Lightning Physics and Lightning Protection

Lightning Physics and Lightning Protection Protection Lightning EM M Bazelyan Bazelyan and Vu Yu P Raizer Raker loP I...

Author: Eduard M. Bazelyan | Yuri P. Raizer

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Lightning Physics and Lightning Protection Protection Lightning

EM M Bazelyan Bazelyan and

Vu Yu P Raizer Raker

loP IOP Institute Institute of Physics Physics Publishing Publishing Bristol Bristol and and Philadelphia Philadelphia

Copyright © 2000 IOP Publishing Ltd.

IOP Publishing Publishing Ltd Ltd 2000 2000 © lOP All rights reserved. No part this publication may be All rights reserved. No part of of this publication may be reproduced, reproduced, stored stored in in aa retrieval system system or or transmitted transmitted in in any any form form or or by by any any means, means, electronic, electronic, retrieval mechanical, photocopying, recording recording or of mechanical, photocopying, or otherwise, otherwise, without without the the prior prior permission permission of the publisher. publisher. Multiple Multiple copying copying is is permitted permitted in in accordance accordance with with the terms of of the the terms licences issued issued by by the the Copyright Copyright Licensing Licensing Agency Agency under under the terms of of its its agreement agreement licences the terms with the the Committee Committee of of Vice-Chancellors Vice-Chancellors and and Principals. Principals. with

British Library Cataloguing-in-Publication Cataloguing-in-PublicationData A catalogue catalogue record for this this book is available available from from the the British British Library. Library. A record for book is ISBN 00 7503 0477 0477 44 ISBN

Library of of Congress Cataloging-in-Publication Data are available

Publisher: Nicki Dennis Publisher: Nicki Dennis Commissioning Editor: Editor: John John Navas Navas Commissioning Production Production Editor: Editor: Simon Simon Laurenson Laurenson Production Control: Control: Sarah Sarah Plenty Plenty Production Cover Design: Design: Victoria Victoria Le Le Billon Billon Cover Marketing Executive: Executive: Colin Colin Fenton Fenton Marketing Published by Institute wholly owned by The Published by Institute of of Physics Physics Publishing, Publishing, wholly owned by The Institute Institute of of Physics, Physics, London London Institute Institute of of Physics Physics Publishing, Publishing, Dirac Dirac House, House, Temple Temple Back, Back, Bristol Bristol BSI BS1 6BE, 6BE, UK UK US Office: Institute Institute of of Physics Physics Publishing, Publishing, The The Public Public Ledger Ledger Building, Building, Suite Suite 1035, 1035, US Office: 150 South South Independence Independence Mall Mall West, West, Philadelphia, Philadelphia, PA PA 19106, 19106, USA USA 150

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Typeset Technical, Bristol Bristol Typeset in in 10/12pt 10/12pt Times Times by by Academic Academic + Technical, Printed the UK W Arrowsmith Arrowsmith Ltd, Ltd, Bristol Bristol Printed in in the UK by by JJ W


Contents Contents

Preface

ix 1 11 55 66 99 11 11 11 11 11 11 12 12 12 12 17 17 18 18 19 19

11 Introduction: Introduction: lightning, lightning, its destructive destructive effects and and protection protection 1.1 1.1 Types Types of of lightning lightning discharge discharge 1.2 1.2 Lightning Lightning discharge discharge components components 1.3 1.3 Basic Basic stages stages of of aa lightning lightning spark spark 104 1.4 Continuous Continuous and and stepwise stepwise leaders leaders 1.5 1.5 Lightning Lightning stroke stroke frequency frequency 1.5.1 1.5.1 Strokes Strokes at at terrestrial terrestrial objects objects 1.5.2 1.5.2 Human Human hazard hazard 1.6 1.6 Lightning Lightning hazards hazards 1.6.1 1.6.1 A direct direct lightning lightning stroke stroke 1.6.2 1.6.2 Induced Induced overvoltage overvoltage 1.6.3 1.6.3 Electrostatic Electrostatic induction induction 1.604 1.6.4 High High potential potential infection infection 1.6.5 1.6.5 Current Current inrush inrush from from aa spark spark creeping creeping along along the the earth's earth’s surface surface 1.6.6 1.6.6 Are Are lightning lightning protectors protectors reliable? reliable? 1.7 1.7 Lightning Lightning as as aa power power supply supply 1.8 1.8 To To those those intending intending to to read read on on References References

20 20 21 21 23 23 24 24 26 26

The streamer-leader process process in a long spark spark 2 The What aa lightning lightning researcher researcher should should know know about about aa long long spark spark 2.1 2.1 What 2.2 A A long long streamer streamer 2.2 2.2.1 The The streamer streamer tip tip as as an an ionization ionization wave wave 2.2.1 2.2.2 Evaluation Evaluation of of streamer streamer parameters parameters 2.2.2 2.2.3 Current and and field field in in the the channel channel behind behind the the tip tip 2.2.3 Current 2.2.4 Gas Gas heating heating in in aa streamer streamer channel channel 2.204

27 28 28 32 32 32 32 34 34 39 39 42 42 vV


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2.2.5 Electron-molecular Electron-molecular reactions reactions and and plasma plasma decay decay in in 2.2.5 cold air air cold 2.2.6 Final Final streamer streamer length length 2.2.6 2.2.7 Streamer Streamer in in aa uniform uniform field field and and in in the the 'absence' ‘absence’ of of 2.2.7 electrodes electrodes 2.3 The The principles of aa leader leader process principles of process 2.3 heating 2.3.1 The The necessity necessity of of gas gas heating 2.3.1 2.3.2 The The necessity of aa streamer streamer accompaniment accompaniment necessity of 2.3.2 mechanism 2.3.3 Channel Channel contraction contraction mechanism 2.3.3 velocity 2.3.4 Leader Leader velocity 2.3.4 2.4 The The streamer streamer zone zone and and cover cover 2.4 2.4.1 Charge Charge and and field field in in aa streamer streamer zone zone 2.4.1 2.4.2 Streamer Streamer frequency frequency and and number number 2.4.2 2.4.3 Leader Leader tip tip current current 2.4.3 2.4.4 Ionization Ionization processes in the the cover cover 2.4.4 processes in 2.5 A A long long leader leader channel channel 2.5 2.5.1 plasma state 2.5.1 Field Field and and the the plasma state 2.5.2 Energy Energy balance and similarity similarity to to an an arc arc 2.5.2 balance and 2.6 Voltage Voltage for for aa long long spark spark 2.6 2.7 A A negative negative leader leader 2.7 References References Available lightning lightning data 3 Available 3.1 Atmospheric Atmospheric field field during during aa lightning lightning discharge discharge 3.1 3.2 3.2 The The leader leader of of the the first first lightning lightning component component 3.2.1 3.2.1 Positive Positive leaders leaders 3.2.2 Negative leaders 3.2.2 Negative leaders 3.3 The The leaders leaders of of subsequent subsequent lightning lightning components components 3.3 3.4 Lightning Lightning leader leader current current 3.4 3.5 3.5 Field Field variation variation at at the the leader leader stage stage 3.6 Perspectives of remote measurements 3.6 Perspectives of remote measurements 3.6.1 3.6.1 Effect Effect of of the the leader leader shape shape 3.6.2 3.6.2 Effect Effect of of linear linear charge charge distribution distribution return stroke 3.7 Lightning Lightning return stroke 3.7 Neutralization wave 3.7.1 Neutralization wave velocity velocity 3.7.1 3.7.2 Current Current amplitude amplitude 3.7.2 3.7.3 Current Current impulse impulse shape shape and and time time characteristics characteristics 3.7.3 3.7.4 Electromagnetic Electromagnetic field field 3.7.4 3.8 Total Total lightning lightning flash flash duration duration and and processes in the the 3.8 processes in intercomponent intercomponent pauses pauses normalized energy 3.9 Flash Flash charge charge and and normalized energy 3.9 3.10 3.10 Lightning Lightning temperature temperature and and radius radius 3.11 3.1 1 What What can can one one gain gain from from lightning lightning measurements? measurements? References References


45 45 48 48

53 53 59 59 59 59 61 61 64 64 66 66 67 67 67 67 70 70 71 71 73 73 75 75 75 75 79 79 81 81 83 83 88 88

90 91 91 94 94 94 94 96 96 98 98 100 100 102 102 107 107 107 107 111 111 115 115 116 116 117 117 122 122 126 126 129 129 131 131 132 132 134 134 136 136

Contents Contents

vii vii

4 Physical processes in aa lightning discharge 4.1 An An ascending ascending positive positive leader leader 4.1 4.1.1 The 4.1.1 The origin origin 4.1.2 Leader 4.1.2 Leader development development and and current current 4.1.3 4.1.3 Penetration Penetration into into the the cloud cloud and and halt halt 4.1.4 Leader Leader branching branching and and sign sign reversal reversal 4.1.4 4.2 Lightning Lightning excited excited by by an an isolated isolated object object 4.2 4.2.1 A 4.2.1 A binary binary leader leader 4.2.2 Binary 4.2.2 Binary leader leader development development 4.3 The 4.3 The descending descending leader leader of of the the first first lightning lightning component component 4.3.1 The 4.3.1 The origin origin in in the the clouds clouds 4.3.2 Negative Negative leader leader development development and and potential potential transport transport 4.3.2 4.3.3 4.3.3 The The branching branching effect effect 4.3.4 4.3.4 Specificity Specificity of of aa descending descending positive positive leader leader 4.3.5 4.3.5 A A counterleader counterleader 4.4 4.4 Return Return stroke stroke 4.4.1 4.4.1 The The basic basic mechanism mechanism 4.4.2 Conclusions Conclusions from from explicit explicit solutions solutions to to long long line line 4.4.2 equations equations 4.4.3 4.4.3 Channel Channel transformation transformation in in the the return return stroke stroke 4.4.4 Return stroke as a channel transformation wave 4.4.4 Return stroke as a channel transformation wave 4.4.5 problems and their solution 4.4.5 Arising Arising problems and approaches approaches to to their solution 4.4.6 The The return return stroke stroke of of aa positive positive lightning lightning 4.4.6 4.5 4.5 Anomalously Anomalously large large current current impulses impulses of of positive positive lightnings lightnings 4.6 Stepwise Stepwise behaviour of aa negative negative leader leader 4.6 behaviour of 4.6.1 The The step step formation formation and and parameters parameters 4.6.1 4.6.2 Energy Energy effects effects in in the the leader leader channel channel 4.6.2 4.7 The The subsequent subsequent components. components. The The M-component M-component 4.7 4.8 Subsequent Subsequent components. components. The The problem problem of of aa dart dart leader leader 4.8 4.8.1 A streamer streamer in in aa 'waveguide'? ‘waveguide’? 4.8.1 4.8.2 The The non-linear non-linear diffusion diffusion wave wave front front 4.8.2 4.8.3 4.8.3 The The possibility possibility of of diffusion-to-ionization diffusion-to-ionization wave wave transformation transformation 4.8.4 The The ionization ionization wave wave in in aa conductive conductive medium medium 4.8.4 4.8.5 4.8.5 The The dart dart leader leader as as aa streamer streamer in in aa 'nonconductive ‘nonconductive waveguide’ waveguide' 4.9 Experimental Experimental checkup checkup of of subsequent subsequent component component theory theory 4.9 References References

138 138 138 138 138 141 141 144 144 148 148 150 150 150 150 152 152 158 158 158 158 161 161 166 166 168 168 169 169 171 171 171 171

5 Lightning Lightning attraction attraction by objects objects 5.1 The The equidistance equidistance principle principle 5.1 5.2 The The electrogeometric electrogeometric method method 5.2 5.3 The The probability probability approach approach to to finding finding the the stroke stroke point point 5.3 5.4 Laboratory Laboratory study study of of lightning lightning attraction attraction 5.4

222 222


175 175 181 181 185 185 190 190 194 194 195 195 197 197 197 197 199 199 202 202 207 207 207 207 209 209 212 212 213 213 215 215 217 217 219 219 223 223 226 226 228 228 232 232

viii Vlll

Contents Contents

..I

Extrapolation Extrapolation to to lightning lightning On the the attraction attraction mechanism mechanism of of external external field field On How point of How lightning lightning chooses chooses the the point of stroke stroke Why more effective Why are are several several lightning lightning rods rods more effective than than one? one? Some technical parameters parameters of Some technical of lightning lightning protection protection 5.9.1 protection zone 5.9.1 The The protection zone 5.9.2 protection angle 5.9.2 The The protection angle of of aa grounded grounded wire wire 5.10 5.10 Protection Protection efficiency efficiency versus versus the the object object function function 5.11 5.11 Lightning Lightning attraction attraction by by aircraft aircraft 5.12 5.12 Are Are attraction attraction processes processes controllable? controllable? 5.13 the lightning misses the the object 5.13 If If the lightning misses object References References

236 236 239 239 24 1 241 241 247 249 249 249 249 25 1 251 252 252 255 255 259 259 263 263 264 264

6 Dangerous Dangerous lightning lightning effects effects on modern structures structures 6.1 6.1 Induced Induced overvoltage overvoltage 6.1.1 6.1.1 'Electrostatic' ‘Electrostatic’ effects effects of of cloud cloud and and lightning lightning charges charges 6.1.2 6.1.2 Overvoltage Overvoltage due due to to lightning lightning magnetic magnetic field field 6.2 6.2 Lightning Lightning stroke stroke at at aa screened screened object object 6.2.1 A A stroke stroke at at the the metallic metallic shell shell of of aa body body 6.2.1 6.2.2 6.2.2 How How lightning lightning finds finds its its way way to to an an underground underground cable cable 6.2.3 6.2.3 Overvoltage Overvoltage on on underground underground cable cable insulation insulation 6.2.4 6.2.4 The The action action of of the the skin-effect skin-effect 6.2.5 6.2.5 The The effect effect of of cross cross section section geometry geometry 6.2.6 Overvoltage Overvoltage in in aa double double wire wire circuit circuit 6.2.6 6.2.7 6.2.7 Laboratory Laboratory tests tests of of objects objects with with metallic metallic sheaths sheaths 6.2.8 Overvoltage Overvoltage in in aa screened screened multilayer multilayer cable cable 6.2.8 6.3 6.3 Metallic Metallic pipes pipes as as aa high high potential potential pathway pathway 6.4 6.4 Direct Direct stroke stroke overvoltage overvoltage 6.4.1 6.4.1 The The behaviour behaviour of of aa grounding grounding electrode electrode at at high high current current impulses impulses 6.4.2 6.4.2 Induction Induction emf emf in in an an affected affected object object 6.4.3 6.4.3 Voltage Voltage between between the the affected affected and and neighbouring neighbouring objects objects 6.4.4 Lines Lines with with overhead overhead ground-wires ground-wires 6.4.4 6.5 6.5 Concluding Concluding remarks remarks References References

265 267 267 267 267 270 270 272 272 272 272 274 274 277 277 283 283 285 285 290 290 29 1 291 294 294 296 296 300 300

5.5 5.5 5.6 5.6 5.7 5.7 5.8 5.8 5.9 5.9


300 300 305 305 307 307 3 14 314 318 318 320 320

Preface Preface

Today, we we know sufficiently much much about about lightning lightning to to feel feel free free from from the the mystic Today, know sufficiently mystic primitive people. to create fears of primitive people. We We have have learned learned to create protection protection technologies technologies fears of and to to make make power transmission lines, lines, skyscrapers, skyscrapers, ships, ships, aircraft, aircraft, and and spacespaceand power transmission craft craft less less vulnerable vulnerable to to lightning. lightning. Yes, Yes, the the danger danger is is getting getting less less but but it it still still exists! technical progress, progress, lightning exists! With With every every step step of of the the technical lightning arms arms itself itself with with new weapon weapon to to continue continue the the war war by its own own rules against the the self-confident self-confident aa new by its rules against we improve improve our our machines machines and and stuff stuff them them with with electronics electronics in in an an engineer. As we engineer. attempt to to replace human beings, lightning acts acts in in an an ever ever refined refined manner. It attempt replace human beings, lightning manner. It takes by surprise takes us us by surprise where where we we do do not not expect expect it, it, making making us us feel feel helpless helpless again again for some some time. for time. We do do not not intend intend to in this set of of universal universal lightning lightning We to present present in this book book aa set protection rules. rules. Such universal antitask would would be be as as futile futile as as advertising advertising aa universal antiprotection Such aa task biotic biotic lethal lethal to to every every harmful harmful microbe. microbe. The The world world is is changeable, changeable, and and today's today’s panacea often becomes becomes aa useless useless pill pill even even before advertising sheet sheet fades. fades. panacea often before the the advertising Technical unawares. Improvement so far far failed failed to to take take lightning lightning unawares. Improvement Technical progress progress has has so and miniaturization of devices devices increase increase our our concern concern about about the the refined refined and miniaturization of destructive behaviour behaviour of of lightning, lightning, but prophet is is able able to to foresee foresee all all of of destructive but no no prophet its its destructive destructive effects. effects. We plan to We do d o not not plan to discuss discuss in in detail detail all all available available information information on on lightlightning. There reference of reference ning. There are are already already some some excellent excellent books books providing providing all all sort sort of data, edited by by R RH H Golde Golde and and data, among among them them the the two two volumes volumes of of Lightning edited by M think it Lightning Discharge by M Uman. Uman. Our Our aim aim is is different. different. We We think it important important up-to-date physical physical concepts to to give give the the reader reader some some clear, clear, up-to-date concepts of of lightning lightning books referred referred to. development, development, which which cannot cannot be be found found in in the the books to. These These will will serve as as aa basis for the the researcher researcher and and engineer engineer to to judge the properties properties of of serve basis for judge the this this tremendous tremendous gas gas discharge discharge phenomenon. phenomenon. Then Then we we shall shall discuss discuss the the nature of various various hazardous manifestations of of lightning, lightning, focusing focusing on on the the nature of hazardous manifestations physical mechanisms of of interaction interaction between lightning and and an an affected affected between lightning physical mechanisms construction. The The results of this this consideration consideration will will further further be used to to estimate estimate be used construction. results of ix

IX


xX

Preface

the the effectiveness effectiveness of of conventional conventional protective protective measures measures and and to to predict predict technical technical means means for for their their improvement. improvement. We We give, give, wherever wherever possible, possible, technical technical advice advice and and recommendations. recommendations. Our Our main main goal, goal, however, however, is is to to help help the the reader reader to to make make his his own own predictions predictions by by providing providing information information on on the the whole whole arsenal arsenal of of potentionally potentionally hazardous hazardous effects effects of of lightning lightning on on aa particular particular construction. construction. We We have have often often witnessed witnessed situations situations when when an an engineer engineer was was trying trying hard hard to to 'impose' ‘impose’ this this or or that that protective protective device device on on an an operating operating experimental experimental structure structure which which resisted resisted his his unnatural unnatural efforts. efforts. Ideally, Ideally, the the designer designer must must be be able able to to foresee foresee all all details details of of the the relationship relationship between between lightning lightning and and the the construction construction being being designed. designed. It It is is only only in in this this case case that that lightning lightning protection protection can can become become functionally protective device be made functionally effective effective and and the the protective device can can be made compatible compatible with with the the construction construction elements. elements. If to follow If an an engineer engineer is is determined determined to follow this this approach, approach, both both expedient expedient and and book useful. useful. It well-grounded, well-grounded, he he will will find find this this book It is is aa natural natural extension extension of of our our previous book Discharge, published previous book Spark Discharge, published by by CRC CRC Press Press in in 1997, 1997, which which dealt dealt with streamer-leader breakdown of with streamer-leader breakdown of long long gas gas gaps. gaps. The The streamer-leader streamer-leader process is part of any any lightning lightning discharge discharge when when aa plasma plasma spark spark closes closes aa process is part of gigantic the destructive primarily gigantic air air gap. gap. Although Although the destructive effect effect of of lightning lightning is is primarily due to the return return stroke stroke which which follows follows the the leader, leader, it it is is the the leader leader that that due to the makes the discharge channel channel susceptible susceptible to to it. it. This This is is why we give give an an overview overview why we makes the discharge of the the streamer-leader streamer-leader process, focusing on on extremal extrema1 estimations estimations and and of process, focusing presenting presenting some some new new ideas. ideas. We We hope hope that that the the second second chapter chapter will will prove prove inforinformative even for for those those familiar familiar with our book of 1997. 1997. mative even with our book of Some results of the lightning investigation investigation run run in in the Krzhizhanovsky Some results of the lightning the Krzhizhanovsky Power used in the book. book. The Power Institute Institute are are used in the The authors authors would would like like to to thank thank Dr B BN Gorin and and Dr Dr A V V Shkilev Shkilev who who kindly kindly allowed allowed us the originals originals Dr N Gorin us to to use use the of to L N Smirnova L N Smirnova for for of lightning lightning photographs. photographs. We We are are also also grateful grateful to translation book. translation of of this this book.


Chapter 1

Introduction: Introduction: lightning, lightning, its its destructive effects and protection protection If you want want to to observe observe lightning, lightning, the the best best thing thing to to do do is to visit visit aa special special lightIf you is to lightning laboratory. laboratory. Such laboratories exist exist in in all all parts parts of of the the globe globe except except the the ning Such laboratories roof of of Antarctic. But But you you can can save on the the travel travel if if you you just just climb climb onto the roof Antarctic. save on onto the your own house to to give give aa good good field vision. Better, Better, fetch fetch your your camera. your own house field of of vision. camera. Even an an ordinary picture can can show details the the unaided unaided human human eye eye often often Even ordinary picture show details misses. You You might might as as well well sit sit back back in in your your favourite favourite armchair, armchair, having having misses. pulled it it up up to to aa window, window, preferably preferably one one overlooking overlooking an an open open space. The pulled space. The camera be fixed on the the window window sill. is nothing nothing else else to to do do but but camera can can be fixed on sill, There There is wait for night. wait for aa stormy stormy night. There is enough time time for the preparations preparations to to be be made made because because the the storm There is enough for the storm will will be be approaching approaching slowly. slowly. At At first, first, the the air air will will grow grow still, still, and and it it will will get get much much darker darker than than it it normally normally is is on on aa summer summer night. night. The The cloud cloud is is not not yet yet visible, visible, but but its its approach approach can can be be anticipated anticipated from from the the soundless soundless flashes flashes at at the the horizon. horizon. They They gradually gradually pull pull closer, closer, and and the the brightest brightest of of them them can can already already be be heard heard as as delayed yet amiable roaring. This may go delayed and and yet amiable roaring. T h s may go on on for for aa long long time. time. It It may may seem that the the cloud cloud has has stopped or turned turned away, away, but but suddenly the sky sky is is seem that stopped still still or suddenly the ripped open open by by aa fire blade. This This is is accompanied accompanied by by aa deafening deafening crash, crash, quite quite ripped fire blade. different from from aa cannon cannon shot because it it takes takes aa much much longer longer time. time. The The first first different shot because lightning discharge discharge is is followed followed by by many many others others falling falling out out of of the the ripped ripped lightning cloud. Some the ground ground while while others others keep keep on on crossing crossing the the sky, sky, competing competing cloud. Some strike strike the with with the the first first discharge discharge in in beauty beauty and and spark spark length. length. This This is is the the right right time time to to start start observations: observations: remove remove the the camera camera shutter shutter and and try try to to take take aa few few pictures. pictures.

1.1 1.1

Types of lightning discharge

The above above recommendation recommendation to to remove remove the the camera camera shutter shutter should should be be taken taken The literally. literally. Much Much information information on on lightning lightning has has been been obtained obtained from from photographs photographs taken taken with with aa preliminarily preliminarily opened opened objective objective lens. lens. It It is is important, important, however, however, that that 1


22

Introduction: lightning, lightning. its destructive effects and protection

Figure 1.1. A static the Ostankino static photograph photograph of of aa lightning lightning stroke stroke at at the Ostankino Television Television Tower Tower in in Moscow. Moscow.

no other other bright light source source should should be within the vision field field of of the the no bright light be present present within the vision camera lens. lens. The The film film can can then then be be exposed exposed for for many many minutes minutes until spark camera until aa spark finds be closed finds its its way way into into the the frame. frame. After After this, this, the the lens lens should should be closed with with the the shutshutter ready for ter and and the the camera camera should should be be set set ready for another another shot. shot. Experience Experience has has shown shown pictures taken that that at at least least one one third third of of pictures taken during during aa good good night night thunderstorm thunderstorm prove successful. prove successful. All be classified, photography, All lightning lightning discharges discharges can can be classified, even even without without photography, into intercloud discharges discharges and and ground ground strikes. strikes. The The frequency frequency into two two groups groups -- intercloud of higher than of the the former former is is two two or or three three times times higher than that that of of the the latter. latter. An An interintercloud cloud spark spark is is never never aa straight straight line, line, but but rather rather has has numerous numerous bends bends and and branchings. Normally, the the spark branchings. Normally, spark channel channel is is as as long long as as several several kilometres, kilometres, sometimes dozens dozens of of kilometres. kilometres. sometimes The length length of of aa lightning lightning spark spark that that strikes strikes the the ground ground can can be be defined defined The kilomore exactly. more exactly. The The average average cloud cloud altitude altitude in in Europe Europe is is close close to to three three kilometres. Spark the same metres. Spark channels channels have have about about the same average average length. length. Of Of course, course, this parameter is this parameter is statistically statistically variable, variable, because because aa discharge discharge from from aa charged charged cloud up to to 10 km and because of cloud centre centre may may start start at at any any altitude altitude up 10 km and because of aa large large number of spark spark bends. bends. The The latter latter are are observable observable even even with with the the unaided unaided number of eye. photograph, they they may may look A photophoto1.1). A eye. In In aa photograph, look strikingly strikingly fanciful fanciful (figure (figure 1.1). graph naked eye the graph can can show show another another important important feature feature inaccessible inaccessible to to the the naked eye -- the main branches which main bright bright spark spark reaching reaching the the ground ground has has numerous numerous branches which have have stopped their their development development at at various various altitudes. altitudes. A A single single branch branch may have stopped may have principal spark aa length 1.2). length comparable comparable with with that that of of the the principal spark channel channel (figure (figure 1.2). Branches used to to define Branches can can be be conveniently conveniently used define the the direction direction of of lightning lightning propagation. Like the direction propagation. Like aa tree, tree, aa lightning lightning spark spark branches branches in in the direction of of Copyright © 2000 IOP Publishing Ltd.

Types Tipes of of lightning discharge

3

Figure 1.2. branches. 1.2. A photograph photograph of of aa descending descending lightning lightning with with numerous numerous branches.

growth. In In addition addition to descending sparks sparks outgrowing outgrowing from from aa cloud cloud toward toward growth. to descending the the ground, ground, there there are are also also ascending ascending sparks sparks starting starting from from aa ground ground construcconstructo aa cloud cloud (figure (figure 1.3). 1.3). Their Their direction direction of of growth growth is is tion and and developing developing up to tion well by branches branches diverging upward. well indicated indicated by diverging upward. In In aa flat flat country, country, an an ascending ascending spark spark can can arise arise only only from from aa skyscraper skyscraper or or of ascending ascending sparks sparks of at at least least 100-200 100-200m and the aa tower tower of m high, high, and the number number of 90% of of all all sparks sparks that that grows with with the the building For example, example, over over 90% grows building height. height. For strike the 530-m high Ostankino Television Tower in Moscow are of the strike the 530-m high Ostankino Television Tower in Moscow are of the ascending was reported similar value value was reported for for the the 41O-m 410-m high high ascending type type [I]. [l]. A similar [2]. Buildings Buildings of of such such aa height Empire State State Building Building in in New City [2]. Empire New York York City height attacked can be be said said to to fire fire lightning lightning sparks sparks up at clouds clouds rather can up at rather than than to to be be attacked In mountainous mountainous regions, regions, ascending ascending sparks sparks have have been been observed observed by them. them. In by an illustration, illustration, we we can can cite cite reports reports of of storm storm for much much lower lower buildings. buildings. As an for observations made on the San Salvatore Salvatore Mount Mount in in Switzerland Switzerland [3]. [3]. The The observations made on the San m high high but most of of the the discharges discharges affecting affecting receiving tower there was only 70 m receiving tower there was only but most it were were of of the the ascending ascending type. type. it Skyscrapers and and television television towers are, however, however, quite quite scarce scarce on on the the Skyscrapers towers are, the researcher researcher has has aa natural desire to to construct, construct, in in the the right Earth. So the natural desire right Earth. place and for for aa short short time, time, aa spark-generating spark-generating tower tower of of his his own. own. For For this, this, place and small probe probe pulling thin grounded grounded wire wire is is launched launched towards towards aa aa small pulling up up aa thin Copyright © 2000 IOP Publishing Ltd.

4 4

Introduction: lightning, lightning, its destructive effects and protection

of an an ascending ascending lightning, lightning. Figure 1.3. A photograph photograph of

storm cloud cloud [4]. When When the the probe rises to to 200-300 200-300m above the the earth, earth, an an storm probe rises m above ascending ascending spark spark is is induced induced from from it. it. A discharge discharge artificially artificially induced induced in in the the atmosphere atmosphere is is often often referred referred to to as as triggered triggered lightning. lightning. To To raise raise the the chances chances for aa successful successful experiment, experiment, the the electric electric field field induced induced by by the the storm storm charges charges at at for the ground prior to the ground surface surface are are measured measured prior to the the launch. launch. The The probe probe is is triggered triggered when the the field field strength strength becomes becomes close close to to 200 200 V V/cm, which provides spark when jcm, which provides spark ignition in in 60-70% 60-70% of of launches launches [5]. [5]. ignition The jcm is the threshThe value value 200 200 V V/cm is two two orders orders of of magnitude magnitude smaller smaller than than the threshold value of E = = 30 30 kVjcm, kV/cm, at at which short air air gap gap with uniform field field is is old value of which aa short with aa uniform broken down broken down under under normal normal atmospheric atmospheric conditions. conditions. Clearly, Clearly, no no spark spark ignition be possible possible without without the the local by electric ignition would would be local field field enhancement enhancement by electric charges the wire. charges induced induced on on the the probe probe and and the wire. Below, Below, we we shall shall discuss discuss the the triggered discharge mechanism mechanism in in more detail. triggered discharge more detail. A field field detector detector on on the the Earth's Earth’s surface surface (it (it might might as as well well be be placed placed on on the the window of room) can the polarity polarity of the charge window of your your own own room) can easily easily determine determine the of the charge transported by the ground. polarity of transported by aa lightning lightning spark spark to to the ground. The The polarity of the the spark spark is by that is defined defined by that of of the the charge. charge. About About 90% 90% of of descending descending sparks sparks occurring occurring in negative charge, these are in Europe Europe during during summer summer storms storms carry carry aa negative charge, so so these are known known as negative negative descending descending sparks. sparks. The The other other descending descending sparks sparks are are positive. The as positive. The Copyright © 2000 IOP Publishing Ltd.

Lightning discharge components

55

proportion of positive sparks has been found found to to be somewhat larger larger in in proportion of positive sparks has been be somewhat tropical and subtropical regions, especially in winter, when it may be as tropical and subtropical regions, especially in winter, when it may be as large as as 50%. 50%. large There is is no no special special name name for for lightning lightning sparks sparks generated generated by aircraft There by aircraft during flights, flights, when when they they are are entirely entirely insulated insulated from from the the ground. ground. Such Such disdisduring charges arise arise fairly fairly frequently. frequently. A A modern modern aircraft aircraft experiences experiences at at least least one one charges lightning stroke stroke every every 3000 3000 flight flight hours. hours. Almost Almost half half of of the the strokes strokes start start lightning from the aircraft itself, itself, not not from from aa cloud. cloud. This This often often happens happens in in heap heap from the aircraft rather than than clouds clouds carrying carrying aa relatively small electric electric charge. charge. The The reason rather relatively small reason for aa discharge discharge from from aa large large ground-insulated ground-insulated object object is is principally principally the the same same for as from from aa grounded grounded object object and and is is due due to to the the electric electric field field enhancement enhancement by by as surface polarization polarization charge. charge. This This issue issue will will be be discussed discussed after after the the analysis analysis surface of ascending ascending sparks sparks in in section section 4.2. 4.2. of

1.2

Lightning discharge discharge components components Lightning

An observer can can notice lightning spark spark flicker flicker which, which, sometimes, sometimes, may may An observer notice aa lightning become become quite quite distinct. distinct. Even Even the the first first cinematographers cinematographers knew knew that that the the human eye could could distinguish distinguish between between two events only only if if they they occurred occurred with human eye two events with aa time s. Since Since lightning lightning flicker flicker is is observable, observable, the the time interval interval longer longer than than 0.1 0.1 s. pause between two current impulses impulses must must be be longer longer than 0.1 s. s. pause between two current than 0.1 A current-free current-free pause can be be measured measured quite quite accurately accurately by by exposing exposing aa A pause can moving film to to aa lightning lightning discharge. discharge. With With up-to-date up-to-date lenses lenses and and photophotomoving film graphic graphic materials, materials, one one can can obtain obtain aa good good 1mm 1 mm resolution resolution of of the the film. film, In In order by 1mm time period period of order to to displace displace an an image image by 1 mm over over aa time of 0.1 0.1 s, s, the the film film speed must must be be about about 1cm/s. 1 cmjs. It It can can be be achieved achieved by by manually manually moving moving the the speed film film keeping keeping the the camera camera lens lens open open (alas, (alas, an an electrically electrically driven driven camera camera is is unsuitable picture like the unsuitable for for this). this). Then, Then, with with some some luck, luck, one one may may get get aa picture like the one in in figure figure 104. 1.4. The The spark spark flashes flashes up and dims dims out out several several times. times. Unless Unless one up and the pause is too long, aa new flash follows follows the the previous previous trajectory; trajectory; otherwise, otherwise, the pause is too long, new flash the spark takes or totally totally new new path. the spark takes aa partially partially or path. multicomponent A with several A lightning lightning spark spark with several flashes flashes is is known known as as aa multicomponent spark. the first spark. One One may may suggest suggest that that the the channel channel of of the first component component formed formed in in unperturbed air air differs differs in in its its basic characteristics from from the the subsequent subsequent chanchanunperturbed basic characteristics nels, path through the ionized heated air. nels, if if they they take take exactly exactly the the same same path through the ionized and and heated air. The formation formation of of subsequent subsequent components components is is considered considered in in sections sections 4.7 4.7 and and The 4.8. Note only both 4.8. Note only that that multicomponent multicomponent sparks sparks are are usually usually negative, negative, both ascending number of ascending and and descending. descending. The The average average number of components components is is close close to to three, number may three, while while the the maximum maximum number may be be as as large large as as thirty. thirty. Generally, Generally, the the average duration duration of of aa lightning lightning flash flash is is 0.2 0.2 ss and and the maximum duration duration is is average the maximum 1-1.5ss [6], [6], so so it it is is not surprising that that the the eye eye can can sometimes sometimes distinguish distinguish 1-1.5 not surprising between between individual individual components. components. Positive Positive sparks sparks normally normally contain contain only only one one component. component.


6


The image image of of aa multicomponent multicomponent lightning lightning in in aa slowly slowly moving film. Figure 1.4. The moving film.

1.3

Basic Basic stages stages of a lightning lightning spark

The to aa spark The affinity affinity of of lightning lightning to spark discharge discharge was was demonstrated demonstrated by by Benjamin Benjamin Franklin as as far far back as the the 18th 18th century. century. Historically, Historically, basic spark elements elements Franklin back as basic spark were first first identified identified in in lightning, lightning, and and only only much later were they observed observed in in were much later were they laboratory sparks. sparks. This This is is easy easy to if one one recalls recalls that lightning laboratory to understand understand if that aa lightning spark has has aa much much greater greater length length and and takes longer time time to to develop, develop, so so spark takes aa longer that require the use of that its its optical optical registration registration does does not not require the use of sophisticated sophisticated equipequipment with with aa high high space space and and time time resolution. The first first streak streak photographs photographs ment resolution. The of lightning, lightning, taken taken in in the the 1930s 1930s by by aa simple simple camera camera with with aa mechanically mechanically of rotated rotated film film (Boys (Boys camera), camera), are are still still impressive impressive [7]. They They show show the the principal principal stages process -- the stages of of the the lightning lightning process the leader leader stage stage and and the the return return stroke. stroke. The The leader leader stage stage represents represents the the initiation initiation and and growth growth of of aa conductive conductive plasma channel between aa cloud between plasma channel -- aa leader leader -- between cloud and and the the earth earth or or between two two clouds. clouds. The The leader leader arises arises in in aa region region where where the the electric electric field field is is strong strong enough by electron However, it propagates enough to to ionize ionize the the air air by electron impact. impact. However, it mostly mostly propagates through region in in which which the the external external field field induced induced by cloud charge charge through aa region by the the cloud does not not exceed exceed several several hundreds of volts centimetre. In In spite spite of of this this it it does hundreds of volts per per centimetre. does propagate, which means means that there is is an an intensive intensive ionization ionization occurring occurring does propagate, which that there in its its tip tip region, region, changing changing the the neutral neutral air air to to aa highly conductive plasma. plasma. This This in highly conductive becomes possible because the leader becomes possible because the leader carries carries its its own own strong strong electric electric field field induced induced by by the the space space charge charge concentrated concentrated at at the the leader leader tip tip and and transported transported together that of metallic together with with it. it. A rough rough analogue analogue of of the the leader leader field field is is that of aa metallic needle connected with thin wire wire to to aa high high voltage voltage supply. supply. If If the the needle needle is is needle connected with aa thin sharp sharp enough, enough, the the electric electric field field in in the the vicinity vicinity of of its its tip tip will will be be very very strong strong Copyright © 2000 IOP Publishing Ltd.

Basic stages of of a lightning spark

77

even even at at aa relatively relatively low low voltage. voltage. Imagine Imagine now now that that the the needle needle is is falling falling down down on on to to the the earth, earth, pulling pulling the the wire wire behind behind it. it. The The strong strong field field region, region, in in which which the the air air molecules molecules become become ionized, ionized, will will move move down down together together with with the the needle. needle. A A lightning lightning spark spark has has no no wire wire at at its its disposal. disposal. The The function function of of aa conductor conductor connecting the the leader leader tip tip to to the the starting starting point point of of the the discharge discharge is is performed performed by by connecting the the leader leader plasma plasma channel. channel. It It takes takes aa fairly fairly long long time time for for aa leader leader to to develop develop up up to to 0.01 0.01 s, s, which which is is eternity eternity in in the the time time scale scale of of fast fast processes processes involving involving an an electric impulse impulse discharge. discharge. During During this this period period of of time, time, the the leader leader plasma plasma electric must be be maintained maintained highly highly conductive, conductive, and and this this may may become become possible possible only only if if must the gas gas is is heated heated up up to to an an electric electric arc arc temperature, temperature, i.e. i.e. above above 5000-6000K. 5000-6000 K. the The of the the channel channel energy energy balance balance necessary necessary for for the the heating heating and and comcomThe problem problem of pensation for for losses losses is is aa key key one one in in leader leader theory. theory. It It is is discussed discussed in in chapters chapters 22 pensation and 4, 4, as as applied applied to to various various kinds kinds of of lightning lightning discharge. discharge. and A leader leader is is an an indispensable indispensable element element of of any any spark. spark. The The initial initial and and all all A subsequent components of aa flash flash begin begin with with aa leader leader process. process. Although Although its its subsequent components of mechanism may may vary vary with with the the spark polarization, propagation propagation direction direction mechanism spark polarization, and the the serial number of the component, component, the the process process remains remains essentially and serial number of the essentially the same. This of aa highly channel due due the same. This is is the the formation formation of highly conductive conductive plasma plasma channel to the the local local enhancement enhancement of of the the electric in the the leader leader tip tip region. region. to electric field field in A return return stroke stroke is is produced produced at at the the moment moment of of contact contact of of aa leader leader with with A the ground ground or or aa grounded grounded object. object. Most Most often, often, this this is is an indirect contact: contact: aa the an indirect counterpropagating leader, commonly commonly termed termed aa counterleader, counterleader, may may start counterpropagating leader, start from an object object to to meet meet the the first leader channel. channel. The The moment moment of their contact contact from an first leader of their initiates aa return return stroke. During the the travel travel from the cloud cloud to to the the ground, ground, the the initiates stroke. During from the lightning leader leader tip tip carries carries aa high high potential potential comparable comparable with with that that of of the the cloud cloud lightning at at the the spark spark start, start, the the potential potential difference difference being being equal equal to to the the voltage voltage drop drop in in the the leader leader channel. channel. After After the the contact, contact, the the tip tip receives receives the the ground ground potential potential and and its its charge charge flows flows down down to to the the earth. earth. The The same same thing thing happens happens with with the the other other parts parts of of the the channel channel possessing possessing aa high high potential. potential. This This 'unloading' ‘unloading’ proprocess cess occurs occurs via via aa charge charge neutralization neutralization wave wave propagating propagating from from the the earth earth up up through through the the channel. channel. The The wave wave velocity velocity is is comparable comparable with with the the velocity velocity of of light light and and is is about about 10 10’8 m/s. mjs. A A high high current current flows flows along along the the channel channel from from the the wave wave front front towards towards the the earth, earth, carrying carrying away away the the charge charge of of the the unloading unloading channel channel sites. sites. The The current current amplitude amplitude depends depends on on the the initial initial potential potential distribudistribution tion along along the the channel channel and and is, is, on on average, average, about about 30 30 kA, kA, reaching reaching 200-250 200-250 kA kA for for powerful powerful lightning lightning sparks. sparks. The The transport transport of of such such aa high high current current is is accomaccompanied panied by by an an intense intense energy energy release. release. Due Due to to this, this, the the channel channel gas gas is is rapidly rapidly heated heated and and begins begins to to expand, expand, producing producing aa shock shock wave. wave. A peal peal of of thunder thunder is is one one of of its its manifestations. manifestations. The The return return stroke stroke is is the the most most powerful powerful stage stage of of aa lightning lightning discharge discharge characterized characterized by by aa fast fast current current change. change. The The current current rise rise can can exceed exceed 1011 10’ A/s, A/s, producing producing aa powerful powerful electromagnetic electromagnetic radiation radiation affecting affecting the the performance performance of of radio radio and and TV TV sets. sets. This This effect effect is is still still appreciable appreciable at at aa distance distance of of several several dozens dozens of of kilometres kilometres from from the the lightning lightning discharge. discharge.


88


Current all components components of of aa Current impulses impulses of of aa return return stroke stroke accompany accompany all descending spark. This means that the leader of every component charges descending spark. This means that the leader of every component charges the but some of the the charge charge becomes becomes the channel channel as as it it moves moves down down to to the the earth, earth, but some of at the return stroke stage. Prolonged peals of of neutralized and redistributed neutralized and redistributed at the return stroke stage. Prolonged peals thunder result from the overlap of sound waves generated by the current thunder result from the overlap of sound waves generated by the current impulses impulses from from all all subsequent subsequent spark spark components. components. An ascending spark is somewhat The leader leader of of the the first first compocompoAn ascending spark is somewhat different. different. The of zero potential. As the channel travels up, the tip tip nent starts at a point nent starts at a point of zero potential. As the channel travels up, the potential changes gradually until the leader development ceases somewhere potential changes gradually until the leader development ceases somewhere deep variation during during this this process; process; as as deep in in the the cloud. cloud. There There is is no no fast fast charge charge variation aa result, However, all all subsequent subsequent result, the the first first component component has has no no return return stroke. stroke. However, spark return strokes strokes and and spark components components starting starting from from the the cloud cloud do do develop develop return behave spark. behave exactly exactly in in the the same same way way as as aa descending descending spark. Of of an an intercloud intercloud discharge. discharge. Its Its Of special special interest interest is is the the return return stroke stroke of loud as as those those of of descending descending existence existence is is indicated indicated by by peals peals of of thunder thunder as as loud sparks. in aa charged charged region region of of aa sparks. Clearly, Clearly, an an intercloud intercloud leader leader is is generated generated in storm towards an an oppositely oppositely charged charged storm cloud, cloud, or or in in aa storm storm cell, cell, and and travels travels towards region. not be be thought thought of of as as aa conconregion. The The charged charged region region of of aa cloud cloud should should not ductive high voltage voltage capacitor. capacitor. Cloud Cloud ductive body, body, something something like like aa plate plate of of aa high charges are are distributed distributed throughout throughout aa space space with of hundreds hundreds of of charges with aa radius radius of metres and and are are localized localized on on water water droplets droplets and metres and ice ice crystals, crystals, known known as as hydrometeorites, having having no no contact contact with with one one another. another. The hydrometeorites, The formation formation of of aa return stroke stroke implies implies that that the the leader leader comes comes in contact with return in contact with aa highly highly conconductive body body of of an an electrical electrical capacitance capacitance comparable comparable with, ductive with, or or even even larger larger than, that that of of the the leader. leader. It It appears appears that that the than, the role role of of such such aa body body in in an an intercloud discharge discharge is is played played by by aa concurrent concurrent spark intercloud spark coming coming in in contact contact with the the first first one. one. with Measurements made made at at the the earth earth surface surface have have shown shown that that the the current current Measurements impulse amplitude amplitude of of aa return return stroke stroke decreases, decreases, on impulse on average, average, by by half half for for 4 lOP4s. This parameter parameter variation variation is is very about 10about s. This very large large - about about an an order order of of magnitude around around the the average average value. value. Current Current impulses impulses of magnitude of positively positively charged charged sparks are are usually usually longer longer than than those those of of negatively sparks negatively charged charged ones, ones, and and the the impulses of of the the first first components components last last longer longer than impulses than those those of of the the subsequent subsequent ones. ones. A return return stroke stroke may may be be followed followed by by aa slightly slightly varying A varying current current of of about about 100A, A, which which may may persist persist in in the the spark spark channel channel for for some some fractions 100 fractions of of aa second. second. At this this final final stage stage of of continuous continuous current, current, the spark channel At the spark channel remains remains electrielectrically conductive conductive with with the the temperature temperature approximately approximately the cally the same same as as in in an an arc arc discharge. The The continuous continuous current current stage stage may follow any discharge. may follow any lightning lightning compocomponent, including including the the first first component component of of an an ascending ascending spark nent, spark which which has has no no return stroke. stroke. This This stage stage may may be be sporadically sporadically accompanied return accompanied by by current current overoverand aa duration shoots with with an an amplitude amplitude up up to to I1 kA and shoots duration of of about about 10- 3 ss each. each. Then the spark light intensity becomes much higher, producing Then the spark light intensity becomes much higher, producing what what is is gengenerally termed termed as as M-components. M-components. erally


Continuous and stepwise leaders

1.4

9

Continuous and stepwise leaders leaders

This introductory introductory chapter chapter contains contains no no theory, and this makes the the discussion discussion This theory, and this makes of very complicated we shall of leader leader details details aa very complicated task. task. So So we shall mention mention only only its its principal be registered principal features features which which can can be registered by by aa continuously continuously moving moving film. film. Continuous photographs show Continuous streak streak photographs show lightning lightning development development in in time. time. One One needs, to be be able needs, however, however, aa certain certain skill skill and and experience experience to able to to interpret interpret them them adequately. Suppose Suppose aa small small light light source source moves perpendicularly to to the the adequately. moves perpendicularly earth earth at at aa constant constant velocity. velocity. It It may may be be aa luminant luminant bomb bomb descending descending with with aa parachute. A film film moving moving horizontally, horizontally, i.e. i.e. in in the the transverse direction, at at aa parachute. transverse direction, constant constant speed speed will will show show aa sloping sloping line line (figure (figure 1.5(a)). lS(u)). Given Given the the film film speed speed (the rate), one velocity from (the display display rate), one can can easily easily calculate calculate the the light light source source velocity from the propagating vertical vertical channel the line line slope. slope. A uniformly uniformly propagating channel will will leave leave on on aa film rather than than aa line. line. From From its its slope, slope, too, too, film aa sloping sloping wedge wedge (figure (figure 1.5(b)) rather one propagation rate. higher the one can can find find the the channel channel velocity, velocity, or or its its propagation rate. The The higher the rate of of the the process process in in question, question, the the display display rate rate in in rate the higher higher must must be be the streak photography. The highest display be obtained streak photography. The highest display rates rates can can be obtained using using an an electron-optical which an electron-optical converter, converter, in in which an image image is is converted converted to to an an electron electron beam beam scanned scanned across across the the screen screen by by an an electric electric field. field. A conventional conventional photophotocamera camera registers registers the the displayed displayed electronic electronic image image from from the the screen screen onto onto an an immobile film. film. Electron-optical Electron-optical converters converters have provided much much information information immobile have provided on but their has been been on long long sparks, sparks, but their application application in in lightning lightning observations observations has limited. main results been obtained mechanical streak limited. The The main results here here have have been obtained using using mechanical streak cameras. cameras. We We described described this this technique technique and and analysed analysed streak streak pictures pictures in in our our book book on on long long sparks sparks [8]. [8]. Figure Figure 1.6(a) shows shows the the leader leader of of an an ascending ascending lightning lightning spark spark going going up the electric up from from the the top top of of aa grounded grounded tower tower in in the electric field field of of aa negatively negatively charged charged cloud cloud cell. cell. The The leader leader carries carries aa positive positive space space charge charge and, and, therefore, therefore, it the bright bright it should should be be referred referred to to as as aa positive positive leader. leader. One One can can clearly clearly see see the trace of nearly continuous trace of the the channel channel tip, tip, which which looks looks like like aa nearly continuous line. line. This This kind of kind of leader leader is is known known in in literature literature as as aa continuous continuous leader. leader. The The changing changing trace trace slope slope suggests suggests that that the the leader leader velocity velocity changes changes during during its its propagation. propagation. These These changes changes are, are, however, however, quite quite smooth, smooth, not not interrupting interrupting the the tip tip travel travel up up to the cloud. to the cloud.

o0

t oo tt t

~r:-' a

b

Figure 1.5. The The analysis analysis of of an an image image of of aa vertically vertically descending descending light light source source in in aa horizontally moving (a) point point source, source, horizontally moving film film (image (image display display in in streak streak photography): photography): (a) (b) (b) elongating elongating channel. channel.


