Prof. Farhad Rachidi and Prof. Carlo Alberto Nucci who helped me in numerous ways, as both friends and colleagues, in compiling the material for this and the previous lightning books published by the IET. I wish to express my sincere thanks to Ms. Helen Langley and Ms. Lisa Reading of the IET publishing department who were at all times prepared to listen to my suggestions and accommodate submission delays and last-minute changes without complaint. My thanks to the donation fund from B. John and Svea Andersson to Uppsala University and to the Swedish Natural Science Research Council (Vetenskapsrådet) for supporting my research over the past 30 years at Uppsala University; without this support I would not be in a position to be the editor of this book. Finally, I express my gratitude to my mother, Dorothy Cooray, who taught me the value of knowledge, to my wife, Ruby Cooray, who provided me with encouragement and ensured that I had ample time to gain knowledge through my research activities, to my sons, Gerald and Charith, and their respective wives, Shamara and Anna, who were always there to share this knowledge and to my granddaughters, Sanduni Cooray and Ella Cooray, who, with their curious young minds, convinced me of the value of documenting the knowledge gathered for the future generations. The work I have done in this book is dedicated to my father, the late Gerard Cooray. Contents 1 Basic electromagnetic theory – A summary 1 1.1 Introduction 1 1.2 The nomenclature 1 1.3 Coordinate systems 2 1.4 Important vector relationships 2 1.4.1 The scalar product of two vectors 2 1.4.2 The vector product of two vectors 4 1.4.3 Vector fi eld 5 1.4.4 The nabla operator and its operations 5 1.4.4.1 The gradient of a scalar function 5 1.4.4.2 The divergence of a vector fi eld 6 1.4.4.3 The curl of a vector fi eld 7 1.4.5 Important vector identities 8 1.4.6 R elationship between the curl of a vector fi eld and the line integral of that vector fi eld around a closed path 8 1.4.7 T he fl ux of a vector fi eld through a surface 8 1.4.8 R elationship between the divergence of a vector fi eld and the fl ux of that vector fi eld through a closed surface 9 1.4.9 Divergence theorem 9 1.4.10 Stokes theorem 10 1.5 Static electric fi elds 10 1.5.1 Coulomb’s law 10 1.5.2 Electric fi eld produced by static charges is a conservative fi eld 12 1.5.3 Gauss’s law 13 1.5.4 Electric scalar potential 14 1.5.5 Poisson and Laplace equations 15 1.5.6 Concept of images 16 1.5.7 Electrostatic boundary conditions 17 1.6 Static magnetic fi elds 19 1.6.1 Electric current 19 1.6.2 Conservation of electric charge 20 1.6.3 Re-distribution of excess charge placed inside a conducting body 22 1.6.4 Magnetic fi eld produced by a current element – Biot–Savart’s law 22 viii Lightning electromagnetics 1.6.5 Gauss’s law for magnetic fi elds 23 1.6.6 Ampere’s law 24 1.6.7 Boundary conditions for the static magnetic fi eld 25 1.6.8 Vector potential 26 1.6.8.1 Vector potential of a current distribution 27 1.6.8.2 Vector potential due to a current element dl 28 1.6.9 Force on a charged particle 29 1.7 Energy density of an electric fi eld 29 1.8 Electrodynamics – Time-varying electric and magnetic fi elds 30 1.8.1 Faraday’s law 31 1.8.2 Maxwell’s modifi cation of Ampere’s law – the displacement current term 33 1.8.3 Energy density in a magnetic fi eld 34 1.9 Summary of the laws of electricity 35 1.10 Wave equation 36 1.11 Maxwell’s prediction of electromagnetic waves 37 1.12 Plane wave solution 39 1.12.1 The electric fi eld of the plane wave 39 1.12.2 The magnetic fi eld of the plane wave 39 1.12.3 Energy transported by a plane wave – Poynting’s theorem 41 1.13 Maxwell’s equations and plane waves in different media (summary) 42 1.13.1 Vacuum 42 1.13.2 Isotropic and linear dielectric and magnetic media 43 1.13.3 Conducting media 44 1.14 Retarded potentials 46 1.14.1 Vector potential of a current element 48 1.15 Electromagnetic fi elds of a current element – Electric dipole 48 1.16 Electromagnetic fi elds of a lightning return stroke 51 Further reading 53 2 Application of electromagnetic fi elds of an accelerating charge to obtain the electromagnetic fi elds of a propagating current pulse 55 2.1 Introduction 55 2.2 Electromagnetic fi elds of a moving charge 55 2.3 Basic equations pertinent to the radiation and velocity fi eld of a propagating current pulse 56 2.4 Electromagnetic fi elds generated by a current pulse propagating with uniform velocity and without attenuation 57 2.4.1 The electric radiation fi eld generated by the initiation of the current 58 2.4.2 The electric radiation fi eld generated by the termination of the current 58 Contents ix 2.4.3 The static fi eld generated by the accumulation of charge at the point of current initiation 58 2.4.4 The static fi eld generated by the accumulation of charge at the point of current termination 58 2.4.5 The velocity fi eld generated by the propagating current pulse 59 2.4.6 The magnetic radiation fi eld generated by the initiation of the current 59 2.4.7 The magnetic radiation fi eld generated by the termination of the current 59 2.4.8 Magnetic velocity fi eld generated by the current pulse 59 2.5 Electromagnetic fi elds generated by a current pulse propagating with non-uniform velocity and with attenuation 60 2.5.1 The electric radiation fi eld generated by the initiation of the current 61 2.5.2 The electric radiation fi eld generated by the termination of the current 61 2.5.3 The static fi eld generated by the accumulation of charge at the point of current initiation 61 2.5.4 The static fi eld generated by the accumulation of charge at the point of current termination 61 2.5.5 The velocity fi eld generated as the current pulse propagates along the channel element 61 2.5.6 The magnetic radiation fi eld generated by the initiation of the current 62 2.5.7 The magnetic radiation fi eld generated by the termination of the current 62 2.5.8 Magnetic velocity fi eld generated as the current pulse propagates along the channel element 62 2.5.9 Electric radiation fi eld generated due to the attenuation of the current along the channel 62 2.5.10 Magnetic radiation fi eld generated due to the attenuation of the current along the channel 63 2.5.11 Static fi eld generated by the charge deposited along the channel by the current pulse 63 2.5.12 Electric radiation fi eld generated due to the variation of current velocity along the channel 64 2.5.13 Magnetic radiation fi eld generated due to the variation of current velocity along the channel 64 2.6 Concluding remarks 65 References 65 3 Basic discharge processes in the atmosphere 67 3.1 Introduction 67 3.2 Electron avalanche 68