10 10

Introduction: lightning, its destructive effects effects and protection

Q-

Figure 1.6. positive ascending 1.6. A schematic schematic streak streak picture picture of of aa positive ascending (a) (a) and and aa negative negative descending (b) lightning lightning leader. leader. descending (b)

An An essentially essentially different different behaviour behaviour is is exhibited exhibited by by the the leader leader shown shown in in manner, covering figure figure 1.6(b). 1.6(b). The The channel channel grows grows in in aa stepwise stepwise manner, covering several several dozens of of metres metres in in each each step. step. Hence, Hence, this this kind kind of of leader leader is is termed termed as as aa dozens stepwise leader. leader. The The new step in in the the photograph is especially especially bright; bright; its its photograph is stepwise new step appearance appearance makes makes the the whole whole channel channel behind behind it it also also aa little little brighter. brighter. The The step step length length varies varies between between 10 10 and and 200m 200m with with an an average average of of 30m. 30m. The The time time ps [9]. [9]. The The stepwise stepwise pattern is characteristic characteristic lapse between two steps steps is is 30-90 30-90 !is pattern is lapse between two of negatively negatively charged charged leaders. leaders. Positive Positive leaders, leaders, both ascending and and descenddescendof both ascending ing, ing, usually usually grow grow in in aa continuous continuous manner. manner. When When averaged averaged over over the the total total time time of of stepwise stepwise and and continuous continuous leaders leaders prove prove nearly nearly of development, development, the the velocity velocity of 5 5 the same, _10 6 mis, with with an m/s. 105m/s. the same, 10 105-106m/s, an average average of of about about 33 x 10 If the next next component of the component moves moves along along the the hot hot track track of of the the If the the leader leader of first first one, one, it it always always develops develops continuously. continuously. The The new new process, process, termed termed aa dart dart leader, leader, differs differs from from the the first first one one exclusively exclusively in in aa high high leader leader velocity, velocity, about about 7 107m/s. It does does not not change change much much along along its its trajectory trajectory from from the (1-4) x 10 (1-4) m/s. It the cloud to the cloud to the earth. earth. Streak Streak photographs photographs clearly clearly show show the the bright bright head head of of aa dart leader, leader, while while the channel light light intensity intensity is is much much lower. lower. If If the the next next dart the channel component path, its component takes takes its its own own path, its leader leader behaves behaves in in the the same same way way as as that that of more slowly of the the first first component, component, i.e. i.e. it it develops develops more slowly and and often often in in aa stepwise stepwise pattern. pattern. Dart leaders leaders have have not not had had aa fair fair share share of of attention attention from from researchers. researchers. Dart There is is neither theory nor nor laboratory laboratory analogue analogue of of this this type type of of gas gas discharge. discharge. There neither theory high leader Still, of discharge discharge developing developing record record high leader Still, it it is is aa most most fascinating fascinating form form of of aa dart dart leader leader with with the the earth earth produces produces the the fastest fastest velocities. The The contact contact of velocities. current reach its This is is current rise, rise, which which can can reach its amplitude amplitude maximum maximum within within 10- 7 s.s. This of record strong electromagnetic electromagnetic fields fields which which exert exert one one of of the the the source source of the record strong most most hazardous hazardous effects effects on on modern modern equipment. equipment. An An attempt attempt at at aa theoretical theoretical treatment be made made in 4.8. treatment of of the the dart dart leader leader will will be in section section 4.8.


Lightning stroke frequency

1.5

11 11

Lightning Lightning stroke frequency frequency

1.5.1 1.5.1 Strokes at terrestrial terrestrial objects objects Experience Experience shows shows that that lightning lightning most most frequently frequently strikes strikes high high objects, objects, especially especially those those dominating dominating over over an an area. area. In In aa flat flat country, country, it it is is primarily primarily attracted by high attracted by high single single objects objects like like masts, masts, towers, towers, etc. etc. In In mountains, mountains, even even low buildings may be affected if they are located on a high low buildings may be affected if they are located on a high hill hill or or on on the the top top of of aa mountain. mountain. Common Common sense sense suggests suggests that that it it is is easier easier for for an an electrical electrical discharge, bridge the highest discharge, such such as as lightning, lightning, to to bridge the shortest shortest gap gap to to the the highest object in in the the locality. locality. In In Europe, Europe, for for example, example, aa 30 30m mast experiences, experiences, on on m mast object the per year years), whereas 0.1 lightning stroke stroke per year (or (or I1 stroke stroke per per 10 10 years), whereas the average, average, O.llightning aa single 00 m construction single IlOOm construction attracts attracts 10 10 times times more more lightnings. lightnings. On On closer closer inspection, inspection, the the strong strong dependence dependence of of stroke stroke frequency frequency on on the the construction construction height does height does not not look look trivial. trivial. The The average average altitude altitude of of the the descending descending discharge discharge origin m height height makes so aa 100 lOOm makes up up only only 3% 3% of of the the distance distance origin is is about about 33 km, km, so between the bendings make make the between the lightning lightning cloud cloud and and the the earth. earth. Random Random bendings the total lightning path much to suggest, total lightning path much longer. longer. One One has has to suggest, therefore, therefore, that that the the near-terrestrial stage processes near-terrestrial stage of of lightning lightning behaviour behaviour involves involves some some specific specific processes which predetermine here. These processes lead of aa which predetermine its its path path here. These processes lead to to the the attraction attraction of descending by high descending leader leader by high objects. objects. We We shall shall discuss discuss the the attraction attraction mechanism mechanism in 5. in chapter chapter 5. Scientific there is Scientific observations observations of of lightning lightning show show that that there is an an approximately approximately quadratic N1 on on the the height height h of of lumped lumped quadratic dependence dependence of of the the stroke stroke frequency frequency NI objects objects (their (their height height is is larger larger than than the the other other dimensions). dimensions). Extended Extended objects objects of power transmission such as as power transmission lines, lines, show show aa different different dependence, dependence, of length length I,I , such N1 '" NI N hI. hl. This This suggests suggests the the existence existence of of an an equivalent equivalent radius radius of of lightning lightning attraction, R eq '" h. attraction, Re, h. All All lightnings lightnings displaced displaced from from an an object object horizontally horizontally at at aa distance ~ Re, R eq are are attracted attracted by by it, it, the the others others missing missing the the object. object. This This distance r 6 primitive pattern of primitive pattern of lightning lightning attraction attraction generally generally leads leads to to aa correct correct result. result. For R eq RZ ~ 3h 3h and and borrow borrow the the stroke stroke frequency frequency For estimations, estimations, one one can can use use Re! per unperturbed area per unit nl, from from meteorological meteorological observaobservaper unit unit unperturbed area per unit time, time, nb tions. used to up lightning tions. The The latter latter are are used to make make up lightning intensity intensity charts. charts. For For example, example, the per I1 km per year nl < I1 per km22 per year for for the the tundra, tundra, the lightning lightning intensity intensity in in Europe Europe is is nl 2-5 up to to 10 regions such the 2-5 for for flat flat areas, areas, and and up 10 for for some some mountainous mountainous regions such as as the Caucasus. m is by Re, R eq == 0.3 Caucasus. A A tower tower of of h == 100 lOOm is characterized characterized by 0.3 km km with with N1 == n17rR& nl'7l"R~q ~ per year ~ 3.5 km- 2 year-I. NI M I1 stroke stroke per year at at the the average average value value ofn] of nl = 3.5 kmP2 year-’. This not very very high high This estimation estimation is is meaningful meaningful for for aa flat flat country country and and only only for for not objects, 150m, which which do do not not generate generate ascending ascending lightnings. lightnings. objects, h < 150m, N

1.5.2 Human hazard It been proved was wrong It has has long long been proved that that Galvani Galvani was wrong suggesting suggesting aa special special 'animal ‘animal electricity'. A human human being being is, just another electricity’. A is, to to lightning, lightning, just another sticking sticking object, object, like like aa tree pole, only tree or or aa pole, only much much shorter. shorter. The The lightning lightning attraction attraction radius radius for for humans humans


12 12


4 2 is km man • If is as as small small as as 5-6 5-6mm and and the the attraction attraction area area is is less less than than 10lop4 km2. If aa man had the middle middle of thousand years years ago, had stopped stopped alone alone in in the of aa large large field field two two thousand ago, he might expected to attract aa direct direct lightning lightning stroke stroke only only by the end end he might have have expected to attract by the of the the third, coming millennium. In actual actual reality, reality, however, the number of of third, coming millennium. In however, the number of lightning lightning victims victims is is large, large, and and direct direct strokes strokes have have nothing nothing to to do do with with this. this. It known from that one It is is known from experience experience that one should should not not stay stay in in aa forest forest or or hide hide under thunderstorm. A A tree under aa high high tree tree in in an an open open space space during during aa thunderstorm. tree is is about about 10 man, and times more 10 times times higher higher than than aa man, and aa lightning lightning strikes strikes it it 100 100 times more frefrequently. under the the tree man has real chance be within quently. When When under tree crown, crown, aa man has aa real chance to to be within the zone the zone of of the the lightning lightning current current spread, spread, which which is is hazardous. hazardous. After lightning strikes strikes the the tree tree top, its current current 1ZM down along along its its runs down After aa lightning top, its M runs stem roots to natural stem and and roots to spread spread through through the the soil. soil. The The root root network network acts acts as as aa natural grounding electrode. electrode. The The current current induces induces in in the the soil soil an an electric electric field field E == pj, grounding where is the the soil soil resistivity resistivity andj and j is is the current density. density. Suppose Suppose the the current current where p is the current spreads the equipotentials will spreads through through the the soil soil strictly strictly symmetrically. symmetrically. Then Then the equipotentials will represent hemispheres with with the the diagonal represent hemispheres diagonal plane plane on on the the earth's earth’s surface. surface. The The 2 current density density at at distance distance r from from the the tree stem is is j == 1IM/(27rr2) the field field is is current tree stem M/ (21fr ) the 2 IMp/(27rr2) and the potential difference between close points r and r Ar IMP/ (21fr ) and the potential difference between close points rand + 6.r is equal equal to A U == (IMP/21f)[r(ZMp/27r)[r-’1 -- (r (Y + 6.rfl] AY)-’] x E(r)6.r. E(r)Ar.Ifa If a person is standstandis to 6.U :::::: person is ing, his side the tree ing, with with his side to to the the tree, tree, at at distance distance r :::::: = I1m m from from the tree stem stem centre centre and and m, the the soil the between his the distance distance between his feet feet is is 6.r Ar :::::: x 0.3 0.3 m, the voltage voltage difference difference on on the soil with resistivity p= = 200 200 [lIm f2/m will will be be 6.U A U :::::: x 220 220 kV kV for for aa moderate moderate lightning lightning of of with resistivity P 1 kA. This nearly ineviZMM == 30 30 kA. This voltage voltage is is applied applied to to the the shoe shoe soles soles and, and, after after aa nearly inevitable and and fast fast breakdown, to the body. There There is is no no doubt doubt that that the the table breakdown, to the person’s person's body. person will suffer or, or, more more likely, likely, will killed -- the applied voltage is too too person will suffer will be be killed the applied voltage is high. Note that high. Note that this this voltage voltage is is proportional proportional to to 6.r. Ar. This This means means that that it it is is more dangerous dangerous to to stand stand with one’s feet feet widely apart than with one's one’s feet feet more with one's widely apart than with pressed pressed tightly tightly together. together. It It is is still still more more dangerous dangerous to to lie lie down down along along the the radius between the points radius from from the the tree, tree, because because the the distance distance between the extreme extreme points contacting becomes equal be much contacting the the soil soil becomes equal to to the the person's person’s height. height. It It would would be much safer to stand still still on on one one foot, foot, like like aa stork. stork. But But it it is, is, of of course, course, easier easier to give to give safer to stand advice than to more advice than to follow follow it. it. Incidentally, Incidentally, lightning lightning strikes strikes large large animals animals more frequently humans, also the distance between their frequently than than humans, also because because the distance between their limbs limbs is is larger. larger. If have aa cottage that If you you have cottage equipped equipped with with aa lightning lightning protector, protector, take take care care that no people people could no could approach approach the the grounding grounding rod rod during during aa thunderstorm. thunderstorm. The The situation here is similar similar to to the the one one just described. situation here is just described.

+

1.6

+

Lightning hazards Lightning

1.6.1 1.6.1 A direct lightning lightning stroke

In In the the case case of of aa direct direct lightning lightning stroke, stroke, the the current current flows flows through through the the conducting the affected with the conducting elements elements of of the affected object, object, with the hot hot channel channel contacting contacting the construction construction element element which has received received the the stroke. stroke. the which has


Lightning Ligk tning hazards

13

Figure 1.7. 1.7. Traces Traces of of lightning lightning strokes strokes at at the the steel steel tip tip of of the the Ostankino Ostankino Television Television in Moscow. Moscow. Tower in Tower

Thermal hazardous at the site Thermal effects effects of of lightning lightning are are most most hazardous at the site of of contact contact of of aa high materials. This high temperature temperature channel channel with with combustible combustible materials. This often often leads leads to to aa fire which becomes probable when fire which becomes most most probable when the the continuous continuous current current stage stage has has aa long duration. duration. A return return stroke stroke is is unlikely unlikely to to cause cause aa fire fire even even in in the the case case of of long aa powerful powerful lightning because the the strong produced lightning discharge, discharge, because strong shock shock wave wave produced blows off the flames In combustible blows off the flames and and combustion combustion products. products. In combustible dielectric dielectric materials lightning stroke stroke contacts contacts on on its its way may first first be down materials aa lightning way may be broken broken down then, in the return by the the strong by strong electric electric field field of of the the leader leader tip tip and and then, in the return stroke stroke and they may may be be melted melted through through at and continuous continuous current current stages, stages, they at the the site site of of contact burn-through or contact with with the the hot hot spark. spark. A burn-through or aa burn-off burn-off often often occurs occurs at at the point where where the the spark spark contacts contacts aa metallic metallic surface surface several several millimetres the point millimetres thick. The holes holes and and burn-offs are usually usually of of the the same same size. size. The The photograph thick. The burn-offs are photograph in figure figure 1.7 1.7 demonstrates demonstrates the of numerous lightning strokes strokes on on the the in the traces traces of numerous lightning steel tip tip of of the the Ostankino Ostankino Television Television Tower. Tower. Slight Slight faults faults cannot cannot disturb disturb steel the mechanical strength of of aa massive construction. Normally, the the mechanical strength massive metallic metallic construction. Normally, the hazards of of burn-offs and fuses fuses are are associated associated with melted metal metal in-flow in-flow hazards burn-offs and with the the melted into an an object object which may contain contain inflammable inflammable and and explosive explosive materials or into which may materials or gas burn-through of gas mixtures. mixtures. Incidentally, Incidentally, not not only only is is aa burn-through of aa metallic metallic wall wall dangerous but but also also the the local local overheating overheating when the temperature of the the dangerous when the temperature of inner metal metal surface surface may may go go up up to to 700-1000°C. 700-1000°C. Unfortunately, Unfortunately, the the surface surface inner often often acts acts as as aa lighter. lighter.


14 14

Introduction: Introduction: lightning, lightning,its destructive effects and protection

Thermal Thermal damage damage of of conductors, conductors, through through which which lightning lightning current current flows, flows, occurs occurs fairly fairly rarely. rarely. It It is is characteristic characteristic of of miniature miniature antennas antennas and and various various detectors mounted mounted on on the the outer outer construction construction surfaces. surfaces. The The probability probability of of detectors emergency increases increases if if lightning lightning current current encounters encounters bolted bolted or or riveted riveted joints. joints. emergency The electric electric contact contact thus thus formed formed always always has has an an elevated elevated contact contact resistance resistance The which may may cause cause aa local local overheating. overheating. This This results results in in the the metal metal release release and and which rivet loosening, loosening, disturbing disturbing the the mechanical mechanical strength strength of of the the joint. joint. Mobile Mobile rivet joints joints (hinges, (hinges, ball ball bearings, bearings, etc.) etc.) are are subject subject to to aa similar similar damage. damage. The The site site of aa sliding sliding contact contact becomes becomes locally locally overheated overheated to to produce produce cavities cavities which which of hamper the the motion motion of of mobile mobile parts. parts. In In extreme extreme conditions, conditions, they they may may hamper become welded. welded. become Electrodynamic effects effects of of lightning lightning current current rarely rarely become become hazardous. hazardous. Electrodynamic Mechanical stress stress arising arising in in electrically electrically loaded loaded and and closely closely spaced spaced metallic metallic Mechanical structures or or in in aa single single structure structure with with an an abruptly abruptly changing changing direction direction of of structures the current current is is not not appreciable appreciable and and lasts lasts less less than than 100 100ms ms (it (it is is the the attenuation attenuation the time of of aa current current impulse). impulse). However, However, lightning lightning current current has has been been repeatedly repeatedly time observed to to narrow narrow down down thin thin metallic metallic pipes, pipes, to to change change the the tilt tilt of of rods rods and and observed to strain strain thin thin surfaces. surfaces. Such Such effects effects are are not not vitally vitally dangerous dangerous in in themselves themselves to but, under under certain certain conditions, conditions, may may lead lead to to an an emergency. emergency. As an an illustration, illustration, but, imagine imagine the the situation situation when when the the lightning-affected lightning-affected pipe pipe is is part part of of an an aircraft aircraft speed control. What will happen if the crew take the readings for granted speed control. What will happen if the crew take the readings for granted and do do not not receive receive corrections corrections from from aa ground ground air air traffic traffic controller? controller? and Electrohydraulic effects effects of of lightning lightning are are much much more more hazardous hazardous than than Electrohydraulic those discussed discussed above. above. Modern Modern machines machines have have parts parts made made from from aa variety variety those of composite composite materials. materials. These These may may include, include, along along with with plastics, plastics, superthin superthin of metallic films films (both (both outer outer and and inner), inner), nearly nearly as as thin thin metallic metallic meshes, meshes, and and minminmetallic iature conductors conductors monolithic monolithic with with aa dielectric dielectric wall. wall. Under Under the the action action oflightof lightiature ning current, current, these these metallic metallic parts parts evaporate, evaporate, the the arising arising arcs arcs contacting contacting the the ning plastic making making itit decompose decompose and and evaporate. evaporate. A shock shock wave wave appears appears which which plastic splits and and bloats bloats the the composite composite wall. wall. A similar similar effect effect arises arises when when aa lightning lightning splits spark partially penetrates through a narrow slit between vaporizable plastic spark partially penetrates through a narrow slit between vaporizable plastic walls (most (most plastics plastics possess possess gas-generating gas-generating properties). properties). No No one one questions questions aa walls great future future of of composite composite materials, materials, but but their their peaceful peaceful coexistence coexistence with with lightlightgreat ning is is still still aa challenge challenge to to the the engineer. engineer. ning Direct Direct stroke stroke overvoltage overvoltage represents represents aa hazardous hazardous rise rise of of voltage voltage when when lightning current current impulse impulse propagates propagates across across the the construction construction elements. elements. aa lightning We shall shall analyse analyse this this very very dangerous dangerous effect effect of of lightning lightning with with reference reference to to We power transmission transmission line, line, because because engineers engineers first first encountered encountered the the phenomphenomaa power enon of of overvoltage overvoltage in in such such lines. lines. Moreover, Moreover, the the problem problem of of electric electric enon insulation for for aa transmission transmission line line can can be be stated stated most most clearly. clearly. Figure Figure 1.8 1.8 insulation shows schematically schematically aa metallic metallic tower tower with with aa ground ground rod rod (the (the grounding grounding shows resistance is is R Rg) and aa high high voltage voltage wire wire suspended suspended by by an an insulator insulator string. string. resistance g ) and Above the the wire, wire, there there may may be be aa lightning lightning conductor conductor attached attached right right to to the the Above tower. It It stretches stretches along along all all the the line line and and is is to to trap trap lightning lightning sparks sparks aimed aimed tower.


Lightning hazards

15

a

Figure 1.8. power transmission 1.8. Lightning Lightning current current as as an an overvoltage overvoltage source source on on aa power transmission line line during the power tower (b). during aa stroke stroke at at the power wire wire (a) (a) and and aa grounded grounded tower (b).

at the the line line wires. wires. A A rigorous solution to to this this problem is given given later later in in at rigorous solution problem is this book. book. Here, Here, we we should should only only like like to to explain explain the the nature nature of of overvoltages overvoltages this phenomenologically. phenomenologically. Suppose Suppose at at first first that that the the lightning lightning conductor conductor has has proved proved unreliable, unreliable, and and discharge has has struck struck the the wire wire (figure (figure 1.8(a)). 1.8(a)). At At the the point point of of the the stroke, stroke, the the aa discharge current will will branch branch to to produce produce two two identical identical waves waves of of the the amplitude amplitude IZM /2, current M /2, where where 1ZM is the the lightning lightning current current amplitude. amplitude. The The two two waves waves will will run run towards towards M is the the ends ends of of the the line line with with aa velocity velocity nearly nearly equal equal to to vacuum vacuum light light velocity, velocity, 8 c= m/s. Until = 33 Xx 10 10sm/s. Until the the end-reflected end-reflected waves waves return, return, the the wire wire potential potential The wave wave resistance resistance relative to to the the ground ground will will rise rise to to UM, == IZ,2/2. relative M Z/2. The Z ) 1/2 C in this expression is defined by the inductance L / Z= = (L (L1/C1)1/2 in this expression is defined by the inductance L1 and the the I and I I capacitance per unit wire length; it varies slightly, between 250 and capacitance C C1 per unit wire length; it varies slightly, between 250 and I 350 n, with 350R, with the the height height and and the the wire wire radius. radius. With With this this wave wave resistance, resistance, the the average average lightning lightning current current with with an an amplitude amplitude 1ZM = 30 30 kA kA will will raise raise the the wire wire M = potential potential up up to to UM, == 3750-5250 3750-5250 kV. kV. The The tower tower potential potential will will practically practically remain remain unchanged unchanged and and equal equal to to zero, zero, so so the the insulation insulation overvoltage overvoltage will will be be close close to to the the calculated calculated value value of of U U,M. . This This will will be be clear clear if if we we compare compare UM, with with the the operating operating line line voltage voltage which which does does not not exceed exceed 1000 1000kV kV even even in in high high power power lines lines but but normally normally is is 250-500 250-500 kV. kV. In In reality, reality, the the distance distance to to the the line line ends ends /I is is as as large large as as many many dozens dozens of of kilometres. kilometres. The The time time it it takes takes the the reflected reflected wave wave of of the the opposite opposite sign sign cutting cutting C, or down down the the overvoltage overvoltage to to arrive arrive back back at at the the stroke stroke point point is is 6.t At == 2// 21/c, or many many hundreds hundreds of of microseconds. microseconds. This This time time is is much much longer longer than than the the strong strong current Ils). For current duration duration in in the the return return stroke stroke (100 (loops). For this this reason, reason, reflected reflected waves, waves, which which become become strongly strongly attenuated, attenuated, do do not not normally normally have have enough enough time time to to interfere interfere with with the the process process so so that that the the overvoltage overvoltage acts acts as as long long as as aa lightning lightning current current impulse. impulse. Practically, Practically, any any lightning lightning stroke stroke at at aa wire wire represents represents aa real real hazard: hazard: the the insulation insulation will will be be broken broken down down to to produce produce short-circuiting. short-circuiting. The The power power line line in in that that case case must must be be disconnected. disconnected. Suppose Suppose now now that that lightning lightning has has struck struck aa tower. tower. More More often, often, this this is is actually actually not not aa tower tower but but rather rather an an overhead overhead grounded grounded wire wire connected connected to to it. it.


16 16

Introduction: Introduction: lightning, lightning, its its destructive destructive effects effects and and protection protection

The The lightning lightning current current will will flow flow down down the the metallic metallic tower tower to to the the ground ground electrodes electrodes to to be be dissipated dissipated in in the the earth. earth. Let Let us us take take point point A A at at the the height height of of the the insulator insulator string string connection. connection. Due Due to to the the lightning lightning current current i(t), i(t), the the potential at at this this point, point, i.pA, pA,will will differ differ from from the the zero zero potential potential of of the the earth earth potential by the the voltage voltage drop drop in in the the grounding grounding resistance resistance R R,g and and in in the the tower tower inducinducby tance L, between the tower base up to the point A: tance L s between the tower base up to the point A: p == Rgi R,i+ i.p

di di

(1.1)

+ LL,-s dt' dt

However, However, the the power power wire wire potential potential will will practically practically remain remain the the same same (in (in this this qualitative qualitative description, description, we we ignore ignore all all inductances inductances between between the the power power wires, wires, tower tower and and grounded grounded wire). wire). The The power power wire wire potential potential is is due due to to the the operating operating voltage source source of of the the power power line: line: i.pw qw == U Uop. Then, the the insulator insulator string string voltage voltage voltage op ' Then, will be be will TT

vU

di

R' L s dt' di = i.pw pw -- i.pA (FA = U Uop = gl -- Ls-, op -- R,i dt

(1.2)

Note that that the the lightning lightning current current and and operating operating voltage voltage may may have have different different Note polarities. As As aa result, result, the the overvoltage overvoltage U may may prove prove to to be be the the sum sum of of the the polarities. three terms terms in in equation equation (1.2). (1.2). three The inductance inductance component component of of the the overvoltage, overvoltage, LL,di/dt, has aa short short The s di/dt, has lifetime: itit acts acts about about as as long long as as the the lightning lightning current current rises. rises. For For aa current current lifetime: impulse with with an an average average amplitude amplitude 1IM 30kA and an an average average rise rise time time impulse kA and M 30 tf == 51ls, 5 p , the the inductance inductance voltage voltage at at LL,s 50 50pH will be be about about 300 300kV. The tf IlH will kV. The resistance component component UM, at at aa typical typical grounding grounding resistance resistance R R,g = = 10 10R will resistance n will have about about the the same same value value but but will will act act an an order order of of magnitude magnitude longer, longer, i.e., i.e., have as long long as as the the lightning lightning current current flows. flows. For For this this reason, reason, this this component component as makes the the principal principal contribution contribution to to the the insulation insulation flashover. flashover. makes The emergency emergency situation situation just described is is not not as as bad bad as as aa direct direct stroke stroke at at The just described power wire wire when when the the same same lightning lightning current current can can induce induce an an order order of of magnimagniaa power tude higher higher voltage. voltage. The The insulation insulation of of aa ultrahigh ultrahigh voltage voltage line line can can withstand withstand tude short overvoltages overvoltages up up to to lOOO-1500kV 1000- 1500kV and and seldom seldom suffers suffers from from lightning lightning short strokes at at aa tower tower or or aa lightning lightning protection protection wire. wire. To To produce produce aa harmful harmful strokes effect, the the lightning lightning current current must must be be 3-5 3-5 times times the the average average value. value. Lightning Lightning effect, strokes of of this this power power do do not not occur occur frequently, frequently, making making up up less less than than 11% of strokes % of all strokes. strokes. Quite Quite different different is is the the effect effect of of aa direct direct stroke stroke at at aa power power network network all with an an operating operating voltage voltage of of 35 35 kV kV and and lower. lower. The The insulation insulation system system will will with suffer equally equally from from aa stroke stroke at at aa power power wire wire or or aa tower. tower. It It is is no no use use protecting protecting suffer such aa line line with with grounded grounded wire. wire. such Insulation flashover due to the the tower tower potential potential rise rise is is referred referred to to as as Insulation flashover due to reverse flashover. flashover. This This name name does does not not imply imply the the definite definite direction direction of of the the reverse discharge development development but but only only indicates indicates the the direction direction from from which which the the discharge potential rises, rises, i.e., i.e., the the grounded grounded end end of of the the insulator insulator string string rather rather than than potential the power power wire. wire. the


Lightning hazards

17 17

The The above above illustration illustration of of overvoltages overvoltages on on the the line line transmission transmission insulainsulation demonstrates, demonstrates, to to some some extent, extent, aa variety of mechanisms mechanisms of of direct direct lightlighttion variety of ning ning current current effects. effects. In In actual actual reality, reality, such such mechanisms mechanisms are are much much more more diverse. remember that modern technologies, technologies, overvoldiverse. It It is is important important to to remember that in in modern overvoltages are not always measured in hundreds of kilovolts, kilovolts, as as for for high-voltage tages are not always measured in hundreds of high-voltage transmission lines. Short Short voltage of only only 100-10 100-1OVV may be hazardous hazardous transmission lines. voltage rises rises of may be to microelectronic devices. devices. Of Of special special interest interest in in this this connection connection are are situasituato microelectronic tions when jackets with tions when lightning lightning current current flows flows across across solid solid metallic metallic jackets with electric electric circuits inside. inside. These These problems are discussed discussed in in chapter chapter 6. 6. circuits problems are

1.6.2 Induced overvoltage overvoltage 1.6.2 Induced overvoltage overvoltage is is the the most common and and dangerous dangerous effect effect of of lightning lightning Induced most common on electric electric circuits circuits of of modern technical equipment. equipment. This This effect effect is is brought on modern technical brought about by electromagnetic induction. induction. The The current current flowing flowing through through the the about by electromagnetic lightning spark spark and and the the metallic metallic structures structures of of an an affected affected object object generates generates lightning an alternating alternating magnetic field which can induce induce an an induction induction emf emf in in any any of of an magnetic field which can the in question. question. The of estimating induced overvoltages the circuits circuits in The procedure procedure of estimating induced overvoltages is simple. If If B BaV(t) induction averaged averaged over circuit is quite quite simple. is the the magnetic magnetic induction over the the circuit av ( t) is cross section S, S , the the induction induction emf emf is is expressed expressed as as cross section Eemf>::;j

dBav S > r,r, this this is is sufficient sufficientto to excite excite aa weak weak counterpropagating counterpropagating leader leader process. process. With II » Of will have have no 10cm cm long, long, it it will no effect effect on on the the Of course, course, if if this this leader leader is is only only about about 10 lightning lightning trajectory. trajectory. Its Its energy energy is, is, however, however, large large enough enough to to ignite ignite an an inflaminflammable gas mixture, if there is mable gas mixture, if there is any any in in the the vicinity, vicinity, since since the the channel channel temperature temperature is is close close to to 5000 5000 K K and and its its lifetime lifetime is is as as long long as as that that of of aa lightning lightning leader. leader.

1.6.4 1.6.4 High potential infection This term has used in This unsuitable unsuitable term has long long been been used in Russian Russian literature literature on on lightning lightning protection. It protection. It means means that that the the surface surface and and underground underground service service lines, lines, which which get get into into aa construction construction to to be be protected, protected, may may introduce introduce in in it it aa potential potential different from from the the zero zero potential of the construction metalwork metalwork connected connected the construction different potential of to possible if not to earth earth connection. connection. This This may may become become possible if aa service service line line is is not linked the grounding but connected linked to to the grounding of of the the construction construction but connected or or passes passes close close to the the earth to earth connection connection of of another another construction construction loaded loaded by by lightning lightning current current during during aa stroke stroke (figure (figure 1.9). 1.9). This This may may also also be be aa natural natural earth earth connection connection formed the moment with the formed at at the moment of of lightning lightning contact contact with the earth earth due due to to an an intense intense breakionization ionization in in it. it. If If the the introduced introduced potential potential is is high, high, it it causes causes aa spark spark breakdown nearby metallic down between between the the service service line line and and aa nearby metallic structure structure of of the the object, object, whose whose potential potential is is zero zero owing owing to to the the earth earth connection. connection. The The scenario scenario of of the the emergency emergency that that follows follows has has been been described described above. above. To potential infection, To avoid avoid sparking sparking induced induced by by high high potential infection, all all metallic metallic service service lines lines subject subject to to explosion explosion zooms zooms are are linked linked to to the the earth earth connection connection of potentials are of the the construction. construction. All All metalwork metalwork potentials are equalized. equalized. The The connection, connection,


20

Introduction: lightning, its destructive effects and protection

I -/ Figure 1.9. high potential 1.9. Schematic Schematic input input of of high potential from from remote remote lightning lightning strokes. strokes.

however, becomes by additional way there however, becomes loaded loaded by additional current, current, which which finds finds its its way there from aa remote remote lightning lightning stroke, stroke, using using the the service service line line as as aa conductor. conductor. from When the earth connection connection resistance is low low and and the the service service line line goes goes through through When the earth resistance is so that that the the current current leakage leakage through through the side the ground ground with with aa high the high resistivity resistivity so the side surface is is not not large, large, nearly all of of the the lightning lightning current current arrives arrives at at the the conconsurface nearly all nection from from the the stroke stroke site. site. This This situation situation appears appears to to be somewhat similar similar nection be somewhat to measures must to aa direct direct lightning lightning stroke. stroke. Sometimes, Sometimes, special special measures must be be taken taken to to restrict restrict the the infection infection current. current. A detailed detailed treatment treatment of of the the problem problem of of current current and potential infections will offered in in chapter chapter 6. 6. and potential infections will be be offered

creeping along the earth's earth’s surface surface 1.6.5 Current inrush from a spark creeping This phenomenon is familiar familiar to to all all communications communications men men who have to repair This phenomenon is who have to repair communications cables cables damaged damaged by lightning. The The damaged damaged site site can can be be communications by lightning. detected easily, easily, because it is is indicated indicated by furrow in in the the ground ground extending extending detected because it by aa furrow furrow may may be be as as long long as as several several dozens dozens far away away from from the the stroke stroke site. site. A furrow far of metres, or IOO-200m 100-200m in in aa high ground. Such Such aa long long gap gap can can of metres, or high resistivity resistivity ground. be bridged bridged by spark because because of of the the electric electric field field created created by by the spark current current be by aa spark the spark spreading out out through through the the ground. ground. The The mechanism of spark spark formation formation along along spreading mechanism of conducting surface surface differs differs from from that of aa 'classical' ‘classical’ leader leader propagating propagating aa conducting that of through air. air. A creeping creeping spark spark can can develop develop in in low low fields fields and and have have aa very very through high high velocity. velocity. Underground cables cables are are not not the the only only objects objects suffering suffering from from creeping creeping Underground spark current. current. Similarly, Similarly, it it can can find find its its way way to to underground underground service service lines lines spark and by lightning and to to the the earth earth connections connections of of constructions constructions well well equipped equipped by lightning protectors. But aa protector protector palisade cannot stop stop lightning. lightning. When convenprotectors. But palisade cannot When the the conventional way from from the the earth earth surface surface is is blocked, it breaks through from from beneath, tional way blocked, it breaks through beneath, making in the ground. Lightning thus behaves much like like aa making aa bypass bypass in the ground. Lightning thus behaves very very much clever general general in in ancient ancient times, times, who ordered his soldiers to make aa secret secret clever who ordered his soldiers to make underground instead of of attacking attacking openly openly the the impregnable impregnable underground passageway passageway instead castle walls. It is is reasonable suggest that the contact contact of of aa creeping creeping spark spark castle walls. It reasonable to to suggest that the with with combustible combustible materials materials is is as as frequent frequent aa cause cause of of aa fire fire as as aa direct direct lightning lightning stroke. stroke. The details details of of the the creeping creeping discharge discharge mechanism mechanism have have been unknown The been unknown until quite recently. recently. They They are are analysed analysed in in chapter chapter 6. 6. until quite


Lightning hazards

21 21

1.10. This This lightning lightning has missed the teletower tip tip by by over over 200m. 200m. Figure 1.10. has missed the te1etower

1.6.6 1.6.6 Are lightning protectors protectors reliable? reliable?

Lightning believed to reliable, since Lightning protectors protectors are are believed to be be reliable, since their their design design has has changed but little Nevertheless, the photochanged but little over over two two and and aa half half centuries. centuries. Nevertheless, the photograph judgement: the graph in in figure figure 1.10 1.10 makes makes one one question question this this judgement: the lightning lightning struck 200 m m below below its its top, top, i.e., i.e., the the Tower Tower struck the the Ostankino Ostankino Television Television Tower Tower 200 could not protect rule. Most could not protect itself. itself. This This is is not not an an exception exception to to the the rule. Most descending descending discharges to what discharges missed missed the the Tower Tower top top more more or or less less closely, closely, contrary contrary to what had had been expected. been expected. This This is is aa serious serious argument argument against against the the vulgar vulgar explanation explanation of of the major major principle principle of the of protector protector operation operation that that lightning lightning takes takes aa shortcut shortcut at at the the final final stage stage of of its its travel travel to to the the earth. earth. There There are are also also other other arguments, arguments, perhaps not not as perhaps as obvious obvious but but still still convincing. convincing. Breakdown under strictly Breakdown voltage voltage spread spread is is registered registered in in long long gaps gaps even even under strictly identical probability 9 \l1 varies varies with with the the pulse pulse identical conditions. conditions. The The breakdown breakdown probability (figure 1.11). 1.11). Deviations Deviations from from the 50% probaprobaamplitude of of test test voltage voltage U (figure amplitude the 50% bility voltage, U bility voltage, Use%,, are appreciable appreciable and and may may be be 10-15% 10-15% either either way. way. SO %, are Curve the probability Curve 22 in in figure figure 1.11 1.11 shows shows the probability function function \l1( !F( U) U ) for for aa shorter shorter gap. ranges, both both curves breakdown probabilities probabilities gap. In In certain certain voltage voltage ranges, curves promise promise breakdown remarkably means that remarkably different different from from zero. zero. This This means that if if two two different different gaps gaps are are tested that any tested simultaneously, simultaneously, there there is is aa chance chance that any of of them them (the (the smaller smaller and and the the larger larger gap) gap) will will be be bridged. bridged. In In general, general, this this situation situation is is similar similar to to that that arisarising point to to strike ing when when aa lightning lightning discharge discharge is is choosing choosing aa point strike at. at. It It does does not not always take the shortest shortest way way to to aa protector but, instead, instead, may may follow follow aa always take the protector but, longer path in order order to to attack attack the the protected protected object. object. longer path in For path problem, problem, one has to For solving solving the the lightning lightning path one has to treat treat aa multielectrode multielectrode system system consisting consisting of of several several elementary elementary gaps. gaps. For For lightning, lightning, all all elementary elementary Copyright © 2000 IOP Publishing Ltd.

22

destructive effects and and protection protection Introduction: lightning, its destructive

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1.2

UIU SO % Distributions of breakdown voltages voltages in in air air gaps gaps of of various various lengths lengths with with aa Figure 1.11. Distributions of breakdown sharply non-uniform electric electric field. sharply non-uniform field.

gaps have have aa common common high high voltage voltage electrode (the leader that has has descended to gaps electrode (the leader that descended to aa certain certain altitude), altitude), while while the the zero zero potential potential electrodes electrodes are are formed by the the formed by earth's with grounded grounded objects objects and protectors distributed distributed on on it. it. The The earth’s surface surface with and protectors problem calculation of of problem of of protector protector effectiveness effectiveness thus thus reduces reduces to to the the calculation breakdown probabilities probabilities for for the the elementary gaps in in aa multielectrode multielectrode breakdown elementary gaps system. The The general general formulation formulation of of this this problem problem is is very very complex, complex, since since the the system. spark spark development development in in the the elementary elementary gap gap in in real real conditions conditions cannot cannot be be taken taken to to be be independent. independent. The The discharge discharge processes processes affect affect one one another another by by redistributing redistributing their their electric electric fields, fields, which which eliminates eliminates straightforward straightforward use use of of statistical statistical relations relations describing describing independent independent processes. processes. We We cannot cannot say say that that the the spark spark discharge discharge theory theory for for aa multielectrode multielectrode system system has has been been brought brought to to any any stage stage of of completion. completion. But But what what has has been been done, done, theoretically theoretically and and experimentally, experimentally, allows allows the the formulation formulation of of certain certain concepts concepts of of the the lightning lightning orientation orientation mechanism mechanism and and the the development development of of engineering engineering approaches approaches to to estimate estimate the the effectiveness effectiveness of of protectors protectors of of various various heights heights (see (see chapter chapter 5). 5). Investigation Investigation of of multielectrode multielectrode systems systems is is also also important important from from another another point point of of view: view: we we must must find find ways ways of of affecting affecting lightning lightning actively. actively. It It would would be be reasonable reasonable to to leave leave the the discussion discussion of of this this issue issue for for specialized specialized chapters chapters of of this this book, book, but but they they will, will, however, however, attract attract the the attention attention of of professionals professionals only, only, or or of of those those intending intending to to become become professionals. professionals. It It is is not not professionals professionals but amateurs amateurs who, who, most most often, often, try try to to invent invent lightning lightning protectors. protectors. They They but have have at at their their disposal disposal aa complete complete set set of of up-to-date up-to-date means: means: lasers, lasers, plasma plasma jets, jets, corona-forming corona-forming electrodes electrodes for for cloud cloud charge charge exchange, exchange, radioactive radioactive


Lightning as a power supply

23 23

sources, high voltage generators generators stimulating stimulating counterpropagating counterpropagating leaders, leaders, etc. etc. sources, high voltage That lightning lightning management management has has aa future future has confirmed by by laboratory laboratory That has been been confirmed studies on on sparks sparks of of multimetre multimetre length. length. These These experiments experiments and and their their implicaimplicastudies tions be analysed analysed below, so there there is is no no point in discussing discussing them them here. here. Still, Still, tions will will be below, so point in it is is hard to resist resist the the temptation make some some preliminary comments hard to temptation to to make preliminary comments it addressed to those who like to to invent invent lightning lightning protection addressed to those who like protection measures. measures. When explaining explaining the the leader leader mechanism mechanism at at the the beginning of this this chapter, chapter, When beginning of we noted that that the the leader leader tip tip carries carries aa strong strong electric electric field field sufficient sufficient for for an an we noted intense air air ionization. ionization. It It is is very difficult to to act act on on this this field field directly, directly, because because intense very difficult it would would be necessary to to create create charged charged regions regions close close by, by, whose whose charge charge dendenit be necessary sity and and amount amount would would be comparable with with those those in in the the immediate immediate vicinity vicinity of of sity be comparable the tip. tip. Pre-ionization Pre-ionization of of the the air air by by radioactive radioactive sources sources is is oflittle of little use use because because the of the the low low air air conductivity conductivity after after radiative radiative treatment. treatment. The The initial initial electron electron of density behind the ionization wave front in the leader process is higher density behind the ionization wave front in the leader process is higher 12 cm than than 10 10l2 ~ m-3,- and and ~ , in in aa 'mature' ‘mature’ leader leader it it is is at at least least an an order order of of magnitude magnitude higher. These and and even even much much lower lower densities densities are are inaccessible inaccessible to to radiation radiation at at aa higher. These distance of dozens of metres from the radiation source which must present no distance of dozens of metres from the radiation source which must present no danger to life. The same is true of a gradual charge accumulation due to danger to life. The same is true of a gradual charge accumulation due to aa slow corona corona formation formation between special electrodes. electrodes. Besides, Besides, one one cannot cannot preslow between special predict the polarity of a particular spark to decide which charge is to to be be dict the polarity of a particular spark to decide which charge is pumped into pumped into the the atmosphere. atmosphere. Quite another another thing thing is is plasma generation. In In principle, we could could create create aa Quite plasma generation. principle, we plasma channel comparable comparable with with the the lightning lightning rod height, thus thus increasing increasing its its plasma channel rod height, length. A high laser could, could, in in principle, used as as aa plasma plasma source. source. It It length. high power power laser principle, be be used is clear clear that that it it should should be be aa pulse pulse source source and and the the plasma produced should should have have is plasma produced short lifetime. lifetime. It It must must be be generated generated exactly exactly at at the the right right moment, when aa aa short moment, when lightning leader leader is is approaching approaching the dangerous region region near the object object to to be be lightning the dangerous near the protected. This is is aa new associated with with synchronization synchronization of of the the protected. This new problem problem associated laser operation operation and and lightning lightning development, development, giving giving aa new to the the task of laser new turn turn to task of lightning protection, which does does not not at at all all become easier. lightning protection, which become easier. Finally, we should always always bear in mind mind that that most lightning discharges discharges are are Finally, we should bear in most lightning multicomponent. In about about half of them, subsequent components components do do not multicomponent. In half of them, the the subsequent not follow the path of of the the first first component. component. In In fact, fact, these these are are new discharges follow the path new discharges which individual handling. To prepare laser light light source source for for which would would require require individual handling. To prepare aa laser operation cycle cycle for for aa fraction fraction of of aa second second is is possible difficult techniaa new new operation possible but but difficult technically. The The cost cost of of such such protection is anticipated anticipated to close to of gold. gold. cally. protection is to be be close to that that of It is is not our intention intention to to intimidate intimidate lightning lightning protection inventors. We It not our protection inventors. We just want want to to warn warn them them against just against excessive excessive enthusiasm. enthusiasm.

1.7

Lightning Lightning as a power supply

The question question of of whether whether lightning lightning could could serve serve as as aa power supply cannot cannot be be The power supply answered positively, how much much we wish it it to to be be one. one. Some Some authors authors answered positively, no no matter matter how we wish


24 24

Introduction: lightning, Introduction: lightning, its destructive effects and protection

of to harof science science fiction fiction books books force, force, quite quite inconsiderately, inconsiderately, their their characters characters to harness lightning lightning in in order order to to use its electric electric power. Even without without this this service, service, ness use its power. Even lightning people by thought. The lightning has has done done much much for for people by stimulating stimulating their their thought. The energy of of aa lightning lightning flash flash is is not not very very high. high. The The voltage voltage between cloud energy between aa cloud and MV even and the the earth earth can can hardly hardly exceed exceed 100 l00MV even in in aa very very powerful powerful storm, storm, the transported transported charge charge is is less less than than 100 lOOC, and maximum maximum energy energy release release is is the C, and 10 10 J. This 10'OJ. This is is equivalent equivalent to to one one ton ton of of trinitrotoluene trinitrotoluene or or 2-4 2-4 ordinary ordinary airborne bombs. A family family cottage cottage consumes consumes more more power power for for heating, heating, illumiillumiairborne bombs. A nation, nation, and and other other needs needs over over aa year. year. Actually, Actually, only only aa small small portion portion of of the the lightning power is accessible accessible to to utilization, utilization, while while most most of of it it is is dissipated dissipated in in lightning power is the the atmosphere. atmosphere. Normally, aa person person lives Normally, lives through through 40-50 40-50 thunder thunder storm storm hours hours during during aa year. per year. All All storms storms send send to to the the earth earth an an average average of of 4-5 4-5 lightning lightning sparks sparks per 2 square kilometre kilometre of of its its surface surface providing power of of less less than than 11 kWjkm kW/km2 square providing aa power 2 per km , this MW, which 500 x 400 400km2, this is is about about 200 200MW, which is is per year. year. In In aa country country of of 500 aa very very small by an small value value compared compared with with the the electrical electrical power power produced produced by an indusindustrial trial country. country. Just Just imagine imagine the the immense immense net net which which would would be be necessary necessary for for trapping lightning lightning discharges discharges in in order order to to collect collect such such aa meagre meagre power! trapping power! Other power sources, Other natural natural power sources, such such as as wind, wind, geothermal geothermal waters, waters, and and tides, tides, are infinitely infinitely more more powerful than lightning, lightning, but but they they are are still still not not utilized utilized are powerful than much. much. Clearly, Clearly, we we should should not not even even raise raise the the problem problem of of lightning lightning power power resources. resources.

1.8

intending to read read on on To those intending

There will will be more popularized stories about about lightning lightning in in this this book. There be no no more popularized stories book. Nor Nor shall we we mention mention ball lightning here. here. The The next chapter will contain aa shall ball lightning next chapter will contain thorough analysis of of available available data data and and theoretical treatments of of the the long long thorough analysis theoretical treatments spark, because because we we believe that without without these these preliminaries preliminaries the the lightning lightning spark, believe that mechanism may not not become become clear clear to to the the reader. reader. Nature Nature has has eagerly eagerly employed employed mechanism may so lightning lightning is is quite quite likely likely to to represent represent the standard solutions solutions to to its its problems, standard problems, so the limiting case case of of the the long long spark. spark. It It would be useful useful for for readers readers to to familiarize familiarize limiting would be p a r k Discharge, because it is is totally themselves with our previous themselves with our previous book book SSpark because it totally concerned with with this this phenomenon. But even even without without it, it, they they will will be be able able to to concerned phenomenon. But find here here basic basic information information on on long long sparks. sparks. We We have tried to to describe describe their their find have tried general mechanisms mechanisms and and to give predictions predictions as as to to their their extension extension to to air air general to give gaps of of extremal extrema1 length. length. Even Even for for this reason alone, alone, the the next next chapter chapter is is not not gaps this reason summary of of the the previous previous book. Lightning is is as as complicated complicated aa phenomenon aa summary book. Lightning phenomenon as the the long long spark spark and and is is definitely definitely more more diverse. diverse. It It is is aa multicomponent multicomponent as process. Since its its subsequent subsequent components components sometimes sometimes take take the the path path of of an an process. Since earlier component, component, we must consider consider the the effects effects of of temperature temperature and and residual earlier we must residual conductivity in in the the spark spark channel channel on on the the behaviour of new new ionization ionization conductivity behaviour of waves. waves.


To those intending to read read on

25

Even Even aa simple simple model model should should not not treat treat aa kilometre kilometre spark spark in in terms terms of of electrical circuits circuits with with lump lump parameters. parameters. A lightning lightning spark spark is is aa distributed distributed electrical system. system. The The time time for for which which the the electric electric field field perturbation perturbation spreads spreads along along the the sparks sparks is is comparable comparable with with the the duration duration of of some some of of its its fast fast stages. stages. The The allowance for for the the delay delay can, can, in in some some cases, cases, change change the the whole whole picture picture allowance radically. This This requires requires new new approaches approaches to to lightning lightning treatments. treatments. ExperimenExperimenradically. tal data data and and theoretical theoretical ideas ideas concerning concerning the the lightning lightning leader leader and and return return tal stroke are are discussed discussed together. together. First, First, there there are are not not many many of of them. them. On On the the stroke other we have have tried tried to to point point out out ideological ideological relationships relationships between between other hand, hand, we experiment experiment and and theory theory and and to to offer offer aa more more or or less less consistent consistent physical physical description. description. Spark discharges discharges in in aa multi-electrode multi-electrode system system are are the the subject subject of of aa special special Spark chapter. We We present present available available data data and and analyse analyse possible possible mechanisms mechanisms of oflightchapter. lightning orientation. orientation. This This is, is, probably, probably, the the most most debatable debatable part part of of the the book. book. ning Field of lightning lightning orientation orientation are are very very difficult difficult to to carry carry out out primarily primarily Field studies studies of because constructions of even 100-200 m high are affected by descending because constructions of even 100-200 m high are affected by descending discharges only once or observer must discharges only once or twice twice aa year. year. The The observer must have have exceptional exceptional patience substantial support support to patience and and substantial to be be able able to to reveal reveal statistical statistical regularities regularities in lightning trajectories. From field observations, one usually borrows the in lightning trajectories. From field observations, one usually borrows the statistics of lightning strokes at objects of various height and, sometimes, statistics of lightning strokes at objects of various height and, sometimes, protected objects, objects, such as power power transmission transmission the statistics the statistics of of strokes strokes at at protected such as lines with with overhead overhead grounding grounding wire wire connections. This material, material, however, however, is lines connections. This is too scarce to build build aa theory. theory. For For this this reason, reason, one has to to refer refer to to laboratory too scarce to one has laboratory experiments on long sparks generated in No one one has has ever ever experiments on long sparks generated in 10-15m 10-15m gaps. gaps. No proved (or will ever so) the the geometrical geometrical similarity of sparks; therefore, proved (or will ever do do so) similarity of sparks; therefore, experimental experimental data data can can be be extended extended to to lightning lightning only only qualitatively. qualitatively. NevertheNevertheless less theoretical theoretical treatments treatments must must be be brought brought to to conclusion conclusion when when one one develops develops recommendations recommendations on on particular particular protector protector designs. designs. We We analyse analyse the the reliability reliability of of engineering engineering designs, designs, wherever wherever possible. possible. The The last last chapter chapter of of the the book book discusses discusses lightning lightning hazards hazards and and protection protection not not only only in in terms terms of of applications. applications. Even Even the the classical classical theory theory of of atmospheric atmospheric overvoltages overvoltages in in power power transmission transmission lines lines required required the the solution solution of of comcomplicated plicated electrophysical electrophysical problems. problems. Thorough Thorough theoretical theoretical treatments treatments are are necessary necessary for for the the analysis analysis of of lightning lightning current current effects effects on on internal internal circuits circuits of of engineering engineering constructions constructions with with metallic metallic casings, casings, on on underground underground cables, cables, aircraft, aircraft, etc. etc. The The range range of of problems problems to to be be considered considered is is not not limited limited to to electromagnetic electromagnetic field field theory. theory. We We shall shall also also discuss discuss gas gas discharge discharge mechanisms mechanisms of of aa spark spark creeping creeping along along aa conducting conducting surface, surface, the the excitation excitation of of leader leader channels channels in in air air with with the the composition composition and and thermodynamic thermodynamic characteristics characteristics locally locally changed changed by by hot hot gas gas outbursts, outbursts, and and the the lightning lightning orientation orientation under under the the action action of of the the superhigh superhigh operating operating voltage voltage of of an an object. object. These These theoretical theoretical considerations considerations will will not not screen screen our our practical practical recommendations recommendations concerning concerning effective effective lightning lightning protection protection and and the the application application of of particular particular types types of of protectors. protectors.


26 26


References References [l] Bazelyan Bazelyan E E M, M, Gorin Gorin B B Nand N and Levitov Levitov V V 11978 I 1978 Physical and Engineering Funda[1] of Lightning Protection (Leningrad: (Leningrad: Gidrometeoizdat) Gidrometeoizdat) p 223 223 (in (in Russian) Russian) mentals of 57 493 [2] McEachron McEachron K K 1938 1938 Electr. Engin. 57493 [2] [3] Berger K K and and Vogrlsanger Vogrlsanger E E 1966 SEV 57 No No 13 1966 Bull. SEV 13 1I [3] Berger [4] Newman Stahmann JJ R, 1967 JJ.. Geophys. Geophys. Res. [4] Newman M M, M, Stahmann R, Robb Robb JJ D, D, Lewis Lewis E EA A et a1 al1967 72 4761 4761 [5] Uman Uman M M A 1987 1987 The The Lightning Discharge (New (New York: York: Academic Academic Press) Press) p377 p 377 [5] [6] R Band B and Kroninger Kroninger H H 1975 1975 Electra 4123 41 23 [6] Berger Berger K, K, Anderson Anderson R [7] Malan D D and and Collens Collens H H 1935 1935 Froc. Proc. Roy. Roy. Soc. Soc. London Ser A 152 152 595 595 [7] Schonland Schonland B, Malan EM M and and Raizer Raizer Yu Yu P P 1997 1997 Spark Discharge (Boca (Boca Raton: Raton: CRC CRC Press) Press) [8] Bazelyan Bazelyan E [8] p 294 p294 191 Schonland Schonland B B 1956 1956 The The Lightning Discharge. Handbuch der Physik 22 (Berlin: (Berlin: [9] Springer) p 576 576 Springer)


Chapter 2

The streamer-leader streamer-leader process process in in a long long spark This will deal with the This chapter chapter will deal with the spark spark discharge discharge in in aa long long air air gap. gap. We We have have already material should be ignored by already mentioned mentioned in in chapter chapter 11 that that this this material should not not be ignored by the know much the reader. reader. But But for for the the long long spark, spark, specialists specialists would would know much less less about about lightning. voltage laboratories lightning. Today, Today, high high voltage laboratories are are able able to to produce produce and and study study long tens and long sparks sparks of of several several tens and even even hundreds hundreds of of metres metres long long [1-3]. [l-31. Many Many of of the the long long spark spark parameters parameters and and properties properties lie lie close close to to the the lower lower boundary boundary of of respective respective lightning lightning values. values. Most Most effects effects observable observable in in aa lightning lightning discharge the laboratory. discharge were, were, sooner sooner or or later, later, reproduced reproduced in in the laboratory. One One exception exception is but the technology rather is aa multicomponent multicomponent discharge, discharge, but the obstacles obstacles lie lie in in the the technology rather than phenomenon. It would be be very very costly than in in the the nature nature of of the the phenomenon. It would costly to to instal instal and and synchronize high voltage power generators, synchronize several several high voltage power generators, making making them them discharge discharge consecutively consecutively into into the the same same air air gap. gap. researchAs for for the the fine fine structure structure of of gas-discharge gas-discharge elements, elements, long long spark spark researchbe otherwise, ers of lightning lightning observers. observers. This This could could not not be otherwise, since since aa ers are are far far ahead ahead of laboratory be reproduced by starting laboratory discharge discharge can can be reproduced as as often often as as necessary, necessary, by starting the the generator generator at at the the right right moment, moment, within within aa microsecond microsecond fraction fraction accuracy, accuracy, and and strictly timing the strictly timing the switching switching of of all all fast fast response response detectors. detectors. But But with with lightning, lightning, the the situation situation is is different. different. It It strikes strikes every every square square kilometre kilometre of of the the earth's earth’s sursurface year. So, face in in Europe Europe approximately approximately 22 to to 44 times times aa year. So, even even such such aa high high conconstruction m) is by lightning struction as as the the Ostankino Ostankino Television Television Tower Tower (540 (540m) is struck struck by lightning only times aa year. these, only only 25-30 25-30 times year. Of Of these, only 2-3 2-3 discharges discharges are are descending, descending, while to aa cloud. Normally, lightning while the the others others go go up up to cloud. Normally, lightning observations observations have have to be be made to made from from afar, afar, so so that that many many details details of of the the process process are are lost. lost. The The be filled gaps gaps in in the the study study of of its its fine fine structure structure must, must, of of necessity, necessity, be filled in in laboratory laboratory conditions. conditions. The The long long spark spark theory theory is is far far from from being being completed, completed, and and there there is is no no the process. process. Still, has lately been some adequate adequate computer computer model model of of the Still, there there has lately been some unwise progress, primarily primarily owing progress, owing to to laboratory laboratory investigations. investigations. It It would would be be unwise to these data try to to use the description to discard discard these data and and not not to to try use them them for for the description of of 21 27 Copyright © 2000 IOP Publishing Ltd.

28 28

The streamer-leader process in a long spark The streamer-leader

lightning. basic lightning. In In this this chapter, chapter, we we shall shall outline outline our our conception conception of of the the basic phenomena in a long spark. We shall present some newer data and ideas phenomena in a long spark. We shall present some newer data and ideas which book [4] which emerged emerged after after the the book [4] on on the the long long spark spark had had been been published. published. We should like to emphasize again that many details of the We should like to emphasize again that many details of the spark spark physics physics are are still still far far from from being being clear. clear.

2.1 2.1

What a lightning lightning researcher should should know know about a long long spark

The how aa spark The key key point point is is how spark channel channel develops develops in in aa weak weak electric electric field, field, by by 1-2 1-2 orders necessary to the electron orders of of magnitude magnitude lower lower than than what what is is necessary to increase increase the electron density under normal Naturally, we density in in air air (E (Eii :::::; x 30 30 kV/cm kV/cm under normal conditions). conditions). Naturally, we speak Near an speak of of aa discharge discharge in in aa sharply sharply non-uniform non-uniform field. field. Near an electrode electrode with this is radius with aa small small curvature curvature radius radius (suppose (suppose this is aa spherical spherical anode anode of of radius x l-lOcm), 1-lOcm), the the field field is is Ea(ra) at the the voltage U x 50-500kV. 50-500kV. rr,a :::::; Ea(r a) == Ea E a > E, Ei at voltage V:::::; = lOr lor,a This is is the the site site of of initiation initiation of of aa discharge discharge channel. channel. At At aa distance distance r = This from the the electrode electrode centre, centre, the channel tip enters the the outer outer gap gap region, region, from the channel tip enters where where the the initial initial value value of of E := =E Ea(r,/r)* is one one hundredth hundredth of of that that on on the the a (r a /r)2 is electrode. This This weak weak field field is is incapable incapable of of supporting supporting ionization. ionization. Nevertheless, Nevertheless, electrode. the moves on, plasma. the channel channel moves on, changing changing the the neutral neutral gas gas to to aa well-ionized well-ionized plasma. There is is no no other other reasonable reasonable explanation explanation of of this this fact fact except except for for aa local local There enhancement the developing enhancement of of the the electric electric field field at at the the tip tip of of the developing channel. channel. The The enhancement enhancement is is due due to to the the action action of of the the channel's channel’s own own charge. charge. Indeed, Indeed, aa conconductive with the the anode be charged much ductive channel channel having having aa contact contact with anode tends tends to to be charged as as much as its its potential U , relative relative to to the the grounded grounded cathode. cathode. Current Current arises arises in in the the as potential Va channel, the positive positive electric channel, which which transports transports the electric charge charge from from the the anode anode (more (more exactly, voltage source, which the the anode exactly, from from the the high high voltage source, to to which anode is is connected). connected). (In (In reality, the channel reality, electrons electrons moving moving through through the channel toward toward the the anode anode expose expose low low mobility positive positive ions.) ions.) Such Such would exactly the the mechanism mechanism of of charging charging aa mobility would be be exactly metallic be pulled pulled out metallic rod rod if if it it could could be out of of the the anode anode like like aa telescopic telescopic antenna. antenna. Then the the strongest strongest field field region region would would move move through through the the gap gap together together with with Then the rod rod tip. tip. We We can can say say that that aa strong strong electric electric field field wave wave is is propagating through the propagating through gap, in in which ionization occurs occurs and and produces new portion of the the plasma plasma aa gap, which ionization produces aa new portion of it as as an an ionization ionization wave, wave, and and tthis h s term term is is commonly commonly channel. We We can can also also name name it channel. used. used. The wave wave mechanism of spark spark formation formation was was suggested suggested as as far far back back as as The mechanism of the by L Raether. The thus formed the 1930s 1930s by L Loeb, Loeb, JJ Meek, Meek, and and H H Raether. The channel channel thus formed was termed termed aa streamer streamer (figure (figure 2.1). 2.1). Experiments Experiments showed showed that that the the streamer streamer was 7 velocity could could be be as as high high as as 10 107m/s. In lightning, lightning, this this velocity is demondemonvelocity m/s. In velocity is strated by by the the dart dart leader leader of of aa subsequent subsequent component. component. Even Even the the mere mere fact fact strated that these these velocities velocities are are comparable comparable justifies our interest interest in in the the streamer streamer that justifies our what determines velocity mechanism. mechanism. It It is is important important to to know know what determines the the streamer streamer velocity and how it with the potential. For have to and how it changes changes with the tip tip potential. For this, this, we we have to analyse analyse


What a lightning lightning researcher should should know about a long spark spark

29 29

grounded cathode

:ni"',>,~ fUlode _ _cathode t ~

t x

Figure 2.1. A schematic (x), external schematic cathode-directed cathode-directed streamer: streamer: U, Uo(x), external field field potential; potential; U(x), potential along along the the conductive conductive streamer streamer axis. axis. U ( x ) ,potential

processes taking taking place place in in the the streamer streamer tip tip region region where where ionization ionization occurs. occurs. It It is is processes necessary to to find out how how the the processes processes of of charged charged particle particle production production are are necessary find out related to to electron electron motion motion in in the the electric due to to which which the the charged related electric field, field, due charged region travels travels through through the the gap gap like like the the crest crest of of aa sea sea wave. wave. region The specific specific nature nature of of spark breakdown is is not not restricted restricted to to the the ionization ionization The spark breakdown wave front, front, because because its its crucial crucial parameter parameter is is the the channel tip potential potential U Its wave channel tip U,. I . Its value may may be be much much smaller than the the potential potential Ua, of of the the electrode, electrode, from which value smaller than from which the streamer has started, started, since the channel channel conductivity is always always finite finite and the streamer has since the conductivity is and the voltage voltage drops drops across across it. it. Therefore, Therefore, the the analysis analysis of of streamer propagation the streamer propagation for large distance will require require aa knowledge knowledge of of the the electron electron density behind for aa large distance will density behind the wave front and the current along the channel in order to eventually the wave front and the current along the channel in order to eventually calculate calculate the the electric electric field field in in the the travelling travelling streamer streamer and and to to derive derive from from it it the voltage voltage drop drop on on the the channel. channel. Incidentally, Incidentally, the the field field and and the the current current the preset preset the the power power losses losses in in the the channel. channel. It It will will become become clear clear below below how how important important this this parameter parameter is is for for spark spark theory. theory. The The streamer streamer creates creates aa fairly fairly dense dense plasma. plasma. Without Without this, this, it it would would be be unable unable to to transport transport an an appreciable appreciable charge charge into into the the gap. gap. A quantitative quantitative description description of of the the ionization ionization wave wave propagation propagation provides provides the the initial initial electron electron density in the channel and defines its initial radius. Behind the density in the channel and defines its initial radius. Behind the wave wave front, front, the streamer channel channel may may the streamer streamer continues continues to to live live its its own own life. life. A streamer expand, expand, through through ionization, ionization, in in the the radial radial electric electric field field of of its its intrinsic intrinsic charge, charge, provided provided that that the the latter latter grows. grows. The The cross cross section section of of the the current current flow then then becomes becomes larger. larger. The The channel channel continuously continuously loses loses the the majority majority curcurflow rent rent carriers carriers -- electrons. electrons. The The rates rates of of electron electron attachment attachment to to electronegative electronegative particles particles and and electron-ion electron-ion recombination recombination strongly strongly affect affect the the fate fate of of the the discharge If the the air air through through which which aa streamer streamer propagates propagates is is discharge as as aa whole. whole. If cold cold and and the the power power input input into into the the channel channel is is unable unable to to increase increase its its temperatemperature ture considerably considerably (by (by several several thousands thousands of of degrees), degrees), the the process process of of electron electron loss loss is is very very fast, fast, since since the the attachment attachment alone alone limits limits the the electron electron average average lifetime lifetime 7 s.s. This to to 10lop7 This is is aa very very small small value value not not only only at at the the scale scale of of lightning lightning but but


30 30

The streamer-leader The streamer-leader process in a long spark

4 also also of of aa laboratory laboratory spark, spark, whose whose development development in in aa long long gap gap takes takes 10lop43 10behind lop3s. s. One One must must be be able able to to analyse analyse kinetic kinetic processes processes in in the the channel channel behind the the ionization ionization wave wave front. front. Without Without the the knowledge knowledge of of their their parameters, parameters, one one will be unable unable to will be to define define the the conditions, conditions, in in which which aa streamer streamer breakdown breakdown in in air will will be air be possible. possible. Here and and below, below, we we shall shall mean mean by of aa gap gap by by Here by aa breakdown breakdown the the bridging bridging of aa channel which, like by aa falling channel which, like an an electric electric arc, arc, is is described described by falling current-voltage current-voltage characteristic. then limited by the the resistance resistance of characteristic. The The channel channel current current is is then limited mostly mostly by of the high voltage source. Such Such aa situation situation in in technology technology is is usually called short short the high voltage source. usually called circuiting. circuiting. Current rise increase of of the gap voltage inevitably suggests suggests aa Current rise without without an an increase the gap voltage inevitably considerable heating of the gas thermal expansion, considerable heating of the gas in in the the channel. channel. Due Due to to thermal expansion, the the molecular density N decreases, decreases, thereby increasing the the reduced reduced electric electric field field thereby increasing molecular density E / N and and the the ionization ionization rate constant (see (see [4]). Another Another consequence consequence of of the the rate constant heating is is aa change change in in the the channel channel gas gas composition composition because because of of aa partial heating partial dissociation of of 02, 0 2 ,N2 and H H 2200 molecules molecules and and the the formation formation of of easily easily dissociation N 2 and ionizable NO The significance significance of of many of charged charged ionizable NO molecules. molecules. The many reactions reactions of particle and loss loss changes. changes. The The importance importance of of electron electron attachment attachment particle production production and decreases, because negative ions ions produced in aa hot hot gas gas rapidly disintegrate to decreases, because negative produced in rapidly disintegrate to set free free the the captured captured electrons. electrons. The The electron-ion electron-ion recombination set recombination rate rate becomes becomes lower. But But of of greater greater importance importance is is associative associative ionization ionization involving involving 0 and and N N lower. atoms. The The reaction reaction is is accelerated accelerated with with temperature temperature rise rise but but it it does does not atoms. not depend directly directly on on the the electric electric field. field. This This creates creates prerequisites for aa falling falling depend prerequisites for current-voltage characteristic. characteristic. current-voltage Clearly, aa researcher dealing with with long long sparks sparks and and lightning lightning cannot cannot Clearly, researcher dealing avoid considering considering the the energy energy balance in the the discharge discharge channel, channel, which which deterdeteravoid balance in mines the the gas gas temperature. It is is here here that that the the final final result result is is most most likely likely to to mines temperature. It depend on on the the scale scale of of the the phenomenon phenomenon and and the the initial initial conditions. conditions. In In the the depend laboratory, aa streamer streamer crossing crossing aa long long gap gap seldom seldom causes causes aa breakdown breakdown laboratory, directly. A streamer streamer propagating propagating through through cold cold air air remains cold. It It will directly. remains cold. will be be shown below below that that the the specific specific energy energy input input into into the the gas gas is is too too small small to to heat heat shown it. Even Even during during its its flight, flight, the the old, old, long-living long-living portions portions of of aa streamer streamer lose lose it. most of of their their free free electrons. electrons. In In actual actual fact, fact, it it is is not not aa plasma plasma channel channel but most but rather its its nonconductive nonconductive trace trace which which crosses crosses aa gap. gap. The The researcher must posrather researcher must possess special special skills skills to to be able to to produce an actual actual streamer streamer breakdown breakdown of of aa sess be able produce an cold air air gap gap in in laboratory laboratory conditions. conditions. cold The situation situation with with lightning lightning may may be different. Most Most lightnings lightnings are are multimultiThe be different. component structures. structures. With With the the next next voltage voltage pulse, pulse, the the ionization ionization wave wave often often component propagates through through the the still still hot hot channel channel of of the component. It It is is not not propagates the previous previous component. cold air air but but quite quite aa different different gas gas with more favourable favourable chemical chemical composicomposicold with aa more tion and and kinetic kinetic properties. Surrounded by by cold cold air, air, the the hot hot tract tract shows shows some some tion properties. Surrounded features of of aa discharge discharge in in aa tube with aa fixed fixed radius radius and, and, hence, hence, with with aa more more features tube with concentrated energy energy release. It seems seems that that the the mechanism mechanism of of the the phenomenon concentrated release. It phenomenon known as as aa dart dart leader leader is is directly directly related streamer breakdown. One should should known related to to streamer breakdown. One

°


What a liKhtning lightning researcher should know lorig spark What knol\' ahout a lonK

31 31

streamer zone

and aa scheme scheme of of aa positive positive leader. leader. Figure 2.2. A photograph photograph and be to give be ready ready to give aa quantitative quantitative description description or or make make aa computer computer simulation simulation of of this process. this process. Long gaps gaps of of cold cold air air are are broken broken down down by by the the leader leader mechanism. During Long mechanism. During the plasma channel (5000- 10 000 000 K) K) is is travelling travelling the leader leader process, process, aa hot hot plasma channel (5000-10 through the the gap. gap. Numerous streamers start start at at high high frequency frequency from from the the through Numerous streamers leader leader tip, tip, as as from from aa high high voltage voltage electrode, electrode, and and form form aa kind kind of of fan. fan. They They fill up up aa volume of several several cubic cubic metres metres in in front front of of the the tip tip (figure (figure 2.2). 2.2). fill volume of This This region region is is known known as as the the streamer streamer zone zone of of aa leader, leader, or or leader leader corona, corona, by analogy analogy with with aa streamer streamer corona corona that that may may arise arise from from aa high high voltage voltage elecelecby trode in in laboratory laboratory conditions. conditions. The The total total current current of of the the streamers streamers supplies supplies trode with energy energy the the leader leader channel channel common common to to the the streamers, streamers, heating heating it it up. up. with with charges charges of of streamers streamers that that are are being being The streamer streamer zone zone is is filled filled up The up with formed formed and and those those that that have have died. died. As As the the leader leader propagates, propagates, the the zone zone travels travels so that the leader leader channel channel enters enters aa space space through the the gap gap together together with with its its tip, tip. so through that the filled with aa space filled with space charge, charge, 'pulling' 'pulling' it it over over like like aa stocking. stocking. A A charged charged leader leader cover thus formed this 2.2). It It is is this cover is is thus formed which which holds holds most most of of the the charge charge (figure (figure 2.2). charge that changes the space charge that changes the the electric electric field field in in the space around around aa developing developing spark and and lightning. lightning. It It is is neutralized neutralized on on contact contact of of the the leader leader channel channel with spark with the earth, creating creating aa powerful powerful current current impulse impulse characteristic characteristic of of the the earth, the return return of interrelated interrelated events, events, stroke of of aa spark. spark. Thus, Thus, we can follow follow aa chain chain of stroke we can which unites the simplest simplest element element of of aa spark spark (streamer) (streamer) with the leader leader which unites the with the process possessing possessing aa complex complex structure structure and and behaviour. behaviour. process All the properties properties All details details of of the the leader leader development development directly directly follow follow from from the of aa streamer streamer zone. zone. In In lightning, lightning, it it is is entirely entirely inaccessible inaccessible to observation of to observation because of the the relatively relatively small small size size and and low low luminosity. luminosity. Today, Today, there is no no because of there is other way way but to study study long long sparks sparks in in laboratory laboratory conditions conditions and and to extrapoother but to to extrapolate the obtained to to extremely extremely long long gaps. gaps. This This primarily concerns aa late the results results obtained primarily concerns stepwise negative leader, whose whose streamer streamer zone zone has has an an exclusively exclusively complex complex stepwise negative leader, Copyright © 2000 IOP Publishing Ltd.

32

The streamer-leader process in a long spark The streamer-leader

structure. polarities, starting structure. It It contains contains streamers streamers of of different different polarities, starting not not only only from from the leader but also them. the leader tip tip but also from from the the space space in in front front of of them. The The leader leader channel channel of of aa very very long long spark, spark, let let alone alone oflightning, of lightning, is is its its longlongAn appreciable part of to the est est element. element. An appreciable part of the the voltage voltage applied applied to the gap gap may may drop drop on on this this element. element. This This is is why why one one should should know know the the time time variation variation of of the the channel properties mostly channel conductivity. conductivity. The The channel channel properties mostly depend depend on on the the current current flowing flowing through through aa given given channel channel cross cross section. section. If If the the current current is is known, known, it it is is not belongs to not particularly particularly important important whether whether it it belongs to aa long long spark spark or or lightning. lightning. The parameter parameter that that changes changes is is the during which which one one observes observes this this The the time time during process: two orders than for process: for for lightning, lightning, it it is is one one or or two orders of of magnitude magnitude longer longer than for aa spark. process ofleader production spark. By By analysing analysing the the self-consistent self-consistent process of leader current current production in the the streamer streamer zone zone and and its its effects effects on on plasma plasma heating heating and and conversion conversion in in the the in channel, one one can can derive derive the the conditions conditions for for an an optimal optimal leader leader development development in in channel, gap of of aa given given length. length. There There are are reasons to believe believe that that these these conditions conditions are are aa gap reasons to realized in in lightning lightning when when it it develops develops in in an an extremely extremely weak weak field. field. Nature realized Nature always strives strives for for perfection, it is is animated animated but because optimal optimal always perfection, not not because because it but because conditions most often lead lead to to the the highest highest probability of an an event. event. conditions most often probability of To to To conclude conclude this this section, section, long long spark spark theory theory is is of of value value in in its its own own right right to specialists in in lightning lightning protection. protection. Lightning Lightning current current is is the the cause cause of of the the most most specialists common type type of of overvoltage overvoltage in in electric electric circuits. circuits. The The amplitude amplitude of of lightning lightning common overvoltages reaches reaches the the megavolt megavolt level. level. In In order order to to design design aa lightninglightningovervoltages resistant circuit, circuit, one one must must be be able able to to estimate estimate breakdown voltages in in air air resistant breakdown voltages gaps of of various various lengths lengths and and configurations. configurations. This This can can be done only only with with aa gaps be done clear understanding of the long spark spark mechanism. mechanism. clear understanding of the long

2.2 2.2.1

A long long streamer The streamer tip as an ionization wave

Let us us consider consider aa well well developed developed 'classical' ‘classical’ streamer, streamer, which has started started from from Let which has high voltage voltage anode anode and and is is travelling travelling towards grounded cathode. cathode. The The aa high towards aa grounded main ionization process occurs in in the the strong strong field field region near the streamer main ionization process occurs region near the streamer tip. We We shall shall focus focus on on this this region. region. The The front front portion of aa streamer streamer is is tip. portion of 2.3 together together with with aa qualitative qualitative axial axial distribudistribushown schematically schematically in in figure figure 2.3 shown E , electron electron density density ne, ne, and and aa difference difference between tion of of the the longitudinal longitudinal field field E, tion between the densities densities of of positive ions and and electrons, electrons, or or the the density density of of the the space space charge charge the positive ions p= = e(n+ e(n+ -- ne) ne) (the (the time time is is too too short short for for negative ions to to be formed). negative ions be formed). The The strong strong field field near near the the tip tip is is created created mostly mostly by by its its own own charge. charge. In In front front of the the tip tip where where the the space space charge charge is is small, small, the field decreases decreases approximately approximately as as of the field E == E ~/r)2, / Y ) ’ ,which is characteristic characteristic of of aa charged charged sphere sphere of of radius Here, Em, ((rTm which is radius .,rYm. Here, E, is the the maximum streamer field field at at the the tip tip front front point. point. In In fact, fact, the at Em is maximum streamer the radius radius at It approxiapproxiwhich the field is is maximum should be termed the the tip tip radius radius .,rYm' It which the field maximum should be termed mately coincides coincides with with the initial radius radius of of the the cylindrical cylindrical channel channel extending extending mately the initial behind the tip. tip. The The front front portion portion of of aa conventionally conventionally hemispherical hemispherical tip tip behind the


A long streamer streamer

33 33

x, Figure 2.3. Schematic Schematic representation representation of of the the front front portion portion of of aa cathode-directed cathode-directed streamer qualitative distributions distributions of of the the electron electron density density ne, ne, the the density density difference difference streamer and and qualitative n, (space charge), along the n+ - ne ne (space charge), and and longitudinal longitudinal field field E along the axis. axis.

should be called the ionization wave front. The streamer tip charge charge is is primarily primarily should be called the ionization wave front. The streamer tip concentrated in in the the region region behind behind the the wave wave front. front. The The field field there there becomes becomes low, low, concentrated dropping to to aa value value E, Ec in the channel, channel, small as compared compared with with E,. Em. The lines of in the small as The lines of dropping force going radially away from the the tip tip in of it become straight lines inside force going radially away from in front front of it become straight lines inside the tip tip and along the the streamer streamer channel. channel. the and align align axially axially along Let us us mentally mentally subdivide the continuous process of subdivide the continuous process of streamer streamer developdevelopLet ment into The strong strong field region in front of of the the tip tip is is the the site ment into stages. stages. The field region in front site of of ionization ionization of of air air molecules molecules by by electron electron impact. impact. The The initial initial seed seed electrons electrons necessary necessary for for this this process process are are generated generated by by the the streamer streamer in in advance. advance. Their Their production of quanta, quanta, accompanying accompanying the the ionization ionization production is is due due to to the the emission emission of process process because because of of electronic electronic excitation excitation of of molecules. molecules. In In our our case, case, highly highly excited molecules are active so that the quanta emitted by them ionize N2 molecules are active so that the quanta emitted by them ionize excited N 2 molecules, whose ionization potential is lower than that of N the O2 molecules, whose ionization potential is lower than that of N2. The the O 2 2 . The radiation is is actively actively absorbed, absorbed, but but its its intensity intensity is is high high enough enough to to provide provide radiation 5 _10 6 cm -3 at an M~ of of about about 10 105-106~m-3 at aa distance distance of of 0.10.1an initial initial electron electron density density no 0.2 cm from 0.2cm from the the tip. tip. Each Each of of these these electrons electrons gains gains energy energy from from the the strong strong field, field, giving giving rise rise to to an an electron electron avalanche. avalanche. Since Since the the number number of of avalanches avalanches developing of developing simultaneously simultaneously is is very very large, large, they they fill fill up up the the space space in in front front of the the streamer streamer tip tip to to form form aa new new plasma plasma region. region. Owing Owing to to the the electron electron outflow outflow towards towards the the channel channel body, body, the the positive positive space space charge charge of of the the plasma plasma becomes becomes exposed. exposed. Simultaneously, Simultaneously, electrons electrons that that have have advanced advanced from from the the ahead ahead region region neutralize neutralize the the positive positive charge charge of of the the 'old' ‘old’ tip tip which which turns turns to to aa new new channel channel portion, portion, thereby thereby elongating elongating the the streamer. streamer. The The gas gas in in the the wave wave front front region region must must be be highly highly ionized ionized for for the the electronelectronion ion separation separation to to produce produce an an appreciable appreciable charge charge capable capable of of creating creating aa strong strong ionizing ionizing field field in in front front of of the the newly newly formed formed tip. tip. For For this this reason, reason, the the region region of of


34 34

The streamer-leader streamer-leader process in a long spark

concentrated concentrated tip tip charge charge is is somewhat somewhat shifted shifted towards towards the the channel channel body body relative relative to to the the intensive intensive ionization ionization site site (figure (figure 2.3). 2.3). Normally, Normally, the the electric electric field field is is pushed pushed out out of of aa good good plasma plasma conductor, conductor, and and the the space space charge charge (if (if the the conductor conductor is is charged) charged) quickly quickly concentrates concentrates near near its its surface surface as as aa 'surface' ‘surface’ charge. The The plasma plasma of of aa fast ('fast' in in the the sense that will will be charge. fast streamer streamer (‘fast’ sense that be specified specified below) possesses possesses aa fairly high conductivity, conductivity, and and these these properties properties apply apply to to below) fairly high such streamer. Therefore, strong field field and space charge such aa streamer. Therefore, the the region region of of strong and space charge in in the looks like like aa thin layer, as as is is shown shown in in figure figure 2.3. 2.3. the tip tip looks thin layer, >> rr,,m, its its velocity velocity and and the the tip tip parameters parameters change change If the the streamer streamer length length is is I » If little during during the the time time the the tip tip travels travels aa distance distance of of its its several several radii. radii. This T h s means means little and the the axial axial coordinate coordinate x, all all parameters parameters that, depending depending on on the the time time tt and that, of the the type type E(x, E ( x . t) t) = = E(x E ( x -- Vst), V,t), and and what what is is shown shown in in are the the functions functions of are figure 2.3 2.3 moves moves to to the the right right as as aa whole, whole, without without noticeable noticeable distortions. distortions. The The figure picture changes changes only only as as the the streamer streamer velocity velocity changes changes relatively relatively slowly. slowly. This This picture kind of of process process represents represents aa wave, wave, in in this t h s case case aa wave wave of of strong strong field field and and kind ionization. ionization. The The external external parameter parameter determining determining the the wave wave characteristics characteristics (its (its , velocity maximum field Em, tip radius r , electron density behind the V,, maximum field E,, tip radius r,, electron density behind the velocity V m s wave is the the tip tip potential potential V U,. It is is indeed indeed an an external external characteristic characteristic of of the the wave nn,) t . It c ) is tip, although although it it partly partly depends depends on on the the properties properties of of the the wave wave itself. itself. The The potenpotentip, Ut, is is equal equal to to the the anode anode potential potential Va U , minus minus the the voltage voltage drop drop on on the the tial V tial streamer channel. channel. The The channel channel properties, properties, however, however, are are initially initially determined determined streamer so that that the the problem problem of of streamer streamer developdevelopby the the ionization ionization wave wave parameters, parameters, so by ment is, is, strictly strictly speaking, speaking, just just one one problem. problem. Still, Still, itit can can be be approximately approximately ment of subdivided into into two two parts: parts: the the ionization ionization wave wave problem problem and and the the problems problems of subdivided voltage drop drop and and current current in in the the channel. channel. Both Both parts parts will will be be related related by by the the voltage dependencies of of V V,( Utt) ) and and current current iilj at at the the channel channel front front on on velocity velocity V V,. dependencies s( V s'

2.2.2 2.2.2 Evaluation Evaluation of streamer streamer parameters parameters

The formulas formulas to to be be derived derived in in this this and and subsequent subsequent sections sections of of this this chapter chapter do do The not claim claim high high accuracy. accuracy. The The streamer streamer and and leader leader problems problems are are very very complex, complex, not and and aa rigorous rigorous solution solution can can be be obtained obtained only only by by numerical numerical computation. computation. But But simplified analytical analytical treatment treatment may may also also be be useful useful because because it it provides provides an an aa simplified understanding understanding of of basic basic laws laws and and relations relations among among the the process process parameters. parameters. of the the phenomphenomIn other other words, words, one one is is able able to to get get aa general general idea idea of of the the physics physics of In enon under under study study and and to to estimate estimate the the order order of of values values of of its its characteristics. characteristics. enon Let us us consider consider aa fast fast streamer, streamer, whose whose velocity velocity is is much much higher higher than than the the Let of electron drift drift velocity velocity in in the the wave. wave. Streamers Streamers are are fast fast in in many many situations situations of electron practical practical interest. interest. The The calculation calculation of of electron electron production production can can ignore ignore the the slight drift drift of of electrons electrons from from aa given given site site in in space space for for the the short short time time the the slight ionization wave wave passes passes through through it. it. In In this this case, case, the the ionization ionization kinetics kinetics ionization along the the streamer streamer axis axis is is described described by by the the following following simple simple equations: equations: along e 3 yn,. -an at == vn e,

at

1


s

f

nc C = exp exp vvidt =exp -n dt = exp no = 1

a0

f

dx

1/'1

V

s

(2.1 )

A long streamer 11:

8

"

35

1011 S-l

6 4

2 E, kVlcm

0

200

400

600

Figure 2.4. Ionization Ionization frequency frequency of of air air molecules molecules by by electron electron impact impact under Figure under normal normal and electron electron drift drift velocity velocity V. V, conditions (from (from the the data data on on ionization ionization coefficient coefficient acy and conditions in [II]). [l 11). in

where vi = Vi vi(E) is the the frequency frequency of of electron electron ionization ionization of of molecules. molecules. Its Its time where Vi = (E) is time integral been transformed x along integral has has been transformed to to the the integral integral over over the the coordinate coordinate x along dt corresponding corresponding to to the the the wave wave axis, axis, according according to to the the equality equality dx dx == V s, dt the coordinate the sharp coordinate system system moving moving together together with with the the wave. wave. Due Due to to the sharp increase increase of the the ionization ionization frequency frequency with field (figure (figure 2.4), 2.4), the the region region where where the field is is of with field the field not not much much less less than than its its maximum maximum makes makes the the largest largest contribution contribution to to the the electron of the the same same order order of of electron production. production. This This region region in in the the wave wave is is of magnitude the tip tip radius radius (figure we can write the 2.3). So So we can write the approximate approximate magnitude as as the (figure 2.3). expressions (2.1) and and streamer streamer velocity: velocity: expressions for for the the integral integral (2.1) (2.2)

This was first by Loeb has been been used used since This type type of of formula formula was first suggested suggested by Loeb [5] [5] and and has since that [6-lo]. The The that time, time, in in this this or or modified modified form, form, in in all all streamer streamer theories theories [6-10]. velocity no and and final final nn,c velocity of of aa fast fast streamer streamer is is weakly weakly related related to to the the initial initial no electron maximum field Em and the electron densities densities and and is is determined determined only only by by the the maximum field E, and the tip radius radius rr,.m' tip The Em and which determine not independent. The quantities quantities E, and rm, which determine V s, are are not independent. They They . For an isolated conductive sphere with aa are by the tip potential potential V are interrelated interrelated by the tip U,. For an isolated conductive sphere with t uniformly distributed Q', we we have have V U == rmE r,E, m == Q' Q 'j47fcorm, /~TTTE~Y,, uniformly distributed surface surface charge charge Qt, 11 12 )-1 ~ F/m is permittivity. A where where co E~ = = (367f ( 3 6 Xx~ 10 lo")-' FZ 8.85 8.85 xX 1010-12F/m is vacuum vacuum permittivity. streamer hemispherical rounded streamer looks looks more more like like aa cylinder cylinder with with aa hemispherical rounded end end (see (see figure perfect conductor 2.3). We We can can show show [4] [4] that that in in aa long long perfect conductor of of this this shape, shape, figure 2.3). the potential potential at one one half half of of the at the the hemisphere hemisphere centre centre is is created created by by charges charges concentrated hemisphere surface half by by those concentrated on on the the hemisphere surface and and the the other other half those on on the the so that that the the tip tip charge charge is is Q Q= = 27fcormVt. ~ T T T E ~The The Y , Ufield field ~ . at at the the tip cylinder surface, surface, so cylinder tip front of that that in in an an isolated isolated front point point is, is, to to good good approximation, approximation, only only one one half half of


36 36

The streamer-leader streamer-leader process in a long spark The

sphere sphere with with the the same same potential, potential, or or

0;t == 2E 2E,r,. U m rm ·

(2.3)

The The tip tip charge charge moves moves because because of of the the electron electron drift drift under under the the field field action. action. The The electron electron density density in in the the wave wave plasma plasma and and the the respective respective plasma plasma conducconductivity tivity must must provide provide the the charge charge transport transport with with the the same same velocity velocity as as that that of of the the wave. wave. This This permits permits estimation estimation of of the the plasma plasma density density in in the the streamer streamer just (2.2),the the electron electron just behind behind its its tip. tip. With With the the same same assumptions assumptions as as those those in in (2.2), density density in in the the strong strong field field region region on on the the streamer streamer axis axis increases increases as as M no noexp exp (Vimt) (vimt)for for the the time time 6.t:::::: At M rm/V rm/Vs. During this this period period of of time, time, the the nnee :::::: s. During electron M no no exp exp (Vim6.t) (vi,&) ,, and and the the electron electron electron density density rises rises to to its its final final value value nncc :::::: Ve, == peE, (where f-te pe is is electron electron drift towards towards the the channel channel with with velocity velocity V drift f-teEm (where mobility taken, taken, for for simplicity, simplicity, to to be be constant) constant) exposes exposes the the charge charge which which mobility E, in in the the region region of of the the new new streamer streamer tip. tip. creates the the field field Em creates The electron electron charge charge that that flows flows through through aa unit unit cross cross section section normal normal to to the the The A t is is axis in in the the wave wave front front region region over over time time 6.t axis At !:>.t

f-teEmnc PeEmnc (2.4) . Jo Vim vim It leaves leaves behind behind it it aa positive positive charge charge of of the the same same surface surface density density q. q. It It is is this this It E,. We We shall shall see see soon soon that that the the effective effective thickness thickness charge that that creates creates the the field field Em. charge Ax « > rraa (l (I isis Let much r ) . We We can can then then neglect neglect the the time time much larger larger than than the the channel channel radius radius r). variation variation of of the the anode anode charge, charge, because because its its capacitance capacitance is is small, small, and and take take the the external external current current to to be be close close to to the the current current ii,a entering entering the the channel channel through through its its base base at at the the anode. anode. Besides, Besides, aa streamer streamer conductor conductor can can be be regarded regarded as as being being solitary, solitary, and and the the unperturbed unperturbed potential potential U U,o far far from from the the anode anode can can be be ignored. ignored. Assume Assume first first that that the the channel channel is is aa perfect perfect conductor. conductor. From From aa well-known well-known electrostatics electrostatics formula, formula, the the capacitance capacitance of of aa long long solitary solitary conductor conductor is is C = 27rc 2 7 ro~llln(llr). ~ l / l n ( l / Its Its ~ ) . charge charge is is Q Q= = CU, CU, because because aa perfect perfect conductor conductor is is C under under only only potential potential U. U . Introduce Introduce now now the the concept concept of of capacitance capacitance per per unit unit length , which is frequently used in electroand radiolength of of the the conductor, conductor, C C1, which is frequently used in electroand radio1 engineering engineering to to analyse analyse long long lines. lines. The The average average capacitance capacitance per per unit unit length length - 5.56 5.56 Xx 10lo-”11 In (llr) F/m ln(I/r) = ln(I/r)

C 27rco 27rq c1=-=-C1 = I = In (llr)

I

(2.8)

has has aa nearly nearly constant constant value value which which only only slightly slightly varies varies with with I and and r.r. CalcuCalculations C1 (x)practically practically coincides coincides with with the the lations show show that that the the local local capacitance capacitance C 1 (x) average average value value from from (2.8) (2.8) along along the the whole whole length length of of aa long long conductor, conductor, except except for for its its portions portions lying lying close close to to its its ends. ends. But But even even at at the the ends, ends, the the local local capacitance capacitance is is less less than than twice twice the the average average value. value. This, This, however, however, does does not not concern concern capacitances capacitances of of the the free free ends ends which which are are much much larger larger (see (see below). below). As an an approximation approximation justifiable justifiable by by calculations, calculations, we we shall shall use use the the capacicapacitance tance per per unit unit length length from from (2.8) (2.8) and and apply apply itit to to aa real real streamer streamer channel. channel. If If aa


The process in aa long spark spark The streamer-leader streamer-leader process

40 40

channel must have have aa longitudinal longitudinal channel possesses possesses aa finite finite conductivity, conductivity, then then it it must potential gradient and U = U(x), when current flows through it. The The = U ( x ) , when current flows through it. potential gradient and charge per unit channel length has the form charge per unit channel length has the form

.(x) ~ % C j, [U(x) [ U ( x )-- Uo(x)] Uo(x)] = T(X) =

27Tco[U(x) - U()] Uo] 27r&O[U(X) In(ljr) Wllr)

(2.9)

which channel raises raises its its potential potential which allows allows for for the the fact fact that that the the local local charge charge of of aa channel relative refinement should should be be Uo(x).A similar similar refinement relative to to unperturbed unperturbed potential, potential. Uo(x). introduced looks like like introduced in in formula formula (2.3), (2.3), which which generally generally looks

U 2Em rm , (2.10) (2.10) U,t -- Uo(l) UO(l) = = 2Emrm. as varies along along its its length, length, aa charactercharacter(2.7). If If the the channel channel radius radius r varies as well well as as in in (2.7). istic value may be substituted into (2.9), because the capacitance varies with r istic value may be substituted into (2.9), because the capacitance varies with only only logarithmically. logarithmically. Now elongates by by dl, d/, its its new new Now turn turn to to channel channel current. current. When When aa channel channel elongates portion 1 parameters of of the the front front E will will denote denote parameters portion acquires acquires charge charge Tt ~ [d/;d lindex index ; channel by local local current current ir it over over = I.1. This This charge charge is is supplied supplied directly directly by channel end, end, xx = time development, we we have have = d/jV dl/V,.s' Therefore, Therefore, at at any any stage stage of of streamer streamer development, time dt dt = . 27Tco[Ut - Uo(l)]Vs It=T/Vs = In (ljr) ,Ut~Ut·

(2.11) (2.11)

The current current at at the the tip tip is is defined defined mainly mainly by The by the the tip tip potential potential and and streamer streamer velocity. velocity. At At the the anode, anode, the the current current is is .

dQ

= dQ

tJ’ /

~ ( xdx dx ) = = T(X)

t 1

(2.12) C1j [[U(x) U ( x )Uo(x)] dx (2.12) C - Uo(x)] dx so should be where, Q Q is is the the total total channel channel charge. charge. Strictly, Strictly, Q where, Q should be supplemented supplemented by by = 27Tcorm[Ut 2 7 r ~ ~ r ~--[ U U,([)], , but the tip tip charge charge Qt Q, = Uo(l)], but it it is is relatively relatively small small in in aa long long the streamer. streamer. and itii at at the the opposite opposite ends ends of Currents i,a and Currents of aa streamer streamer channel channel do do not not generally coincide. coincide. Of Of course, course, their their values values may generally may be be very very close close or or differ differ conconsiderably, depending depending on on particular particular conditions. conditions. For siderably, For example, example, if if the the electrode electrode is raised raised during during the the streamer streamer development, development, the voltage is voltage the potential potential and and charge charge distributed along along the the channel channel increase. increase. Some Some of of the the anode anode current current is is used used to to distributed supply an an additional additional charge charge to to the the old old channel channel portions, that only only the the supply portions, so so that remaining current current reaches reaches its its front front end: end: ii,a > it. il. But remaining But if if the the electrode electrode voltage voltage is decreased, the ‘excess’ charge of the old channel goes back is decreased, the 'excess' charge of the old channel goes back to to the the supply supply through the the anode anode surface, surface, so so that that the the current current decreases decreases nearer nearer to to the the through anode (positive current is created by charges moving away from anode (positive current is created by charges moving away from the the if. anode): i,a < if. anode): A long long streamer streamer can can develop develop at at constant constant voltage A voltage when when the the electric electric field field in in the channel, channel, E(x, E ( x .t), t ) ,does does not not vary vary much much with with time. time. The The potential potential at at any any point point the of the the existing existing channel channel U(x) U ( x )= =U U,a -- f6' E ( x )dx ( xvary ) slightly of E(x) dx and and ~ T(X) vary slightly with with time, which which means means that that current current does does not not branch branch off off on on the the way way from from the the time, 1,. la

dt, =ili'

Q= = Q

0

Jt


A long streamer

41 41

anode case, the anode to to the the channel channel tip. tip. In In this this case, the anode anode current current is is close close to to the the end end U,t which current (2.11) including current defined defined by by (2.11) including potential potential U which may may be be much much lower lower than U,. Many average channel channel field than Ua' Many experiments experiments have have shown shown that that the the average field for air air in must exceed aa certain about 55 kV/cm must exceed certain minimum minimum value value of of about kV/cm for in normal normal conditions conditions (see (see sections sections 2.2.6 2.2.6 and and 2.2.7) 2.2.7) to to be be able able to to support support aa long long positive positive 600 kV kV at at the the anode anode and and the the streamer streamer length length streamer. For For instance, instance, if if Ua, == 600 streamer. is I1 ;::::: 1m, 1 m, nearly nearly all all voltage voltage drops drops in in the the channel channel and and U U,t « 500 kV), we have ln(l/r) = 6.9, illustration in section 2.2.2 (with Ua 500 kV), we have In(ljr) = 6.9, 12 7 F/m, T{ C/m, and A. Streamer C C1 =8 8 Xx 1010-12F/m, r1== 2.7 2.7 Xx 10lOP7C/m, and ii,a == i{i, == 0.46 0.46A. Streamer j = currents currents of of this this order order of of magnitude magnitude (as (as well well as as much much higher higher or or much much lower lower currents) have have been been registered registered in in many many experiments. experiments. These These values values can can also also currents) be of possible possible streamer streamer velocities velocities be obtained obtained from from calculations calculations with with the the account account of from from 10 lo55 to to 10 lo77 m/s m/s in in air, air, which which have been found found in in some some experiments experiments to to be have been be even higher higher [4]. even [4]. In aa simple simple model model of of potential potential and and current current evolution evolution in in aa developing developing In streamer channel, the the latter latter can can be be represented represented as as aa line line with with distributed distributed streamer channel; parameters: the the capacitance capacitance C1 C 1 and and resistivity resistivity R1 Rj = = (7rr2ei-Lenetl per unit unit (xr2epene)-' per parameters: length. The The electron electron density density n,(x. ne(x, tt)) should should be be calculated calculated in in terms terms of of the the length. plasma plasma decay decay kinetics kinetics (see (see section section 2.2.5). 2.2.5). Estimations Estimations show show that that self-induction self-induction effects effects are are not not essential essential in in streamer streamer development development [4]. [4]. Then, Then, the the process process is is described described by by the the following following equations equations for for current current and and voltage voltage balance: balance: [JT d r 8i di -+--0, at+ax=O' at

dx

aU dU . j, - - = zR1, -ax=iR dX

7 = C1(U - Uo). T=Cj(U-Uo)·

(2.13) (2.13)

A boundary boundary condition condition in in the the set of equations at x x= = I1 is the equality equality A set of equations (2.13) (2.13) at is the

4

= c1[U1 -

UO(4l Vs

(2.14) (2.14)

equivalent to to (2.11). Formula (2.12) automatically follows follows from and equivalent (2.1 1). Formula (2.12) automatically from (2.13) (2.13) and Another boundary condition may be the setting of anode potential, (2.14). (2.14). Another boundary condition may be the setting of anode potential, since U ( 0 ,t)t ) == Ua(t). U a ( t ) .Equations Equations (2.13) (2.13) and and (2.14) (2.14) will will be be used used in in the the next next since U(O, section section to to evaluate evaluate the the heating heating of of aa streamer streamer channel. channel. Illustrations Illustrations of of streamer streamer development development calculations calculations will will be be given given in in sections sections 2.2.6 2.2.6 and and 2.2.7 2.2.7 plasma decay A complete after after aa discussion discussion of of the the plasma decay mechanism. mechanism. A complete set set of of equations for for aa long long line, line, generalized generalized by by taking taking self-induction into account, account, equations self-induction into will be be applied applied in in section 4.4 to to the the treatment treatment of return stroke. will section 4.4 of aa lightning lightning return stroke. of longitudinal longitudinal field E c in in the the channel Equality (2.11) (2.11) allows channel Equality allows evaluation evaluation of field E, is still as high high as as that that behind the the streamer tip, where where the the electron electron density density is behind streamer tip, still as


42 42

The The streamer-leader streamer-leader process in a long spark

behind the the tip tip is is conduction conduction created created by by an an ionization ionization wave. wave. The The current current behind current i, = Jrr~encf.LeEc' By equating this expression to (2.11) and using using current il = 7rrfnencpeEc.By equating this expression to (2.11) and and (2.10) with V = V" we find (2.6) (2.6) and (2.10) with Ut, = U,, we find

/3Em

E c i'::j In (nmlno) In (fIrm)

i3

=

4k 2k _ 1 i'::j 2.5.

(2.15)

For the denominator denominator of of (2.15) (2.15) is is For aim a 1 m streamer, streamer, the the product product of of logarithms logarithms in in the close of aa streamer streamer channel channel in in to 100. 100. Therefore, Therefore, the the field field in in the the front front end end of close to normal kV jcm from from section section 2.2.2). 2.2.2). normal density density air air is is E E,c i'::j M 4.2 4.2 kVjcm kV/cm (Em (E, = = 170 170 kV/cm Within not contradict contradict the the average average Within the the theory theory accuracy, accuracy, this this value value does does not measured kV jcm necessary necessary to to support support the the streamer. streamer. measured channel channel field field of of 55kV;cm There therefore electrons electrons are are lost lost in in There is is no no ionization ionization in in such such aa weak weak field, field, therefore attachment processes. attachment and and electron-ion electron-ion recombination recombination processes. Current than that that of of the the tip tip adjacent adjacent to to near the the channel channel end end is is lower lower than Current i,il near the tip length =Q Qtlrm is larger larger than than TT J r , is the channel, channel, because because the the charge charge per per unit unit tip length TTtt = in end effects effects for for long long conducconducin the the channel. channel. This This is is aa typical typical consequence consequence of of end tors, charge density density at at the the free free tors, well-known well-known from from electrostatics. electrostatics. The The surface surface charge end its lateral lateral surface. surface. In In our our end of of aa conductor conductor is is much much higher higher than than on on its simple been replaced replaced by by aa hemisphere hemisphere of of simple model, model, in in which which aa channel channel tip tip has has been radius the average average charge charge radius ,rYm and and charge charge Qt Q, written written after after formula formula (2.12), (2.12), the M 2m:o[Ut 2 7 r ~ ~ -[ UVo(f)]. U~O ( l ) ]It It. is is In (l/rm) times per unit unit length length is is TT,t i'::j per In (fIrm) times larger larger than than T, T, at the the channel channel end end (see (see (2.9)). (2.9)). The The tip tip current current i,it much at much exceeds exceeds il. it. This This does does not because the the charge charge Q Q of of aa long long channel channel not affect affect the the total total charge charge balance, balance, because Q,. is much much larger larger than than the the tip tip charge charge Qt. is Note that that current current perturbation perturbation in in the the tip Note tip region region has has aa local local character. character. It It cannot be be detected detected by by current current registration registration from from the cannot the anode anode side. side. The The streamer streamer the charge charge of here makes makes use use of of its its own own resources resources -- the here of the the ‘old’ 'old' tip tip has has moved moved on on into the the gap gap with with the the elongating elongating streamer. streamer. It It is into is the the charge charge overflow overflow that that it. If If aa current current detector detector were creates current current it. creates were placed placed at at the the site site of of aa newly newly born portion portion of of the the channel, channel, it it would would register born register current current ii M i'::j it for for aa very very 9 At i'::j = rrm/Vs M 10lop9 s; then short period period of of time time D.t short ml Vs i'::j s; then the the current current would would decrease decrease and evolve evolve as as the the solution solution of of equations equations (2.13) (2.13) and to i,il and to and (2.14) (2.14) indicates. indicates.

2.2.4 Gas Gas heating heating in a streamer streamer channel 2.2.4

streamer process process is is accompanied accompanied by by current current flow flow and, A streamer and, hence, hence, by by Joule Joule heat heat was mentioned mentioned above, above, the the viability viability of release. As was release. of aa plasma plasma channel channel depends depends primarily on on temperature, temperature, so so this this issue issue is is of of principal primarily principal importance. importance. The The initial initial heating of of aa given given gas gas volume volume occurs occurs when when aa streamer streamer tip heating tip with with its its high high current current the channel channel develops, develops, the and field field passes passes through through it. it. As the and the gas gas is is heated heated further further by streamer streamer current current flowing flowing through through it. it. Let Let us us evaluate evaluate both both components components of of by released energy. energy. released 1 cm33 per per second second is is jjE E= The energy energy released released in in 1cm The = aE2, aE 2 , where where jj = = aE is is the the is the the plasma plasma conductivity conductivity in current density density and and aa is in aa given given site site in in aa given given current


A long streamer

43 43

3 moment of of time. time. The The energy energy released released in in 1cm 1 cm3 as aa result result of of ionization ionization wave wave moment as passage passage is is

s 2 Js

J

2 W = aE2 w == aaEE 2 dt dt = aE dx/Vs dx/Vs

(2.16) (2.16)

--x to to +00 +xbut but actually actually over over the the where the the integrals integrals are are formally formally taken taken from from -00 where ionization wave wave region. region. The The principal principal contribution contribution to to energy energy release release is is made made ionization by aa thin thin layer layer behind behind the the wave wave front front where where the the electron electron density density and and field field are are by ~ ~ E i / using using 2, high. The The integral integral of of (2.16) (2.16) was was found found rigorously rigorously to to be be foE~/2, high. equations for for this this wave region [4]. [4]. This This value of value has has the the physical physical meaning meaning of equations wave region electrical energy at maximum maximum field. field. The The contribution of the the region region electrical energy density density at contribution of (nm/no)times, smaller before before the the wave wave front front is is In In (nm/no) times, or or an an order order of of magnitude, magnitude, smaller than is as front, than this this value. value. Although Although the the field field there there is as high high as as that that behind behind the the front, the is of of n, and the conductivity (J0 is the electron electron density density is of the the order order of nm and the conductivity is (nm/no)times smaller (section In In (nm/no) times smaller (section 2.2.2). 2.2.2). Therefore, Therefore,

Wx oEk/2 ~ x 2.6 x 10- 3 J/cm J/cm33 W ~ efoE~/2

(2.17) (2.17)

where = 170 170 kV. where the the numerical numerical value value corresponds corresponds to to E, Em = kV. The fact that the density of energy release in a gas is of of the the same same order order of of The fact that the density of energy release in a gas is magnitude as the energy density of of the the electric electric field field is is quite quite consistent consistent with with magnitude as the energy density electricity electricity theory. theory. When When aa capacitor capacitor with with capacitance capacitance C is is charged charged through through resistance R to to voltage voltage V of aa constant constant voltage voltage supply, half of the work work U of supply, half of the resistance 2 QV = done by by the the supply supply is is stored stored by by the the capacitor capacitor as as electrical electrical QU = CV CU2 done energy, and and the the other other half half is is dissipated dissipated due due to to resistance, resistance, irrespective irrespective of of its its energy, value. The The value value of of R determines determines only only the the characteristic characteristic time time of of the the capacitor capacitor value. charging, RC. Something like this this is is valid valid for the case case in in question question but, but, of of Something like for the charging, course, without without both both energies being rigorously rigorously equal equal to to each other, because because course, energies being each other, this situation situation is is much much more more complicated. complicated. Indeed, Indeed, according according to to the the results results this C,t = = Q/ Q/U 27re0rm, of section 2.2.2, of section 2.2.2, the the tip tip capacitance capacitance is is C Vt, % ~ 27rfor m, volume volume V, , E Ut/rm, Vt x ~ 4rrLI3, 47rr~/3, and and field field E Em ~ Vt/r so that that the the energy energy dissipation dissipation per per unit unit m, so tip tip volume volume is is W F~Z CtU:/2Vt Ct vf /2 Vt z ~ eOEk foE~ (we (we have have ignored ignored the the unessential unessential term term Uo(l)). Vo(l)). Joule heat heat is is released released directly directly in in aa current current carrier carrier gas, gas, or or an an electron electron gas. gas. Joule Then electrons electrons give give off off their their energy energy to to molecules molecules in in collisions. collisions. An An appreciable appreciable Then portion of of electron electron energy energy (even (even most most of of it it in in aa certain certain range range of of E E/I N) is used used N ) is portion for the excitation excitation of relaxing vibrations vibrations of nitrogen molecules. molecules. Some for the of slowly slowly relaxing of nitrogen Some energy is is used used for for ionization ionization and and electron electron excitation excitation of of molecules, molecules, about about energy W = ~ 100 eV per per pair pair of of charged charged particles particles produced, produced, i.e., i.e., nneW ~ 10- 3 J/cm3 J/cm 3 at at U’ 100 eV , ~ ’= ne F~Z 1014 10 14 cmP3. cm- 3 . But But even even without without the the account account of of these these ‘losses’, 'losses', the the gas gas temperatemperan, ture rise rise in in the the wave wave front front region region appears appears to to be be negligible: negligible: AT 6.T < Wlcv W / Cv = ~ 3 3 K. K. ture Here, Here, cv Cv = = qkBN ~kBN = = 8.6 8.6 x loP4 10- 4 J/(cm3/K) J/(cm 3 /K) is is the the heat heat capacity capacity of of cold cold air air and and kB kB is is the the Boltzmann Boltzmann constant. constant. Let us us see see what what subsequent subsequent gas gas heating heating can can provide provide by by the the moment moment it it Let is somewhere somewhere in in the the middle middle of of aa long long streamer streamer channel. channel. We We multiply multiply the the is


spark The streamer-leader streamer-leader process in a long spark

44

second by ii and whole channel second equation equation of of (2.13) (2.13) by and integrate integrate over over the the whole channel length, length, assuming, the anode. U,a is is applied applied to to the anode. assuming, for for simplicity, simplicity, that that constant constant voltage voltage U After by-part integral After taking taking aa by-part integral in in the the left left side side of of the the equation, equation, we we substitute substitute ai/ox ailax from from (2.13) (2.13) and and i(l) i(1)== ii,l from from (2.14), (2.14), followed followed by by simple simple transformatransformaresult, we tions. tions. As As aa result: we have have

fSo' 1

I

f'

+

2

+

[y

I d 0-2C j U dx U,i, = = 0 i2 R R Ij dx dx + d x + [C1U Cij UIUO(l) U I U O (]/ )V Vs ]s Uaia dt -2- -- C dt

(2.18) (2.18)

which describes power balance balance in which describes the the power in the the system; system; here, here, Uo(l) Uo(1) is is unperturbed unperturbed potential of the external x == l.1. The power The input input power potential of the external field field at at the the streamer streamer tip tip point point x Uai used for Uai,a into into aa discharge discharge gap gap is is used for Joule Joule heat heat release release in in the the channel channel (the (the first first term on on the the right), right), for for increasing increasing the the electric electric energy energy stored stored in in its its capacitance capacitance term (the to the (the second second term), term), and and for for the the creation creation of of new new capacitance capacitance due due to the channel channel elongation third term). elongation (the (the third term). Joule Joule heat heat associated associated with with the the ionization ionization wave wave is is not represented represented here. here. The The field field burst burst and and the impulse current current that that make make up up not the tip tip impulse W W calculated calculated above above are are absent absent from from equations equations (2.13) (2.13) and and (2.18). (2.18). Having Having integrated period of integrated equality equality (2.18) (2.18) over over the the period of time time from from the the moment moment of of channel has acquired length length l,I , we we channel initiation initiation to to the moment t the channel the moment the channel has acquired get balance in get the the equation equation for for the the energy energy balance in the the system system at at the the moment moment t:t :

UQ a

= Kd,s + II0 Cdu

2 -

2

U5(x)] dx

+ \/

Cj

ul _ C

2

U U (l)) l j

lOt

(2.19)

where where charge charge Q Q is is given given by by (2.12). (2.12). The The energy energy input input into into the the channel, channel, UaQ, U,Q, is is used right), partly partly stored used to to create create capacity capacity (the (the last last term term on on the the right), stored in in this this capacity braces ( )1 (&Is). dis ). The The braces ) t indicate indicate capacity (the (the integral) integral) and and partly partly dissipated dissipated (K process. In the time time averaging the averaging of of the the process. In case case of of aa long long channel, channel, much much of of the the applied potential UI, is so the the tip tip potential is small small applied voltage voltage drops drops across across its its length, length, so most time, as potential U U,,av of of about about most of of the the time, as compared compared with with average average channel channel potential U,.a . Then, Then, the the last last term term in in (2.19) (2.19) can can be be neglected. neglected. U Q If we compare compare the the left left side side of of (2.19) (2.19) with with the substituted expression expression for for Q Ifwe the substituted from (2.12) (2.12) and and the the integral integral in in the the right side of of (2.19), (2.19), we we can can conclude conclude that that from right side the difference between these values cannot be be much much smaller smaller than than their their own own the difference between these values cannot values the same same order order of of magnitude. magnitude. Therefore, Therefore, the the energy energy values but but rather rather have have the KdlS dissipated in in the the channel channel is is equal, equal, in in order order of of magnitude, magnitude, to to the the gained gained K dis dissipated electrical electrical energy, energy, which which is is in in agreement agreement with with aa similar similar situation situation discussed discussed above. above. The average average energy energy dissipated dissipated per unit channel channel length length is is W W,,jav ~ xC CIj Uiv/2 The per unit U~v/2 and per unit unit channel volume is and the the average average energy energy contributed contributed per channel volume is I

C1U~v

W~--

2JTr~v

(2.20) (2.20)

where rav is the the average average channel channel radius. radius. With With the formation of of every every new new where ray is the formation portion of the the channel, channel, its its radius radius was approximately proportional to the was approximately proportional to the portion of


A long streamer

45 45

tip potential owing that the the maximum maximum tip tip potential owing to to the the fact fact that tip field field remained remained approxiapproxiSo we we have have Uav/r Uav/r,,av i=::j M E,. Substituting this expression and and mately constant. constant. So Em. Substituting this expression mately (2.8) (2.8) into into (2.20), (2.20), we we find find ~ Em co 2

/

W

i=::j

In (f/r m ) < W.

One can can see see that that subsequent subsequent heating of the channel gas gas adds adds little little to the initial initial One heating of the channel to the heating by by an an ionization ionization wave wave passing passing through through the channel site. site. heating the particular particular channel To heating due To conclude, conclude, gas gas heating due to to streamer streamer development development is is negligible negligible if if the the gap gap voltage voltage remains remains constant. constant. Higher Higher voltage voltage does does not not change change the the situation situation because because the the energy energy dissipated dissipated in in the the channel channel grows grows in in proportion proportion with with the the channel cross cross section section and and the the air air volume volume to to be be heated. heated. Specific Specific heating channel heating remains volume remains unchanged, unchanged, since since it it is is determined determined by by aa more more or or less less fixed fixed volume density density of of electric electric energy. energy. 2.2.5 2.2.5 Electron-molecular Electron-molecular reactions and plasma decay in cold air

Electron to attachment molecules and Electron loss loss in in cold cold air air is is due due to attachment to to oxygen oxygen molecules and dissociative dissociative recombination. recombination. The The main main attachment attachment mechanism mechanism in in dry dry air air at at moderate fields fields is is aa three-body three-body process moderate process

+

0 22 + + ee + O 0 22 O

kat k,,

= (4.7 (4.7 -- 0.25 0.257) 1 ) xX

i=::j

°

+ 0 22,, 2+

+O 0; ----+

10~31 cm /s, cm6/s, 6

16 /N X x 10 10'6V.cm2 1y == E E/N y .cm 2

(2.21)) (2.21

where kat kat is the rate K. In where is the rate constant constant at at T = = 300 300K. In higher higher fields, fields, the the dominant dominant rate constant process is process is dissociative dissociative attachment attachment O O2 e ----+ -+ 0- + 0 with with the the rate constant 2+e

°

-9.42 h 12.717 -9.42 -- 12.7 log k,a == { logk -10.215.7h -10.21 - 5.7,'~

at 1y < 99 at 9. at 17 > 9. at

(2.22) (2.22)

In high fields kY/cm at In not not excessively excessively high fields of of E < 70 70 kV/cm at 11 atm, atm, air air is is ionized ionized at at the the rate constant constant kkii == Vi/ q /N N rate

logkii = = -8.31 -8.31 -- 12.7 12.7,'~ at 1y < 26. 26. logk h at

(2.23) (2.23)

Since Since the the rate rate of of electron electron loss loss through through attachment attachment is is proportional proportional to to electron electron density through recombination to y12, n~, the density ny1,e and and that that through recombination is is proportional proportional to the latter latter is is unimportant beginning of unimportant at at the the beginning of ionization. ionization. The The equality equality kkii == k a, valid valid at at 1y i=::j M 12 12 determines determines the the minimum minimum field field mentioned mentioned above, above, which which is is necessary necessary to E i i=::j to initiate initiate the the growth growth of of electron electron density density in in unperturbed unperturbed air; air; Ei E 30 30 kYat kV at p == 11 atm atm and and room room temperature. temperature. Oxygen possessing aa lower N 2 are Oxygen molecules molecules possessing lower ionization ionization potential potential than than N2 are ionized the ionization threshold. Electrons ionized in in fields fields not not much much exceeding exceeding the ionization threshold. Electrons recombine with with 0; at usually termed recombine at the the rate rate constant constant 3, 0, usually termed aa recombination recombination coefficient: coefficient:

ot

0; +e----+ + e -+ 0+0, 0 + 0, ot


/? M 2.7 x 10-7(300/Te)'/2cm3/s

(2.24) (2.24)

46 46

The The streamer-leader streamer-leader process in in a long spark

where where TT,e is is electron electron temperature temperature in in Kelvin Kelvin degrees. degrees. However, However, complex complex ions ions are are more more effective effective with with respect respect to to electron-ion electron-ion recombination. recombination. The The most most important ions, while while in in atmosphere atmosphere saturated saturated by by important ions ions in in dry dry air air are are 0; ions, water water vapour, vapour, as as in in thunderstorm thunderstorm rain, rain, H H 330+ 0 +(H ( H220h 0 ) 3cluster cluster ions ions are are more more important. important. For For these, these, the the recombination recombination coefficients coefficients

ot

ot 0; + + ee ----+ O O2 02. 2 + 02,

6 3/s D = 1.4 1.4 x 1O10-6(300/T,)'/2 cm3/s /3;:::: (300/Te )I/2 cm

+

+

H + e ----+ H H + 4H 4 H220, 0, H 330+(H 0 + ( H20h 2 0 ) 3+ ---f

(2.25) (2.25)

3/s 6 B 6.5 6.5 Xx 1O10-6(300/T)'12 cm3/s /3;:::: (300/T)1/2 cm

(2.26) (2.26) are are an an order order of of magnitude magnitude larger larger than than for for simple simple ions. ions. ions are are formed formed from from simple simple ions ions in in the the conversion conversion Complex 0; ions Complex reaction reaction

ot

0;

+0 2 +0 2

--f

0;

+ 02,

k = 2.4 x 10-30(300/Te)'/2cm6/s. (2.27)

Chains of of hydration hydration reactions reactions lead lead to to the the production production of of H H30'(H20)3 ions. A A Chains 30+ (H 20h ions. typical chain chain looks looks like like this: this: typical 3/s 9 1.5 Xx 10lop9 cm3/s ot 0) + 02, k == 1.5 cm 0; + + HH20 -+ 0i(H O;(H20) 02> 20 ----+ 3/s Ot(H H OH k == 3.0 3.0 x 10lo-'' cm cm3/s O i ( H 20) 0)+ + HH20 H30' +O H + 02, 02? 20 30+ + 3/s = 3.1 3.1 x 10- cm H kk = cm3/s H30+ -tH H20 + (M) (M) ----+ H H30+(H20) + (M), (M), 30+ + 20 + 30+(H 20) + 3/s 2.7 x 10lop9 cm cm3/s H H30+(H20) + HH20 -t (M) (M) ----+ H H30+(H20)2 (M). kk == 2.7 30+(H 20) + 20 + 30+(H 20h + (M), 2

10

X

--+ +

2

9

X

+

X

+

9

3/s cm3/s H (M) H3O+(H20)2 -t H H 2200 + i(M) ----+ -+ H H 330+(H 0 + ( H220)3 0 ) 3+ -t (M), (M), k == 2.6 2.6 Xx 10-9 cm 30+(H 20h + (2.28) (2.28)

(M is is any any molecule, molecule, kk correspond correspond to t o pp == 1atm, 1 atm, T == 300 300 K); K); here, here, aa hydrated hydrated (M ion replaces replaces aa 0; ion. ion. ion Another, similar similar chain chain begins begins with with the the production production of of an an H H 220+ 0 + ion ion in in Another, ionization of of water water molecules molecules by by electron electron impact. impact. Then Then comes comes the the conversion conversion ionization reaction reaction

ot

+

+

H2200 --+ -+ H H 330+ 0 ++ OH, OH, H 220+ 0 ++ H H

3/s 9 kk == 1.7 cm 1.7 Xx 10lop9 cm3/s

producing an an H H30+ ion, followed followed by by the the reaction reaction chain chain of of the the type type (2.28). (2.28). producing 30+ ion, The production production of of complex complex ions ions is is accompanied accompanied by by their their decay. decay. For For an an The 0; ion, ion, this this is is the the reaction reaction

ot

0; + + O0 22 ot 0; + i-O 0 2 2+ -t02, 0 2 , ot + --+

6 3/s. 3.3 Xx 1O10p6(300/T)4exp (-504O/T) cm cm3/s. k == 3.3 (300/T)4 exp (-5040/T)

(2.29) (2.29)

It is is greatly greatly accelerated accelerated by by gas gas heating, heating, but but in in cold cold air air the the reaction reaction effect effect It is negligible. negligible. The The same same is is true true of of other other complex complex ions, ions, including including hydrated hydrated ions. ions. is


A long streamer

47 41

In aa cold cold streamer streamer channel, channel, simple simple positive ions turn turn to to complex complex In posItIve ions 8 ions very very quickly, quickly, for for the the time time T,eonv , x 1010-8-10-7s. It is is these these ions ions that that ions ~ _10- 7 S. It determine the rate of electron-ion recombination in cold air, except for determine the rate of electron-ion recombination in cold air, except for aa very very short short initial initial stage stage with with t :::;; 6T r,,,,. eonv . If the ionization rate is too low and if if the the detachment-decay detachment-decay of of negative If the ionization rate is too low and negative ions is slow, as in a cold streamer channel, the plasma decay is described ions is slow, as in a cold streamer channel, the plasma decay is described by by the equation equation the

3 = -vane - pn,2 dt

(2.30) (2.30)

where va is electron electron attachment attachment frequency. frequency. Its Its solution solution at at initial initial electron electron where Va is density equal equal to the plasma plasma density density behind the ionization ionization wave, wave, nn,, is density to the behind the e , is n (t) e

-

ne exp (-vat) -,-----;-::-----;:----:-=:-:-,----...::.....c..-,--------,-,1 + (,8ne /va )[1 - exp (-vat)]

(2.31)) (2.31

where where the the time time is is counted counted from from the the moment moment the the streamer streamer tip tip passes passes through through aa particular point point of of space. space. particular According to (2.15), wehaveE andE/ N ~ x 4.2kV/cm 4.2kV/cmandE/N x 1.7 1 . X7 1010-16V/cm ~ 16 V/cm Accordingto(2.15), wehaveE ~ for aa streamer streamer channel channel just behind the the tip tip at at p = = 1atm. 1 atm. The The electron electron for just behind 7 attachment S-I va ~ x 1.2 1.2 Xx 10 lo7 sC1 and and the the characteristic characteristic attachment frequency frequency from from (2.21) (2.21) is is Va attachment attachment time time is is T 7, = v;1 v;’ ~ x 0.8 0.8 Xx 10- 7 . Over Over this this time, time, most most simple simple oj 0; a = ions ions in in dry dry air air turn turn to to complex complex 0; ions. ions. Electrons Electrons recombine recombine with with them them 7 3 with cm /s corresponding with the the coefficient coefficient ,8 ,8 ~ x 2.2 2.2 Xx 10lop7 cm3/s corresponding to to electron electron temperatempera4 ture / N. The ture TTe x 11 eV eV == 1.16 1.16 xx 10 lo4K K at at the the above above value value of of E E/N. The initial initial electron electron e ~ 14 3 x 10 1014 cmP3 is so so high high that that the the parameter parameter ,8n @nc/va x 22 determining determining density nn,e ~ density cmis e /va ~ the the relative relative contributions contributions of of recombination recombination and and attachment attachment is is larger larger than than unity. This This means means that that at at an an early early decay decay stage stage with with t < T ra x 10lop77 s,s, electrons electrons unity. a ~ are lost lost primarily primarily due due to to recombination, recombination, with with attachment attachment playing playing aa lesser lesser role. role. are 7 Later, s, the Later, at at t > 2T 2raa ~ x 22 Xx 10lO-’s, the electron electron density density decreases decreases exponentially, exponentially, as as is is inherent inherent in in attachment, attachment, but but as as if if starting starting from from aa lower lower initial initial value value nj nl = = ne/(l nJ(1 + ,8n @ ne Tca~) a~ x) 0.3n 0.3n,; nee ~ x nl nl exp exp (-vat). (-vat). e; n The The plasma plasma conductivity conductivity decreases decreases by by two two orders orders of of magnitude, magnitude, as as compared with with the the initial initial value, value, over over t ~ x 3 3 Xx 10-7 S.s. At At the the streamer streamer velocity velocity compared 6 V mis, Vs, ~ x l0 lo6 m/s, this this occurs occurs at at aa distance distance of of 30 30 cm cm behind behind the the tip. tip. A A micromicrosecond later, later, the the conductivity conductivity drops drops by by six six orders orders of of magnitude. magnitude. The The streamer streamer second plasma in in cold cold humid humid air air decays decays still still faster faster because because of of aa several several times times higher higher plasma of an an rate of of recombination recombination with with hydrated hydrated ions ions and and due due to to the the appearance appearance of rate additional attachment attachment source source involving involving water water molecules. molecules. These These estimations estimations additional indicate indicate aa low low streamer streamer viability. viability. It It is is only only very very fast fast streamers streamers supported supported by by megavolt megavolt voltages voltages that that are are capable capable of of elongating elongating to to I1 ~ x 1m 1 m in in cold cold air air without without losing losing much much of of their their galvanic galvanic connection connection with with the the original original electrode. electrode. This is is supported supported by by experiments experiments with with aa single single streamer streamer and and aa powerful powerful This streamer [4]. streamer corona corona [4].

ot

+


48 48

The streamer-leader streamer-leader process in a long spark

Note that where Note that aa streamer streamer plasma plasma has has aa longer longer lifetime lifetime in in inert inert gases, gases, where attachment attachment is is absent absent and and recombination recombination is is much much slower. slower. This This makes makes it it possible to heat by flowing possible to heat the the plasma plasma channel channel by flowing current current for for aa longer longer time time after after the the streamer streamer bridges bridges the the gap gap (the (the estimations estimations of of section section 2.2.4 2.2.4 do do not not extend to these these conditions). extend to conditions). Such Such aa process process sometimes sometimes leads leads to to aa streamer streamer the formulation (leader-free) breakdown [17]. (leader-free) gap gap breakdown [ 171. Still, Still, the formulation of of the the streamer streamer breakdown problem is justified for hot air and is related to lightning breakdown problem is justified for hot air and is related to lightning (see (see section section 4.8 4.8 about about dart dart leader). leader). 2.2.6

Final streamer length

When When aa streamer streamer starts starts from from the the smaller smaller electrode electrode (anode) (anode) of of radius radius ra, Y,, to to which Va, » Eir a is propagates in which high high voltage voltage U >> Elya is applied, applied, it it propagates in aa rapidly rapidly decreasing decreasing external but then external field. field. It It is is first first accelerated accelerated but then slows slows down down after after it it leaves leaves the the the voltage region region of of length length rY,a where where it it senses senses aa direct direct anode anode influence. influence. If If the voltage is is too too low, low, the the streamer streamer may may stop stop in in the the gap, gap, without without reaching reaching the the opposite opposite electrode (say, (say, aa grounded grounded plane plane placed placed at at aa distance distance d). d). The The higher higher is is Va' U,, electrode the longer is the distance the streamer can cover; at a sufficiently high voltage, the longer is the distance the streamer can cover; at a sufficiently high voltage, it the gap. to estimate it bridges bridges the gap. In In order order to estimate the the sizes sizes of of the the streamer streamer zone zone and and leader leader cover cover in in aa long long spark spark or or lightning lightning -- aa task task important important for for their their theory we need that would theory -- we need aa criterion criterion that would allow allow estimation estimation of of maximum maximum streamer No direct measurestreamer length length under under different different propagation propagation conditions. conditions. No direct measurements of of this this kind have been been made made for for single single long long streamers streamers in in air, air, because because ments kind have there however, is there is is always always aa burst burst of of numerous numerous streamers. streamers. This, This, however, is quite quite another matter (see below). below). So So we we shall shall use indirect experimental experimental results another matter (see use indirect results and physical considerations, and invoke invoke physical considerations, theory, theory, and and calculations. calculations. It has has been established experimentally experimentally that that streamers streamers comprising comprising aa It been established streamer burst burst are are able able to to cross cross an an interelectrode interelectrode gap gap of of length length dd only only if if the the streamer relation which varies varies with relation E E,,av = Val U,/dd exceeds exceeds aa certain certain critical critical value value E E,,cr which with the the kind kind of of gas gas and and its its state. state. Under Under normal normal conditions conditions in in air, air, this t h s critical critical value value is wide range ~ OJ-10m. is E E,,cr ~ e 4.5-5kV 4.5-5kV in in aa wide range of of dd = 0.1-10m. The The data data spread spread does does not the measurement not exceed exceed the measurement error. error. Bazelyan Bazelyan and and Goryunov Goryunov [18] recommend recommend E,,cr = = 4.65 4.65 kV/cm kVjcm for for positive streamer, averaged averaged over over various various the value the value E positive streamer, measurements. the voltage bridge aa measurements. Therefore, Therefore, the voltage necessary necessary for for aa streamer streamer to to bridge is U,,, or more. more. For For example, example, aa gap gap of of I1 m m length length gap of of length length dd is Vamin == E,,d Ecrd or gap requires about about 500 500 kV (Ec, e 10 kV/cm for negative streamer in air). requires kV (E ~ 10 kV Icm for negative streamer in air). cr At moment of voltage Va to the At the the moment of crossing crossing aa gap, gap, all all voltage U , is is applied applied to the streamer, is also the average field in the streamer. If a gap is long streamer, so so E Ea" is also the average field in the streamer. If a gap is long av E,,av can can be be identified identified with the average average channel channel field. field. Indeed, Indeed, in in criticritienough, E enough, with the cal with E a streamer crosses a gap at its limit parameters. cal conditions conditions with E,,av = =E Ecr, a streamer crosses a gap at its limit parameters. cn It approaches approaches the the opposite opposite electrode electrode at at its its lowest lowest velocity velocity corresponding corresponding to to It the minimum excess of of the the tip tip potential potential VI U, ~ e VI U , over over the the external external potential, potential, the minimum excess AUlI == VI U , -- Vo(d) Uo(d)~ = 5-8 5-8 kV, kV, below the streamer streamer practically stops. In In D..V below which which the practically stops. U o ( d )= = O. 0. If If aa gap gap is is so so long long (say, (say, 1m) 1 m) that that the case case of of aa grounded grounded electrode, electrode, Vo(d) the


A long streamer

49

t1U, in the the channel. channel. Therefore, Therefore, A U , :::: =U U,t « o t~E

0.15

~

...

....l

ffi

0

} > > > >

1

Ian 0 10

...s:: U

-2

-4 Figure 3.17. Polarization Polarization charge charge distribution distribution along along aa straight straight conductor conductor (a (a leader leader Figure system) in in the the cloud cloud dipole dipole field, field, with with allowance allowance for for aa dipole dipole reflection reflection in in the the conductconductsystem) ing earth. earth. ing


Perspectives remote measurements measurements Perspectives of of remote

113 113

the the rod rod radius) radius) rises rises rapidly. rapidly. The The the charge charge density density here here (along (along aa length length of of the = ax, charge be approximated as Tr(x) charge distribution distribution at at the the rod rod base base can can be approximated as (X)= where The reader reader will will soon soon see see where the the coordinate coordinate origin origin x is is at at the the rod rod centre. centre. The that small at at larger larger distances, distances, if if that the the contribution contribution of of the the end end charges charges ±q fq is is small radius r. r. the the rod rod length length 2d 2d is is much much greater greater than than its its radius For is concentrated concentrated on on the the outer outer For simplicity, simplicity, let let us us assume assume that that the the charge charge is and that that the the potential potential rod rod surface, surface, as as is is the the case case when when it it has has aa conductivity, conductivity, and will The rod rod centre centre will will be be taken taken as as will be be calculated calculated along along the the longitudinal longitudinal axis. axis. The the zero zero point point of of the the external external field field Eo Eo potential. potential. The The potential potential p 'P at at the the point point x the is -Eox, the the end end charges, charges, pq, 'P q, is aa sum sum of of potentials potentials created created by by the the external external field, field, -Eox, and and the the charges charges distributed distributed along along the the rod, rod, 'PT: pr:

'PT(X) = _a_Jd ydy 47f€0 -d [(y - x)2 + r 2]1/2 = _a_ {[(d _ x)2

+ r 2]1/2

47f€0

+ + + + +

1

2 '12 (d - x ) [(a- x12 + r2] _- [(d [(d + x)2 x12 + r2]1/2 r2]1/2 + xln x In (d - x) + [(d - x)2 + r ]1/2 } -(d [(d + X)* x)2 + rr22]1/2 -(d + xx)) + [(d p2

+ +

';::j

~ [In 4(d

47f€0

+

2

2

r

2

x

) _

(3.10) (3.10)

2].

Here, to the the rod rod sites sites lying lying far far from from its its Here, the the last last approximate approximate expression expression refers refers to ends, Id ± fxl X I »>> r.r. Here, Here, the the term term 'Pq pqcan can be be neglected, neglected, and and we we shall shall approxiapproxiends, mately have have 'PT pT -- Eox Eox ';::j = O.0 . With With the the actual actual charge ( xand ) the charge distribution distribution ~r(x) and the mately end charges charges providing providing 'Pq' pq, the the rigorous rigorous equality equality 'P(x) p(x)= = 0 0 should should be be valid valid end along the the whole whole rod rod length. length. By By relating relating the approximate equality the approximate equality to to the the along centres of of the the semi-axes semi-axes x = = ±d/2, f d / 2 , we we find find centres a';::d

27f€oEo 27f€oEo = In (V3d/r) - 1 In V3d/er

(3.11) (3.11)

The potential potential at at the the rod rod ends ends must must be calculated with The be calculated with the the account account of of their higher higher charges. charges. Assuming Assuming this this charge charge to their to be be concentrated concentrated along along the the end circumference, circumference, the the potential potential at at the the centre centre of of the the end end plane plane (at (at the the end points x = = ±d f d on on the the axis) axis) can can be be described described as as points 'P(±d) = 'Pq

+ 'PT(±d) - (±Eod)

';::j

0,

q

'Pq = ±-4-' 7f€or

The potential potential 'PT pr must must now now be be calculated calculated from from the The the unsimplified unsimplified (3.12) that the end charge is (3.10). It follows from (3.10). It follows from (3.12) that the end charge is approximately approximately q x 27f€orE 27r~orEod and by by aa factor factor of of q';::j od and ad22 ad d K=-NN ';::j 2rln (V3d/er) K = 24 2q 2r In (&/er)


(3.12) (3.12) formula formula equal equal to to (3.13) (3.13)

114 114

A Available vailable lightning lightning data

smaller smaller than than the the charge charge distributed distributed over over each each half half (it (it is is an an order order of of magnitude magnitude smaller d/r~ z 100). 100). Therefore, Therefore, the the account account oflocalized of localized tip tip charge charge may may be be smaller for for d/r necessary to the the tip, tip, necessary only only for for the the calculation calculation of of electric electric field field in in the the region region close close to at aa distance distance less less than than lOr 10r from from it. it, In In aa remote remote region, region, where where measurements measurements at it is is sufficient sufficient to to consider consider are usually usually made, made, such such aa subtlety subtlety is is unnecessary unnecessary -- it are only only the the charge charge distribution distribution along along the the leader leader channel. channel. Clearly, Clearly, this this is is not not aa uniform uniform distribution, distribution, taken taken for for granted granted by by some some researchers. researchers. It It is is time time to to look look at at the the shape shape of of aa field field strength strength pulse pulse at at the the earth, earth, deterdetermined mined by by the the charge charge of of aa linearly linearly polarized polarized vertical vertical axis axis with with charge charge ~ ( x==) ±ax f a x per per unit unit length. length. It It is is defined defined by by the the algebraic algebraic sum sum of of terms terms T(X) from the the positively positively and and negatively negatively charged charged semi-axes semi-axes and and is is equal equal to to from

=ra

AE(L) _ !J.E(L) -

JL

_a_

-L - L 27TEO 2rEo

x ( H -- x) X) dx dx x(H 2]3/2 [(H [ ( H-- x)2 x)’ + R R2I3/’

where where LL are are the the lengths lengths of of leader leader sections sections which which have have moved moved away away from from the the starting point point to to the the earth earth and and upwards. upwards. Integration Integration with with (3.11) (3.1 1) gives gives starting

L L !J.E(L) Eo [ L _ L A E ( L ) _= 2 -In(V3L/er) [(H_L)2+ ]1/2 - [(H+L)2+ 2]1/2 In (fiL/er) [ ( H - L)’ R R2]‘/’ [ ( H L)’ + R R2I1/’ Eo 2)1/2f [H + (H (H’2 + R R2)”’]2 [H ] -- In In -----------;;-'-----:--';--;,::-----'-------'-------;;--~.,._;;;_ 2 2 {H L + [(H L)2 + R ]1/2}{H + L + [(H + L)2 + R ]1/2} {H [ ( H- L)* R2I1/’}{H [ ( H L)’ R2]’/’J

[

+

+

+

+

+

+

+ +

+

+

(3.14) (3.14)

Here, H is is the the height height of of the the leader leader start, start, r IS is ItS its radius, radius, and and R is is the the Here, distance between between the the leader leader axis axis and and the the observation observation point. point. It It can can be be distance A E ( L ) of of (3.14) (3.14) rises rises smoothly smoothly with with L and and has has shown that that the the function function !J.E(L) shown no extrema. extrema. no The linear linear charge charge distribution distribution assumed assumed in in the the above above illustration illustration is, is, of of The course, another another speculation speculation (section (section 4.3). 4.3). Moreover, Moreover, aa leader leader goes goes up up and and course, down non-uniformly, non-uniformly, and and the the field field in in the the earth-cloud earth-cloud gap gap is is far far from from down being uniform: uniform: its its strength strength decreases decreases towards towards the the earth. earth. This This limits limits the the being linear linear charge charge growth growth from from the the start start downward. downward. The The finite finite channel channel conductivconductivis impossible impossible to to ity exhibits exhibits similar similar behaviour, behaviour, reducing reducing the the tip tip potential. potential. So itit is ity find the the actual actual charge charge distribution distribution exactly exactly without without knowing knowing these these paramparamfind eters. eters. Thus, Thus, aa processing processing of of field field oscillograms oscillograms can can give give nothing nothing more more than than what they they actually actually show. show. The The field field at at aa point point is is an an integral integral effect effect of of the the what whole combination combination of of charges charges created created or or transported transported by by aa given given moment moment whole of time. time. It It is is probably probably worth worth speculating speculating about about registrations registrations but but one one of should assess assess the the results results soberly, soberly, considering considering all all possible possible variants variants and and should is, of of course, course, to to insuring oneself oneself whenever whenever possible. possible. The The best best insurance insurance is, insuring increase the the number number of of registration registration points points and and parameters parameters determined determined by by increase independent independent methods. methods.


Lightning return stroke

3.7

115 115

Lightning Lightning return return stroke

All lightning lightning hazards hazards are are associated associated with with the the return return stroke, and this this accounts All stroke, and accounts for the of investigators as much investigators to to learn learn as much as as possible possible about about this this for the great great effort effort of discharge stage. stage. It It has has been been established established that that the the contact contact of of aa descending descending discharge or aa grounding grounding electrode electrode produces produces aa return return lightning leader leader with with the the earth earth or lightning wave wave of of current current and and voltage. voltage. It It travels travels up up along along the the leader leader channel, channel, partially partially neutralizing neutralizing and and redistributing redistributing the the charge charge accumulated accumulated during during the the leader leader development development (figure (figure 3.18). 3.18). The The travel travel is is accompanied accompanied by by an an increased increased light light the channel, wave front. intensity intensity of of the channel, especially especially at at the the wave front. At At the the earth, earth, the the wave front front intensity intensity acquires its maximum maximum over over 3-41ls As the the wave wave wave acquires its 3-4ps [31]. [31]. As goes up up to to the the cloud, cloud, the the wave wave intensity intensity steepness steepness and amplitude decrease decrease goes and amplitude many-fold, indicating indicating aa considerable decay. Judging Judging by by streak pictures, the the many-fold, considerable decay. streak pictures, region of of aa high high light light intensity intensity at at the the wave wave front extends to to 25-110 m. The The region front extends 25-1 10m. whole wave wave travel travel takes takes 30-50 30-5Ops. This time time is is especially especially convenient convenient for for IlS. This whole electron-optical electron-optical methods methods of of streak streak photography. photography. However, However, available available attempts attempts to to use use such such methods methods can can hardly hardly be be considered considered successful. successful. A A serious serious obstacle obstacle is is the the exact exact synchronization synchronization of of aa streak streak camera camera and and lightning lightning contact contact with with the the earth. earth. Although Although there there are are many many synchronization synchronization methods, methods, they they have have no simple no simple technical technical solutions solutions and and are are seldom seldom used used in in lightning lightning experiments. experiments. Continuous (e.g. (e.g. sinusoidal) electron streak photography has has not not justified justified Continuous sinusoidal) electron streak photography hopes. Basic Basic results results on on return return stroke velocities have have been been obtained obtained using using hopes. stroke velocities cameras with with aa mechanical mechanical image image processing, processing, which which do do not not need need synchronisynchronicameras zation (Boyce (Boyce camera). camera). zation

Eo

Eo

1

1 ...+... +

:t 1

M

3.18. Scheme of the the return return stroke propagation after after the the contact contact of descendFigure 3.18. Scheme of stroke propagation of aa descend= 0). A leader leader brings brings potential potential U < 0; 0; 1ZM is ing leader leader with with the the earth earth (at moment t = ing (at moment M is return stroke current. return stroke current.


116

A vailable lightning data Available

3.7.1 Neutralization Neutralization wave velocity velocity The measurements measurements made made half half aa century century ago ago [25,39] [25,39] and and those those performed performed The recently recently [40] [40] indicate indicate aa high high velocity velocity of of aa return return current-voltage current-voltage wave. wave. The The 7 minimum measured measured values values are are close close to to (1.5-2) (1.5-2) x 10 minimum lo7 mls mjs and and the the maximum maximum ones an order 0.5-0.8 of speed c. A ones are are an order of of magnitude magnitude higher, higher, reaching reaching 0.5-0.8 of light light speed velocity speed does does not deal with velocity comparable comparable with with light light speed not mean mean that that we we deal with relativistic relativistic particles particles or or purely purely electromagnetic electromagnetic perturbations. perturbations. The The wave wave velocity is is the the phase phase velocity velocity of of the the process. process. velocity There are are not not so There so many many successful successful optical optical registrations registrations of of the the return return stroke, stroke, the number number of of really really good good ones ones being being about about 100. 100. Most Most of of the the available available data data the concern subsequent subsequent lightning lightning components. components. This This is is natural natural because because every every concern successfully registered registered discharge discharge includes includes the the return return strokes strokes of of several several compocomposuccessfully nents. The The wave wave velocities velocities of of subsequent subsequent components components are are somewhat somewhat higher higher nents. than those those of of the the first first ones. ones. According According to to [40], [40], the the first first component component has has an an averaverthan 7 age velocity velocity V V,r :::::; x 9.6 9.6 Xx 10 lo7 mjs while while the the subsequent subsequent ones ones are are aa factor factor of of 1.25 1.25 age mls higher. Similar Similar data data are are cited cited by by other other authors authors for for subsequent subsequent components components of of higher. lightning discharges discharges triggered triggered from from aa grounded grounded wire wire elevated elevated by by aa rocket. rocket. lightning To illustrate illustrate the the statistical statistical velocity velocity spread spread in in individual individual measurements measurements To and in in those those made made by by different different researchers, researchers, figure figure 3.19 3.19 shows shows integral integral and distribution curves curves for for the the data data of of [25] [25] and and [40]. [40]. The The first first and and subsequent subsequent distribution

1.0 0.8 0.8

g.--O

3 :E cd ro D .L:J

e&

Q.,

0.6 0.6 -

0.4 0.4 0.2

-

0.01 0.0

- .

0.05 0.05

*

.

1

I

0.1 0.1

0.2 0.2 vv,ic ,Ie

0.5 0.5

1

3.19. Velocity Velocity distribution distribution of of the the lightning lightning return return stroke: stroke: (1) (1) averaged averaged over over the the Figure 3.19. [25];(2) (2) averaged averaged over over 1.3 1.3km km above above the the earth earth [40]. [40]. visible channel channel length length [25]; visible



117 117

components were were not separated. Within Within aa 50% probability, there is is aa 2-fold 2-fold components not separated. probability, there difference between between the the velocities. velocities. Earlier Earlier measurements measurements generally generally give give lower lower difference return stroke stroke velocities. velocities. The The point is that that most most measurements measurements performed performed return point is during the the 1980s 1980s were were two-dimensional, two-dimensional, usually usually providing providing higher velocities, higher velocities, during whereas the earlier data data had had allowed allowed conclusions conclusions only only about about the the vertical vertical whereas the earlier component of of velocity. velocity. Moreover, Moreover, the the application application of of improved improved optics optics and and component photographic materials, materials, as as well well as as higher higher relative motion rates of the the photographic relative motion rates of image and and film, film, improved improved the time resolution of streak streak photographs. As aa image the time resolution of photographs. As result, the value obtained obtained in in the the 1980s 1980s was was more accurate and and result, the velocity velocity value more accurate higher because the measurements were averaged over the initial stroke higher because the measurements were averaged over the initial stroke length of of about about 1km 1 km at at the the earth's earth’s surface, surface, where the wave 1.5-2 length where the wave moves moves 1.5-2 times times faster, faster, rather rather than than over over the the whole whole stroke stroke length. length. All return stroke All measurements measurements show show that that the the return stroke velocity velocity gradually gradually decreases the wave wave front decreases and and that that the the velocity velocity V V,r drops drops abruptly abruptly when when the front passes through the point branching. The passes through the point ofleader of leader branching. The latter latter fact fact suggest suggest aa certain certain relation between stroke velocity velocity and and the the current current transported by the the wave: relation between the the stroke transported by wave: so the the at the branching point, the current is divided among the branches, at the branching point, the current is divided among the branches, so velocity becomes lower. The knowledge of this relation could improve the velocity becomes lower. The knowledge of this relation could improve the calculation calculation accuracy accuracy of of overvoltages overvoltages in in electrical electrical circuits circuits during during lightning lightning disdischarges. Unfortunately, the charges. Unfortunately, the available available data data are are insufficient insufficient to to allow allow finding finding this relation relation reliably. this reliably. Simultaneous Simultaneous registrations registrations of of current current and and velocity velocity have been made made only return strokes have been only for for return strokes of of subsequent subsequent components components of of triggered lightnings but they triggered lightnings but they cannot cannot provide provide aa representative representative statistics. statistics. With With reference reference to to [12,41], [12,41], there there is is note note in in [1] [l] about about aa satisfactory satisfactory agreement agreement between these registrations registrations and between these and Lundholm's Lundholm’s semi-empirical semi-empirical formula formula Vr/c V,/c == (1 (1 + 40/I 40/1M)-1’2, where 1lM is aa return return stroke stroke current current amplitude amplitude M )-1/2, where M is expressed 3.7.2). The The lack lack of of factual factual data data is is sometimes sometimes expressed in in kA kA (see (see section section 3.7.2). compensated compensated by by aa superposition superposition of of distribution distribution statistics. statistics. It It is is assumed assumed that values of probability that the the values of current current and and velocity velocity characterized characterized by by an an equal equal probability correspond to each this correspond to each other. other. There There are are no no serious serious arguments arguments in in favour favour of of this operation but it the lack better method. method. operation but it is is used used for for the lack of of aa better 3.7.2 3.7.2 Current amplitude amplitude

The parameter. Most hazards of The current current amplitude amplitude is is an an important important lightning lightning parameter. Most hazards of lightning whose lightning are are associated, associated, directly directly or or indirectly, indirectly, with with stroke stroke currents, currents, whose registration has much time time and Very few were made registration has taken taken much and effort. effort. Very few of of them them were made by direct using aa shunt belt [28,29,42-46]. by direct methods, methods, using shunt and and aa Rogovski Rogovski belt [28,29,42-461. Still Still fewer by equipment fewer direct direct measurements measurements were were made made by equipment with with aa wide wide dynamic dynamic range, range, which which can can register register both both powerful powerful impulses impulses with with an an amplitude amplitude to to 200 which are 200 kA kA and and low low currents currents of of aa few few hundreds hundreds of of amperes, amperes, which are equally equally physics. important important for for the the understanding understanding of of the the lightning lightning physics. have been been made A of measurements measurements have made by by magnetic magnetic detecdetecA large large number number of tors. tors. Such Such aa detector detector represents represents aa rod rod several several centimetres centimetres in in length, length, made made


118 118

Available A vailable lightning data

from rod detectors from magnetically magnetically hard hard steel. steel. Preliminarily Preliminarily demagnetized demagnetized rod detectors were were placed at placed at aa fixed fixed distance distance from from aa conductor conductor aimed aimed at at leading leading lightning lightning current current to to the the earth. earth. This This could could be be aa grounding grounding lead lead of of aa lightning lightning conductor conductor or or aa metallic tower metallic tower of of aa power power transmission transmission line. line. With With the the appearance appearance of of lightning lightning current, the detector be within within the range of current, the detector proves proves to to be the range of its its magnetic magnetic action action and and becomes magnetized. magnetized. One the residual residual steel becomes One measures measures the steel magnetization magnetization and and calculates the current problem. The calculates the current by by solving solving the the inverse inverse problem. The advantages advantages of of this method magnetic detectors this method are are its its simplicity simplicity and and low low cost. cost. Usually, Usually, magnetic detectors are are installed by the to obtain the necessary necessary statistics. they installed by the thousand thousand to obtain the statistics. However, However, they can yield nothing but aa current can yield nothing else else but current impulse impulse amplitude. amplitude. Of Of course, course, by by markmarking the ends the direction ing the ends of of the the detector, detector, one one can can also also determine determine the direction of of current current and to the negative). The and attribute attribute it it to the lightning lightning type type (positive (positive or or negative). The accuracy accuracy of of current very low reasons. current measurements measurements is is very low for for several several reasons. First, with aa simple First, there there are are few few objects objects with simple system system of of current current spread spread over over metallic constructions. metallic constructions. A single single conductor conductor would would be be ideal ideal in in this this respect, respect, because because it it excludes excludes current current branching. branching. In In reality, reality, lightning lightning current current is is distribudistributed among among many many conductors, conductors, the the distribution distribution pattern being unpredictable ted pattern being unpredictable since it it varies varies with with temporal temporal parameters of the the impulse. impulse. We We shall shall illustrate illustrate since parameters of this situation situation with with reference reference to to aa simple simple system system consisting consisting of of two two parallel inducthis parallel inductively connected branches with their own inductances L1 and L2, mutual tively connected branches with their own inductances L 1 and L 2 , mutual inductance M and resistances resistances R1j and and R2. Suppose aa rectangular rectangular current current inductance M,, and R 2 . Suppose short risetime risetime is is applied applied to to the system. The The current current distribudistribuimpulse I with impulse with aa short the system. tion between the two two branches is described described as as tion between the branches is . di2 di2 di2 di j . Rjl j +L j - +M- = R 2 12 +L 2 -d +M -d . dt dt t t

Initial currents currents iilo and ii20 at the the stage stage when when current current II(t) ( t ) is is stabilized stabilized to to ZI Initial lO and 20 at are generated generated over over aa very very short short time time equal equal to to the the I risetime. The branch are risetime. The branch currents, therefore, therefore, rise from zero zero very very quickly. quickly. The The reactive reactive components components of of currents, rise from voltage drop drop rv -di/dt produced by them are are much much larger larger than than the the ohmic ohmic voltage di/dt produced by them ones "'-'i -i that can be neglected for for the the time time being. Hence, we we have have ones that can be neglected being. Hence, ii1o/i20 /i = (L M)/(L M), and the initial current, say, in the first branch lO 20 = (L2 2 - M ) / ( L11- M ) , and the initial current, say, in the first branch = Z(L2 ) / ( Lj 1+ L2 2 M ) . When When the transitional process, whose is iilo is I(L 2 -- M M)/(L L 2 -- 2M). the transitional process, whose lO = duration is is defined defined by the time time constant constant ti.t At = = (L] (Li+ L2 2 M ) / ( R I+ R 22),) , duration by the L 2 -- 2M)/(R] is over, over, currents currents iilx = IR2/(R1 and ii2x = ijocR ilxR1/R2 are established established is IR 2 /(R 1 + R2) R 2 ) and 1cc = 20c = 1 / R 2 are in the circuits. The The durations durations of of lightning lightning currents currents are are usually usually comparable comparable in the circuits. with the constant ti.t. At. Therefore, Therefore, aa magnetic magnetic detector detector placed in one one of of with the time time constant placed in the branches branches will will register register aa current current intermediate intermediate between the initial initial and and estabestabthe between the lished values values having having aa maximum maximum amplitude, amplitude, since since the residual magnetization lished the residual magnetization of the rod contains contains information information only only about about the the maximum maximum magnetic field of of of the rod magnetic field current. For this reason, one can can calibrate calibrate aa magnetodetector for deriving deriving aa current. For this reason, one magnetodetector for full current current amplitude amplitude only only if if the impulse shape shape is is known. known. This This cannot cannot be full the impulse be done in in aa real experiment, so so one one has has to to resort to aa rough rough estimation estimation of of current current done real experiment, resort to distribution over over metallic constructions and and use it in in data data processing. distribution metallic constructions use it processing.

+

+



119 119

Second, the the operating operating range range of of the magnetization curve curve is is not not large, large, Second, the rod rod magnetization a linear to saturation region may produce additional and the transition from and the transition from a linear to saturation region may produce additional errors in in data data processing. processing. To To avoid avoid saturation, saturation, the the magnetodetector magnetodetector is is placed errors placed far from from the the conductor, conductor, which which creates creates difficulties difficulties in in data data processing of lightlightprocessing of far nings with low current current and and low low magnetic field. Besides, Besides, when the distance distance nings with low magnetic field. when the between conductor and and aa detector detector is is large, large, the the magnetic magnetic field field effects effects of of between aa conductor other metallic metallic elements elements with current are are hard hard to to take into account. account. So So aa other with current take into 100% error error does does not seem too too high high for for magnetodetectors, magnetodetectors, even even when several 100% not seem when several detectors are are placed at different different distances distances from from aa current current conductor. conductor. Their Their detectors placed at records sufficient material for engineering engineering estimations estimations or or for for aa records provide provide sufficient material for qualitative comparison comparison of of storm storm intensity intensity in in different different regions, regions, but they are are qualitative but they insufficient for for theory. theory. Organization Organization of of direct direct registrations takes much much time insufficient registrations takes time and effort. effort. There There are are no no more more than than aa hundred successful registrations and hundred successful registrations made over aa decade. decade. Let Let us see what what information information can can be be derived derived from from them. them. made over us see Current impulse impulse amplitudes amplitudes vary vary widely, widely, from from 2-3 2-3 to to 200-250 200-250 kA. Current kA. Some magnetodetector magnetodetector measurements give even even 300-400 300-400 kA, kA, but but these these Some measurements give amplitudes seem seem doubtful. doubtful. According According to to [42,46], [42,46], the the integral integral amplitude amplitude disdisamplitudes tributions for for the the first first and and subsequent subsequent lightning lightning components components obey obey the the sosotributions called lognormal lognormal law, law, in in which it is is current current logarithms, logarithms, rather rather than currents called which it than currents themselves, probability of themselves, that that meet meet the the normal normal distribution distribution criterion. criterion. The The probability of lightning lightning with with aa current current larger larger than than 1ZM, is defined defined as as M , is I [kA] (3.16) (3.16) is where where (lg (lg I)av I)av is an an average average decimal decimal logarithm logarithm of of the the currents currents measured measured and and (JIg olg is is the the mean mean square square deviation deviation of of their their logarithms. logarithms. This This approximation approximation cannot be considered cannot be considered accurate. accurate. The The relative relative deviation deviation of of the the value value of of (3.16) (3.16) from be several from the the real real one one may may be several tens tens percent; percent; it it may may be be even even more more for for pracpractically ranges. Nevertheless, Nevertheless, lognormal tically important important current current ranges. lognormal distributions distributions allow allow measurement measurement comparison comparison and and serve serve as as aa guide guide to to engineering engineering estimations. estimations. For example, example, about about 200 200 current current oscillograms oscillograms for for lightnings lightnings that that struck struck the the For 70m tower on on the the San San Salvatore Salvatore Mount Mount in in Switzerland Switzerland [42] [42] satisfactorily satisfactorily 70 m tower obey the the lognormal lognormal law law with with (lg (lgZ)av 1.475 and and (JIg olg= = 0.265 0.265 for for the the first first obey I)av == 1.475 means that component component currents currents of of aa negative negative lightning lightning discharge. discharge. This This means that the the 50% current value is be 30 current value is estimated estimated to to be 30 kA; kA; 95% 95% of of lightnings lightnings must must have have kA and kA. The probability of currents 5% of of lightnings lightnings 80 80kA. The probability of currents exceeding exceeding 44kA and 5% 100 kA kA is is expected expected in in 2% 2% of of cases cases and and higher currents currents rapidly rapidly decreases: decreases: 100 higher 200 kA in % of 200kA in less less than than 0.1 0.1% of cases cases (figure (figure 3.20). 3.20). It It should should be be emphasized emphasized the distribution be treated with caution. again again that that the distribution boundaries boundaries must must be treated with caution. The The curve shape shape in in the low current current range range strongly strongly depends depends on on the the sensitivity sensitivity of of curve the low the measuring instruments used used (its (its left-hand left-hand limit limit is is usually usually taken taken to to be be the measuring instruments 1-3 measurements in 1-3 kA kA in in distribution distribution plots). plots). There There are are few few measurements in the the high high current current


120 120

Available lightning data

11

0.01 0.01

I

50

'

I

100 100

.

I

.

I

150 150 200 Lightning Lightning current, current, kA

.

1

250

Lognormal distributions distributions of of return stroke currents: currents: A, the the first first comcomFigure 3.20. Lognormal return stroke ponent of ponent of aa negative negative lightning lightning with with (lg (lg I).v I)," == 1.475 1.475 and and O"lg clg= = 0.265; 0.265; B, B, subsequent subsequent components with with (lg (lgZ),v 1.1 and and O"lg olg= = 0.3; 0.3; C, C, positive positive lightnings lightnings with components I).v == 1.1 with (lgZ)," == 1.54 1.54 and and O"lg clg== 0.7. 0.7. (lgI).v range: it it is is considered considered as as good good luck luck if if they provide aa reliable order of of magnimagnirange: they provide reliable order tude. Note that negative lightning been tude. Note that negative lightning currents currents above above 200 200 kA kA have have never never been registered registered reliably. reliably. The The approximation approximation of of data data on on subsequent subsequent lightning lightning components components in in [42] [42] gives high currents. A lognormal much lower lower integral integral probability probability for for high currents. A lognormal distridistrigives aa much = 1.1 1.1 and and (/lg clg= = 0.3. 0.3. The The bution can be satisfactorily described described by by (lg (lg& bution can be satisfactorily I)av = kA, 5% kA, and calculated calculated 50% 50% current current is is 12.5 12.5kA, 5% current current is is only only 39 39kA, and the the kA is % chance chance for for aa subsequent subsequent component component to to exceed exceed 100 lOOkA is close close to to 0.1 0.1% (figure (figure 3.20). 3.20). The positive lightnings, whose number number is 10% of of the the The statistics statistics for for positive lightnings, whose is about about 10% total registrations, is All descending positive lightnings total registrations, is less less representative. representative. All descending positive lightnings are are one-component. one-component. The The integral integral current current distribution distribution for for them them has has aa large large spread. probabilities of spread. The The probabilities of both both low low and and high high currents currents are are larger larger without without to 35 an an essential essential change change of of the the 50% 50% value. value. The The 50% 50% value value is is close close to 35 kA, kA, i.e., the same negative lightning. i.e., it it is is nearly nearly the same as as for for the the first first component component of of negative lightning. An the lognormal positive curAn approximate approximate description description of of the lognormal distribution distribution of of positive currents in be made made with with (lgI)av 1.54 and and (/lg clg= = 0.7 0.7 (figure (figure 3.20). 3.20). rents in [42] [42] can can be (lgI)av== 1.54 Positive high current Positive high current lightnings lightnings are are more more frequent frequent than than negative negative ones. ones. A A kA can with aa 5% 100 kA can be be expected expected with 5% probability probability corresponds corresponds to to 250 250 kA, kA, and and 100



121 121

20% probability. 26 successful successful registrations registrations of of positive positive lightnings lightnings in in 20% probability. Among Among 26 [43], only only one one showed showed 300 300 kA current. It It seems seems likely likely that positive lightlight[43], kA current. that many many positive nings were, under the the observation observation conditions conditions [43], [43], ascending ascending ones, ones, which nings were, under which may account account for for the the large large spread. spread. Such Such lightnings lightnings have practically no no may have practically return stroke, stroke, and and the the equipment equipment seems seems to to have registered the the relatively relatively return have registered low leader leader current current of of the the final final development development stage. stage. These These data data were low were used used to to derive the integral distribution distribution extending extending to the low low current current region. region. derive the integral to the The great great importance importance of of lightning lightning current current statistics statistics to to applied applied lightning lightning The protection necessitated necessitated aa unification of theoretical theoretical distribution distribution curves. curves. OtherOtherprotection unification of wise, engineers engineers would would have unable to to compare compare the the frequency frequency of of harmful harmful wise, have been been unable lightning protection efficiency. being done within the lightning effects effects and and protection efficiency. This This work work is is being done within the frame of of the the CIGRE CIGRE (Conference (Conference Internationale Internationale des des Grands Grands Reseaux Reseaux frame Electrique haute tension) Electrique 11a haute tension) -- an an operating operating international international conference conference on on high-voltage networks. Data high-voltage networks. Data on on current current from from all all over over the the globe globe are are collected collected and analysed. analysed. However, However, there there is is no no unified unified approach approach to to these these data: data: different different and data are are discarded discarded for for different different reasons, reasons, so so that that the the distributions distributions obtained obtained data differ markedly. markedly. For For example, example, aa report report submitted submitted to to [47] [47] compares compares two two loglogdiffer normal laws laws with with (lgI)av (lg&" == 1.4 1.4 and and 1.477 1.477al, = 0.39 0.39 and and 0.32. 0.32. The The latter latter is is normal O"lg = preferable for power transmission lines, lines, since since the measurements for for objects objects preferable for power transmission the measurements higher than were excluded higher than 60 60 m m were excluded from from this this derivation derivation (power (power transmission transmission lines his book, book, Uman Urn an [1] table of lines are are usually usually lower). lower). In In his [l] gives gives aa table of lightning lightning currents currents mostly mostly based based on on the the measurements measurements of of [42]. [42]. Attention to Attention to details details is is inevitable, inevitable, since since slight slight corrections corrections in in parameter parameter distributions may cause probabilities of distributions may cause manifold manifold changes changes in in the the calculated calculated probabilities of currents above above 100 100 kA, especially important important in in lightning lightning protection of currents kA, especially protection of important Both theory important objects. objects. Both theory and and applications applications suffer suffer from from aa lack lack of of lightlightning measurements. We We shall here some key issues be discussed ning current current measurements. shall list list here some key issues to to be discussed in below. in more more detail detail below. We have mentioned height. It We have mentioned the the importance importance of of an an object's object's height. It has has been been known since that h high known since Benjamin Benjamin Franklin's Franklin's experiments experiments that g h constructions constructions attract more lightnings. the process process of attract more lightnings. It It seems seems likely likely that that the of attraction attraction depends depends on the statistics on the the potential potential of of aa descending descending leader. leader. If If this this is is so, so, the statistics of of descenddescending various height may prove prove different: there ing leader leader currents currents for for objects objects of of various height may different: there will be be aa kind this case, reliable will kind of of lightning lightning separation. separation, In In this case, aa comparison comparison of of reliable current various objects resolve the the much current statistics statistics for for various objects could could help help resolve much debated debated problem of problem of lightning-object lightning-object interaction interaction mechanism. mechanism. Of this connection Of interest interest in in this connection is is the the following following fact. fact. In In the the case case of of aa very very high construction, of the the ascending ascending type type having having high construction, many many first first components components are are of no return the first no return stroke. stroke. But But the first component component is is followed followed by by subsequent subsequent descenddescending ing components, components, whose whose average average stroke stroke currents currents are are lower lower than than in in subsequent subsequent high components buildings. This components affecting affecting low low buildings. This suggests suggests an an involvement involvement of of high that an ground the formation ground constructions constructions in in the formation of of storm storm clouds. clouds. It It appears appears that an ascending high construction before the has ascending leader leader starts starts from from aa high construction before the cloud cloud has matured. Its potential are, therefore, lower. lower. This matured. Its charge charge and and potential are, therefore, This accounts accounts for for


122 122

Available lightning lightning data

the the fact fact that that subsequent subsequent components components discharging discharging an an immature immature cloud cloud will will transport lower lower stroke stroke currents currents than than aa mature mature cloud. cloud. transport Finally, Finally, an an important important issue issue is is the the effect effect of of grounding grounding resistance resistance of of objects objects on on lightning lightning current. current. This This may may provide provide information information on on the the resistance resistance of of lightning itself. itself. This This resistance resistance is is to to be be introduced introduced in in equivalent equivalent circuits, circuits, lightning when calculating calculating overvoltages overvoltages affecting affecting various various electrical electrical circuits. circuits. This This when problem is is still still much much debated: debated: some some investigators investigators suggest suggest the the substitution substitution of of problem lightning channel channel by by aa current current source source with with an an 'infinite' ‘infinite’ resistance, resistance, others others aa lightning ascribe to to the the channel channel the the wave wave resistance resistance of of aa common common wire wire (about (about 3000). 300 0). ascribe It would would not not be be hard hard to to solve solve this this problem problem if if we we had had at at our our disposal disposal reliable reliable curcurIt rent rent statistics statistics for for objects objects of of various various height height but but different different grounding grounding resistances. resistances. No such such statistics statistics exist exist yet. yet. To To speed speed up up the the work work in in this this area area and and to to reduce reduce its its No cost, various various remote remote registration registration techniques techniques are are being being employed. employed. They They register register cost, electromagneticfields fields and and coordinates coordinates ofpoints of points where where the the lightning lightning strikes strikes (ide(ideelectromagnetic ally, ally, the the lightning lightning trajectory), trajectory), followed followed by by the the solution solution of of the the inverse inverse problem problem for the the field field source, source, i.e., i.e., lightning lightning current current (see (see section section 3.7.4). 3.7.4). for There is is also also an an increasing increasing number number of of direct direct current current registrations registrations from from There lightnings triggered triggered from from aa wire wire lifted lifted by by aa rocket rocket to to the the height height of of 150150lightnings 250m. The The first first component component of of aa triggered triggered lightning lightning (ascending (ascending leader) leader) has has 250m. no return return stroke; stroke; therefore, therefore, one one deals deals only only with with subsequent subsequent components. components. A A no comparison of of such such registrations registrations with with natura11ightning natural lightning currents currents was was made made comparison in Alabama, Alabama, USA USA [15]. [15]. The The statistics statistics were were not not particularly particularly representative representative in (45 measurements), measurements), so so no no principal principal differences differences were were revealed. revealed. The The lognormal lognormal (45 distribution of of currents currents corresponded corresponded to to the the parameters parameters (lg (lg& = 1.08 1.08 and and I)av = distribution olg== 0.28, 0.28, nearly nearly the the same same as as those those obtained obtained in in Switzerland Switzerland for for subsequent subsequent O"lg components of of natural natural lightnings lightnings [42]. [42]. We We should should like like to to warn warn the the reader reader components against aa possible possible overestimation overestimation of of this this coincidence. coincidence. The The comparison comparison against involved measurements measurements from from geographical geographical points points separated separated by by large large disdisinvolved tances, whereas whereas the the global global variation variation of of lightning lightning parameters parameters still still remains remains tances, unclear. A more more important important thing thing is is that that the the lightnings lightnings studied studied in in [42] [42] unclear. cannot be be regarded regarded as as totally totally natural. natural. They They struck struck aa 70-m 70-m tower tower on on aa cannot mountain elevated elevated at at 600 600m above the the earth's earth’s surface surface close close to to aa lake. lake. The The mountain m above conditions here here are are more more similar similar to to those those of of lightning lightning triggering triggering than than to to its its conditions natural development development in in aa flat flat country. country. Lightning Lightning parameters parameters are are known known to to natural differ with with altitude: altitude: currents currents registered registered by by magnetodetectors magnetodetectors at at an an altitude altitude differ of 1-2 1-2km were two two times times lower lower than than in in aa flat flat country, country, for for less less than than 50% 50% of km were probabilities [48]. [48]. probabilities

3.7.3 Current impulse impulse shape shape and and time characteristics characteristics 3.7.3

Records of of lightning lightning current current impulses impulses look look more more like like abstractionists' abstractionists’ Records pictures -- they they are are so so diverse diverse and and fanciful. fanciful. The The conventional conventional approximation approximation pictures Z(t) = =1 Io[exp (-at) -- exp exp (-!3t)], which which is is of aa impulse impulse by by two two exponents exponents I(t) of 0 [exp (-at) suggested in in various various guides guides to to equipment equipment testing testing lightning lightning resistance, resistance, is is suggested

,I)@-(


123 123

Lightning return stroke Lightning

10

t, f.iS

o

20

40

60

80

100

t, f.iS

Figure 3 .21. A schematic 3.21. schematic oscillogram oscillogram of of aa current current impulse impulse in in the the first first component component of of aa negative lightning. lightning. negative

intended intended for for currents currents of of laboratory laboratory sources sources simulating simulating lightning, lightning, rather rather than than for natural natural lightning. lightning. Let Let us us try try to to identify identify the the main main features features of of the the time time for of current, current, essential essential for for the the understanding understanding of of the the return return stroke stroke variation of variation mechanism and and applications. applications. mechanism The most most reliable reliable data have been been obtained obtained for for first component currents The data have first component currents of aa negative negative lightning. lightning. This current is is easy easy to to register, register, since the impulse impulse front front of This current since the takes several several microseconds microseconds and an oscillographic oscillographic record record reproduces reproduces it in takes and an it in detail. A A sketch sketch of of aa current current impulse impulse averaged averaged over over many many oscillograms oscillograms is is detail. shown in in figure figure 3.21 3.21 in in two two time time scales. Note the the concave concave shape shape of of the the shown scales. Note front. An An expression expression of of the the type type 11 -- exp looks least for its its front. exp (-{3t) (-pt) looks least suitable suitable for description. The The first current peak peak is is often often followed followed by by aa higher higher one, and description. first current one, and evaluation of the impulse impulse risetime risetime tf is is associated with some some reservations. reservations. evaluation of the associated with For example, measured the the time time of of current rise from 2 kA, aa value value For example, [42] [42] measured current rise from 2kA, close to to the the resolution resolution threshold, threshold, to to the the first maximum ., 1 In this this case, case, close first maximum ZM , In of negative lightnings had the risetime of the first component about 50% about 50% of negative lightnings had the risetime of the first component over over 5.5Ils, 5.5 ps, 5% 5% exceeded exceeded 181ls, 18 ys, and and another another 5% 5% less less than than 1.81ls. 1.8 1s. The The knowlknowledge of the risetime allows calculation of the average impulse edge of the risetime allows calculation of the average impulse slope slope AI AI av == 1Z,/tf, However, the the calculation calculation of of electromagnetic electromagnetic fields fields of of lightning lightning M /tf' However, and and the the evaluation evaluation of of possible possible hazards hazards require require aa maximum maximum slope slope AI,,,max == (dl/ (dZ/dt),,,, rather than than an an average average one. one. The The error error in in evaluations evaluations AI dt)max, rather of of this this parameter parameter from from current current oscillograms oscillograms may may be be very very large, large, because because one has has to to replace replace the the tangent tangent to to the the I(t) one Z ( t ) curve curve by by aa secant. secant. Nevertheless, Nevertheless, this this operation operation has has aa sense sense for for aa fairly fairly long long impulse impulse of of the the first first component. component. The integral integral distribution distribution of of the the values, values, like like the the current current itself, itself, is is described described The by the the lognorma11aw lognormal law with with the the parameters parameters (lg (lgZ)," = 1.1 1.1 and and CTlg glg= 0.255, 0.255, if if by I)av = the slope slope is is expressed expressed in in kAIIlS. kA/ps. It It results results in in 12 12kAIIlS kA/ps for for 50% current, current, and and the the the slope slope exceeds exceeds 33 33 kAIIlS kA/ys with with aa 5% 5% probability. probability. To To describe describe the the electromagnetic electromagnetic effect effect of of lightning lightning current, current, let let us us find find 2 placed in aa frame frame of of area area SS = 1m 1 m2 placed at at distance distance D = = 1m 1m the induced induced emf emf UM, in the from from the the channel channel or or aa grounding grounding conductor, conductor, when when the the first first component component current current flows flows through through itit (the (the frame frame is is in in aa plane plane normal normal to to the the current current AI max == 33 33 kAIlls, kA/ps, we we have have magnetic field). field). Even Even for for aa moderate moderate steepness steepness AI magnetic U M, = = = /-LaAImaxS(211"D)-1 p0AImaxS(2~D)-' = 6.6kV, 6.6 kV, where where /-La p0 = = 411" 47r x 10lo-' 7 Him Hjm is is vacuum vacuum


124 124


magnetic permeability. permeability. The The role role of of aa frame frame can can be be performed performed by by any any metallic metallic magnetic structure within within the the construction construction affected affected by by lightning-wires, lightning-wires, wall wall fittings, fittings, rails, rails, structure metallic stripes stripes touching touching each each other, other, etc. etc. At At the the site site of of aa poor poor contact, contact, induced induced metallic emf will will produce produce aa spark, spark, much much more more effective effective than than that that in in an an electric electric lighter. lighter. emf This is is dangerous dangerous because because the the spark spark may may come come in in contact contact with with an an explosive explosive gas gas This mixture. mixture. Sometimes, lightning lightning current current behaves behaves in in aa dual dual way, way, creating creating the the inducinducSometimes, at the the resistance resistance of of the the grounding grounding electrode electrode UR, = = IR. /R. It It tion emf emf and and voltage voltage at tion is important, important, therefore, therefore, to to have have knowledge knowledge about about the the relation relation between between the the is current amplitude amplitude and and maximum maximum slope. slope. Although Although both both parameters parameters obey obey current the same same lognormal lognormal law, law, no no correlation correlation has has been been found found between between them. them. the is bad bad for for engineering engineering applications, applications, for for one one has has to to calculate calculate the the This is This probabilities of each current with a whole set of possible slopes. probabilities of each current with a whole set of possible slopes. There have have been been attempts attempts at at aa more more detailed detailed description description of of the the current current There impulse front. front. They They were were initiated initiated by by the the CIGRE CIGRE mentioned mentioned above above to to impulse handle hazardous hazardous effects effects of of lightning on power power transmission transmission lines. lines. A A set of handle lightning on set of additional parameters parameters has has been been suggested suggested to to reduce reduce errors errors in in current oscilloadditional current oscillogram processing processing and and some regions of of the the impulse impulse front have been been described gram some regions front have described quantitatively. This This is illustrated in in figure figure 3.22 requires no no comment. quantitatively. is illustrated 3.22 and and requires comment. [47]. The processing technique used did Some of the results were cited in did Some of the results were cited in 1471. The processing technique used not lead to considerable data refinement, since the 50% maximum slope not lead to considerable data refinement, since the 50% maximum slope AI max = kA/llS is only two two times times larger than the the 50% 50% average AI,,, = 12 12kA/ps is only larger than average slope slope A ~o% = 5 o y O / t f 5 o= 30/5.5 = 5.5kA/llS. 5.5 kA/ps. But of 22 is is essential A5O'io = Z/50'/0/00% =~ 30/5.5::::;: But the the factor factor of essential to to of ultrahigh ultrahigh voltage voltage insulation. the electrical electrical strength the strength of insulation.

-I

Eo

-

t

~O%

~O%

------\-

\ . .\ \ '

1·moxl

.\

\'.\

\'.\,

Figure 3.22. 3.22. The The distribution distribution of of current current impulse impulse parameters parameters in in the the return return stroke, stroke, Figure based based on on oscillograms. oscillograms.


Lightning return stroke Lightning

125

Current have aa shorter shorter risetime. risetime. In In Current impulses impulses of of subsequent subsequent components components have the probability and and 0.2 0.2).!s for aa the work work cited cited above above [42], [42], tf 9 < 1.1 1.1 ).!S ps for for aa 50% 50% probability ps for 5% with caution caution because because this this value value 5% probability. probability. The The latter latter should should be be treated treated with is equipment. The The impulse impulse front front is close close to to the the resolution resolution limit limit of of the the measuring measuring equipment. in is mentioned mentioned in in [l] [1] with with in subsequent subsequent components components is is likely likely to to rise rise faster. faster. It It is reference many digital digital registrations registrations the the reference to to other other publications publications that that in in many current the first detector reading reading (for (for 0.2 0.2 ps). ).!s). to aa maximum maximum during during the first detector current could could rise rise to The in subsequent subsequent components components The maximum maximum slope slope of of aa impulse impulse front front in obeys, law: (lg (lg AI,,),,,max)av = = 1.6 1.6 obeys, in in the the first first approximation, approximation, the the lognormal lognormal law: and 40 kA/ps kA/).!s with with aa 50% 50% probability probability and O'lg glg= = 0.35. 0.35. The The value value of of AI AI max exceeds exceeds 40 and probability. When When affected affected by by such such aa 5% probability. and is is larger larger than than 120 120 kA/).!s kA/ps with with aa 5% would exceed exceed 25 25 kV kV in in the the steep steep impulse, impulse, the the amplitude amplitude of of induced induced voltage voltage would above above example example of of aa frame. frame. Current In 55% of cases, cases, the the front front Current of of positive positive lightnings lightnings rises rises slowly. slowly. In % of duration ).!s. With of air air duration was was tf 9 > 200 200ps. With these these impulses, impulses, the the electric electric strength strength of gaps 2.6, formula formula gaps of of several several metres metres in in length length is is close close to to aa minimum minimum (section (section 2.6, (2.51)). than aa (2.51)). The The voltage voltage with with tf::::: tr 200).!s 200ps is is much much more more dangerous dangerous than 'common' micro‘common’ lightning lightning overvoltage overvoltage impulse impulse with with aa risetime risetime of of several several microseconds. nonseconds. Minimum Minimum breakdown breakdown voltage voltage in in air air gaps gaps with with aa sharply sharply nonuniform than in in aa standard standard uniform field field (see (see formula formula (2.52)) (2.52)) is is about about 1.5 1.5 times times lower lower than lightning with the the conconlightning overvoltage overvoltage impulse impulse of of 1.2/50).!s 1.2/50 ps (in (in accordance accordance with is the the ventional ventional way way of of presenting presenting time time characteristics characteristics of of aa impulse, impulse, 1.2 1.2 is risetime ).!s). risetime and and 50 50 is is the the impulse impulse duration duration at at 0.5 0.5 amplitude, amplitude, all all in in ps). The protecThe duration duration of of aa current current impulse impulse is is as as important important for for lightning lightning protection practice practice as as the the risetime. risetime. Impulse Impulse duration duration is is usually usually characterized characterized as tion as aa time span between its beginning and the moment its amplitude decreases time span between its beginning and the moment its amplitude decreases by half. half, Since Since current current is is related related to to the the neutralization neutralization wave by wave travelling travelling along along the the time time of of the the the channel, channel, the the impulse impulse duration duration tt,p is is comparable comparable with with the 8 wave mls and length wave travel. travel. If If its its velocity velocity is is V V,r ::::: E 10 108m/s and the the average average channel channel length the value value of of ttpp will will be be several several tens tens of of microseconds. microseconds. A isis 33km, km, the A similar similar value isis derived derived from from experimental experimental data. data. The The impulse impulse duration duration in value in the the first first component of of aa negative negative lightning lightning is is above above 30, 30, 75 75 and and 200).!s 200 ps for component for the the probprobabilities 95, 95, 50 50 and and 5%, 5%, respectively. respectively. For For subsequent subsequent components, components, the the abilities 6, 32 32 and and 140 140 IlS ps for for the the same same probabilities. impulse is is much much shorter: shorter: 6, impulse probabilities. Positive Positive lightnings must must be be longer longer because because most most of of the the positive positive charge charge of of aa storm storm lightnings cloud is located 2-3km higher than the negative charge. Indeed, is cloud is located 2-3 km higher than the negative charge. Indeed, ttpp is above 230).!s 230ps with with aa 50% 50% probability. probability. The The shortest shortest durations durations for for positive above positive lightnings are are the the same same as as for for the the first first component component of of aa negative negative one. one. 'Anom‘Anomlightnings alously’long long impulses impulses stand stand out out against against this this background background -- about about 5% 5% ofposiof posialously' tive currents decreased to half the amplitude for 2000 ps. tive currents decreased to half the amplitude for 2000 ).!s. Today, we we know know nothing nothing about about the the nature nature of of superlong superlong positive Today, positive impulses. One One thing thing is is clear: clear: they they are are unrelated unrelated to to the the wave wave processes processes in in impulses. the lightning lightning channel. channel. One One may may suggest suggest that that hydrometeor hydrometeor charge charge is is accumuaccumuthe lated and and descends descends to to the the earth earth due due to to an an ionization ionization process process in in the the positively positively lated


126 126

A vailable lightning data Available

charged region of nature of this charged region of aa cloud. cloud. But But we we can can only only speculate speculate about about the the nature of this process producing producing final process final current current of of 100 100 kA kA and and ask ask why why it it is is manifested manifested only only in in positive lightnings. positive lightnings. 3.7.4 3.7.4 Electromagnetic Electromagnetic field field

Electromagnetic to those Electromagnetic field field of of lightning lightning is is familiar familiar to those leaving leaving aa TV TV or or radio radio set set on on during during aa thunderstorm. thunderstorm. Sound Sound and and video video noises noises inform inform about about aa storm begins. Lightning storm long long before before it it actually actually begins. Lightning was was the the first first natural natural radio radio station by the radio engineering of radio engineering for for testing testing their their receivers. receivers. station used used by the founders founders of The by A S Popov in S Popov in 1885 1885 is is still still Russia's Russia’s The lightning lightning detector detector designed designed by national pride. pride. For meteorologists surveyed national For many many years years meteorologists surveyed approaching approaching storm by registering pulses of of electroelectrostorm fronts fronts by registering so-called so-called atmospherics atmospherics -- pulses magnetic of magnetic radiation radiation from from lightning lightning discharges discharges occurring occurring hundreds hundreds of kilometres away. the late kilometres away. In In the late 1950s, 1950s, much much interest interest in in atmospherics atmospherics was was due due to the nuclear to the nuclear weapon weapon race: race: suspiciously suspiciously similar similar to to radiation radiation pulses pulses from from nuclear explosions, explosions, they they interfered interfered with the diagnostics diagnostics of of the latter. nuclear with the the latter. It is is clear clear from from the the foregoing foregoing that that in in aa return stroke the charge accumuaccumuIt return stroke the charge lated by aa leader varies and redistributed rapidly rapidly along lated by leader cover cover varies and is is redistributed along the the channel, channel, producing variation producing variation of of the the static static component component of of the the electric electric field. field. Charge Charge variation occurs simultaneously with the propagation of a current wave variation occurs simultaneously with the propagation of a current wave along along the the channel, channel, inducing inducing aa magnetic magnetic field. field. The The induction induction emf emf varying varying in in time time gives gives rise rise to to an an induction induction component component of of the the electric electric field. field. Finally, Finally, variation in the the current current dipole dipole moment moment (a (a leader leader channel channel can can be variation in be regarded regarded as mirror reflection by the of its its mirror reflection by the earth) earth) gives gives rise rise as aa dipole, dipole, with with the the account account of to radiation component to an an electromagnetic electromagnetic wave wave producing producing aa radiation component of of the the electric electric field with with aa concurrent concurrent magnetic radiation component. component. There There is is another another magmagfield magnetic radiation magnetostatic one one proportional proportional directly directly to to current. current. netic component component -- aa magnetostatic netic It is is common common practice to distinguish distinguish between between the near and and far far regions regions of of It practice to the near electromagnetic radiation. radiation. In In the the near near region, static field field components components may may be be electromagnetic region, static dominant: the the electric electric component, component, damped damped in in proportion proportion to to the the cubic cubic dominant: distance, to the magnetic component, with the dipole dipole centre, centre, and and the the magnetic component, varying varying with distance Y to distance be neglected F2.These These can can be neglected for for the the far far region, region, because because they they are are distance as as ,-2. much than the E,, H ::::; Now, after much smaller smaller than the radiation radiation components components E cz ,-1. Y - ’ . Now, after these these preliminary how preliminary remarks, remarks, we we shall shall turn turn to to experimental experimental data data showing showing how much the shape of of aa registered registered pulse pulse varies varies with with distance distance between lightning much the shape between aa lightning discharge and and aa field field detector. detector. discharge The shapes shapes of of return return stroke stroke radiation radiation pulses pulses are are shown shown schematically schematically in in The 3.23 for for the the near near and and far far regions. regions. At At large large distances, distances, where where the the static static figure 3.23 figure components of of magnetic and electric electric fields fields are are nearly completely damped, damped, components magnetic and nearly completely E(t) H(t) the pulses E( t ) and and H ( t ) become become geometrically geometrically similar. similar. Both Both are are bipolar bipolar the pulses and high front well defined and have have aa high front slope, slope, aa well defined initial initial maximum maximum and and several several smaller pulse slope, producing the smaller ones ones along along the the slowly slowly falling falling pulse slope, producing the effect effect of of damping oscillations. oscillations. Note that the oscillation period period is is smaller smaller than than the the damping Note that the oscillation



E, Vim

127 127

H,Alm

r =2km

2000 1500

2.0 "-

1000 500

,, "" ,

1.0

" t, f.JS

0

6

50

100

E,Vlm

150 H,A/m

r =100 km

0.01

4 2 0

0.005

t, JLs 50

Figure pulses of Figure 3.23. Schematic Schematic oscillograms oscillograms of of electromagnetic electromagnetic pulses of lightning lightning in in the the near near (top) (top) and and far far (bottom) (bottom) zones zones at at the the distances distances 2km 2 km (top) (top) and and lOOkm 100 km (bottom). (bottom).

double time of the the wave wave run run along along the channel. After After passing the zero zero point, double time of the channel. passing the point, the part opposite nearly the the pulse pulse part opposite in in sign sign rises rises and and then then decreases decreases with with nearly the same same rate; its its amplitude amplitude is is 2-3 2-3 times times smaller smaller that that the the first first 'half ‘half period’. rate; period'. The proportionality of The inverse inverse proportionality of radiation radiation components components to to the the distance distance from was the reason why why measurements presented from the the radiation radiation source source was the reason measurements are are presented Ybas = = 100 100 km km in the above form: form: they they are are normalized normalized to to the distance rbas in the above the basic basic distance 5 as E:,, = 1010-5Em,,r in metres. metres. For For the the first first lightning lightning component, component, as E:nax = Emaxr with with r in the average average values values of of the the initial initial pulse peak of of the vertical component, component, pulse peak the vertical the EA,,, lie within within 5-l0Y/cm 5-10 V/cm [49-54] [49-541 (compare: (compare: radio radio receivers receivers detect detect well well E:nax, lie signals of of 1mY/m lmV/m in in the the medium bandrange). The The electric electric component component of of signals medium bandrange). subsequent is 1.5-2 1.5-2 times times smaller. smaller. The The spread spread of of subsequent lightning lightning components components is measurements measurements is is as as large large as as that that oflightning of lightning currents. currents. The The standard standard deviation deviation (JE range 35-70% oE is is in in the the range 35-70% for for the the first first lightning lightning component component and and 30-80% 30-80% for for


128 128


subsequent subsequent ones. ones. The The horizontal horizontal component component of of magnetic magnetic field field strength strength 12 (po/~O)-'/2E~ax varies respectively. respectively. Magnetic Magnetic induction induction H':nax == (/Lo/cor varies / E':nax Bmax = /LoHmax is km from B,,, = poHmax is about about 10lo-*8 T T at at aa distance distance of of 100 100 km from the the lightning. lightning. The pulse of The radiation radiation pulse of the the first first lightning lightning component component rises rises to to the the initial initial peak processing, the risetime is peak with with an an increasing increasing rate. rate. In In oscillogram oscillogram processing, the risetime is arbitrarily the initial arbitrarily subdivided subdivided into into two two components: components: the initial slow slow one one of of 3-51ls 3-5 ps duration the final duration and and the final fast fast one one taking taking 1-0.llls. 1-0.1 ps. The The standard standard deviation deviation is is also also large large here: here: 30-40% 30-40% of of the the average average value value for for the the slow slow front front and and about the fast about 50% 50% for for the fast front. front. In In the the final final stage, stage, the the signal signal rises rises for for about about 0.5-1.0E~,,. With With some some reservations, subdivision into into aa slow slow and and fast fast 0.5-1.0E':nax. reservations, aa subdivision component component can can be be also also made made for for radiation radiation pulse pulse of of the the return return stroke stroke of of subsequent be more subsequent lightning lightning components. components. But But it it would would be more correct correct to to consider consider that the the rise to the initial peak occurs quickly quickly there, there, for for 0.15-0.6Ils. 0.15-0.6 ps. Note that that rise to the initial peak occurs Note that the risetimes for the first and and subsequent subsequent components components are are close close to to those those of of the risetimes for the first their their current current impulses impulses in in aa return return stroke. stroke. The moment of sign sign reversal reversal for for radiation radiation pulses of the the first first components components The moment of pulses of is delayed, relative to the onset of a return stroke, by 50ps in temperate is delayed, relative to the onset of a return stroke, by 50llS in temperate latitudes the tropics reversal for latitudes [54] [54] and and by by 90 90 IlS ps in in the tropics [52]. [52]. The The sign sign reversal for subsequent subsequent components occurs occurs by factor of of 1.3-1.5 1.3- 1.5 earlier. earlier. The The time time for for maximum maximum field field components by aa factor to be established after the sign reversal is of the same order of magnitude as to be established after the sign reversal is of the same order of magnitude as that prior to the reversal. that prior to the reversal. The radiation components near The radiation components E and and H are, are, naturally, naturally, present present in in the the near region, too, but they than the region, too, but they are are much much smaller smaller than the static static component. component. One One exception is is the the initial initial moments of time. The initial initial peaks peaks in in oscillograms oscillograms exception moments of time. The E(t) H(t) be attributed to radiation, radiation, since E ( t ) and and H ( t ) should should be attributed to since the the static static field field components did not have enough time to reveal themselves. The monotonic components did not have enough time to reveal themselves. The monotonic rise of of electric electric field field over over 20-50IlS, 20-50ps, the the time time long long enough enough for for the the radiation radiation rise component to to be damped, is is nearly due to to electrostatic electrostatic effect. effect. The The component be damped, nearly totally totally due induced powerful, because because the induced electrostatic electrostatic field field is is quite quite powerful, the charge charge accumulated accumulated by the the stepwise by stepwise leader leader of of the the first first component component or or by by the the dart dart leader leader of of subsequent components components is is neutralized neutralized during during the the return return stroke. stroke. For For subsequent example, the the electrostatic electrostatic field field changes changes by by several several kV/m kV/m at at the the distance distance of of example, 1km IlS (for 1 km from from the the channel channel lightning lightning during during the the first first 50 50ps (for the the subsequent subsequent times lower than for component, component, the the signal signal is is 2-3 2-3 times lower than for the the first first one one ); aa slower slower field rise may field rise may continue continue for for about about 100 100 Ils. ps. All All in in all, all, the the field field of of the the first first lightlightradiation ning component ning component is is an an order order of of magnitude magnitude higher higher than than the the initial initial radiation rise. With km, the to 15-20 15-20km, the radiation radiation component component rise. With increasing increasing distance distance r to becomes dominant over over the the others, others, and and the the initial initial radiation radiation peak becomes dominant peak becomes becomes an an absolute absolute maximum maximum of of the the registered registered signal. signal. The magnetostatic component the near region is not so The magnetostatic component in in the near region is not so important. important. Still, Still, at at aa distance distance of of 1km, 1 km, it it contributes contributes as as much much to to the the signal signal as as the the radiation component here is high radiation component (figure (figure 3.23). 3.23). The The magnetic magnetic induction induction here is as as high 5 as T. maximum is lop5 T. The The absolute absolute magnetic magnetic field field maximum is achieved achieved later later than than the the as 10because stroke stroke current current peak peak registered registered at at the the earth's earth's surface. surface. This This is is clear clear because


Total Total lightning flash duration duration and processes in the intercomponent pauses

129 129

the the magnetostatic magnetostatic component component is is proportional proportional not not only only to to the the current current but but to to the conductor length. the conductor length. The The length length increases increases as as aa neutralization neutralization wave wave travels travels from the earth up same reason, for the from the earth up to to the the cloud. cloud. For For the the same reason, the the times times for the first first and subsequent subsequent components components do and do not not differ differ much. much. The The duration duration of of pulse pulse B(t) in in the the near near region region is is comparable comparable with with that that of of current current inducing inducing aa B(t) magnetic field. field. magnetic

3.8

Total lightning lightning flash flash duration duration and and processes processes in in the Total intercomponent pauses pauses intercomponent

A descending descending negative negative lightning lightning flash flash has has on on average average two two or or three three compocompoA nents, each terminated by by aa more more or less powerful powerful current the nents, each terminated or less current impulse impulse of of the return stroke. stroke. The number of of components components in in an an ascending lightning return The average average number ascending lightning is four. The The maximum lightning flash flash may may be be as as is four. maximum number number of of components components in in aa lightning large large as as 30. 30. The The pauses pauses between between the the components components ~tcom At,,, vary vary from from several several milliseconds milliseconds to to hundreds hundreds of of milliseconds. milliseconds. With With aa 50% 50% probability, probability, their their duration duration exceeds exceeds 33 33 ms; ms; the the integral integral distribution distribution curve curve is is described described by by the the lognormal law law with with the the parameters parameters (lg (lg ~tcom)av At,,,)av == 1.52 1.52 and and (jIg olg== 0.4, 0.4, at at lognormal ~tcom [ms]. total flash with the the number At,,, [ms]. The The total flash duration duration varies varies with number of of components. components. Negative one-component one-component flashes the shortest ones, since their current current Negative flashes are are the shortest ones, since their often ceases right after the return return stroke, less than than aa millisecond. millisecond. An An often ceases right after the stroke, for for less ascending one-component one-component positive positive flash pass current ascending flash can can pass current for for aa longer longer time, time, 0.5 s, in spite of the the absence absence of of aa return return stroke. course, this this is low current, current, 0.5 s, in spite of stroke. Of Of course, is aa low than 1kA. The average average flash duration is is close close to to 0.1-0.2s 0.1-0.2 s and and the the less than 1 kA. The flash duration less maximum s. These maximum is is 1.5 1.5s. These large large times times are are discernible discernible by by the the naked naked eye, eye, so so lightning lightning flickering flickering is is not not aa physiological physiological by-product by-product of of vision vision but but aa physical physical reality. reality. Intercomponent pauses pauses take take most most of of the the flash flash time. time. They They cannot cannot be be said said Intercomponent to be A lightning by current nearly all the to be current-free. current-free. A lightning leader leader is is supplied supplied by current nearly all the time, and and this this current current is is high high enough enough to to support plasma in in aa state close to to time, support plasma state close that of steady-state arc. arc. Current of an an intercomponent intercomponent pause pause is is referred referred that of aa steady-state Current of to as continuous to as continuous current, current, which which is is aa fairly fairly ambiguous ambiguous term. term. Average Average continuous current varies varies between between 100 and 200A. 200 A. Nearly Nearly as high current current continuous current 100 and as high supplies an an arc arc in in aa conventional welding set set used used for for cutting cutting metal metal sheets supplies conventional welding sheets or or for for welding welding thick thick pipes. pipes. Most Most thermal thermal effects effects of of lightning lightning are are associated associated with with its its continuous continuous current, current, rather rather than than with with return return stroke stroke impulses impulses which which are are more more powerful powerful but but shorter. shorter. The The highest hghest continuous continuous current current measured measured [55] A. Continuous [55] was was 580 580A. Continuous current current usually usually slowly slowly decreases decreases with with time. time. In In aa one-component having no return stroke, one-component ascending ascending lightning lightning having no return stroke, the the contact contact of of the leader leader with with the the cloud cloud is is terminated terminated by by charge overflow from the cloud to the charge overflow from the cloud to the same value. the earth as aa decreasing continuous current the earth as decreasing continuous current of of about about the same value. Cloud discharging by continuous current can can be be easily registered by by an elecCloud discharging by continuous current easily registered an electric field Field varies varies monotonically, monotonically, as as long flows through through tric field detector. detector. Field long as as current current flows


130 130

A vailable lightning data Available data

the current of of 100 A extracts, extracts, the channel. channel. These These are are appreciable appreciable changes, changes, since since current lOOA from the time time 0.1 0.1 s. s. The The field field on on the the earth earth AQ ~ x lOC 1OC over over the from aa cloud, cloud, charge charge DoQ DoE = AQ/(27qH2) DoQ/(27rc oH 2) M~ 200V/cm 200V/cm if if right right under under aa cloud cloud changes changes by by the the value value A E= the km; at at distance distance rr = = 10 10 km km from from the height height of of the the charged charged cell cell centre centre is is H H= = 3 km; the r2)3/2] zz ~ 5V/cm. 5V/cm. Similar Similar values values the lightning lightning axis, axis, DoE AE = = DoQH/[27rco(H2 A Q H / [ 2 q , ( H 2+ Y~)~'~] were were registered registered during during observations. observations. by slowly slowly rising rising and and as as slowly slowly Continuous Continuous current current flow flow is is accompanied accompanied by decreasing up to to 11 kA. kA. These These are are MMdecreasing current current impulses impulses with with an an amplitude amplitude up components typical M-component M-component is is about about components of of lightning. lightning. The The risetime risetime of of aa typical 0.5ms, the level level 0.5) 0.5) is is twice twice as as much, much, an an 0.5ms, an an average average impulse impulse duration duration (on (on the average M-components with with current current up up A, although although M-components average amplitude amplitude is is 100-200 100-200 A, to Pulsed current current rise rise is is always always to 750 750 A A have have also also been been registered registered [56,57]. [56,57]. Pulsed accompanied intensity of of the the whole whole channel, channel, accompanied by by an an increase increase in in light light emission emission intensity from photographs (even (even taken taken slowly) slowly) do do from the the cloud cloud down down to to the the earth. earth. Streak Streak photographs not wave front front similar, similar, say, say, not show show the the propagation propagation of of aa well well defined defined emission emission wave most of of the the channel channel flares flares up up to to the the tip tip of of aa dart dart leader. leader. It It seems seems as as if if most simultaneously, propagates down down from from aa simultaneously, although although excitation, excitation, no no doubt, doubt, propagates cloud measurements of of [58]). [58]). cloud with with aa high high velocity, velocity, (2.7-4)x (2.7-4)x 10 lo77 mls mjs (from (from measurements Two as ascending ascending ones. ones. In In later later Two M-components M-components were were identified identified in in [58] [58] as measurements, the the existence existence of of ascending ascending processes processes were were questioned, questioned, because because measurements, there the appearance of an an inducing inducing there were were no no clear clear physical physical reasons reasons for for the appearance of perturbation perturbation at at the the earth's earth's surface. surface. Variations in in current current and and electric electric field field ofM-components of M-components were Variations were registered registered in triggered triggered lightning lightning flashes flashes at at aa short short distance distance from in from the the channel channel (r = = 30 30 m) m) [57].The The field field variation variation of of aa vertical vertical component component at [57]. at the the earth earth is is shown shown in in figure figure 3.24. The The pulse pulse DoE A E rises rises to to its its maximum maximum 70/is 70ps earlier 3.24. earlier than than the the current current impulse. The The field field rises rises and and decreases decreases at at nearly impulse. nearly the the same same rate. rate. The The pulse pulse component of of field field perturbation perturbation is is nearly nearly completely completely damped component damped while while the the current still still has has aa high high amplitude. amplitude. current

+

I- --

o 0.5

2?0 200

I

400

600 m

b

f.ls t, PJ-

,e-

1.0

loO1 1.5

E, kV/m E,kV/m Figure 3.24. 3.24. Superimposed Superimposed schematic schematic oscillograms oscillograms of M-component electric Figure ofM-component electric field field and and [57]. current at at the the earth earth [57]. current


Flash charge and normalized normalized energy

131 131

The number number ofM-components of M-components in in aa flash flash may may even even be be larger larger than than that that of of The subsequent components, components, but they are are of of little little interest interest to to lightning lightning protection protection subsequent but they practice their charge charge and and current current are are too too low. low. Theoretically, Theoretically, however, however, these these practice -- their components are are of of great great interest, interest, because because they they seem seem to to contain contain information information components on unobservable occurring in in storm storm clouds. clouds. It It is is quite quite likely likely that that on unobservable processes processes occurring these give rise to aa dart dart leader leader with with aa return stroke or or to to aa these processes processes give rise to return stroke stroke-free M-component. M-component. Some Some authors authors [27] [27] believe that an an M-component M-component stroke-free believe that is always always formed formed against against the the background background of of continuous continuous current, current, whereas is whereas aa necessary prerequisite for aa dart dart leader leader is is aa current-free current-free pause, pause, during during necessary prerequisite for which the grounded lightning lightning channel channel partly loses its its conductivity. conductivity. This This is is which the grounded partly loses very important important detail detail shedding shedding light light on on processes processes occurring occurring in in aa storm storm aa very cloud after after aa grounded grounded plasma plasma channel channel of of the the first first lightning lightning component component cloud has penetrated it. it. The The transport transport of of the earth’s zero zero potential potential to cloud by by has penetrated the earth's to aa cloud conducting channel, channel, resulting in aa rapid rapid increase increase in in the the cloud cloud electric electric aa conducting resulting in field in in the the vicinity vicinity of of the the channel channel top, top, is is aa powerful powerful stimulus stimulus for for gas gas discharge discharge field processes there there (for (for details, details, see see sections sections 4.7 4.7 and and 4.8). 4.8). processes

3.9

Flash Flash charge charge and and normalized normalized energy

During During intercomponent intercomponent pauses, pauses, charge charge is is transported transported from from aa cloud cloud to to the the earth earth by by both both powerful powerful return return stroke stroke impulses impulses and and continuous continuous current, current, the the latter being being much much lower lower but but longer-living. longer-living. The The contributions contributions of of these these currents currents latter to to the the total total charge charge effect effect are are comparable. comparable. With With aa 50% 50% probability, probability, the the stroke stroke charge transported transported by by the the first first component component of of aa negative negative flash flash is is over over 4.5C, 4.5C, charge while while 5% 5% of of flashes flashes transport transport over over 20C 20C and and another another 5% 5% less less than than l.1C 1.1C [42]. [42]. The The lognormal lognormal law law described described above above is is suitable suitable for for an an approximate approximate reprerepre0.653 sentation of of the the integral integral distribution distribution curve curve with with the the values values (lg (lg Q)av == 0.653 sentation and (Jlg olg== 0.4. 0.4. The The return return strokes strokes of of subsequent subsequent components components have, have, for for the the and same probabilities, probabilities, five five times times smaller smaller charges charges due due to to their their shorter shorter duration duration same and lower lower currents. currents. The The largest largest spread spread of of charge charge measurements measurements is is charactercharacterand istic of positive lightning, in agreement with the diversity of their shape and and istic of positive lightning, in agreement with the diversity of their shape duration. Positive pulse charges exceed 16C with a 50% probability, 150C duration. Positive pulse charges exceed l6C with a 50% probability, l50C with aa 5% 5% probability, probability, and and are are less less than than 2C with with aa 5% 5% probability. probability. These These with seem to be positive lightning with no return stroke. For the description of seem to be positive lightning with no return stroke. For the description of integral integral charge charge distribution distribution for for positive positive pulses, pulses, the the lognormal lognormal parameters parameters may be be taken taken to to be be (lg (lg Q)av Q)av== l.2 1.2 and and (JIg rlg== 0.6. 0.6. may We have have already already mentioned mentioned that that the the charge charge of of aa lightning lightning flash flash is is always always We larger than than the the sum sum of of charges charges transported transported by by the the return return strokes strokes of of the the first first larger and and subsequent subsequent components, components, since since aa substantial substantial contribution contribution to to the the total total charge is is made made by by continuous continuous current. current. The The total total negative negative flash flash charge charge exceeds exceeds charge 7.5C with with aa 50% 50% probability, probability, 40C 40C with with aa 5% 5% probability, probability, and and is is nearly nearly the the 7.5C same same as as the the first first negative negative pulse pulse charge charge in in the the least least powerful powerful flashes. flashes. The The total with 95%, 50% 50% and and 5% 5% total positive positive charge charge is is appreciably appreciably larger larger -- with


132 132

Available lightning data data

probabilities, probabilities, it it exceeds, exceeds, respectively, respectively, 20, 20, 80 80 and and 350C. 350C. One One cannot cannot say say that that the the charge charge transported transported by by aa flash flash is is very very large. large. For For comparison, comparison, even even aa very very large large lightning lightning charge charge of of 350C 350C flows flows through through the the arc arc of of aa conventional conventional welding welding unit unit for for 3-5 3-5 s.s. Charge Charge transport transport is is accompanied accompanied by by energy energy release. release. An An average average negative negative flash flash with with aa charge charge Q Q == 10C 1OC and and gap gap voltage voltage 50 50 MV MV dissipates dissipates about about QU == 8 55 Xx 10 J, which 10sJ, which is is equal equal to to the the energy energy released released by by aa 100 100 kg kg trinitrotoluene trinitrotoluene explosion. explosion. While While most most energy energy is is released released within within the the lightning lightning trace, trace, the the problem problem of of energy energy release release and and heating heating of of metal metal constructions constructions is is of of much much interinterest. est. Normally, Normally, the the resistance resistance of of metallic metallic conductors conductors and and that that of of aa grounding grounding electrode much less electrode are are much less than than the the equivalent equivalent resistance resistance of of aa lightning lightning channel channel R[ RI ~ = U / 1IMM (1M (IM is is the the impulse impulse amplitude amplitude of of aa return return stroke); stroke); R[ RI ~ = I1 kn kR if if U ~ = 50 kA. Therefore, lightning can be regarded as a current x 50 50 MV MV and and 1ZM = 50 kA. Therefore, lightning can be regarded as a current M the object's source, is independent independent of of the object’s resistance. resistance. Any Any source, assuming assuming that that current current 1IM M is conductor with lightning conductor with lightning current current flow flow releases releases the the energy energy

1;

2 dt ( K / R ) R . K= = R J~ ii2 dt == (K/R)R,

KIR =

1;

i2 dt

proportional practical calculations, proportional to to the the conductor conductor resistance resistance R. For For practical calculations, data data on ‘normalized’ energies K / R characterizing lightning only are published. on 'normalized' energies / R characterizing lightning only are published. According the According to to [42], [42], 95%, 95%, 50% 50% and and 5% 5% probabilities probabilities correspond correspond to to the 4 5 7 2 measured values exceeding 2.5 x lo4, 6.5 x IO5 and 1.5 x 107A2s for measured values exceeding 2.5 10 , 6.5 X 10 and 1.5 10 A s for 3 5 2 positive flashes , 5.5 A s for positive flashes and and 6.0 6.0 x 10 lo3, 5.5 Xx 10 IO44 and and 5.5 5.5 x 10 105A2s for negative negative flashes, flashes, respectively. respectively. For For subsequent subsequent components components of of negative negative flashes, flashes, the the respective values magnitude smaller respective values are are an an order order of of magnitude smaller and and do do not not contribute contribute much to to the total energy potency of much the total energy release. release. To To get get an an idea idea about about thermal thermal potency of lightning, evaluate the heat of a steel conductor with a cross section of lightning, evaluate the heat of a steel conductor with a cross section of 5 S == 1cm nR em, 1 cm’.2 . With With resistivity resistivity p == 10lop5 cm, the the energy energy density density released released by by aa 7 2 3 powerful positive positive flash flash (K/R ( K / R == 1.5 1.5 x 10 lo7 A’s)s) is is (K/R)(p/S2) ( K / R ) ( p / S 2== ) 150J/cm 150 J/cm3, A powerful , with with the the conductor conductor temperature temperature increasing increasing by by 40°C. 40°C. Owing Owing to to Joule Joule heat, heat, aa lightning lightning flash flash is is capable capable of of burning burning down down only only aa very very thin thin conductor conductor with with aa cross cross section section less less than than 0.1 0.1 cm’. cm2 . In In many many cases, cases, however, however, heating heating just just by by several several hundred degrees degrees may may become become hazardous. hazardous. hundred

3.10

Lightning temperature and radius radius Lightning

Plasma temperature temperature is is usually usually measured measured by by spectroscopic spectroscopic methods. methods. LightLightPlasma ning spectroscopy spectroscopy is is aa hundred hundred years years old, old, and and it it was was used used even even before before ning photography photography and and field-current field-current measurements. measurements. Reviews Reviews of of spectroscopic spectroscopic 591 together results results can can be be found found in in Uman’s Uman's books books [l, [1,59] together with with extensive extensive data on on lightning lightning plasma plasma are are still still very very scarce. scarce. references. However, However, direct direct data references. Lightning Lightning spectra, spectra, naturally, naturally, contain contain lines lines of of molecular molecular and and atomic atomic oxygen oxygen and and nitrogen, nitrogen, as as well well as as singly singly charged charged ions ions N2, N 2 , argon, argon, cyane cyane and and some some


Lightning temperature and radius

133 133

other impurities. impurities. No doubly charged charged ions ions have have been detected, indicating indicating that that other No doubly been detected, the does not not exceed exceed 30000 30 000 K. K. Measurements Measurements of of time time resolved resolved the temperature temperature does NI1 (N') line line intensities intensities show show that that the the return stroke temperature temperature reaches reaches N II (N+) return stroke 30 000 K for for the the first first lOllS 10 ps [59,62] [59,62] and and drops drops to to 20 20 000 000 K K in in 20IlS. 20 ps. Average 30000K Average temperatures are are estimated estimated to be about about 25000 25 000 K. K. These These results results are are obtained obtained temperatures to be assuming that that aa plasma channel is is optically optically transparent transparent and and that that the the assuming plasma channel excitation of of atoms atoms in in the the plasma plasma is is equilibrium equilibrium (of (of the the Boltzmann Boltzmann type). type). excitation The estimations estimations justify this assumption. assumption. The justify this Electron densities densities found found from from the the Stark Stark broadening of the lines Electron broadening of the Ha He> lines are 10 1 018' cm* ~ m3 -for for ~ the the first first 51ls 5ps of of the the stroke stroke life. life. Under Under thermodynamic thermodynamic are equilibrium conditions conditions at at T == 30000 30 000 K, K, this this value value of of nnee corresponds corresponds to equilibrium to the pressure pressure of atm [63]. About 10 IlS later, lops later, nnee decreases decreases to to 10 l O17' 'cm~ m -3 ,~ , the of 88atm [63]. About corresponding to to the drop down down to to the atmospheric pressure. pressure. corresponding the pressure pressure drop the atmospheric Then value of the N N IIII line Then the the value of n nee remains remains unchanged unchanged over over the the time time of of the line registration. This This does does not not seem seem strange. strange. Equilibrium Equilibrium electron electron density density in in registration. air wide temperature air at at p == const const == 1atm 1 atm changes changes only only slightly slightly in in aa wide temperature 17 range 15 cmrange 15000-30 000-30 000 000 K, K, remaining remaining about about 10 1017 ~ m3-. As As ~ . the the channel channel cools cools down, the ionization x = when the down, the ionization degree degree x = nee/ /NN certainly certainly decreases, decreases, but but when the pressure reaches reaches the the gas pressure the atmospheric atmospheric value, value, the gas density density N rises rises simul· simulN does does not not change change much. much. High High intensity intensity taneously. For this this reason, reason, nnee == xxN taneously. For radiation is IlS (from to 1000 Ils). The is observed observed for for about about 100 loops (from 40 40 to lOOOps). The first first radiation peak is by another hundreds of microseconds peak is often often followed followed by another one one several several hundreds of microseconds later. later. Spectroscopic measurements were mostly mostly made during aa return return stroke, stroke, Spectroscopic measurements were made during but some register the but some authors authors (64] [64] managed managed to to register the spectrum spectrum of of aa 2-m 2-m portion portion of of aa stepwise N II stepwise leader. leader. The The leader leader tip tip temperature temperature calculated calculated from from the the N 11 lines lines lies lies within 20000-35000 the radiation within 20 000-35 000 K. K. The The diameter diameter of of the radiation region region is is less less than than 35 unavailable. It unlikely that 35 cm. cm. More More accurate accurate evaluations evaluations are are unavailable. It seems seems unlikely that this temperature temperature is is characteristic characteristic of of the whole leader leader channel. channel. Rather, Rather, the the this the whole experiment temperature rise rise during experiment registered registered aa short short temperature during aa powerful powerful step step which was akin akin to to aa miniature stroke (section (section 2.7). 2.7). The The step-induced step-induced which was miniature return return stroke perturbation involving involving part of the the channel channel region region is is most most likely likely to to be be perturbation part of damped damped rapidly rapidly along along the the leader leader length. length. It is is not only the the plasma dynamics but the channel channel radius, radius, too, too, which which It not only plasma dynamics but the still the radius, still remains remains enigmatic. enigmatic. In In making making evaluations evaluations of of the radius, one one usually usually relies photographs. But But in this case, to agree the relies on on photographs. in this case, it it is is very very important important to agree on on the kind be the kind of of radius radius being being evaluated. evaluated. This This may may be the radius radius of of the the channel, channel, through which current flows flows during during the the leader leader and and stroke stroke stages. stages. Clearly, Clearly, through which current such aa radius radius will will include include the the best best conducting conducting and, and, hence, the hottest hottest core core such hence, the of the the plasma channel. Or, Or, one one can can follow follow another another approach. approach. When When solving solving of plasma channel. the problem problem of of electric electric field field variation variation during during the the lightning lightning development, development, one one the of the the space space charge charge has to to deal deal with with the the radius radius of of the the leader leader cover cover where where most most of has is concentrated. concentrated. This This is is the the charge charge radius radius of of lightning. lightning. Therefore, Therefore, each each time time is we radius, we we speak speak of of radius, we must must define define exactly exactly what what we we mean. mean.


134 134

Available A vailable lightning data

Here, we we shall shall use use the the concept concept of of channel channel radius radius as applied to to the the region region Here, as applied where the the lightning lightning current current is is accumulated accumulated and and the the concept concept of where of cover cover radius radius to to the region region where where most most of of the the space space charge charge is is concentrated. concentrated. The The former former can.be can.be the determined, to to some some extent, extent, by by using using optical optical methods, methods, although although this this is is aa comcomdetermined, plicated plicated task. task. With With reference reference to to the the optical optical measurements measurements [65], [65], one one usually usually is accessible to deals with radii radii of of several centimetres. This resolution is deals with several centimetres. This resolution accessible to modern cameras cameras at distance of kilometre, but but the the cameras cameras must must modern at aa distance of about about aa kilometre, have the highest highest resolution resolution possible. possible. Anyway, Anyway, we have the we have heard about about have never never heard the application application of of such such perfect perfect optical optical equipment equipment in in lightning lightning research. research. the In addition addition to to using using special-purpose special-purpose optics, optics, the the experimentalist experimentalist must must In match perfectly perfectly the the sensitivity sensitivity of of photographic photographic materials materials and and exposures. exposures. match A longer longer exposure exposure produces produces aa halo, halo, increasing increasing the the actual actual radius. radius. Unless Unless A special special measures measures are are taken, taken, the the error error may may be be very very large, large, especially especially for for flashes with with aa high high light light intensity. intensity. For For some some reasons, reasons, the the optical optical radius radius of of flashes lightning channel channel may may exceed exceed manifold manifold the the thermal thermal radius. radius. Such Such an an aa lightning effect effect was was observed observed in in studies studies of of spark spark leaders leaders in in laboratory laboratory conditions conditions [66]. [66]. Registration Registration of of the the thermal thermal radius radius appears appears problematic problematic even even for for trigtriggered lightning, lightning, with with aa fixed fixed point point of of contact contact with with the the earth. earth. For For natural natural gered is much much more more complicated. complicated. As As for for the the cover cover radius, radius, lightning, this this task task is lightning, there is is no no reliable reliable technique technique for for its its registration registration at at all. all. So So lightning lightning radius radius there measurements cannot cannot provide provide unquestionable unquestionable data, data, and and the the researcher researcher is is measurements to rely rely on on theoretical theoretical evaluations evaluations only. only. to

3.1 1 What can can one one gain gain from from lightning lightning measurements? measurements? 3.11 It was was not not our our task task to to review review all all experimental experimental studies studies on on lightning: lightning: this this has has It [l, 591. We We believe believe that that the the latest latest experimental experimental data data will will been well well done done in in [1,59]. been

be presented presented in in aa new new Uman Uman book book now now in in preparation. preparation. But But the the basic basic facts facts be have been been discussed discussed here, here, and and we we can can now now ask ask ourselves ourselves whether whether the the have available data data are are sufficient sufficient to to build build lightning lightning theory theory and and to to check check itit by by available experiment. experiment. The situation situation with with lightning lightning is is somewhat somewhat similar similar to to that that for for aa long long The laboratory spark, spark, i.e., i.e., experiments experiments give give mainly mainly external external parameters parameters of of aa disdislaboratory charge. In In the the laboratory, laboratory, these these are are velocities velocities of of the the major major structural structural elements elements charge. (streamers and and leaders), leaders), their their initiating initiating voltages, voltages, currents, currents, transported transported (streamers charges, and, and, possibly, possibly, some some other other characteristics characteristics Sometimes, Sometimes, we we have have charges, some information information on on channel channel radii, radii, or or on on the the time time variation variation of of radii, radii, or or some scarce data data on on plasma plasma parameters. parameters. But But that that is is all. all. scarce The arsenal arsenal oflightning of lightning researchers researchers is is much much smaller. smaller. First, First, they they have have no no The information about about the the voltage voltage in in the the cloud-earth cloud-earth gap gap at at the the lightning lightning start, start, information and there there are are no no data data on on the the initial initial distribution distribution of of electric electric field. field. Both Both literally literally and and figuratively, figuratively, the the bulk bulk of of aa storm storm cloud, cloud, where where aa descending descending leader leader and originates, isis obscure. obscure. Measurements Measurements made made at at the the earth's earth’s surface surface cannot cannot originates,


What can one gain from lightning measurements?

135 135

help much, much, because because the the number of registration registration points points is is too too small, small, so so it it is is help number of impossible to to reconstruct the initial initial field field distribution distribution along along the the whole whole impossible reconstruct the lightning path. path. lightning The fine structure of of aa lightning lightning flash flash is is not clear either. either. Observations Observations The fine structure not clear give no information about the size of the streamer zone in lightning give no information about the size of the streamer zone in aa lightning leader, and even the existence of such a zone is largely speculative. do leader, and even the existence of such a zone is largely speculative. Nor Nor do we know the origin and structure of volume leaders, which are responsible we know the origin and structure of volume leaders, which are responsible for the stepwise pattern of aa negative leader, at at least, least, observable observable in in laboratory laboratory for the stepwise pattern of negative leader, conditions. There is no information the track track of conditions. There is no information on on the the gas gas state state in in the of aa preceding component, preceding component, when when aa dart dart leader leader travels travels along along it. it. The The only only dart dart leader parameter is its its velocity. leader parameter that that has has been been measured measured is velocity. What What has has just just been been that space listed refers primarily primarily to to the the return return stroke. listed refers stroke. It It appears appears that space charge charge neutralization the basic basic process occurring in in it it -- is is related to the the fast fast neutralization -- the process occurring related to radial propagation propagation of of streamers streamers away away from from the the channel. channel. This This is is the the way way radial the there are the cover cover charge charge is is supposed supposed to to change. change. But But there are no no experimental experimental data data on this process, nor nor can can we we hope obtain any any in in the near future. future. on this process, hope to to obtain the near Most Most available available findings findings concern concern lightning lightning currents currents and and transported transported As in usually registered charges. charges. As in aa laboratory laboratory spark, spark, lightning lightning currents currents are are usually registered at the the earth's earth’s surface, surface, so so we we have have data data on on leader leader currents currents for for ascending ascending disdisat charges only. only. There There are are no no direct direct measurements measurements of of currents currents for for descending descending or or charges dart dart leaders, leaders, the the latter latter fact fact being being especially especially disappointing. disappointing. There There are are more more or or less return strokes, but the less detailed detailed descriptions descriptions of of currents currents for for return strokes, but the measurements measurements made with the restrict the the possibilities possibilities of made at at one one point point (that (that of of contact contact with the earth) earth) restrict of both theoretical physicist physicist and practical engineer. the current both aa theoretical and aa practical engineer. Data Data on on the current wave the leader because then wave damping damping along along the leader are are important important for for the the former former because then he may try try to reconstruct the the plasma plasma conductivity he may to reconstruct conductivity variation. variation. The The latter latter needs needs them them to to be be able able to to calculate calculate the the lightning lightning electric electric field field at at the the earth earth and to both both ground and in in the the troposphere, troposphere, because because it it is is hazardous hazardous to ground objects objects and and aircraft. aircraft. Lightning Normally, they Lightning current current statistics statistics deserves deserves special special attention. attention. Normally, they the occurrence probability of are used in are used in calculations calculations of of the occurrence probability of lightning lightning with with the impulse hazardous parameters, e.g., hazardous parameters, e.g., aa critically critically fast fast rise rise of of the impulse front front and/or and/or amplitude. requirements on amplitude. The The practical practical requirements on the the calculation calculation reliability reliability are are provide the extremely extremely high. high. Indeed, Indeed, it it is is impossible impossible to to provide the necessary necessary accuracy, accuracy, using lognormal parameter distributions. Any approximation using lognormal parameter distributions. Any approximation of of an an actual actual distribution would be be approximate, the range distribution lognormally lognormally would approximate, especially especially in in the range of of large protection. The may be be as high large values values important important for for lightning lightning protection. The error error may as high as mind when as 100%. 100%. One One should should keep keep this this in in mind when comparing comparing calculations calculations of of hazardous lightning the available protection. hazardous lightning effects effects and and the available experience experience in in object object protection. by aa theorist This This is is the the reality reality not not to to be be ignored ignored either either by theorist attempting attempting to to protection. create create aa lightning lightning model model or or by by an an engineer engineer working working on on lightning lightning protection. No matter theorist may be, he No matter how how ingenious ingenious aa theorist may be, he will will not not be be able able to to check check his his model, filling by general model, filling the the gaps gaps by by laboratory laboratory spark spark data data or or by general physical physical considerations. practical lightning usually gained one usually gained considerations. As As for for practical lightning protection, protection, one


136 136

Available lightning data Available

the unfortunate unfortunate experience experience from from analyses of emergencies emergencies that that resulted resulted from the analyses of from the lack lack of of knowledge knowledge of of atmospheric atmospheric electricity. electricity. the

References References [1] Uman M M 1987 (New York: York: Academic Academic Press) Press) p p 377 377 [l] Uman 1987 The Lightning Discharge (New (London, New York: Academic 1, 2 [2] Golde Golde R RH (ed) 1977 1977 Lightning (London, [2] H (ed) New York: Academic Press) Press) vols vols 1,2 [3] Clouds and Precipitation (Lenin(Lenin[3] Imyanitov Imyanitov II M M 1970 1970 Aircraft Electrization in Clouds grad: Gigrometeoizdat) Gigrometeoizdat) pp 210 210 grad: [4] [4] Gunn Gunn R R 1948 1948 J. J . Appl. Phys. 19481 19 481 [5] Gunn Gunn R R 1965 1965 J. J . Atmos. Sci 22 498 498 [5] [6] Evans Evans W WH H 1969 1969 J. J . Geophys. Geophys. Res. Res. 74 939 939 [6] J . Geophys. Geophys. Res. Res. 79 1761 1761 [7] Winn Winn W W P, P, Schwede Schwede G G Wand W and Moore Moore C CB B 1974 1974 J. [7] B and and Holmes Holmes C CR R 1981 1981 J. J. Geophys. Geophys. Res. 861187 86 1187 [8] Winn Winn W W P, P, Moore Moore C CB [8] [9] Kazemir Kazemir H H Wand W and Perkins Perkins F F 1978 1978 Final Report, Kennedy Space Center Contract Contract [9] CC 69694A [lo] Newman Newman M M M, M, Stahmann Stahmann JJ R, R, Robb Robb JJ D D et al1967 all967 J. J . Geophys. Geophys. Res. Res. 72 4761 4761 [10] [ l l ] Kito Kito Y, Y, Horii Horii K, K, Higashiyama Higashiyama Y Y and and Nakamura Nakamura K K 1985 1985 J. J . Geophys. Geophys. Res. 90 90 [11] 6147 6147 [12] Hubert Hubert P P and and Mouget Mouget G G 1981 1981 J. J . Geophys. Geophys. Res. Res. 865253 86 5253 [12] [13] Hubert Hubert P, P, Laroche Laroche P, P, Eybert-Berard Eybert-Berard A A and and Barret Barret L L 1984 1984 J. J . Geophys. Geophys. Res. Res. 89 [13] 251 1 2511 [14] Idone Idone V V PP and and Orville Orville R RE E 1984 1984 J. J . Geophys. Geophys. Res. Res. 89 7311 7311 [14] a1 1993 J. J . Geophys. Geophys. Res. Res. 9822887 98 22887 [15] Fisher Fisher R R G, G, Schnetzer Schnetzer G G H, H, Thottappillil Thottappillil Ret R et al1993 [15] [16] Wang Wang D, [16] D, Rakov Rakov V V A, A, Uman Uman M MA A et al1999 a1 1999 J. J . Geophys. Geophys. Res. Res. 1044213 104 4213 [17] Malan Malan D D JJ and and Schonland Schonland F FG G 1951 1951 Proc. R. R. Soc. Soc. London Ser. Ser. A 209158 209 158 [17] [18] Malan Malan D D JJ 1963 1963 Physics of of Lightning (London: (London: English English Univ. Univ. Press) Press) pp 176 176 [19] Malan Malan D D JJ 1963 1963 J. J . Franklin Inst. Inst. 283 526 526 [19] (Boca Raton: Raton: CRC CRC Press) Press) [20] Bazelyan Bazelyan E EM M and and Raizer Raizer Yu Yu PP 1997 1997 Spark Discharge (Boca [20] 294 pp 294 (2nd edn) edn) (Oxford: (Oxford: Pergamon) Pergamon) pp 418 418 [21] Chalmers Chalmers JJ A A 1967 1967 Atmospheric Electricity (2nd [21] [22] Berger Berger K [22] K and and Fogelsanger Fogelsanger E E 1966 1966 Bull. Bull. SEV S E V 5713 57 13 11 [23] Schonland Schonland B B 1956 1956 The The Lightning Discharge. Handbuch der Physik 22 (Berlin: (Berlin: [23] Springer) Springer) pp 576 576 [24] B, Malan Malan D D and and Collens Collens H H 1938 1938 Proc. Proc. Roy. Roy. Soc. Soc. London Ser. Ser. A 168 168 [24] Schonland Schonland B, 455 455 D and and Collens Collens H H 1935 1935 Proc. Proc. Roy. Roy. Soc. Soc. London Ser. Ser. A 152 152 [25] Schonland Schonland B, B, Malan Malan D [25] 595 595 D M, M, Rakov Rakov V V A, A, Beasley Beasley W W Hand H and Uman Uman M MA A 1997 1997 J. J . Geophys. Geophys. Res. Res. [26] Jordan Jordan D [26] 102 22.025 10222.025 G, Schnetzer Schnetzer G G H, H, Thottappilli1 Thottappillil Ret R et al1992 a1 1992 Proc. Proc. 9th Intern. Intern. Con! Conf. on on [27] Fisher Fisher R R G, [27] Atmosph. Electricity 3 (St (St Peterburg: Peterburg: A A II Voeikov Voeikov Main Main Geophys. Geophys. Observ.) Observ.) pp 873 873 [28] McCann McCann G G 1944 1944 Trans. Trans. AlEE A I E E 63 63 1157 1157 [28] S E V 56 56 No No 11 22 [29] Berger Berger K K and and Vogrlsanger Vogrlsanger E E 1965 1965 Bull SEV [29] [30] Baze1yan Bazelyan E E M, M, Gorin Gorin B B Nand N and Levitov Levitov V V 11978 I 1978 Physical and Engineering Funda[30] of Lightning Protection Protection (Leningrad: (Leningrad: Gidrometeoizdat) Gidrometeoizdat) pp 223 223 (in (in Russian) Russian) mentals ofLightning


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[31] Extrait de de la 1a Revue Revue Generale Genera1e de de [31] Saint Privat dAllier d’Allier Research Group 1982 1982 Extrait

l'E1ectricite, I’Electricite, Paris, Paris, September September Gorin B Nand Elektrichestvo 2 29 29 Shkilev A AV V 1974 1974 Elektrichestvo Gorin B N and Shkilev Idone 6159 Idone V V P, P, Orville Orville R RE E 1985 1985 J. J . Geophys. Geophys. Res. 90 6159 Rubinstein P 1992 Froc. 9th Intern. Intern. Conf. Con! on Rubinstein M, M, Uman Uman M M A A and and Thomson Thomson P 1992 Proc. Atmosph. Voeikov Main Main Geophys. Geophys. Observ.) Observ.) pp 276 276 Atmosph. Electricity 1 (St (St Peterburg: Peterburg: A A II Voeikov [35] J. Geophys. Res. 100 8863 8863 M, Rachidi Rachidi F, F, Uman Uman M M A et al1995 a1 1995 J. [35] Rubinstein Rubinstein M, [36] [36] Thomson Thomson E EM M 1985 1985 J. J . Geophys. Geophys. Res. 90 8125 8125 [37J Equipment [37] Kolechizky Kolechizky E E C C 1983 1983 Electric Field Calculation for High-Voltage Equipment (Moscow: Russian) (Moscow: Energoatomizdat) Energoatomizdat) pp 167 167 (in (in Russian) [38] 6555 [38] Jordan Jordan D DM M and and Uman Uman M MA A 1983 1983 J. J . Geophys. Geophys. Res. 88 6555 [39] Froc. Roy. Roy. Soc. Soc. London Ser. A 143 143654 H 1934 1934 Proc. 654 [39] Schonland Schonland Band B and Collens Collens H [40] 879703 [40] Idone Idone V V PP and and Orville Orville R RE E 1982 1982 JJ.. Geophys. Geophys. Res. 87 9703 [41] 1385 R E, E, Hubert Hubert P P et al1984 a1 1984 J. J . Geophys. Res. 89 1385 [41] Idone Idone V.P, V.P, Orville Orville R [42] 4123 [42] Berger Berger K, K, Anderson Anderson R R Band B and Kroninger Kroninger H H 1975 1975 Electra 41 23 [43] Schweiz. Elekrtotech. Elekrtotech. Ver.63 Ver. 63 1403 1403 [43] Berger Berger K K 1972 1972 Bull. Bull. Schweiz. [44J and Lightning Protection [44] Gorin Gorin B B Nand N and Shkilev Shkilev A AV V 1979 1979 in in Lightning Physics andLightning (Moscow: Inst.) pp 99 (Moscow: Krzhizhanovsky Krzhizhanovsky Power Power Engineering Engineering Inst.) [45] 29 [45] Gorin Gorin B B Nand N and Shkilev Shkilev A V V 1974 1974 Elektrichestvo 2 29 [46] 8) 238 238 Trans. South Afr. lEE ZEE 69 (Pt (Pt 8) [46] Eriksson Eriksson A A JJ 1978 1978 Trans. [47] 65 [47] Anderson Anderson R RB B and and Eriksson Eriksson A A JJ 1980 1980 Electra 69 65 [48] in Lightning Lightning Physics and Lightning Lightning a1 1974 1974 in [48] Alizade Alizade A A A, A, Muslimov Muslimov M M M M et al (Moscow: Krzhizhanovsky Krzhizhanovsky Power Power Engineering Engineering Inst.) Inst.) pp 10 10 Protection (Moscow: [49] Darveniza M M 1984 IEEE Trans. Trans. P PAS [49] Master Master M M J, J, Uman Uman M M A, A, Beasley Beasley W W Hand H and Darveniza 1984 ZEEE AS Pas-103 2519 Pas-l032519 [50] 8471 [50] Krider Krider E E PP and and Guo Guo C C 1983 1983 J. J . Geophys. Geophys. Res. 88 8471 [51] [51] Cooray Cooray V V and and Lundquist Lundquist S S 1982 1982 J. J . Geophys. Geophys. Res. 87 11203 11203 V and and Lundquist Lundquist S S 1985 1985 JJ.. Geophys. Geophys. Res. 90 6099 [52] Cooray Cooray V [52] 6099 [53] McDonald McDonald T T B, B, Uman Uman M M A, A, Tiller Tiller JJ A A and and Beasley 1979 JJ.. Geophys. [53] Beasley W WH H 1979 Geophys. Res. 84 1727 841727 [54] Lin Lin Y T, T, Uman Uman M MA A et al1979 a1 1979 JJ.. Geophys. Geophys. Res. 84 6307 [54] 846307 [55] J. Geophys. Res. 84 842432 [55] Krehbiel Krehbiel PP R, R, Brook Brook M M and and McCrogy McCrogy R R A 1979 1979 J. 2432 [56] Thottappilli1 Thottappillil R, R, Goldberg Goldberg JJ D, D, Rakov Rakov V A, Uman 1995 JJ.. Geophys. [56] V A, Uman M MA A et a1 al1995 Geophys. Res. 100 100 25711 25711 Res. [57] Rakov Rakov V V A, A, Thottappillil Thottappillil R, R, Uman Uman M MA and Barker 1995 JJ.. Geophys. Res. [57] A and Barker P PP P 1995 100 25701 10025701 [58] Malan Malan D D JJ and and Collens Collens H H 1937 1937 Proc. 175 [58] Froc. R. R. Soc. Soc. London A 162 175 [59] Uman Uman M M A 1969 1969 Lightning (New (New York: York: McGraw-Hill) [59] McGraw-Hill) [60] Orvill Orvill R RE E 1968 1968 J. J . Atmos. Sci. Sci. 25 827 827 [60] [61] Orvill Orvill R RE E 1968 1968 J. J. Atmos. Sci. Sci. 25 839 839 [61] [62] Orvill Orvill R RE E 1968 1968 J. J . Atmos. Atmos. Sci. Sci. 25 852 852 [62] [63] Kuznetsov Kuznetsov N NM M 1965 1965 Thermodynamic Functions and Shock Adiabata [63] Adiabata for High (Moscow: Mashinostroenie) Mashinostroenie) (in Temperature Air (Moscow: (in Russian) Russian) [64] Orvill Orvill R RE E 1968 1968 J. J. Geophys. Geophys. Res. Res. 73 73 6999 6999 [64] E 1977 1977 in in Lightning, Lightning, vol vol 1, 1, R R Golde Golde (ed) (ed) (New [65] Orvill Orvill R RE [65] (New York: York: Academic Academic Press) Press) 281 pp 281 [66] Positive Discharges in Air Gaps at Les Renardieres 197s 1977 1977 Electra 53 31 [66J - 1975 31 [32] [32] [33] [33] [34] [34]


Chapter 4

Physical Physical processes processes in in a lightning Iight ning discharge discharge Here Here we we shall shall discuss discuss the the basic basic phenomena phenomena occurring occurring in in aa lightning lightning discharge: discharge: descending negative negative leader, leader, an an ascending ascending positive positive leader, leader, the the return return strokes strokes aa descending of the the first first and and subsequent subsequent components, components, aa dart dart leader, leader, and and some some others. others. of Lightning may may travel travel not not only only from from aa cloud cloud towards towards the the earth, earth, or or from from aa Lightning grounded object object towards towards aa cloud, cloud, but but it it may may also also start start from from aa body body isolated isolated grounded plane, aa rocket, rocket, etc. etc. About About 90% 90% of of all all descending descending from the the earth earth -- aa plane, from discharges are are negative negative and and about about as as many many ascending ascending discharges discharges are are positive. positive. discharges For this this reason, reason, an an ascending ascending leader leader is is said said to to be be positive. positive. Available Available experiexperiFor mental data data on on lightning lightning as as such such are are oflittle of little use use in in our our attempts attempts to to explain explain the the mental mechanisms underlying underlying the the above above processes. processes. There There are are very very few few observations observations mechanisms So, one one has has to to resort resort to to specspecthat might might shed shed light light on on their their physical physical nature. nature. So, that ulations, invoking invoking both both theory theory and and experimental experimental data data on on aa long long laboratory laboratory ulations, spark, which which relate relate primarily primarily to to aa positive positive leader. leader. Since Since this this process process is is most most spark, simple (to (to the the extent extent aa lightning lightning process process may may be be considered considered simple), simple), we we simple of an an ascending ascending positive positive leader. leader. shall begin begin with with the the discussion discussion of shall

4.1 4.1

An ascending ascending positive positive leader

4.1.1 The The origin origin 4.1.1

The lightnings lightnings people people observe observe most most frequently frequently are are descending descending discharges, discharges, The which originate originate among among storm storm clouds clouds and and strike strike the the earth earth or or objects objects located located which on its its surface. surface. However, However, constructions constructions over over 200 200m high and and those those built built in in m high on of mountainous regions regions suffer suffer mostly mostly from from ascending ascending lightnings. lightnings. These These are are of mountainous nearly as as much much interest interest to to the the physicist physicist as as the the seemingly seemingly common, common, descending descending nearly discharges. An An ascending ascending leader leader is is initiated initiated by by aa charge charge induced induced by by the the elecelecdischarges. tric field field of of aa storm storm cloud cloud in in aa conducting conducting vertically vertically extending extending grounded grounded tric If aa metal metal conductor conductor of of height height h with with aa characteristic characteristic radius radius of of the the object. If object. r > Eo is created created by by the the induced induced charge charge at at external Eo, aa field E) ~ Eoh/r » Eo is the conductor conductor top top (see (see section section 2.2.7). 2.2.7). This This field field rapidly rapidly decreases decreases in in air air (for (for aa the distance of of several several r values), values), creating creating aa potential potential difference difference between between the the distance conductor end end and and the the adjacent adjacent space, space, 6AU When the the cloud cloud bottom bottom conductor Ux ~ Eoh. Eoh. When is charged charged negatively negatively and and the the vector vector Eo is directed directed from from the the earth earth up to is Eo is up to the cloud, cloud, the the grounded grounded conductor conductor becomes charged, since since the the the becomes positively positively charged, field makes makes some some of of the negative charges charges leave leave the the metal metal to to go go down down to to field the negative the earth. earth. the No stringent conditions conditions are are necessary for the the field field E) E , to to initiate initiate the the air air No stringent necessary for ionization (at (at sea sea level level E) El ~ xE E,i ~ x 30 30 kV/cm) kV/cm) or or for for aa corona corona discharge discharge to to ionization arise at at the the pointed parts of of aa high high structure structure (it (it is is necessary necessary to to have have arise pointed parts El ~ x40-31kV/cm for r = = l-10cm). 1-10cm). The The conditions conditions for for aa leader leader to to be E) 40-31 kV/cm for be initiated in the streamer corona stem are much more rigorous. The energy initiated in the streamer corona stem are much more rigorous. The energy estimations made in section section 2.6 2.6 show show that that there there is is no no chance chance for for aa leader leader to to estimations made in arise if if the the leader leader tip U,, or, more exactly, its excess over the external arise tip potential potential U or, more exactly, its excess over the external , I potential at at the the tip, AU = =U U,I -- U U,, is less less than than !:lUI. AVrm,,~ x 300-400kV. 300-400 kV. This This potential tip, !:lU o, is estimate is is supported supported by experiments with with leaders, leaders, ';hose whose streamer streamer zones zones estimate by experiments have contact with with the the electrode electrode of of opposite opposite sign sign at at the the initial initial moment moment of of have no no contact time. Therefore, for the desired potential difference A Ut,, to be produced, time. Therefore, for the desired potential difference !:l U tmin to be produced, x !:lU AUrm,,/EO x 20-30m 20-30m if if the the average average the structure structure must must have, have, at at least, least, h ~ ) Eo ~ the lmi field of of the the storm storm cloud cloud at at the the site site of of the the grounded grounded object object is is "" -150 V/cm. 150 V/cm. field On the the other other hand, hand, even even if if aa leader leader is is produced produced at at such such aa low low potential, potential, On AVfm,,, it can can hardly hardly travel travel for for aa large large distance. distance. The The leader leader current current will will be be too too !:l Ulmin , it low to to heat the channel channel to to aa sufficiently sufficiently high high temperature. temperature. As result, the the low heat the As aa result, channel resistance resistance will will be be too too high high so so that that aa very very strong strong field field will will be be required required channel to E e is, to support support the the current current in in the the channel. channel. The The channel channel field field E, is, however, however, limited by by the the external external field field Eo. Indeed, aa grounded grounded body of height height h, h, from from Eo. Indeed, body of limited which leader has started, possesses zero potential. potential. Having Having covered covered which aa positive positive leader has started, possesses zero the leader leader tip tip acquires acquires the the potential = -EeL. -E,L. Here, Here, the the the distance distance L, the L, the potential UI, = potential of the the unperturbed external fields fields is is U U.o = = -Eo(L + h), h), and and we we have have potential of unperturbed external

+

+

AUt = =U UtI -- U U0o = = !:lUi AU, + (Eo (Eo -- E,)L. !:lUI Ee)L,

AU, = = Eoh. !:lUi Eoh.

(4.1) (4.1)

For aa leader leader to to develop develop from from the the initial initial threshold threshold conditions, conditions, the the potential For potential difference UI, should not decrease relative to Ui . For AU should not decrease relative to the the initial initial value value of of !:l AU,. For difference !:l this, the the average average channel channel field field E, be lower lower than than the the external external field field Eo. this, Ee must must be Eo. However, aa mature mature channel channel possesses falling current-voltage current-voltage characteristic characteristic However, possesses aa falling E, (i). A A decrease decrease in in E, to "" 100 100 V/cm V/cm requires requires aa channel channel current current higher than Ee(i). Ee to higher than We discussed discussed this this issue issue in in sections sections 2.5.2 2.5.2 and and 2.6. 2.6. With With the the approximation approximation A. We 11A. accepted there there (E (E,e ~ x b/i b / i and and b =300 =300 VAjcm), leader current current is is to exceed accepted VA/cm), the the leader to exceed ,,iimin = = b/E b/Eo x 2 A at Eo x 150 V/cm. o ~ 2A at Eo ~ l50V/cm. see how large the the potential difference !:l AU u It should should be to make make the the Let us how large potential difference be to Let us see current exceed exceed imino i,. In chapter chapter 2, 2, we we derived derived formula formula (2.35) (2.35) relating relating the the current In channel current current behind leader tip to the tip potential U,I and and the the leader leader channel behind the the leader tip to the tip potential U vL. That That formula formula was was applicable applicable to the laboratory laboratory conditions conditions velocity VL' velocity to the N


140 140

Physical processes in a lightning discharge

considered in in that that chapter, chapter, when when aa leader leader travelled travelled through through the considered the rapidly rapidly decreasing field field of of aa high-voltage high-voltage electrode. electrode. Having Having covered covered aa distance distance of of decreasing only aa few few radii radii of of the electrode curvature, curvature, usually usually very small, the the leader leader only the electrode very small, tip potential, U tip found found itself itself in in aa space space with with aa nearly nearly zero zero potential, U,o « > U U,.L . becomes very very high high only only when when the the tip tip reaches reaches the the region region with with 1 Uo(L)1 becomes If If one one desires desires to to refine refine these these simple simple results results by by taking taking account account of of the the voltage drop, drop, charge charge redistribution, redistribution, and and current current variation variation along along the the channel, channel, voltage one one should should regard regard itit as as aa long long line, line, as as was was done done with with the the streamer streamer in in section section 2.2.3. U ( t . x ) ,charge charge per per unit unit length length r(x, r(x,t), t), 2.2.3. The The distributions distributions of of potential potential U(t,x), and current current i(x, i ( x ,t)t ) along along the the line line can can be be described described by by equations equations similar similar to to and (2.13) (2.13) and and (2.14): (2.14):

+

1

d r ai di _ 0 fh at ax at + dx -= 0,,

+

dU

dX = E,(i),

--

i(L) i(L) == r(L)vL r(L)vL

(4.6) (4.6)

where E E,e is is the the longitudinal longitudinal channel channel field field expressed expressed through through current current i(x, i ( x , t) t) where from the the current-voltage current-voltage characteristic characteristic (the (the field field in in (2.13) (2.13) was was expressed expressed from


An ascending positive leader

143 143

through current current and and linear linear resistance, E,e = = iR,). iR1). The The leader leader velocity velocity VL uL is is through resistance, E uL. given, for for example, example, by by formula formula (4.3): (4.3): dL/dt dL/dt == VL' given, Equations (4.6) (4.6) must involve an an electrostatic electrostatic relation charges Equations must involve relation between between charges and potentials. In aa simple simple approximation, approximation, expressions expressions (2.13) (2.13) and and (4.2) (4.2) and potentials. In were allowed allowed to to contain contain aa local local relation, ~ ( x= = ) C, C1[U(x) [ U ( x )-- Uo(x)], U o ( x ) ] through through , were relation, T(X) linear capacitance capacitance C,. C1.This This approximation approximation was was shown shown by by many many calculations calculations linear to be quite the case to be quite acceptable acceptable to to the case of of aa uniform uniform or or weakly weakly non-uniform non-uniform external field, field, but it appears appears insufficiently insufficiently rigorous rigorous for for aa strongly strongly nonnonexternal but it uniform field, which which aa leader leader crosses crosses on on entering entering aa storm storm cloud. cloud. In In fact, fact, uniform field, the potential at every every point along the the channel channel length length is is also also created created by the potential at point along by charges located located at at adjacent adjacent channel channel sites. sites. To To simplify simplify the the non-local relation, charges non-local relation, the leader leader charge charge can can be assumed to to be be concentrated concentrated on on aa cylindrical cylindrical surface surface the be assumed with an effective effective cover cover radius radius R; R; then desired relation relation takes takes the the form form with an then the the desired 1

JL

7rco

0

A U ( x , t)t) = = U(x, U ( X ,t)t) -- Uo(x) U ~ ( X= =)-46.U(x,

r ( z ,t) t) dz dz T(Z, 2 '/2 . [(z -- x) [(z x ) +~ R2] R2I1l2 *

(4.7) (4.7)

The boundary boundary conditions conditions for for the set of of integral integral differential differential equations equations (4.6) (4.6) The the set and (4.7) (4.7) are are described described by the third third equality equality in in (4.6) (4.6) and and U(O, U ( 0 ,t) t) = = 0, 0, since since and by the the leader leader base base is is grounded. grounded. Practically, Practically, it it is is convenient convenient to to subdivide subdivide the the the channel channel into into N fragments fragments and and consider consider the the charge charge density density in in each each fragment fragment to be dependent time, thus by to be dependent only only on on time, thus replacing replacing the the integral integral equation equation of of (4.7) (4.7) by set of of linear linear algebraic algebraic equations. equations. Each Each of of them them will relate the the potential will relate potential aa set U(xk) at at the the middle middle point xk of of the fragment to to the the intrinsic intrinsic and and all all U(Xk) point Xk the kth kth fragment other linear linear charges. charges. After After integrating integrating (4.7), (4.7), one one can can easily easily see see that that radius radius other with R enters enters logarithmically logarithmically the the factors factors of of the the set set of of equations equations (compare (compare with (4.2)), justifying the (4.2)), thereby thereby justifying the use use of of linear linear leader leader charge charge Tr instead instead of of its its cover space space charge. charge. The The set set of of algebraic algebraic equations equations for for U(Xk) U(xk) and and T(Xk) '(Xk) is is cover solved in in time time at at each each step, step, and and the the progress is made equations solved progress is made by by using using equations (4.6). presenting the (4.6). We We are are presenting the result result of of this this solution. solution. As the the tip As the leader leader tip tip approaches approaches the the cloud, cloud, the the external external field field at at the tip site site becomes stronger stronger and and the ever increasing increasing portion portion of of the the channel channel finds finds itself itself becomes the ever in non-uniform field. the velocity in aa strongly strongly non-uniform field. Since Since the velocity and and current current are are largely largely defined by by the Uo(L) at at the the tip site, formulae formulae (4.5), (4.5), in in which is defined the potential potential Uo(L) tip site, which Eo Eo is an average average field, field, remain remain valid. valid. In In aa simple simple model of aa cloud cloud with with aa spherical spherical an model of unipolar charged region, region, the the potential distribution in in the the space space free free from from unipolar charged potential distribution charges point charge. height of charges is is the the same same as as for for aa point charge. If If H is is the the height of the the spherical spherical charge centre, centre, Qe, Q,, the the potential potential at at height at the the point displaced from from the the height x at point displaced charge vertical vertical charge charge axis axis for for distance distance r (with (with the the account account of of the the mirror mirror reflection reflection by the plane) is by the earth's earth's plane) is

"{~ {

}.

1 1 _ 1 } 2]'/2 - [(H 2]'/2 . = 47rco 4T&o [[(H ( H-- xx)2 ) 2 + rr2I1l2 [ ( H+ x)2 + ry2I1l2

Uo(x) =

(4.8) (4.8)

Figure presents the the parameter parameter calculations Figure 4.1 4.1 presents calculations for for an an ascending ascending leader leader propagating were made made from propagating in in such such aa non-uniform non-uniform field. field. The The calculations calculations were from


144 144

lightning discharge Physical processes in aa lightning 3300 00

- 1.4 1.4

33

-
:: .r:;:."

200

s Ii

2i

5 U U

-g 3

-

EE 0 1.0

*z

6 i-8.... 3 -0.8 0.8 'g 'u .

8t:

~

M 8MV, 8MV, if if Qc Q, == -13.3C -13.3C and and H H1 3 km, as as in in the the U : :> -Qc/201TcoHj j == 3km, previous examples. examples. Even Even the the maximum maximum potential potential transported transported to to the the earth earth previous of aa descending descending positive positive is small. small. This This means means that that the the return return stroke stroke current current of is leader travelling travelling along along the the dipole dipole axis axis will will be be low. low. The The potential potential and and the the leader


The descending leader of first lightning component of the the$rst

169 169

current will be still still lower, lower, with with the the real real voltage drop U U,c across across aa channel channel of of current will be voltage drop 4H x 10 lOkm taken into into account. account. Even Even for for the the channel channel field field total length 4H:::::; total length km taken E x 10 10 V/cm, V/cm, the the value value of of U U,c :::::; 10 10 MV MV is is comparable comparable with with U U,max ,. ' This This lightlightE:::::; so weak weak that that it it has has little little chance chance of of being and included included in in the the ning is is so being registered registered and ning statistics. statistics. Vertical channels channels demonstrate demonstrate maximum maximum positive positive potentials transported Vertical potentials transported to the the earth. earth. They They go go through through more more or or less less identical identical regions regions of of negative negative (at (at to the bottom) and positive positive (at (at the the top) top) external external potentials, and the the respective respective the bottom) and potentials, and contributions to to the the integral integral of of (4.12) (4.12) are are mutually mutually compensated. compensated. Positive Positive contributions lightnings, however, can possess possess very very high high currents. currents. With With the foregoing lightnings, however, can the foregoing taken into into account, account, one one can can suggest suggest at at least least two reasons for for this. One is is aa taken two reasons this. One favourable random random deviation deviation of of the the channel channel path from the the vertical line. favourable path from vertical line. Suppose the the ascending ascending leader leader of of aa binary system, starting starting from from the Suppose binary system, the upper upper positive charge point point closest closest to earth, xo = H2 moves up vertically, positive charge to the the earth, Xo = H 2 -- R,, R c , moves up vertically, while the the other other leader, leader, having having descended descended to to the the zero zero potential potential point while point between between the charges, charges, turns turns aside aside and and goes goes along along the the zero zero equipotential equipotential line. line. After After it it the has deviated for for aa large large distance distance r from from the the dipole dipole axis, axis, it it turns down has deviated turns down vertically to to contact contact the the earth earth this this time. time. In In this this case, case, the the descending descending leader leader vertically misses to misses the the region region of of high high negative negative potential, potential, and and positive positive contribution contribution to the integral of (4.12) remains uncompensated. Calculations with formulae the integral of (4.12) remains uncompensated. Calculations with formulae (4.12), made at km, H (4.12), (4.10) (4.10) and and (4.17) (4.17) made at HI H I == 3 km, H22 == 6km, 6km, R R,c == 0.5 0.5 km, km, and and r == 1km transport to 1 km show show that that the the descending descending leader leader will will transport to the the earth earth aa potenpotential 4.3 times greater than than that to be be transported transported along along the the dipole dipole axis. axis. tial 4.3 times greater that to Another principal possibility possibility is Another principal is the the deviation deviation of of the the dipole dipole axis axis itself itself from with the vertical leader path preserved. from the the vertical vertical line, line, with the vertical leader path preserved. The The centres centres of top and bottom charges be shifted vertical line of the the top and bottom charges can can be shifted from from the the same same vertical line because of the difference the wind wind forces the because of the difference in in the forces at at different different heights. heights. Then Then the leader that has started started up up vertically vertically from from the the top top charged charged region region passes leader that has passes through the region of of high high positive positive potential, potential, while its twin, twin, descending descending ververthrough the region while its tically, will will appear appear to to be shifted aside aside relative to the the bottom bottom charge charge and and go go tically, be shifted relative to through the the region of low low negative negative potential. potential. The The effect effect will will be the same same as as through region of be the in the the first first case. case. Quantitatively, Quantitatively, it it may may even even appear appear to to be be stronger, stronger, since since in the length length and and capacitance capacitance of of the the descending descending leader leader are are smaller smaller due due to to the the the lack of of an an extended extended path path along along the the zero zero potential potential line. line. lack 4.3.5

A counterleader counterleader

The descending descending lightning lightning leader leader does does not not reach reach the the earth earth or or aa grounded grounded body, body, The the electric because because it it is is captured captured by by the the ascending ascending leader leader developing developing in in the electric field field of of cloud cloud and and earth-reflected earth-reflected charges. charges. This This field field is is enhanced enhanced by by the the charge charge of of the descending descending leader leader approaching approaching the the earth. earth. This This can can also also happen happen in in laboralaborathe tory conditions, conditions, especially especially if if the the descending descending leader leader is is negative. negative. Then Then the the tory counterleader is is positive and requires requires aa lower lower field field for for its its development. development. positive and counterleader Streak Streak pictures pictures of of laboratory laboratory sparks sparks clearly clearly show show the the counterleader counterleader start start


170 170

Physical processes in a lightning discharge Physical

Figure 4.11. A streak photograph of with aa counterleader streak photograph of aa long long spark spark with counterleader coming coming from from aa grounded electrode. electrode. grounded

and motion towards the the descending descending leader leader (figure (figure 4.11). 4.11). The The altitude altitude at at which which and motion towards their encounter encounter occurs occurs depends depends on on the the descending descending leader leader sign sign and and charge. charge. their The length length of of the the counterleader counterleader at at the the moment of their their contact contact is is important important The moment of for lightning lightning protection protection practice, because it it defines defines the the number number of of strikes strikes at at for practice, because bodies bodies of of different different heights heights and, and, to to some some extent, extent, the the current current rise rise parameters parameters of body. of the the return return stroke stroke from from the the affected affected body. Let which the Let us us estimate estimate the the altitude altitude z, which the descending descending leader leader tip tip is is to to reach to to be be able produce aa reach able to to create create aa field field at at the the earth earth high high enough enough to to produce viable counterleader. The The latter latter does does not not differ differ from from any any other other ascending ascending viable counterleader. leader, body of requires that that the the leader, and and its its development development from from aa body of height height d requires near-terrestrial field should should exceed exceed the the value value of of Eo Eo from from formula formula (4.11). (4.1 1). near-terrestrial field For premises, the For the the height height d == 30m 30m characteristic characteristic of of industrial industrial premises, the field field must Eo ~ x 480 480V/cm. If the the cloud cloud field field is is '" -lOOV/cm, field must be be Eo Vfcm. If 100 Vfcm, the the field AE == 380Vfcm 380V/cm must must be created by the descending descending leader leader with with charge. charge. t::..E be created by the The main main contribution contribution to the near-terrestrial near-terrestrial field field is is made made by by the the charge charge The to the concentrated at at the the leader leader channel channel bottom. bottom. Therefore, Therefore, the the calculation calculation of of concentrated the field field t::..E AE under very long long vertical vertical conductor conductor should should utilize constant the under aa very utilize the the constant value of of 7r averaged averaged over over this this bottom bottom of of length length "'z, -z, rather than the the linear linear value rather than the charge by the density density of of the the non-uniform non-uniform charge charge 7(X). r(x).With With the charge reflected reflected by the earth, earth, we we have have

-

r "dx r_ ~ U u-uo - _7 t::..E z ~ ~Jx dx = - Uo ~ N

*E(z) ()

U U

=GIZ - z l n ( zLIn(L/ / R ) R) -zln(H/R) 27rco = T x - G 27rcoz z In(H/ R) 2


(4.21)) (4.21

Return stroke

171 171

where where U is is the the channel channel potential potential and and LL :::::; xH H is is its its length, length, which which is is about about the the tf 9 at!

possessing aa linearly possessing linearly rising rising front front and and exponentially exponentially decreasing decreasing tail. tail. The The calculation was made j..ls, calculation in in [23] [23] was made with with moderate moderate parameters parameters tf == 55ps, ZMM == 20 20 kA, kA, and and ttpp == 50 50 ps corresponding to to aa moderate moderate power power lightning. lightning. 1 j..lS corresponding The other other calculation calculation [24] [24] was was for for tf tf == 55 ps, = 100 100 kA, kA, and and ttpp = = 100 100 ps The j..lS, 1ZM j..lS M = of aa very very powerful lightning. It It is is generally generally believed that the the air air conductivity conductivity of powerful lightning. believed that a0 corresponds by corresponds to to its its thermodynamic thermodynamic equilibrium equilibrium and and is is determined determined by temperature temperature (figure (figure 4.15). 4.15). Figure Figure 4.16 4.16 shows shows the the evolution evolution of of pressure, pressure, gas gas density, density, temperature, temperature, and behind the powerful and radial radial velocity velocity distributions distributions behind the shock shock front front for for aa powerful current impulse impulse [24]. [24]. The The curves curves for for moderate moderate current current impulses impulses are are qualitaqualitacurrent tively tively similar. similar. problem of of aa short short laboratory laboratory spark spark is is much much simpler. simpler. The The set set should should include include aa simple simple ttThe The problem discharge bank as high-voltage source discharge equation equation for for aa capacitor capacitor bank as aa high-voltage source for for aa spark spark gap gap with with the the desired desired circuit usually usually resistance and and allowance allowance for for the the circuit circuit inductance. inductance. Note that this this kind of LRC circuit resistance Note that kind of registers damped unobservable in registers damped oscillations oscillations unobservable in lightning. lightning.


Return stroke stroke 2525 20

35

t = 511S 5p t=

25

~20

i$15a 15 r--.

f-24 >24 5.5 5.5 44 22 11

0.1 0.1 0.3 0.3 0.7 0.7 1.2 1.2 1.4 1.4

1600 1600 3.1 3.1 0.38 0.38 0.20 0.20 0.27 0.27

88 26 26 38 38 46 46

Current impulse impulse 100 100 kA kA with p,ttpp = 100 100 Ils, ps, Q Q = lOC 1OC Current with tf = 55 Ils, 55 20 20 50 50 100 100 200 200 300 300

35 35 22 22 18 18 14 14 12 12 10 10

25 25 20 20 15 15 12 12 11 11 10 10

180 180 140 140 90 90 70 70 40 40 20 20

16 16 5.7 5.7 2.6 2.6 1.8 1.8 1.0 1.0 1.0 1.0

0.8 0.8 22 33 44 55 55

0.28 0.28 0.057 0.057 0.039 0.039 0.028 0.028 0.032 0.032 0.064 0.064

-

-

",ISO -150 -

Note. T T,,,max is is the the temperature temperature along along the the channel channel axis, axis, T Teff is the the average average temperature temperature in in the the Note. eff is conductive channel, channel, O'eff oeffis is an an average average channel channel conductivity, conductivity, p is is channel channel pressure, reffis is the effecconductive pressure, reff the effective radius of total energy tive radius of the the conductive conductive channel, channel, W is is the the total energy released released (no (no data data for for the the second second variant; the the given given values values was was estimated estimated as as W W >::; x ii~axRl ~ a x Rt1p t),p )and and r Q Q is is the the charge charge transported transported variant; during the the current current impulse. impulse. during

-

-

3 2 rise. with ions / , but rise. In In aa strongly strongly ionized ionized plasma, plasma, with ions of of constant constant charge charge IJ' o '" T T3I2, but 2 doubly with increasing doubly charged charged ions ions appear appear with increasing T. T . Since Since IJ' U '" ZiZC2,, where where Zj Zi is is the the ion charge charge multiplicity, multiplicity, the the two two effects effects compensate compensate each each other. other. The The resistance resistance ion of aa highly highly heated heated channel channel decreases decreases with time due due to to its its expansion expansion only. only. of with time Some pressure at Some time time later, later, however, however, the the pressure at the the channel channel centre centre drops drops to to atmospheric pressure, and atmospheric pressure, and the the expansion expansion ceases. ceases. The The conductive conductive channel channel cross reduced gradually because of the gas by cross section section is is reduced gradually because of the gas cooling cooling caused caused by thermal radiation. radiation. The resistance begins begins to because of thermal The channel channel resistance to rise rise slowly slowly because of decreasing reff and T Teff. decreasing reff and . eff The expansion expansion time time of of the the channel channel becomes longer and and its its minimal minimal The becomes longer resistance decreases for for the the stronger stronger current current impulses. impulses. Physically, Physically, the the linear linear resistance decreases W, resistance is affected affected by the energy energy released released per unit channel channel length, length, W, by the per unit resistance is rather than by the unambiguously by rather than by the current. current. This This value value is is not not described described unambiguously by the what is the current current amplitude; amplitude; what is more more important important is is the the amount amount of of the the transported charge charge Q: Q: WI W1 '" P i 2RRIt'" l t QiR QiRlI '" QE and and the the field field does does not not transported vary much. The but vary much. The calculations, calculations, however, however, deal deal with with the the current current impulse impulse but W1. Semi-quantitatively, Semi-quantitatively, the the time time dependence dependence of of resistance resistance can can be not with WI' not with be powerful cylindrical cylindrical understood using the for the the shock shock wave wave of of aa powerful understood using the relations relations for explosion. be strong explosion. The The explosion explosion can can be be considered considered to to be strong as as long long as as energy energy is is released released in in aa thin thin channel channel and and the the pressure pressure of of the the explosion explosion wave wave does does not to the pressure. In not fall fall close close to the atmospheric atmospheric pressure. In this this case, case, the the flow flow is is self-similar. self-similar.

- - -


Return stroke

185

The shock shock front front radius radius rs r s and and pressure pressure pp in in the the affected affected region region depend, depend, within within The the accuracy accuracy of of numerical numerical factors, factors, on on W WI1 and and t as as r, r s rv ((WJ!PO)I/4 and W 1 / p o ) 1t/l/2 4 t and 1 /2 the pp rv Wl/rz Wlpo)'/2t-', where po is cold air density. The channel Wl/r; rv ((W PO)I/2 -l, where Po is cold air density. The channel t I expansion expansion is is completed completed when when the the pressure pressure drops drops to to aa certain certain value value close close to to atmospheric pressure. pressure. This This sets sets the the limit limit to to the the validity validity of of formulae formulae for for selfselfatmospheric similar motion. motion. This This means means that that they they are are still still applicable, applicable, and and the the duration duration of of similar the resistance resistance reduction reduction then then is is t rv ((W It can be be shown shown [12] [12] that that for for the W IlPO)I/2. p o ) 1 / It 2 . can aa self-similar self-similar cylindrical cylindrical explosion explosion in in the the central central region region with with the the pressure pressure equalized equalized along along the the radius radius (figure (figure 4.16), 4.16), the the internal internal specific specific energy energy depends depends and W1 WI as as E€ rv ((W?/2 where y 'Y is is the the adiabatic adiabatic exponent. exponent. on r, t and t2- rr -2)1/(r-I), on Wy/2t2-Tr-2)1/(T'I, where A point point with with fixed fixed temperature, temperature, e.g., e.g., T T:::::: K, can can be be regarded regarded as as the the M 10000 10000 K, A conductive conductive channel channel boundary, boundary, since since the the plasma plasma conductivity conductivity below below this this point ( T ) varies point is is relatively relatively low. low. The The radius radius of of aa point point with with fixed fixed T and and E€(T) varies as r rv W:'4t1-y/2, W?/4 t l- r /2, reaching reaching aa value value proportional proportional to to r,,, rmax rv W;I2 W11/ 2 with time time as with I 2 by the the moment moment the the channel channel stops stops expanding, expanding, t rv W:12. WI / . Therefore, Therefore, the the linear linear by -2 r;;;;x rv Wand channel resistance drops to aa value value proportional proportional to to Rl,, R lmm rv rmax W-'I ,, and channel resistance drops to I 2 this occurs occurs for the time time t rv W / . These relationships are qualitatively this for the W'/'. These relationships are qualitatively consistent with the the calculations calculations for for the the two two variants variants described described in in table table 4.1. 4.1. consistent with N

N

N

N

N

N

N

N

N

N

N

4.4.4 Return stroke as a channel transformation wave 4.4.4 channel transformation The first first substantiated attempt to to make make aa numerical numerical simulation of the the lightlightThe substantiated attempt simulation of ning return return stroke with allowance allowance for for the the resistance resistance variation variation was was undertaken undertaken ning stroke with as far back back as as the the 1970s The most most important important features features of of the the process, process, as far 1970s [25,26]. [25,26]. The which are due to to an abrupt conductivity conductivity rise rise at the site of intensive intensive Joule Joule heat heat which are due an abrupt at the site of release, release, became became evident evident at at once. once. The The simulation simulation showed showed that that aa weak weak initial initial perturbation perturbation (precursor) (precursor) propagating propagating up up along along the the channel channel at at an an electroelectromagnetic magnetic signal signal velocity velocity close close to to light light velocity velocity does does not not change change the the plasma plasma state state and and cannot cannot be be treated treated as as the the return return stroke stroke wave wave front front visible visible in in streak streak photographs. photographs. The The main main wave wave of of current current and and decreasing decreasing potential potential travels travels several several times times slower; slower; its its velocity velocity is is defined defined by by the the transformation transformation of of the the low low conductivity conductivity leader leader to to the the low low resistivity resistivity stroke stroke channel. channel. This This conclusion conclusion was was formulated formulated explicitly explicitly in in [25-28]; [25-281; it it reflects reflects the the nature nature of of the the lightning lightning return return stroke. stroke. Turning Turning to to numerical numerical simulation simulation today, today, we we should should like like to to formulate formulate this this problem problem in in aa simple simple and and clear clear physical physical language language and and to to try try to to outline outline problems problems to to be be solved solved within within this this model. model. An An obviously obviously essential essential aspect aspect of of the the theory theory still still is is the the resistivity resistivity dynamics dynamics of of the the lightning lightning channel. channel. An An exhaustive exhaustive formulation formulation of of this this problem problem would would involve involve aa simultaneous simultaneous solution solution of of equations equations describing describing the the propagation propagation of of aa current-voltage current-voltage wave wave and and the the channel channel dynamics dynamics at at every every point point along along its its length, length, affected affected by by the the ever ever varying varying energy energy release. release. So So we we shall shall restrict restrict the the discussion discussion to to aa simple simple model, model, having having accepted accepted aa probable probable law law for for the the linear linear conductivity conductivity rise, G == Ri R:',l , and and focusing focusing on on the the qualitative qualitative results results of of the the solution. solution. rise, G


186 186


Let Let us us describe describe G G in in the the simplest simplest way way reflecting reflecting the the main main qualitative qualitative be assumed features features of of the the channel channel evolution. evolution. It It will will be assumed that that the the linear linear conducconductivity resistance tivity increases increases with with current. current. This This partly partly reflects reflects the the fact fact that that resistance decreases with with increasing increasing charge charge through through aa particular particular channel channel site. site. But But the the decreases resistance is is stabilized stabilized with time, even even though though the current continues continues to resistance with time, the current to flow. principle, the flow. In In principle, the stable stable state state of of aa lightning lightning channel channel hardly hardly differs differs in aa hhigh g h current current arc arc only only slightly slightly varies varies from that that of of an an arc. arc. The The field field E in from with current; in in other other words, words, the linear conductivity conductivity of of the the channel channel is is with current; the linear i /EE rv i. Assume equal to to the the field field E EL in the lightning leader, G,,st == i/ G Assume E to to be be equal in the lightning leader, L then the the conductivity conductivity is is whose current not low whose current is is not low on on the the arc arc scale; scale; then i / E LL. . In In aa mature mature gas-dynamic gas-dynamic process process when when the the shock shock wave is still still G,,st == i/ G wave is strong, the will drop drop with with time. As the the shock shock wave wave becomes strong, the resistance resistance will time. As becomes R11 and and the the increase increase in in G G become slower. These These weaker, the the decrease decrease in in R weaker, become slower. tendencies are described described by the relaxation-type relaxation-type formula formula tendencies are by the N

dG i/EL dG EL -- G(t) Gst(i) - G(t) G(t) - - i/ - Gst(i) - G(t) dt T T dt Tgg Tgg

(4.38) (4.38)

is the the characteristic characteristic time time oflinear of linear conductivity conductivity variation variation (relaxation (relaxation where TTg where g is const, T Tg = const, and G(0) = 0, we have have time). In In aa simple simple case case with with i == const, time). = const, and G(O) = 0, we g -t/Tg )]. G GSt[1-- exp( exp(-t/T,)]. G == Gsdl U ( x ,0) == Vi U, and and Equations (4.24) (4.24) are are solved solved with with the the initial initial conditions conditions V(x,O) Equations i ( x ,0) == 0, 0, R RI1(x, (x30) 0) == R RI,-; the reactive parameters are taken to be constant: i(x,O) ; the reactive parameters are taken to be constant: 1L C1 = 10 10 pF pF/m and LL1 = 211Hjm. 2 pH,”. The The channel channel does does not not close close on on the the earth earth C 1= jm and 1 = so through through the the time-decreasing time-decreasing resistance resistance of of the the in an an instant instant but but does does so in commutator (similarly (similarly to to the the real real lightning lightning length length decreasing decreasing through through the the commutator R,,, = R(0) exp(-cut), R(0) 10 R streamer zone). zone). The The accepted accepted values values of of R streamer = R(O) exp( -at), R(O) = 10 fl eom and acu == IIlS-1 1 ps-’ provide provide aa typical typical duration duration of of the the negative negative current current impulse impulse and FZ 511S. 5 p . The The boundary boundary condition condition at at the the grounded grounded end end of of the the line line front tf ~ front no doubt: doubt: V(O, U ( 0 .t)t ) == i(O, i(0,t)R t)R,,,. The problem of the far end up in the raises no raises The problem of the far end up in the . eom H , is is much much more more complex. complex. Conventionally, Conventionally, it it is is considered considered as as clouds, xx == H, clouds, i ( H ,t)t) == O. 0. In In reality, reality, the the situation situation is is far far from from being being being open, open, assuming assuming i(H, being self-evident. When When the the line line gets gets discharged discharged and and its its end end in in the the clouds clouds takes takes self-evident. zero potential, potential, aa high high electric electric field field must must arise arise near near itit due due to to the the voltage voltage zero AU = = -Vo(H). - U o ( H ) . This This gives gives impetus impetus to to very very intensive intensive ionization ionization difference 6.V difference processes, processes, probably probably involving involving high high current. current. This This situation situation will will be be partly partly to have have no no current. current. discussed below. below. Now, Now, we we shall shall assume assume the the upper upper end end to discussed The results results to to be be presented presented were were obtained obtained for for aa vertical vertical unbranched unbranched channel channel The H == 44 km. km. This This is is the the height height the the ascending ascending leader leader tip tip with the the total total length length H with reaches when when the the descending descending leader, leader, which which has has started started from from the the point point closest closest reaches to the the earth earth in in the the bottom bottom negative negative sphere sphere with with the the centre centre 33 km km high, high, contacts contacts to the earth. earth. It It is is normal normal practice practice to to use use the the following following averaged averaged parameters parameters the ELL == 10 10V/cm, = 100 lOOA, and of the the leader leader prior prior to to the the contact: contact: E of Vjcm, iiL A, and L = RIL = Ed EL/iL 10fljm. O/m. For For aa realistic realistic description description of of the the resistivity resistivity dynamics dynamics R lL = i L == 10 (section 4.4.3), 4.4.3), the the relaxation relaxation time time should should be be taken taken to to be be TTg = 40 40 llS, ps, when when the the (section g =


Return stroke

187 187

current at at this this channel channel site site rises, rises, and and T Tg = 200IlS, 2 0 0 p , when when the current current the current g = decreases. The The model calculation reproduces the distributions distributions of of current current decreases. model calculation reproduces the i(x, potential U(x, i ( x , t) t ) and and potential U ( x ,t) t ) along along the the channel; channel; the the linear linear charge charge is is r(x, ~ ( xt) t ), == C1[ U ( x ,t) t ) -- Uo(x)]. U o ( x ) ]Generally, Generally, . the external external field field potential potential Uo(x) Uo(x) can also also CdU(x, the can vary vary in in time, time, because because we we should should not not discard discard aa possible possible partial partial neutralization of of the the charge charge in in one one of of the the regions regions of of the the cloud cloud dipole. dipole. The The neutralization latter point will not not be discussed for for the the time time being. latter point will be discussed being. The precursor traThe calculations calculations are are presented presented in in figures figures 4.17-4.21. 4.17-4.2 1. The The precursor travelling velocity 0.64e this is the velling with with velocity 0 . 6 4 ~is is damped damped so so fast fast that that this is not not shown shown in in the plots after noticeable break break from the chanplots after aa noticeable from the the principal principal wave wave re-charging re-charging the channel nel (we (we shall shall term term it it aa discharge discharge wave wave for for simplicity). simplicity). The The wave wave in in figure figure 4.17 4.17 travels along the the channel channel with velocity VU,r ~ x OAe, 0.4c, i.e., i.e., 1.6 1.6 times times slower slower than than travels along with velocity the precursor. precursor. This the This velocity velocity somewhat somewhat decreases decreases as as the the wave wave moves moves up. up. Its Its variation be conveniently well-defined variation can can be conveniently followed followed from from the the change change in in the the well-defined maximum power of the Joule R 1 (figure i2Rl (figure 4.17 4.17 (centre)). (centre)). The The maximum linear linear power of the Joule losses losses P wave wave front front power power rises rises abruptly abruptly along along aa 100-200 100-200 m m length, length, then then it it decreases decreases towards the with intensive towards the earth, earth, making making the the channel channel tip tip with intensive energy energy release release stand stand out region which out clearly. clearly. It It seems seems that that it it is is this this region which is is clearly clearly discernible discernible in in streak streak photographs. The power proportional photographs. The linear linear power proportional to to the the squared squared current current drops drops remarkably on way up up the maximum becomes remarkably on the the way the cloud, cloud, and and the the maximum becomes smeared. smeared. This with observations This is is also also consistent consistent with observations of of radiation radiation intensity intensity [14,29]. [14,29]. A A photometric study photometric study has has shown shown that that the the radiation radiation from from the the wave wave front front is is attenuated the front boundary. The attenuated and and the front loses loses its its clear clear boundary. The current current wave wave is is not not attenuated attenuated so so rapidly rapidly (figure (figure 4.17 4.17 (top)). (top)). For For the the time time of of its its earth-cloud earth-cloud the channel travel travel lasting lasting for for 341ls, 3 4 p , the the current current at at the channel base base drops drops from from the the maximum of to 24 that maximum of 35 kA kA to 24 kA. kA. This This agrees agrees with with observations observations indicating indicating that an to an average average current current impulse impulse duration duration in in aa negative negative lightning lightning is is close close to 751ls 75ps on on the the 0.5 0.5 level. level. The The wave wave front front deformation deformation depends depends on on the the initial initial potential Uii delivered delivered by by the the leader leader to to the the earth. earth. The The higher the value Uil, potential U higher the value II U i I, the the discharge the higher higher the discharge current. current. The The rate rate of of resistivity resistivity decrease decrease at at the the wave front wave front grows grows respectively, respectively, so so the the front front steepness steepness increases. increases. This This is is evident evident from from aa comparison comparison of of figures figures 4.18 4.18 (top) (top) and and 4.18 4.18 (bottom). (bottom). At At I1 Uil Uii = = 50 50 MV, MV, the the channel practically without the current current wave wave goes goes along along the channel practically without elongating elongating the the lUil = = 10 10MV it has has aa lower lower velocity and aa smooth smooth front. front. front? while at lUiI frontt while at MV it velocity and Unfortunately, there there have have been been no Unfortunately, no registrations registrations of of current current and and streak streak photographs return stroke photographs of of the the return stroke taken taken simultaneously. simultaneously. A A comparison comparison of of the provide aa good the relationships relationships between between current current and and wave wave velocity velocity could could provide good test test for for the the return return stroke stroke theory. theory. the current current impulse impulse amplitude amplitude rises, rises, the the linear linear resistance resistance falls falls more more As the quickly wave is quickly and and to to aa lower lower level, level, so so the the wave is damped damped more more slowly slowly during during its its The motion motion of of aa high high current current wave with attenuation attenuation but but without without noticeable noticeable distortions distortions t The wave with facilitates the electromagnetic field field calculation calculation necessary in many applied problems of lightning lightning facilitates the electromagnetic necessary in many applied problems of protection and methods. protection and in in substantiation substantiation of of remote remote current current registration registration methods.


188 188

Physical processes in a lightning discharge 35 30 25

:s

20

cf

~~ 15

U

10 5 2 X, Ian

3

4

2

3

4

30 25

:> 20 ~ oJ

E 15

'0

:>

10 5

1

0

X, X,2Ian km

8p t == 81ls

600

~

18

:E 400 ..... ~ 0

~

200

o+-~-+-~-..........~-..,......->-,---..l,

o0


1

2

x,

X, Ian km

3

4

Return stroke

189 189

1.0

67 kA kA II,,,= max = 67 0.8

3 >-< i:l

0.6

E

U

\ ........

.3

0.4 0.4

-

0.2 0.2

-

0.0 0.0

.

I

00

,

.

.

, 2

1

.

, 3

, 4

x, kIn lan

X,

1.0 t = 8).15

I1-max = 8,15 kA

0.8 0.8 i:l

~-..e 0.6 0.6 1 ........ .I

0.4 0.4

0.2 0.2 0.0 0.0 00

2

1 X,

km

3

4

Deformation of of the the current current wave wave front front at at leader leader potential potential (top) (top) Figure 4.18. Deformation -50MV and (bottom) (bottom) -10 -10 MY; MV; for for the the other other parameters, parameters, see see figure figure 4.17. 4.17. Uii == -50 MY and

propagation along along the the channel. channel. There There is is no no damping damping at at aa very very high high current current and and propagation the impulse front front becomes steeper, as as was discussed in in section section 4.4.2 4.4.2 (figure (figure the impulse becomes steeper, was discussed 4.18). is also also observed observed in in the the current current amplitude amplitude dependence dependence 4.18). Non-linearity Non-linearity is U,i at at the the earth. earth. If If the the commutator commutator were were perfect perfect on the the initial initial potential on potential U (R,,, = 0), 0), the the current current at at the the earth earth at at the the moment moment of of contact contact would instantly (R would instantly eom = Figure 4.17. (Opposite) (Opposite) Numerical Numerical simulation simulation of of the the return return stroke stroke excited excited by by aa Figure -30 MY: MV: (top) (top) current current and and (centre) (centre) voltage disdescending leader leader with with potential potential -30 descending voltage dis10 n/m. tributions; tributions; (bottom) (bottom) the the power power of of Joule Joule losses. losses. The The initial initial leader leader resistance, resistance, 10 rl/m. V/cm. Steady Steady state state field field in in the the channel channel behind behind the the wave, wave, 10 lOY/em.


190 190


r 0.6

200 1

V "...... '"

< ..I
1

(4.59)

Axi is is the the extension extension of of the the ionization ionization region region from from the the initial initial to to the the final final where ~Xi where point of of the the wave. wave. For For aa wave wave to to survive, survive, its its parameters parameters must must meet meet the the last last point of (4.59). (4.59). Otherwise, Otherwise, the the field field within within the the wave wave will will be be unable unable to to inequality of inequality E*, so so no no ionization ionization will will occur. occur. exceed E*, exceed The capabilities capabilities of of an an ionization ionization wave wave are are limited, limited, and and this this limit limit increases increases The with increasing increasing initial initial conductivity conductivity of of the the medium. medium. For For example, example, if if the the initial initial with 12 7 2 ne0 M 10 lOl2cmP3 and x0 x 10 107m2/s were even even lower lower than than the the parameters neo::::; parameters cm- 3 and Xo::::; m Is were kV/cm, critical values values found found in in section section 4.8.3 4.8.3 and and if if the the threshold threshold field field was was 33 kV/cm, critical 6 it would would be be necessary necessary to to have have the the ionization ionization frequency frequency Vi 4 == 2.1 2.1 Xx 10 lo6 s-' S-1 it and potential potential U U2 = 300MV in order to increase ne and x by three orders of and = 300 MV in order to increase n and X by three orders of 2 e (Ax = 1m, 1 m, U I1 = 0.3 0.3 MY) MV) and and U U2 30 MV by two orders. The magnitude (~x magnitude = 30 MV by two orders. The 2 wave width width in in this t h s case case is is ~Xi Axi ::::; E 22 22m, i.e., it it is is very very extended. extended. Only Only when when m, i.e., wave Sometimes, it it seems seems better better to to describe describe !lj vi the the function function of of E and, and, on on the the contrary, contrary, to to remove remove U t Sometimes, from (4.48) (4.48)and and (4.57). (4.57). Instead Instead of of (4.48), (4.48), we we then then get get from

~~=E(;-~),


!lj=!lj(E)

Subsequent components. of a dart leader components. The The problem of

215

the lower (nCo (neo = = 10" 1011 cmP3, cm -3, the initial initial conductivity conductivity is is still still an an order order of of magnitude magnitude lower 6 2 5 Xo Is, and wave width width begins begins to to approach approach what what xo == 10 lo6m m2/s, and R R= = 10 lo5Dim), fl/m), the the wave the same medium at at the the same same Uv would would be be desired desired for for aa dart dart leader. leader. In In the same medium and the ratio ratio xX2/XO = lo3 103 would would be be and E*, E*, the the parameters parameters necessary necessary for for the 2 / x 0= 7 IIi = 30 MV, D.x = 10 cm, D.xi = 5 m, and U\ = 30 kV. vi = = lAx 1 . 410 lo7 ~ S-I, sP1, U U2 = 30 MV, Ax = 10 cm, Axi = 5 m, and U , = 30 kV. 2 A rise would would be be obtained obtained at at aa still still A still still narrower narrower region region of of the the potential potential rise the applicability applicability limits limits of of lower lower initial initial conductivity. conductivity. But But then then we we approach approach the the propagation in in aa conductive conductive the basic basic concepts concepts of of the the theory theory of of perturbation perturbation propagation we are probably coming coming closer closer to to the the medium medium and and of of the the long long line line theory, theory, and and we are probably production. understanding understanding of of criteria criteria for for the the dart dart leader leader production. 4.8.5 4.8.5 The dart dart leader as a streamer in a 'nonconductive waveguide'

The channel, or or in in aa long long line, line, is is The diffusion diffusion mechanism mechanism of of field field evolution evolution in in aa channel, incompatible hence, with with strong strong fields. fields. If If incompatible with with abrupt abrupt potential potential changes changes and, and, hence, abrupt by diffusion. diffusion. We We believe believe abrupt changes changes do do arise, arise, they they are are rapidly rapidly smeared smeared by for wave nor nor aa dart dart leader leader can can for this this reason reason that that neither neither aa narrow narrow ionization ionization wave be find the the conditions, conditions, in in which which aa be formed formed in in aa well-conducting well-conducting channel. channel. To To find remind ourselves ourselves of of the the prerequiprerequivery very strong strong field field can can be be induced, induced, we we should should remind sites for for the the long long line line equations. equations. sites The geometry has has the the form: form: The electrostatics electrostatics equation equation for for cylindrical cylindrical geometry

-+--rE dEx 1 d -P aEx+~~fEr=.f!... (4.60) (4.60) rax r or EO dx r dr EO where p is is space space charge charge density. density. By By integrating, integrating, in where in the the cross cross section, section, aa conducconductor of of radius radius ro ro and and neglecting neglecting the the dependence dependence of of the the longitudinal longitudinal field field E, Ex on on tor we obtain obtain r,r, we

a::

1;

2 8Ex m-0 -+ + 2rrroEro:a 27rroEr, = - , rrr~ 7-

(4.61) r= = LO 27rrpdr T 2rrrp dr (4.61) dX EO where E Er0 is the the radial radial field field on on the the surface surface of of aa conductor where conductor of of length length II > »> yo: ro: ro is E

U

ro ~ ro In(l/ro) ,

C

2rrEo

1

= In(l/ro) .

(4.62) (4.62)

If the the longitudinal longitudinal field field varies varies along along the the channel channel so so slowly If slowly that that the the axial axial diverdivergence can can be be neglected neglected (the (the characteristic characteristic length length for of E, Ex is is gence for the the variation variation of Ax » >> fO), ro), we we arrive arrive at at one one of of the the basic basic conceptions conceptions of D.x of the the long long line line theory, theory, ~ ( x= =) C\ C1U(x), U(x), whose whose implication implication is is the the potential T(X) potential diffusion diffusion mechanism. mechanism. It It is suggested suggested implicitly implicitly that that the the resistance resistance varies is varies very very slowly slowly along along the the channel, so so this this variation variation cannot cannot be be an an obstacle obstacle to to aa charge charge flux, flux, making making channel, the flux flux velocity velocity decrease decrease abruptly abruptly and and create create aa space space charge charge due due to to its its the local accumulation accumulation (a (a long long line line has has no no 'jams'). 'jams'). local However, space space charge charge does does accumulate accumulate at at aa sharp However, sharp boundary boundary between between aa poorly- and and aa well-conducting well-conducting channel channel portion. is formed formed at at poorlyportion. A charged charged tip tip is the end end of of an an ideal ideal (or (or non-ideal) non-ideal) conductor, conductor, the the potential potential in in front front of of it it the


216 216


drops drops abruptly, abruptly, at at distances distances about about equal equal to to ro, ro, inducing inducing there there aa strong strong field field capable capable of of sustaining sustaining an an ionization ionization wave. wave. This This is is what what happens happens in in aa common medium. It what is common streamer streamer in in aa non-conductive non-conductive medium. It is is then then clear clear what is necessary to the ionization necessary to support support aa sharp sharp potential potential drop drop at at the ionization wave wave front front for for aa long the perspective perspective trajectory long time. time. The The conductivity conductivity along along the trajectory must must drop drop to to aa value be unable unable to the value low low enough enough for for the the diffusion diffusion field field tongue tongue to to be to smear smear the sharp potential drop. sharp potential drop. Therefore, Therefore, the the tongue tongue length length must must become become comparable comparable with the the channel channel radius radius 6.x Ax ~ x xo/v N roo yo. Because Because of of the the strong strong temperature temperature with Xolv '" dependence dependence of of the the degree degree of of equilibrium equilibrium ionization ionization in in air air at at low low temperatures, temperatures, aa drop K would would be be sufficient. drop to to T ~ x 3000 3000K sufficient. The The equilibrium equilibrium electron electron density density 10 3 established for for the the long long zero-current zero-current pause pause will will be be neo'" neo 10 10'o-lO1l cmP3; _10 11 cmestablished ; 5 2 6 5 6 hence, RIO'" _10 ll/m, and Xo '" 10 _10 m Is. But Rlo 10 105-1060jm, and xo 106-105m2/s. But the the air air density density in in hence, the cooled cooled channel channel of of the the previous previous component component at at T ~ x 3000 3000K is by by an an order order the K is of that the of magnitude magnitude lower lower than than that that of of cold cold air, air, so so that the conductivity conductivity drop drop will will not interfere properties of not interfere with with the the 'waveguide' 'waveguide' properties of the the track. track. The The velocity velocity of of aa dart dart leader leader as as an an ionization ionization wave wave is is defined, defined, in in order order of the streamer of magnitude, magnitude, by by the the same same formula formula (2.2) (2.2) as as the streamer velocity. velocity. But But the the 10 'pre-ionization' this case _10 11 cm -3) is 'pre-ionization' in in this case (neo (nee '" 10 10'o-lO'l cmP3) is considerable, considerable, and and aa x 5) 5 ) is is to much smaller smaller number number of of electron electron generations generations (In(n (1n(ne2/neo) to be be much e2In eo) ~ produced in the the wave. wave. With With the the account account of of the similarity law law for for Vi vi at at produced in the similarity an an order order of of magnitude magnitude lower lower gas gas density, density, the the ionization ionization frequency frequency is is Vi '" 1010 we obtain vi 10" S-l sC1 and and ro ro '" 1cm; 1 cm; then then we obtain aa correct correct order order of of the the velocity velocity qro/ In(ne2/ne0) io77 m/s. mjs. vU = ~ viral In(n e2Ineo) '" 10 One cannot cannot say say that that all all the the details details of of the the dart dart leader leader behaviour have been been One behaviour have clarified by the the dart clarified by the above above considerations. considerations. For For the dart leader leader channel channel to to be be well well conductive, the electron density density in in it it must must be at least least 5-6 5-6 orders orders of of magnitude magnitude conductive, the electron be at higher than than the initial value for the the track. track. But But the the capabilities capabilities of of the the ionizaionizahigher the initial value for tion wave wave to to produce more electrons electrons are are limited. limited. The The maximum maximum conductivity conductivity tion produce more of an an ionization ionization wave wave propagating propagating through through aa non-conductive non-conductive medium medium is is of defined, in in order order of of magnitude, magnitude, by the relation relation umaxl ~ F co~ '" ~Vi vi (section (section ~ / E 2.2.2), 2.2.2), ~ defined, by the because space charge charge of of the the streamer streamer tip, tip, providing strong ionization ionization because the the space providing aa strong field is is dissipated dissipated with the Maxwellian Maxwellian time time TM == col Eo/cF.t After the the wave wave with the u. t After field

-

-

-

-

-

--

-

It also also determines determines the the rate rate at at which the linear linear charge charge 7T == C C1 U is is established, established, if if it it is, is, in in the the t It which the j U channel. Let Let us us integrate integrate the relation for for charge charge conservation conservation in in the conductor cross cross section. section. channel. the relation the conductor along the the channel channel length, length, we we obtain obtain Neglecting, for simplicity, simplicity, the variation in in a0 along Neglecting, for the variation op

oE _ 0

at + aa;x - ,

07

2

oE,

at + 1rroaa; = O.

Using (4.61) we arrive relation between between 7T and Using (4.61) and and (4.62), (4.62), we arrive at at aa refined refined equation equation for for the the relation and U: U: co 07

--; at + 7

= C j U.

The postulate of the the long long line line theory, theory, 7T = = C j, U, U, is is valid valid if if the changes in in the the system, system, which which also also The postulate of the changes ~ ( t )occur occur , slower than than with T ,= = ~ co/a. co/u. When When applied applied to to the wave front front moving moving at at define 7(t), slower with 7M the wave define in aa line line with with conductivity conductivity aD, uo,this happens at at aD 00 » >> vco/ro u i o / r o and and xo >> vro. vr0. velocity velocity vU in this happens Xo »


Experimental checkup of of subsequent component theory

217

has and ionized, ionized, but but both both proprohas passed, passed, the the channel channel still still needs needs to to be be heated heated and cesses as in in aa classical classical leader leader channel. channel. cesses are are to to occur occur in in aa moderate moderate electric electric field, field, as Besides, radial field field makes makes the the channel channel Besides, this this must must take take place place before before aa strong strong radial become enveloped enveloped by by expand expand beyond beyond the the hot hot gas gas tube, tube, or or if if it it has has already already become aa stabilizing stabilizing charge charge cover cover (section (section 4.8.1). 4.8.1). There the processes processes in in aa dart dart leader leader that that There are are still still many many questions questions about about the remain quantitative theory theory is is also also aa remain to to be be answered; answered; the the development development of of its its quantitative task task of of further further research. research. To features of of aa current current To conclude, conclude, it it is is worth worth noting noting some some specific specific features components. Generally, Generally, the the impulse impulse in in the the return return stroke stroke of of subsequent subsequent components. takes the the return return stroke stroke to to run run impulse impulse duration duration is is related related to to the the time time it it takes components, this this time time must must along along the the whole whole channel. channel. For For the the subsequent subsequent components, be to the the attached attached intercloud intercloud be longer longer than than for for the the first first component component due due to the subsequent components is is about about leader. leader. But But the the impulse impulse duration duration in in the subsequent components twice wave velocities velocities are are generally generally the the same. same. twice as as short, short, although although the the return return wave The to be be the the absence absence of of branches branches in in aa The reason reason for for this this difference difference is is likely likely to relatively slow slow process process of of their their reredart dart leader. leader. It It is is quite quite possible possible that that the the relatively charging elongates the current impulse tail of the first component. The charging elongates the current impulse tail of the first component. The do not not reverse reverse the the sign, sign, similarly similarly impulses impulses of of the the subsequent subsequent components components do to branches, the the action action of of the the reflected reflected to those those of of the the first first one. one. In In the the absence absence of of branches, the randomly randomly reflected reflected waves waves of of the the wave wave can can no no longer longer be be screened screened by by the of ‘white 'white noise’ noise' should, should, numerous hypothesis of numerous branches branches (section (section 4.4.5). 4.4.5). The The hypothesis probably, This problem, problem, like like the the others others probably, be be discarded discarded as as being being inadequate. inadequate. This above, awaits awaits its its solution. solution. above,

4.9

Experimental component theory Experimental checkup checkup of subsequent subsequent component

The theoretical theoretical treatment treatment of of processes processes occurring occurring in The in the the channel channel of of the the previous component component has has been been reduced reduced to previous to the the various various wave wave propagation propagation modes -- the the diffusion diffusion mode mode in in the the M-component M-component and modes and the the ionization ionization wave wave mode in in the the dart dart leader. leader. The The former former has has aa strongly strongly elongated mode elongated front front with with aa slowly varying varying potential, potential, and and the the latter latter must must possess slowly possess aa tip tip with with aa concentrated concentrated charge, producing producing an an abrupt abrupt potential potential change. change. Indirect charge, Indirect evidence evidence for for the the significant difference in the field distribution is the registrations of current significant difference in the field distribution is the registrations of current impulses at at the the earth. earth. The The impulse impulse front front durations durations are impulses are found found to to differ differ by by 2-4 orders of magnitude between an M-component and a dart 2-4 orders of magnitude between an M-component and a dart leader. leader. There is is aa possibility possibility for for aa direct direct experimental experimental evaluation There evaluation of of the the potential potential distribution in in aa wave wave approaching approaching the the earth. earth. This be done done by by measuring measuring distribution This can can be the electric electric field field gain gain at at the the earth earth during during the the wave wave motion. motion. If If the the potential potential the slowly rises rises along along the the whole whole wave wave length, length, as slowly as in in an an M-component M-component (figure (figure 4.24(a)), the the distributions distributions of of the the potential potential and and linear 4.24(a», linear charge charge from from the the initial initial h, to to the the cloud cloud can front point, point, located located at at height height h, can be be considered considered to to be be front linear, T(X) ~ ( x= = ) aq(x a,(x -- h) h ) (x (x isis counted counted from from the 2 h). linear, the earth earth and and xx> h). For For the the


218 218


field at distance distance r from from aa vertical vertical channel. channel, we we find find field at

[

aq

AE(r)= 27rEO

+

+r 2 y 2

]

lnH ( H 2 H-h h (h2 + r2)1'2 ( H 2 r2)1/2

+

+

(4.63 (4.63))

where of the at least, several times where H is is the the height height of the grounded grounded channel. channel. If If H is, is, at least, several times r, the of field A E on on the larger than larger than 1', the dependence dependence of field 6.E the distance distance between between the the registration point and the channel be only registration point and the channel line line will will be only logarithmic. logarithmic. The The same same is is true true of of the the front front duration duration of of aa field field pulse. pulse. The situation situation must must be be quite quite different different for for aa dart dart leader leader with with the the abrupt abrupt The potential potential drop drop at at the the wave wave front, front, since since the the first first approximation approximation in in the the field field calculation calculation may may assume assume aa uniform uniform potential potential along along the the channel channel and and ~ ( x= = ) const const at at xx > h. h. This This gives gives formulae formulae (3.6) (3.6) and and (3.7), (3.7), which which yield yield the the T(X) maximum (r) rv r~]. Such a large difference in the field variation AE,,,(r) r ' . Such a large difference in the field variation maximum value value 6.E max is easily easily detectable detectable experimentally, experimentally, especially especially if if we we remember remember that that it it concerns concerns is not not only only the the field field pulse pulse amplitude amplitude but but also also its its front front rise rise time. time. To To see see that that this this is so, it it is is sufficient sufficient to to introduce introduce into into (4.63) (4.63) and and (3.6) (3.6) the the h-coordinate h-coordinate for for the the is so, wave ut. wave front, front, expressed expressed through through the the respective respective velocities: velocities: h == H -- vt. Triggered lightning lightning is is aa perfect perfect source source for for such such measurements. measurements. A triggered triggered Triggered lightning is is initiated initiated by by launching launching aa small small rocket rocket raising raising aa very very thin thin wire wire which which lightning evaporates during during the the development development of of the the first first component. component. The The point point of of evaporates so itit is is easy easy to to contact of of the the lightning lightning with with the the earth earth is is strictly strictly defined, defined, so contact position current current detectors detectors at at the the necessary necessary distances. distances. Besides, Besides, the the channel channel at at position the earth earth follows follows the the wire wire track track and and is is strictly strictly vertical, vertical, as as is is implied implied in in the the numerical numerical formulae. formulae. Such Such measurements measurements have have been been partly partly made made [44-45]. [44-451. the A E at at distances distances In section section 3.5, 3.5, we we discussed discussed the the measurements measurements of of field field 6.E In r1 == 30 30m and r2 r2 == 500 500m from the the channel channel during during the the dart dart leader leader developdeveloprj m and m from ment. These These measurements measurements were were not not synchronized. synchronized. However, However, the the ratio ratio ment. 17.4 for for approximately approximately equal equal currents currents is is nearly nearly the the AE(30)/AE(500) == 17.4 6.E(30)/6.E(500) r2/r1 == 16.7. 16.7. same as as r2lr] same The field field measurements measurements for for M-components M-components have have been been reported reported only only for for The = 30m 30m [42]. [42]. The The oscillogram oscillogram of of 6.E(t) A E ( t ) is is accompanied accompanied by by aa simultaneous simultaneous I'r = 800A and the the front front registration of of aa current current impulse impulse with with the the amplitude amplitude of of 800 registration A and ps. The The duration duration of of the the impulse impulse front front 6.E A E is is approximately approximately rise time time rv -100 rise 100 /-lS. the same, same, but but the the field field reaches reaches its its maximum maximum of of l350Vjm 1350 V/m earlier, earlier, when when the the the current has has reached reached half half of of its its maximum maximum amplitude amplitude (until (until the the potential potential current wave arrives, arrives, the the current current at at the the earth earth is is zero, zero, whereas whereas the the field field begins begins to to wave rise and rise since since its its start). start). Figure Figure 4.26 4.26 shows shows the the calculated calculated functions functions i(t) and 6.E(t) m and m. The A E ( t ) at at the the observation observation points points with with r = 30 30m and 500 500m. The long long line line C1 = lOpFjm, 10pF/m, model described described in in section section 4.7.1 4.7.1 was was used used with with the the same same C model j = L1j == 2.7 2.7 /-lH/m, pH/m, and and R jI(0) (0) == 10 10 [lIm. njm. The The length length of of the the grounded grounded channel channel L was 4000 4000m and that that of of the the intercloud intercloud leader leader contacting contacting itit was was 2000 2000m. The was m and m. The of 800 800 A was was reproduced reproduced in in the the calculation calculation experimentally observed observed current current of experimentally at the the leader leader potential potential U U,i = = 9.7 9.7MV. Under these these conditions, conditions, the the field field MV. Under at N


References

219 219

~PO)

1.5

800

,-

,

I

\

• 600 is

.§

.,

\,

1.0

,, ,

-0

!.;: (,,)

,,

,,

400

~

1)

,, ,,

~:::l

U

.,,

'.6 ,,

(,,) ~

,,

,,

200

0.5 ~ ,,

' ..........

o+---,-.........---r-.,.....--,-~-"T"""--+-

o

100

200 300 Time, IlS

0.0

400

Calculated variations variations of of the the electric electric field field at at the earth’s surface surface due due to to the Figure 4.26. Calculated the earth's the M-component under under the conditions of of figure figure 4.24. The dashed dashed curve curve shows shows the the M-component the conditions 4.24. The current impulse impulse II.. current amplitude of of 1500 1500V V/m at the the point point r == 30 30 m m is is close close to to the measured value. value. It It amplitude 1m at the measured 4.26 that that the the temporal temporal parameters of the the current current impulse impulse follows from from figure figure 4.26 follows parameters of = 500 500m, the are also also consistent consistent with the measurements. At the the point point r = are with the measurements. At m, the calculated field field amplitude amplitude is is aa factor factor of of three smaller and and the for the the calculated three smaller the time time for maximum amplitude m. Both 30m. Both parameters parameters maximum amplitude is is nearly nearly the the same same as as for for r == 30 would differ by an order order of of magnitude magnitude in in aa dart dart leader leader with with this this increase increase in in would differ by an r.r . Therefore, Therefore, the the diffusion diffusion model model of of the the M-component M-component reproduces reproduces fairly fairly well available observations. observations. It It would, would, certainly, certainly, be most desirable desirable to well the the available be most to make simultaneous field field registrations registrations at at different different distances distances from from aa grounded grounded make simultaneous lightning channel. channel. lightning

References References [I] [l] Berger Berger K, K, Anderson Anderson R R Band B and Kroninger Kroninger H H 1975 1975 Electra 41 41 23 23 [2] Idone Idone V VP P and and Orville Orville R RE E 1985 1985 J. J. Geophys. Geophys. Res. 90 6159 6159 [2] [3] Antsurov Antsurov K K V, V, Vereschagin Vereschagin II P, P, Makalsky Makalsky L LM M et al1992 a1 1992 Proc. 9th Intern. Con! Con$ [3] on Atmosph. Electricity 11 (St Voeikov Main (St Peterburg: Peterburg: A I Voeikov Main Geophys. Geophys. Observ.) Observ.) 360 360 [4] A, Makalsky Izvestiya. [4] Vereschagin Vereschagin II P, P, Koshelev Koshelev M M A, Makalsky L LM M and and Sysoev Sysoev V VS S 1989 1989 Izvestiya. 100 Akad. Nauk SSSR, S S S R , Energetika i transport 4 100 [5] Simpson Simpson G GC C and and Robinson Robinson G GD D 1941 1941 Proc. R 117 281 [5] R.. Soc. Soc. London A 117281 [6] Res. 65 1873 [6] Kasemir Kasemir H HW W 1960 1960 J. J . Geophys. Geophys. Res. 1873


220 220

Physical processes in a lightning discharge Physical

[7] [7] [8] [8] [9] [9] [lo] [10] [ll] [11] [12] [12]

Gorin Nand Gorin B BN and Shkilev Shkilev A AV V 1976 1976 Elektrichestvo 6 31 31 Geophys. Res. 76 1078 1078 Proctor D DA A 1971 1971 J. Geophys. Proctor Mazur V, Gerlach Gerlach JJ C C and and Rust Rust W WD D 1984 1984 Geophys. Geophys. Res. Lett. 11 11 61 61 Mazur V, V, Rust Rust W WD D and and Gerlach Gerlach JJ C C 1986 1986 J. J. Geophys. Geophys. Res. 91 91 8690 8690 Mazur V, Mazur Raizer Yu P 1991 1991 Gas Discharge Physics (Berlin: (Berlin: Springer) Springer) p 449 449 Raizer Yu P of Shock Waves Waves and HighZel’dovich Ya Ya Band B and Raizer Raizer Yu Yu P P 1968 1968 Physics of Zel'dovich (New York: York: Academic Academic Press) Press) p 916 916 Temperature Hydrodynamic Phenomena (New [13] [13] Schon1and Schonland B B 1956 1956 The The Lightning Discharge. Handbuch der Physik 22 (Berlin: (Berlin: Springer) 576 576 Springer) E 1999 1999 J. Geophys. Res. 104 104 [14] Orvill Orvill R RE [14] J. Geophys. [15] Gorin Gorin B BN and Shkilev Shkilev A AV V 1974 1974 Elektrichestvo 2 29 29 [15] Nand EM M and and Raizer Raizer Yu Yu P P 1997 1997 Spark Discharge (Boca (Boca Raton: Raton: CRC CRC Press) Press) [16] Bazelyan Bazelyan E [16] pp 294 294 Gegechkori N M, Drabkina Drabkina S II and and Mande1'shtam Mandel’shtam S L L 1947 1947 Zh. Zh. [17] Abramson I S, Gegechkori [17] Abramson IS, N M, Eksper. i Teor. Teor. Fiz. 17862 17 862 [18] Drabkina Drabkina S II 1951 1951 Zh. Zh. Eksper. ii Teor. Teor. Fiz. Fiz. 21 473 473 [18] [19] Dolgov Dolgov G GG G and and Mandel'shtam Mandel’shtam S L L 1953 1953 Zh. Zh. Eksper. Eksper. ii Teor. Teor. Fiz. Fiz. 24691 24 691 [19] [20] N 1958 1958 Soviet Phys. JETP 7 (Eng. (Eng. Trans.) Trans.) 1068 1068 [20] Braginsky Braginsky S N [21] Zhivyuk Zhivyuk Yu Yu Nand N and Mandel'shtam Mandel’shtam S L L 1961 1961 Soviet Phys. Phys. JETP 13 13 (Eng. (Eng. Trans.) Trans.) [21] 338 338 14 2111 and and 2124 2124 [22] P100ster Plooster M MN N 1971 1971 Phys. Fluids 142111 [22] [23] Paxton Paxton A A N, N, Gardner Gardner R RL L and and Baker Baker L L 1986 1986 Phys. Fluids 29 2736 2736 [23] [24] [24] Sneider Sneider M MN N 1997 1997 Unpublished report [25] Gorin Gorin B BN N and and Markin Markin V V II 1975 1975 in in Research of of Lightning and High-Voltage [25] (Moscow: Krzhizhanovsky Krzhizhanovsky Power Power Engineering Engineering Inst.) Inst.) pp 114 114 (in (in Discharge (Moscow: Russian) Russian) [26] Baze1yan Bazelyan E E M, M, Gorin Gorin B B Nand N and Levitov Levitov V V II 1978 1978 Physical and Engineering [26] of Lightning Protection (Leningrad: (Leningrad: Gidrometeoizdat) Gidrometeoizdat) pp 223 223 (in (in Fundamentals of Russian) Russian) [27] 10 [27] Gorin Gorin B BN N 1985 1985 Elektrichestvo 4 10 Proc. 9th Intern. Intern. Con! Conf. on Atmosph. Electricity 1 (St (St Peterburg: Peterburg: [28] Gorin Gorin B BN N 1992 1992 Proc. [28] Voeikov Main Main Geophys. Geophys. Observ.) Observ.) 206 206 A I Voeikov A J. Geophys. Geophys. Res. Res. 88 88 6555 6555 [29] Jordan Jordan D DM M and and Uman Uman M MA A 1983 1983 J. [29] [30] Rakov Rakov V VA A and and Uman Uman M M A 1998 1998 IEEE Trans. Trans. on EM E M Compatibility 40 40 403 403 [30] in Lightning, vol. vol. 1, 1, Physics Lightning (R (R Golde Golde (ed) (ed) New New York: York: [31] Berger Berger K K 1977 1977 in [31] Academic Academic Press) Press) pp 119 119 E 1966 1966 Bull. Bull. SEV S E V 57(13) 57(13) 11 [32] Berger Berger K K and and Vogrlsanger Vogrlsanger E [32] [33] Schonland Schonland B, B, Malan Malan D D and and Collens Collens H H 1935 1935 Proc. Proc. Roy. Roy. Soc. Soc. London Ser A 152 152 [33] 595 595 [34] [34] Schon1and Schonland B, B, Malan Malan D D and and Collens Collens H H 1938 1938 Proc. Proc. Roy. Roy. Soc. Soc. London Ser A 168 168 455 455 [35] [35] Orvill Orvill R RE E 1968 1968 J. J . Geophys. Geophys. Res. Res. 736999 73 6999 RE E and and Idone Idone V V PP 1982 1982 J. J . Geophys. Geophys. Res. Res. 87 87 11177 11177 [36] Orvill Orvill R [36] E PP 1974 1974 J. J. Geophys. Geophys. Res. Res. 794542 79 4542 [37] Krider Krider E [37] [38] (New York: York: McGraw McGraw Book Book Company) Company) pp 300 300 [38] Uman Uman M MA A 1969 1969 Lightning (New A, Uman Uman M M A et al1993 a1 1993 J. J . Geophys. Geophys. Res. 9822887 98 22887 [39] Fisher Fisher R R G, G, Rakov Rakov V V A, [39] V A, A, Uman Uman M MA A et al1992 a1 1992 Proc. Proc. 9th Intern. Con! Con$ on on Atmosph. [40] Fisher Fisher R R G, G, Rakov Rakov V [40] (St Petersburg: Petersburg: A A II Voeikov Voeikov Main Main Geophys. Geophys. Observ.) Observ.) pp 873 873 Electricity 3 (St


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[41] Phys. Rep. 21 21 [41] Bazelyan Bazelyan E EM M 1995 1995 Fiz. Plazmy 21497 21 497 (Engl. (Engl. transl.: transl.: 1995 1995 Plasma Phys. 470) 470) [42] Rakov Rakov V V A, A, Thottappillil Thottappillil Rand R and Uman Uman M MA A 1995 1995 J. J . Geophys. Res. 100 100 25701 25701 [42] [43] N 1999 transl.: 1999 [43] Sinkevich Sinkevich 00 A A and and Gerasimov Gerasimov D DN 1999 Fiz. Plazmy 25376 25 376 (Engl. (Engl. transl.: 1999 Plasma Phys. Rep. 25 339) 339) [44] Uman M A et al1995 [44] Rubinstein Rubinstein M, M, Rachidi Rachidi F, F, Uman MA a1 1995 J. J . Geophys. Geophys. Res. 100 100 8863 8863 [45] Uman M J. Geophys. [45] Rakov Rakov V V A, A, Uman M A, A, Rambo Rambo K K JJ et al1998 a1 1998 J. Geophys. Res. 103 14117 14117


Chapter 5

Lightning Lightning attraction by objects

In this this chapter, chapter, we we shall shall describe describe the the way lightning channel channel chooses chooses aa point In way aa lightning point to terrestrial or to strike strike (a (a terrestrial or aa flying flying body). body). This This is is the the principal principal issue issue for for lightning lightning protection technology. technology. In In any any case, case, aa direct direct stroke stroke is is more more hazardous hazardous than than aa protection remote lightning lightning effect effect via the electromagnetic electromagnetic field field or or shock shock wave wave in in the air. remote via the the air. Historically, direct direct lightning lightning strokes strokes were observed earlier earlier than indirect ones, ones, Historically, were observed than indirect and research into protection problems was associated with and the the first first research into lightning lightning protection problems was associated with direct strokes. strokes. direct Everyday experience experience and and scientific scientific observations, observations, including including those those made Everyday made as far far back back as as the the 18th 18th century, century, indicate indicate that that lightning lightning most most often often strikes strikes as individual the earth. towers, individual structures structures elevated elevated above above the earth. These These may may be be towers, churches, high open just high high trees. this list churches, houses houses on on high open hills, hills, and and just trees. Today, Today, this list is is much longer longer and and includes includes power lines, transmitting transmitting and and much power transmission transmission lines, receiving antennas, receiving antennas, and and the the like. like. The The experience experience in in maintaining maintaining such such structures that the the frequency with the structures indicates indicates that frequency of of strokes strokes increases increases with the object's height. This was used used as basis for the most most common object’s height. This observation observation was as aa basis for the common lightning protection protection techniques. techniques. A grounded the object grounded rod rod higher higher than than the object to to lightning lightning rod rod -- put up in in the the vicinity of the object is is be protected -- aa lightning be protected put up vicinity of the object supposed the object. supposed to to attract attract most most strokes, strokes, thus thus protecting protecting the object. The The underlying underlying principle has not not changed was principle of of this this approach approach has changed since since the the first first lightning lightning rod rod was constructed has changed constructed two two and and aa half half centuries centuries ago. ago. What What has changed is is the the requirerequirement the protection become extremely ment for for the protection reliability, reliability, which which have have become extremely stringent. stringent. For reason, the with exceptions rather than than the For this this reason, the specialists specialists have have to to deal deal with exceptions rather the rules, focusing focusing on on the the rare cases of of lightning lightning breakthroughs to the the object object rules, rare cases breakthroughs to being they lead lead to to emergencies emergencies and and sometimes sometimes to being protected, protected, because because they to catastrophes. catastrophes. The timeThe study study of of lightning lightning attraction attraction mechanisms mechanisms is is extremely extremely timeconsuming the number number of consuming and and expensive. expensive. Even Even aa simple simple measurement measurement of of the of lightning various heights heights is very hard hard to lightning strokes strokes at at objects objects of of various is very to arrange. arrange. Most houses and Most apartment apartment houses and industrial industrial premises premises in in Europe Europe are are less less than than 222 Copyright © 2000 IOP Publishing Ltd.

The The equidistance principle

223

50 m high. high. On On the the average, average, aa lightning lightning strokes strokes aa 50m 50 m building building once in five five 50m once in years. attracts years. Every Every kilometre kilometre of of aa power power transmission transmission line line 30m 30 m high high attracts approximately lightning discharge of approximately one one lightning discharge per per year. year. Long-term Long-term observations observations of aa large and multi-kilometre large number number of of buildings buildings and multi-kilometre transmission transmission lines lines are are necessary to to accumulate accumulate aa representative representative statistics. statistics. The The difficulties difficulties increase increase necessary many-fold when when one one needs needs to to extract extract information information on on the the protection protection reliabilreliabilmany-fold ity from from the the observational observational statistics. statistics. To To illustrate, illustrate, 10-20 10-20 years years of of continuous continuous ity observations of of aa 50 50 m m building building would would be be required required to to obtain obtain information information on on observations the lightning lightning discharge discharge frequency, frequency, and and at at least least 1000 1000 years years would would be be necessary necessary the to check whether its lightning lightning rod can really really provide provide aa '99% ‘99% protection’ to check whether its rod can protection' promised by by the the rod rod producers. producers. promised In situation like evaluations, In aa situation like this, this, one one has has to to resort resort to to theoretical theoretical evaluations, and this one reason attraction theory focal and this is is one reason why why lightning lightning attraction theory has has been been the the focal point of research specialists. Here, research for for many many lightning lightning specialists. Here, as as in in many many other other point of lightning there is an acute lack of of factual factual data. data. The The available available lightning problems, problems, there is an acute lack evidence evidence obtained obtained from from laboratory laboratory investigations investigations on on long long sparks sparks does does not not always always provide provide an an unambiguous unambiguous interpretation, interpretation, and and this this makes makes one one treat treat with with caution caution many, many, even even generally generally accepted, accepted, concepts. concepts. We We shall shall focus focus on on the the most most advanced advanced approaches, approaches, discussing, discussing, where where necessary, necessary, alternative alternative hypotheses. hypotheses.

5.1 5.1

The equidistance principle principle

This approach is oldest oldest and in its theoretical formulation. formulation. This approach is and clearly clearly correct correct in its theoretical Suppose is located located on on aa flat flat earth's earth’s Suppose that that an an object object of of small small area area and and height height h is surface surface (it (it is is aa rod rod electrode electrode in in laboratory laboratory simulations). simulations). Let Let us us assume assume further further that that aa lightning lightning channel channel is is shifted shifted from from it it horizontally horizontally at at aa distance distance r, r, and and the the channel channel tip tip is is at at an an altitude altitude H Hoo (figure (figure 5.1). 5.1). In In order order to to predict predict whether whether the the object the earth, we shall the lightning lightning will will strike strike the object or or the earth, we shall take take into into account account the breakdown breakdown voltage voltage measurements measurements of long air with aa sharply nonthe of long air gaps gaps with sharply nonuniform electric They show show that that the the longer longer the the gap, gap, the the higher higher the the uniform electric field. field. They average voltage required required for its breakdown breakdown and and the the longer longer the the time time necessary necessary average voltage for its for the discharge discharge formation. This means means that that the the shortest shortest gap gap has has the the best best for the formation. This chance to experience experience aa breakdown, breakdown, provided provided that that the the same voltage is is applied applied chance to same voltage simultaneously simultaneously to to several several gaps. gaps. Let Let us us keep keep in in mind mind that that the the distance distance from from the the 2]1/2, is h)2 + r shorter than that to the lightning [(Ho h)’ r2I1/’, is shorter than that to the lightning tip tip to to the the object, object, [(H o earth's earth’s surface surface H Hoo at at (5.1 )

The distance distance Re, R eq is is known known as as the the equivalent equivalent attraction radius for for an an object of The attraction radius object of height h. h. It indicates the the surface from which which lightning lightning discharges that height It indicates surface area, area, from discharges that


224

Lightning attraction by objects

Estimating the the equivalent equivalent attraction attraction radius. radius. Figure 5.1. Estimating

Hoo are are attracted attracted by by the the object. object. For For aa compact compact have descended descended to to the the altitude altitude H have is aa circle circle of of area area Seq Seq~ M 7rR~q; T R ; ~for for ; an an object of of small small cross cross section, section, this this is object >> h and and width width b « hIo Zlwo is is found found as as number

+

Nj({',.fo) = nj


J:,

5 s(I)r..p(I) dl

(5.4)

228


500l-----+----+----+-~+_f_+_t

400 l-------+---+--~~I'_+~f_+-+l 300 l----+------:~-+--,'---i--:---r-I_+_r_+___+_7'--i

200 J-----+-..".4-f---'--++----r---t-+-+-+--r-t----t-t-t-i

1m, 50

kA

150

100

Figure 5.4. The The dependence dependence of of striking striking distance distance on on lightning lightning currents. currents. The The lower lower curve using [5] curve is is plotted plotted using using [4] [4] data, data, the the upper upper using [5] data. data. The The spread spread region region isis hatched. hatched.

where where 'P(I) p(Z) is is the the probability probability density density of of current current of of amplitude amplitude IZ found found from from natural measurements. To total number natural measurements. To find find the the total number of of strokes, strokes, the the lower lower limit limit of (5.4) should should be be taken taken to to be be zero. zero. of the the integral integral in in (5.4) The in current current The generally generally correct correct idea idea of of differentiating differentiating distances distances rr,s in amplitude to refine refine the There is is no no factual factual amplitude actually actually fails fails to the calculation calculation of of N 1I•. There ( Z M ) experimentally, experimentally, while while information to to determine determine the the function function rrss == ff(IM) information so that that theoretical evaluations suffer suffer from from an an unacceptably unacceptably large large spread, spread, so theoretical evaluations the values times (figure the values obtained obtained by by different different authors authors differ differ several several times (figure 5.4). 5.4). The heights could, The stroke stroke statistics statistics for for objects objects of of various various heights could, to to some some extent, extent, but it it proves proves be used used for for fitting fitting the the calculated calculated total total number of strokes strokes N bI , but be number of unsuitable for finding finding the the function function rr,s ==f(Z,w). unsuitable for f(IM)' The procedures in The calculation calculation procedures in the the electrogeometric electrogeometric approach approach do do not not involve involve aa strong strong dependence dependence of of stroke stroke frequency frequency on on an an object's object's height. height. Indeed, for for aa single single construction, construction, like like aa tower, tower, the the capture capture region region projection projection Indeed, on on to to the the earth's earth's plane plane is is aa circle circle of of radius radius 2)1/2 at R == (2r (2r,h h2)'I2 at sh -- h

R == rr,s

at at

rrss ~ 2h rr,s

h must frequency powerful discharges height for compact compact objects objects and and as as h l/2 for for extended extended ones. ones. Actually Actually slower than than h for slower both of both of these these dependencies dependencies are are steeper. steeper.

5.3

The probability probability approach approach to finding finding the stroke stroke point point

choice of of the the discharge discharge path path through through an an air air gap gap contradicts contradicts A predetermined predetermined choice the Neither aa spark the experience experience gained gained from from long long spark spark investigations. investigations. Neither spark nor nor aa


The finding the stroke point The probability approach to Jinding

229 229

lightning path. When voltage is is simultaneously simultaneously lightning travel travel along along the the shortest shortest path. When voltage various lengths, lengths, it it is is the the longest longest applied applied in in parallel parallel to to several several air air gaps gaps of of various gap This is is supported supported by by the the large large gap that that is is sometimes sometimes closed closed by by aa spark. spark. This deviation u a for for multi-metre multi-metre spread spread of of breakdown breakdown voltages: voltages: the the standard standard deviation of the the average average breakdown breakdown gaps gaps with with aa sharply sharply non-uniform non-uniform field field is is 5-10% 5-10% of voltage. voltage. If the distributions distributions of of breakdown breakdown If two two gaps, gaps, tested tested individually, individually, possess possess the (U), then, provided provided that that the the voltage voltage of of the the probability probability densities densities 'PI cpl (U) ( U ) and and 'P2 p2( U ) ,then, the breakdown breakdown probprobvoltage voltage of of aa common common source source is is applied applied simultaneously, simultaneously, the is described described as as ability ability for for one one of of the the gaps, gaps, say, say, the the first first one, one, is (5.6) the probability probability of of the the gap gap breakbreakwhere where
b = Nb/N1 shows the fraction of lightnings, attracted attracted by by the the whole whole system system of of grounded grounded electrodes, electrodes, Nj, N I , which which have have missed the the lightning lightning rod rod to to strike strike the the object. object. The The calculations calculations were were made made missed ra== 0.1 0.1 for for objects objects of of height height hho 30- 150 m. with the the attraction attraction standard standard O"a with o == 30-l50m. The actual actual protective protective effect effect is is achieved achieved only only if if the the height height of of the the lightning lightning The rod considerably considerably exceeds exceeds that that of of the the object. object. For For short short constructions constructions with, with, rod 2 say, hhoo == 30m, 30m, aa 99% 99% protection protection reliability reliability (q>b (ab = = 10lo-*) requires the the lightlightsay, ) requires ning rod rod height height excess excess of of !:!..h Ah ~ x 0.2h 0.2ho, which is is quite quite feasible feasible technically technically ning o, which because m high because itit is is equal equal only only to to 66 m m above above the the object. object. An An object object 150 150m high will will require m higher ), which require aa lightning lightning rod rod 50 50m higher than than the the object object (!:!..h (Ah ~ x 0.3h 0.3ho), which o

1.2

1.3

1.4

1.5

1.6

h/h r 0 Figure 5.16. The The evaluated evaluated probability probability of of aa lightning lightning breakthrough breakthrough to to an an object object of of Figure height hho, protected by by an an adjacent adjacent lightning lightning rod rod of of height height hh,r > > hho. height a, protected a.


W h y are several lightning rods more effective than one? Why

247 247

will be be more more expensive expensive and and complicated. complicated. In In technical applications, the the tentenwill technical applications, dency of the Qb(Ah) curves to saturation is very important. This tendency dency of the if>b(6.h) curves to saturation is very important. This tendency becomes greater greater with with increasing increasing construction construction height, that aa becomes height, which which means means that single rod will be be ineffective ineffective for for aa high high protection protection reliability. It is is hard hard to single rod will reliability. It to lOOm with aa reliability reliability above above 0.999% 0.999% protect an an object object with protect with hhoo > 100 m with 3 (ab== 10lop3), an object object with with h hoo > 150m 150m above above 0.99%, 0.99%, etc. etc. The The higher higher the the (if>b ), an construction, the more complicated is the problem, and this is the reason construction, the more complicated is the problem, and this is the reason why the Ostankino Tower Tower is is unable unable to to protect protect itself. itself. Nine lightning strokes strokes why the Ostankino Nine lightning 200 m from the top [18]. were registered photographically along its length of were registered photographically along its length of 200 m from the top [18]. The protection efficiency decreases as the distance between the top of aa The protection efficiency decreases as the distance between the top of of an object of similar height increases, reducing high lightning rod and that high lightning rod and that of an object of similar height increases, reducing the the mutual mutual effect effect of of counterleaders. counterleaders. Formally, Formally, this this manifests manifests itself itself as as aa larger larger U,, in in accordance accordance with (5.19). Sooner Sooner or or later, later, its its effect effect begins choice standard standard (Jc, choice with (5.19). begins to dominate to dominate over over that that of of lengths lengths in in formula formula (5.17), (5.17), so so the the upper upper limit limit of of the the stops rising. rising. probability integral integral A c, stops probability

5.8

Why are several lightning lightning rods rods more effective effective than than one? one?

The be found us consider The answer answer to to this this question question can can be found geometrically. geometrically. Let Let us consider two two lightnings the same plane going lightnings which which travel travel in in the same vertical vertical plane going through through an an object object and rod in both leader and its its lightning lightning rod in opposite opposite directions. directions. Suppose Suppose both leader tips tips are are at at an the same Hoo at at the same distance distance from from the the rod. rod. They They have, have, theretherean attraction attraction altitude altitude H fore, to be be attracted the object-rod fore, an an equal equal chance chance to attracted by by the object-rod system. system. The The only only will approach difference on the the lightning lightning rod rod side side difference is is that that one one leader leader will approach it it on (version protected (version 2). (version 1) 1) and and the the other other on on the the side side of of the the object object to to be be protected (version 2). Assume, for that the the displacement Assume, for definiteness, definiteness, that displacement of of the the lightnings lightnings relative relative to to the the rod radius Re, R eq = = 3h 3h,r (i.e., (i.e., an an average average displacedisplacerod axis axis is is equal equal to to the the attraction attraction radius ment), Ho Ho = between the the object = 5h 5h,,n and and the the horizontal horizontal distance distance between the rod rod and and the object ment), is upper limit the probability probability integral AY== hh,r -- hhoo « ~ (Jc u,25fih, 25 V2h r ' ~

=

A

~ =

Ar 6.r

4 c22 ~ (J 25V2h u,25&hr

r’

Consequently, probability integral version 22 also Consequently, the the Probability integral from from (5.16) (5.16) for for version also decreases, decreases, increasing increasing sharply sharply the the probability probability of of striking striking the the object. object. To To Ar = = 0.2h 0.2hrr and and (Jc uc = = 0.01, 0.01, the takes the the values values illustrate, for for 6.r the parameter parameter A c, takes illustrate, and 0.57, 0.57, respectively. When the lightning approaches approaches on on the the lightning lightning of 44 and of respectively. When the lightning rod side, the probability probability of of striking striking the the object object is, is, according according to to (5.16), (5.16), rod side, the nearly but on on the the object object side side it it is is 0.28. 0.28. Therefore, Therefore, aa single single lightning lightning nearly zero, zero, but rod protect an reliably only the 'back', rod can can protect an object object reliably only from from the ‘back’, while while its its protection protection if efficiency from from the the 'front' ‘front’ is is much much lower. lower. This This situation situation can can be efficiency be rectified rectified if between two the the object object to to be be protected protected is is placed placed half-way half-way between two rods; rods; it it is is still still better if so on on -- this this becomes becomes only only aa matter matter of of better if there there are are three three rods rods and and so


248 248


cost. cost. No No rod rod palisades palisades are are known known from from the the protection protection practice; practice; nevertheless, nevertheless, it of interest interest with with aa protecting protecting wire, wire, it is is tempting tempting to to surround surround the the object object of especially especially if if itit is is not not very very high high but but occupies occupies aa large large area. area. As us consider consider aa case case simple simple for for the the calculations. calculations. This This As an an illustration, illustration, let let us will us to to get get numerical numerical results results and and demonstrate demonstrate the the calculation calculation will allow allow us procedures. m is Ro lOOm is densely densely filled filled by by procedures. Suppose Suppose aa circle circle of of radius radius R o == 100 constructions = 10m. All of them must be protected with aa constructions of of height height hho = 10m. All of them must be protected with o 0.99% 0.99% reliability, reliability, i.e., i.e., the the probability probability of of aa lightning lightning stroke stroke should should not not exceed exceed R . )2 < Rand R and ddo,,, = [ ( r R0)2 ( H o hO)2]1’2 at > Ro. Omin O o O o The The calculation calculation procedure procedure reduces reduces to to finding, finding, for for every every value value of of r,r , the the and Act Act in in the the integrals integrals of of (5.7) (5.7) and and (5.16) (5.16) to to calculate calculate upper limits limits A a, and upper P,(Y)and and Pc(r), P , ( r ) , and and to to (extract from from tables) tables) these these integrals, integrals, which which give give Pa(r) (extract calculate (5.20). Practically, Practically, it it is is sufficient sufficient to to make make the the calculate the the integrals integrals of of (5.20).

+

The value value of of the the choice choice standard standard aoc necessary for for the the calculation calculation of of A c, is is found found from from formula formula tf The c necessary (5.19). taking taking into into account account the the distance distance D between between the the protector protector top top and and the the point point on on the the object's object’s (5.19), ua "" 0.1. surface nearest nearest to to the the lightning lightning with with the the instantaneous instantaneous coordinate coordinate r;r; aa surface 0.1.


Some technical parameters of of lightning protection

249

45 40

S 35 ~

30 25 20 I

.1

2 10'3 1010'\ 10” 1O‘* 10’‘ Lightning breakthrough breakthrough probability 0 Lightning probability

The object object of of 10 10 m and of of 100 100 m m radius is protected by aa bounding bounding Figure 5.17. The m height height and radius is protected by circular wire. In presented the wire height height h necessary necessary to circular wire. In the the graph graph is is presented the evaluated evaluated wire to of aa lightning lightning breakthrough to the object up to the value decrease the probability of decrease the probability breakthrough to the object up to the value of of CJshown shown on on the the abscissa abscissa axis. axis.

Ar M (0.1 (0.1 -- 0.2)h 0.2)hrr and and finish finish them them when when P,(r) calculations with with the the step step 6.r:::::: calculations Pa(r) 6 drops _10- 7 with probability integral drops to to 1010-6-10-7 with growing growing r.r . If If the the probability integral is is given given reasonreasonably by borrowing ably (by (by an an empirical empirical formula formula or or by borrowing it it from from aa table, table, e.g., e.g., using using aa spline), proves so with spline), the the volume volume of of calculations calculations proves so small small that that they they can can be be made made with modern computer, computer, the the time time necessary necessary for for calculator. With With aa modern aa programmed programmed calculator. numerical computations is only that for the data input. numerical computations is only that for the data input. The The calculations calculations made made for for the the above above example example are are shown shown in in figure figure 5.17. 5.17. The probability probability of of aa lightning lightning breakthrough breakthrough to to the the object object decreases decreases to to the the The 2 given of 10lo-* when when the the protective protective wire wire is is suspended suspended at at aa reasonable reasonable given value value of height m. Note, Note, for that aa single placed height hh,r :::::: 34 34m. for comparison, comparison, that single lightning lightning rod rod placed at provides the at the the centre centre of of aa similar similar area area provides the same same protection protection reliability reliability only only if its its height height is is h, 150m. Even if if one one builds builds such such aa rod, rod, the the result result may may if hr > ISO m. Even prove disappointing. prove disappointing. Quite Quite often, often, it it is is impossible impossible to to provide provide aa safe safe delivery delivery to the earth high lightning to the earth of of aa high lightning current current impulse, impulse, when when conductors conductors with with current being protected. protected. Electromagnetic current pass pass close close to to structures structures being Electromagnetic induction, induction, sparking capable capable of of setting setting aa fire, fire, etc. etc. may may also also be dangerous. sparking be dangerous.

5.9 5.9.1 5.9.1

Some technical parameters of lightning lightning protection protection The protection zone

It follows follows from from the the foregoing foregoing that that aa lightning-rod lightning-rod has has aa better chance of of It better chance intercepting descending descending lightnings lightnings if if it it has greater height above the intercepting has aa greater height above the


250


object object and and is is closer closer to to it. it. Practically, Practically, it it is is important important to to identify identify aa certain certain area area around protected. This around aa protector, protector, which which would would be be reliably reliably protected. This is is the the protection protection be considered be protected protected zone. within this this zone zone. Any Any object object located located within zone must must be considered to to be with higher than used for with aa reliability reliability equal equal to to or or higher than that that used for the the calculation calculation of of the the zone boundary. There technically constructive. zone boundary. There is is no no doubt doubt that that this this idea idea is is technically constructive. When When the the configuration configuration of of the the protection protection zone zone is is known, known, the the determination determination of the grounded reduces to or geogeoof the grounded rod rod or or wire wire height height reduces to aa simple simple calculation calculation or metrical this was was an the recent an important important factor factor in in the recent age age of of metrical construction construction -- this ‘manual’ protection designing. At At that that time, time, the general tendency tendency was was to to 'manual' protection designing. the general simplify the the zone zone configuration configuration as as much much as as possible. In Russia, Russia, for for instance, instance, simplify possible. In aa single usually aa circular vertex single lightning lightning rod rod zone zone was was usually circular cone, cone, whose whose vertex coincided with top [10]. [lo]. When When lightning lightning protection engineers realized realized protection engineers coincided with the the rod rod top that the height of the to exceed exceed that that of of the the object object to be protected protected that the height of the rod rod was was to to be 5.7), the the cone cone vertex vertex was on the the rod rod axis axis under its top top [19]. [19]. (section 5.7), (section was placed placed on under its The greater the protection reliability required, the more pointed and lower The greater the protection reliability required, the more pointed and lower was was the the zone zone cone. cone. For For aa grounded grounded wire, wire, the the protection protection zone zone had had aa double double pitch symmetry; symmetry; when when intersected intersected transversally transversally by by aa plane, it produced an pitch plane, it produced an isosceles isosceles triangle triangle with with nearly nearly the the same same dimensions dimensions as as those those of of aa vertical vertical cross protection manuals cross section section made made through through the the rod rod cone cone half. half. Lightning Lightning protection manuals give protection zones give aa set set of of empirical empirical formulas formulas to to design design protection zones for for simple simple types types of of lightning protector protector [2,4]. [2,4]. lightning The long-term long-term practice has somewhat somewhat screened screened the ambiguity The practice has the principal principal ambiguity the of the the notion notion of of protection protection zone. zone. Indeed, Indeed, having having only only one one parameter of parameter -- the abmaX one is is unable to determine determine admissible probability of aa lightning lightning stroke stroke y-' converges converges very very rapidly, so one one can can restrict The rapidly, so restrict Ei, rises with the time oneself to to the first term term only. only. Therefore, Therefore, the the field field E oneself the first rises with the time in f10iJd2l1r2; its constant constant T~ T: = = ,-I y-l = p0ad2/7r2; its value value is is 6)ls 6 ps at at iJcr ~ FZ 5 5 xX 10 lo77 (0. (Cl. m)-l m)-' and d M~ 1mm. 1 mm. This This permits the neglect of the the skin-effect skin-effect action action on on overoverand permits the neglect of voltages in long long underground cables, in in which which the current diffusion diffusion along along voltages in underground cables, the current the sheath sheath and, and, hence, time of of the the overvoltage overvoltage rise the maximum maximum rise to to the the hence, the the time T : . However, However, the the skin-effect skin-effect take 11 or or 22 orders orders of of magnitude longer than take magnitude longer than T~. in objects objects located located on on the the earth's earth's surface surface and and having having relatively relatively short short sheaths, sheaths, in


Lightning stroke at a screened screened object 1.0

285 285

IIImax

0.8 0.6

0.4

0.2

'

,I 0.0 .......-.--r-----.---,--..----,.-.......... I 200 100 150 100 150 50 Time, llS p Time,

Figure 6.7. pulse deformation 6.7. Overvoltage Overvoltage pulse deformation in in aa cable cable sheath sheath due due to to skin-effect skin-effect with with 10 IlS. p. An exponential current current impulse impulse is is duration duration of of lOOIlS 100 ps the time constant constant TT,s == 10 the time An exponential (dashed (dashed curve). curve).

in which which lightning lightning current current propagates propagates over over the the time t y-'. (6.32) loR j [-at) -- -it) ] , i-a i-a 7-a The The results results of of the the calculations calculations presented presented in in figure figure 6.7 show show that that the the skin skin effect effect elongates the overvoltage T i and and the the amplitude amplitude decreases decreases by by elongates the overvoltage pulse pulse front front to to T~ percent. several several dozens dozens of of percent. E in

6.2.5

~

geometry The effect of cross section geometry

We have have assumed assumed so so far far that that the the sheath sheath has has the the shape shape of of aa circular circular cylinder. cylinder. We In that case, uniformly along the cross In that case, the the current current is is distributed distributed uniformly along the cross section section perimeter and magnetic field this model perimeter and there there is is no no magnetic field inside. inside. But But this model is is inapplicable inapplicable to real objects, to many many real objects, for for example, example, the the fuselage fuselage or or wing wing of of an an aircraft, aircraft, having having very very complex complex cross cross section section profiles profiles with with different different curvatures. curvatures. The The current current flowing flowing through through aa non-circular non-circular sheath sheath is is distributed distributed non-uniformly non-uniformly along along its its perimeter perimeter and and the the magnetic magnetic field field is is present present inside. inside. These These factors factors affect affect the the mechanism mechanism of of overvoltage overvoltage excitation excitation by by lightning lightning current. current. Let Let us us consider consider aa two-dimensional two-dimensional sheath sheath shaped shaped as as aa cylinder cylinder of of aa nonnoncircular when the the end end effects effects are are circular cross cross section section and and aa considerable considerable length length I,I , when weak weak and and the the current current and and field field distributions distributions are are plane-parallel. plane-parallel. The The sheath sheath is uniform resistivity us subdivide subdivide is considered considered to to have have aa uniform resistivity and and thickness. thickness. Let Let us


286 286

Dangerous Dangerous lightning lightning effects tlffects on on modern modern structures structures

the f:1rk along along the the the sheath sheath into into aa set set of of N N parallel parallel conductors conductors of of aa short short length length Ark unit length length in in cross cross section section perimeter, perimeter, such such that that the the current current JJ kk per per perimeter perimeter unit time (the (the total total the the kth kth conductor conductor could could be be regarded regarded as as varying varying only only with with time mode when when current = Jkf:1rk)' JkArk). In In aa steady-state steady-state mode current in in the the kth kth conductor conductor is is iikk == they are are the the current current becomes becomes direct, direct, all all JJ kk values values are are the the same, same, since since they mere determined determined by by equal equal ohmic ohmic voltage voltage drops drops in in all all of of the the conductors. conductors. A mere that aa summing summing of of the the magnetic magnetic fields fields of of the the conductors conductors will will indicate indicate that cross section section magnetic magnetic field field may may be be present present inside inside aa sheath sheath of of an an arbitrary arbitrary cross geometry. geometry. very quickly, quickly, the the When When lightning lightning current current is is introduced introduced into into the the sheath sheath very magnetic greatly exceeds exceeds the the ohmic ohmic voltage voltage magnetic induction induction emf emf in in the the conductors conductors greatly drop. the conductors conductors will will indeed indeed form form an an drop. But But in in this this approximation, approximation, all all of of the integral namely, they they will will be be connected connected in in integral 'perfectly ‘perfectly conducting' conducting’ sheath, sheath, namely, will have have equal equal potentials. potentials. Hence, Hence, the the parallel. parallel. This This means means that that all all of of them them will is the the same. same. This This provides provides aa magnetic magnetic flux Aux coupling coupling @ for for each each conductor conductor is at the initial stage of the process: set set of of equations equations for for finding finding the the currents currents iik at the initial stage of the process: k

Lkik(0) + + Lkik(O)

N

L

M k m i m ( 0 ) == Mkmim(O)

,

m # k,

m= 1 m=l

N

M L im(0) im(O) = == III,{

(6.33) (6.33)

kk=l = 1

where inductance of of the the conductors conductors k where LLkk is is inductance, inductance, M Mkm is the the mutual mutual inductance km is and is the lightning input current. Now, in contrast to the the and m, m, and and 1IM is the lightning input current. Now, in contrast to M be different in identical identical conductors conductors steady-state steady-state mode, mode, the the currents currents will will be different even even in if We shall shall illustrate illustrate this this with with if they they are are located located at at different different sites sites of of the the sheath. sheath. We reference length I and and radius radius r located located in in the the reference to to three three parallel parallel conductors conductors of of length same plane plane so so that that the the distances distances between between the the adjacent adjacent conductors conductors are are same identical and and equal equal to to D. D.If If iill is is the the current current in in the the central central conductor conductor and and izi z identical is that that in in the the end end conductors, conductors, with with M 1122as as the is the mutual mutual inductance inductance of of the the of remote ones, we shall have adjacent adjacent conductors conductors and and M M2 3 of remote ones, we shall have 23

Lil

+ 2M12iz = Liz + M12il+ M23i2.

i1 t + 2i2 2i z = == IAM IM il

L-Mlz 1 M 3L - 4M 12 + M Z3 M Z3

~ ~~ In (2~).

The current current in in the the central central conductor The conductor is is lower lower than than in in the the end end conductors conductors because of of M M12 M23. because > M ' Z3 lz It is is easy easy to to solve solve aa set set of It of equations equations of of the the type type (6.33) (6.33) even even for for aa large large number of of conductors conductors simulating simulating aa sheath. sheath. Only Only the the calculation calculation of of interinternumber conductor distances distances is is somewhat somewhat cumbersome, cumbersome, requiring requiring knowledge knowledge of of the the conductor cross section section profile profile coordinates. coordinates. We cross We shall shall leave leave this this problem problem to to the the reader reader and illustrate, illustrate, instead, instead, the analytical solution and the analytical solution for for the the current current distribution distribution in aa long long cylindrical cylindrical sheath sheath with [9]. This This solution solution in with an an elliptical elliptical cross cross section section [9].


Lightning stroke at a screened object

287 287

is useful useful for for the the evaluation evaluation of of many many real real profiles profiles and and for for testing testing computation computation is programmes: programmes: J(x) J ( x ) ==

1M

(6.34) (6.34)

.

2 ( I1 _ 2 I a2 )] 1/2 ‘ 27r[a 2 4 a 22 -- xx2( -b b2/a2)]’I2

is the the large large and and b the the small small semiaxis semiaxis of of the the ellipse ellipse and and xx is is the the distance distance Here, a is Here, between the the ellipse ellipse centre centre and and the the calculation calculation point point projection projection on on the the large large between axis. The The ratio ratio of of the the minimum minimum linear linear current current density density (on (on the the plane plane part part of of axis. the ellipse) ellipse) to to the the maximum maximum one one (on (on its its tip) tip) is is Jrnax/lrnin Jmax/Jmin = alb. a / b . The The current current the = non-uniformity non-uniformity may may be be great great in in real real structures, structures, such such as as the the aircraft aircraft wing, wing, a / b > 100. 100. alb> There is is no no magnetic magnetic field field in in the the sheath sheath at at the the moment of time time tt == O. 0. This This There moment of is the result result of is the of the the initial initial current current distribution distribution among among the the conductors conductors owing owing to to the magnetic magnetic induction induction emf. emf. With With the the redistribution of the the currents currents under under the the the redistribution of action of of ohmic ohmic resistance, resistance, aa magnetic field will will gradually gradually arise arise in in aa nonnonaction magnetic field circular sheath. sheath. The The field field becomes the source source of of overvoltages overvoltages in in the the inner inner circular becomes the circuits of of the the object. object. By By integrating integrating numerically numerically the the set set of of equations equations circuits N

m

i:- k,

2:: i

k

= i(t)

(6.35)

k=l

where the unknown unknown voltage the length U (t) is is also also the voltage drop drop along along the length of of the the sheath sheath where U(t) 'made up' of conductors, one can find the variation in the current ‘made up’ of conductors, one can find the variation in the current distribudistribution along perimeter. The the integration tion along the the sheath sheath perimeter. The initial initial condition condition for for the integration is is the with the the solution solution to to (6.33). (6.33). The The calculation calculation accuracy accuracy increases increases with the number number N of of simulating simulating conductors. conductors. But But the the limiting limiting case case of of N = = I1 is is also also suitable suitable for the evaluation of the time constant of a transient process: T for the evaluation of the time constant of a transient process: TI, = L j1I /Rj, R1, tr = where L1 L 1 and R 1 are resistance. The where and RI are the the linear linear sheath sheath inductance inductance and and resistance. The current current is is redistributed redistributed slowly, slowly, T T,, 0.1 s, s, in in the the sheaths sheaths of of large large objects objects with with radius radius tr '" 0.1 Im m and and thickness thickness d '" 1I mm. mm. During During the the action action of of aa common common lightning lightning Yr '" 1 current impulse impulse with with t, tp ' " 100 J.lS, the the current current distribution distribution along along the the sheath sheath current 100 ps, perimeter differs differs but but little little from from the the initial initial distribution distribution profile. profile. The The results results perimeter of aa computer computer simulation simulation support support this this conclusion. conclusion. The The computation computation for for of sheath of of complex complex geometry geometry (figure (figure 6.8) 6.8) with with L 1 = 0.57 0.57 pH/m J.lH/m and and aa sheath 1= 5 R1 = = 1.05 [lIm (T = 54ms) 54ms) has has shown shown that that the the linear linear current current RI 1.05 xX 10lop5 Rjm (Ttz tz = density at at all all characteristic characteristic points points of of the the sheath sheath takes takes the the steady-state steady-state value value density for a time about 20 ms. During the first 200 J.lS typical of lightning current, for a time about 20 ms. During the first ps typical of lightning current, the density density cannot cannot change change appreciably. appreciably. the us consider consider overvoltages overvoltages across across the the insulation insulation between between an an inner inner Let us Let is placed placed very very close close to to conductor and and the the sheath. sheath. Suppose Suppose the the conductor conductor is conductor the inner inner sheath sheath surface. surface. The The contour contour area area between between the the conductor conductor and and the the the be very very small, small, and and the the internal internal magnetic magnetic flux flux will will be be unable unable to to wall will will be wall create an an appreciable appreciable induction induction emf. emf. The The voltage voltage between between the the conductor conductor create will be be equal equal to to the the integral integral of of the the ohmic ohmic component component of of the the and the the sheath sheath will and N

pv

N

N


288 288

Dangerous lightning effects on modern structures

1 2.0

2.0 1.0 1--

_

2 3

o.....-~50~~I'!'"OO~-1""!5-0-7f,,~IJS

1.0 J.:;.:.~-------

---

2

Ie aa:=::=::-" 3

OL...-----::::r::--::"':"""--=-"""""":':::--~~

O’

10 fo

20

30

40 4-0

t,t,ms ms

Evaluated evolution evolution of of aa linear linear current current density density at at indicated indicated points of the Figure 6.8. Evaluated points of the J ( t -+ + (0). XI). wing-like sheath sheath shown. shown. JIx, = wing-like == l(t longitudinal E in (x) at longitudinal electric electric field field &(x) at the the site site of of the the conductor conductor location location x. x. But But now, E in should be based based on now, the the evaluation evaluation of of Ein should not not be on the the average average current current density density in the the sheath, sheath, using using the total current current and and linear linear resistance resistance R II. .For For aa sheath sheath of of in the total thickness d, d , we thickness we have have

(x)~/d. Ein(x) == JJ(x)pld.

(6.36) (6.36)

The nearer the the maximum the higher The nearer the current current line line with with the maximum linear linear density, density, the higher the the overvoltage to the overvoltage across across the the conductor conductor insulation insulation relative relative to the object's object’s shell. shell. One One practical conclusion manifold the practical conclusion is is quite quite evident. evident. To To reduce reduce manifold the overvoltage overvoltage in in the electrical back end, the electrical line line inside inside an an aircraft aircraft wing wing and and along along its its thinnest thinnest back end, where wire closer where the the current current density density is is maximal, maximal, it it is is sufficient sufficient to to shift shift the the wire closer to the upper wing wing plane better, to the lower where the to the upper plane or, or, better, to the lower one, one, where the current current density density is minimal minimal due due to the wing wing curvature curvature (figure (figure 6.8). 6.8). Laboratory Laboratory measurements is to the measurements have confirmed this suggestion have confirmed this suggestion [10]. [lo]. Note Note the the seemingly seemingly ambiguous ambiguous character character of of the the evaluations. evaluations. The The sheath sheath cross so the the inner inner conductor conductor must must be be cross section section is is practically practically equipotential, equipotential, so under the voltage with point of under the same same voltage with respect respect to to any any point of the the sheath sheath in in aa particular particular cross section. section. However, However, the the ohmic ohmic overvoltage overvoltage component component for for aa conductor conductor cross inside inside an an elliptical elliptical cylinder cylinder (figure (figure 6.9) 6.9) with with respect respect to to points points 11 and and 2 of of the the large and and small small semiaxes semiaxes differ differ by by aa factor factor of of Jmax/Jmin = alb, a/b, in in agreement agreement large Jrnax/lrnin = with (6.34). (6.34). This This contradiction contradiction is is superficial. superficial. In In the the presence of aa magnetic magnetic with presence of field, there is the magnetic the electrical field, there is the magnetic component, component, in in addition addition to to the electrical one, one, =U U,e + U U,.M' The The distance distance between the conductor conductor and and current current line line 11 is is U = between the practically narrow practically zero, zero, and and the the magnetic magnetic flux flux induces induces nothing nothing in in such such aa narrow circuit, U M, = 0. On On the the contrary, contrary, aa wide wide circuit, circuit, made made up of aa conductor conductor circuit, ~ O. up of and by most and remote remote current current line line 2, is is affected affected by most of of the the internal internal magnetic magnetic flux. flux.

+


Lightning stroke at a screened object

289 289

6.9. The The conductors conductors inside inside an an elliptic elliptic cylinder. cylinder. Figure 6.9.

flux adds adds the the ohmic ohmic voltage voltage to the necessary necessary value value U. U. The emf emf induced induced by the flux The by the to the The evaluation evaluation of of the magnetic flux direction direction will will show show that signs of of U U,e The the magnetic that the the signs and UM, coincide coincide if if the circuit, in in which is induced, induced, is is composed composed by the and the circuit, which UM, is by the current line line with linear density density less less than the average average value; value; otherwise, otherwise, U U,e current with aa linear than the and UM U, have have opposite opposite directions. directions. Therefore, Therefore, the the values values of of U U,e and and UM U, vary vary and with the the design design circuit circuit chosen, chosen, but but the the sum sum remains remains the the same. same. with We shall shall make make use use of of this this circumstance circumstance to to find find the the time time variation variation of of the the We magnetic field field inside inside the the sheath. sheath. It It has has been been pointed pointed out out above above that that lightning lightning magnetic current i(t) i ( t ) acts acts for for such such aa short short time time that that it it cannot cannot be be redistributed redistributed radically radically current along the the sheath sheath perimeter; perimeter; therefore, therefore, we we have have J(t) J ( t ) rv w i(t) i ( t )at at any any point. point. Hence, Hence, along Ein(t) i (t)t ) for for aa thin t h n sheath sheath where where the the skin slun effect effect is is inessential. inessential. ChoosChooswe get get E we in (t) rv i( ing aa design design circuit circuit with with U U,M = = 0, 0, we we find find U(t) U (t ) = Ue(t) U, ( t ) rv Ein(t) E,, ( t ) rv i(t). i( t ). But But in in ing the general general case, case, this this is is U(t) U ( t )= = Ue(t) U e ( t )+ UM(t), U M ( t ) ,with with the the values values of of U U,e and and UM , the being being comparable; comparable; hence, hence, UM(t) U , ( t ) rv i(t). i( t ). Thus, Thus, the the induction induction component component of of overvoltages overvoltages in in aa sheath sheath is is proproportional to to lightning lightning current current rather rather than than to to its its derivative! derivative! Therefore, Therefore, the the portional magnetic magnetic flux flux penetrating penetrating into into the the sheath sheath varies varies in in time time as as the the integral integral of of current i(t). i( t ) . This This remarkable remarkable result result has has been been confirmed confirmed by by experiments. experiments. current The oscillograms oscillograms in in figure figure 6.10 6.10 illustrate illustrate aa test test current current impulse, impulse, similar similar in in The shape to to aa lightning lightning current current impulse, impulse, and and aa magnetic magnetic pulse pulse H(t) H ( t ) inside inside aa shape N

N

N

N

i -'==:t OV

0.2

0.'4

0.'6 0.6

' t,,ms ms

Figure 6.10. 6.10. Oscillograms Oscillograms of of the the test test current current and and magnetic magnetic field field inside inside the the wing-like wing-like sheath. sheath.


290 290


sheath response time time of magnetic sheath simulating simulating an an aircraft aircraft wing wing [11]. [l 11. The The response of the the magnetic field detector did not exceed 0.5I..ls, so that the H(t) pulse front to close to field detector did not exceed 0.5ps, so that the H ( t ) pulse front close 300 I..lS and an order of magnitude higher than the current impulse front 300ps and an order of magnitude higher than the current impulse front causes causes no no doubt. doubt. 6.2.6 Overvoltage Overvoltage in a double wire circuit circuit

Although metal, most most Although the the use use of of aa metallic metallic sheath sheath as as aa reverse reverse wire wire saves saves on on metal, internal be protected because they internal circuits circuits of of objects objects to to be protected consist consist of of two two wires, wires, because they are the magnetic are better better screened screened from from noises. noises. When When the magnetic field field inside inside an an object object is is zero, the case perfectly circular the lightning zero, as as is is the case with with aa perfectly circular sheath, sheath, the lightning current current raises the the potential potential of wire relative the shell raises of each each wire relative to to the shell but but no no overvoltage overvoltage arises the wires. wires. This because the arises between between the This is is important important because the electromagnetic electromagnetic induction induction can can damage damage the the insulation insulation and and produce produce noises noises in in information information transmission transmission systems. systems. The The consequences consequences of of an an information information line line disorder disorder are hazardous as unit. are often often as as hazardous as aa failure failure in in an an electronic electronic unit. It follows follows from from the the previous section that induction emf emf It previous section that the the magnetic magnetic induction inside the the sheath sheath strongly strongly depends depends on on the the wire wire location. location. The The emf emf value value is is inside maximal wires goes near the the inner maximal when when one one of of the the wires goes near inner sheath sheath surface surface along along the current line line of of maximum linear density density and and the other wire is immediately immediately the current maximum linear the other wire is adjacent to the line with J,,,. The overvoltages U1 and U, of the wires adjacent to the line with Jmin . The overvoltages U1 and U2 of the two two wires relative to relative to the the shell shell are are determined determined only only by by the the ohmic ohmic components, components, since since the wires immediately adjacent to the inner sheath produce with it zero the wires immediately adjacent to the inner sheath produce with it zero area circuits: circuits: U U11 == J,,,pl/d and U U2 = J,,,pl/d. The voltage voltage between between the area Jmaxpl/d and Jminpl/d. The the 2 = wires ~Umax AU,,, == U U11 -- U22 == (Jmax (J,,, -- J,,,)pl/d is due due to to the internal magnetic magnetic wires Jmin)pl/d is the internal field, so so the variation rate rate of of the magnetic flux flux penetrating circuit field, the variation the magnetic penetrating through through the the circuit = (J (J,,,max -- Jmln)pl/d. can again again conconcomposed of of the the wires is dcI>in/dt dQ,,/dt = composed wires is Jmin)pl/d. We We can clude that magnetic field field pulse pulse in in the the sheath sheath is is not similar to to the lightning clude that the the magnetic not similar the lightning current but to its its time integral. During During aa current current impulse impulse of of aa negative negative lightlightcurrent but to time integral. ning 100 I..ls ys (on (on the the 0.5 0.5 level) level) with with J,J,max ~ = const, const, the the magnetic magnetic field field within ning ttpp = ~ 100 within non-circular sheath sheath rises rises as as H ( t ) '" t (for (for aa circular circular sheath, sheath, Jmax and aa non-circular H(t) J max = J,,, J min and = 0). 0). At At this lightning current, current, the higher the the conductive conductive sheath sheath resistivity resistivity H = this lightning the higher and the the greater greater the the non-uniformity non-uniformity of of the the initial initial current current distribution distribution along along and the sheath sheath perimeter, the higher higher is is the the internal internal magnetic magnetic field. field. the perimeter, the To will be made for To illustrate, illustrate, an an estimation estimation will be made for an an elliptical elliptical cylinder cylinder of of length I == 100 100 m. m. The The following following parameters be used: = 1m, 1 m, b == I1 cm, cm, length parameters will will be used: aa = the thickness d = the aluminium aluminium sheath sheath thickness = I1mm, mm, and and p = =3 3 x 10- 8 n. fl-m. m. The The lightning current current amplitude amplitude will taken to to be 200kA, value used lightning will be be taken be 1,M == 200 kA, aa value used in in aircraft aircraft tests tests for for lightning lightning resistance. resistance. Using Using formula formula (6.34), (6.34), we we obtain obtain kA/m, J,,, J min NN ~ 32kA/m, real construcJ,J,max ~ x 3200 3200 kA/m, 32 kA/m, and and ~Umax AU,, ~ 9.5 9.5 kV. kV. In In aa real construction, such such aa great great overvoltage overvoltage could could have have resulted resulted from from aa poor poor design design of of the the tion, internal electrical electrical network. network. Wires Wires running running to to the the same same electronic electronic unit unit must must internal not separated so so much much from from each each other, other, nor nor should should they they be on the not be be separated be placed placed on the inner side side of of aa metallic metallic shell shell at at places differing much much in in the the surface surface curvature curvature inner places differing N


Lightning stroke at a screened screened object

2911 29

and, A compact packing of of cable cable assemassemand, hence, hence, in in the the linear linear current current density. density. A compact paclung blies good and and nearly nearly free free means means blies at at sites sites of of minimum minimum surface surface curvature curvature is is aa good of with metallic metallic shells. shells. of limiting limiting overvoltages overvoltages in in internal internal circuits circuits of of objects objects with is made made from from aa plastic plastic and and if if Overvoltages Overvoltages rise rise considerably considerably if if the the shell shell is its metallic jacket jacket extending extending its electric electric circuits circuits are are located located in in aa special special outer outer metallic of the the jacket jacket may may be be 1-2 1-2 from resistance of from the the head head to to the the tail. tail. The The linear linear resistance orders The ohmic ohmic component component orders higher higher than than that that of of the the totally totally metallic metallic fuselage. fuselage. The of the magnetic magnetic compocompoof overvoltage overvoltage will will increase increase respectively. respectively. To To eliminate eliminate the magnetic field field into into the the jacket, jacket, nent, nent, associated associated with with the the penetration penetration of of the the magnetic circular cross cross section. section. it it is is very very desirable desirable to to make make it it as as aa pipe pipe with with aa circular 6.2.7 6.2.7 Laboratory Laboratory tests of objects objects with metallic metallic sheaths

The many technical technical problems problems The lightning lightning protection protection practice practice involves involves aa great great many measurement of of all all associated associated with with the the formation formation of of test test current current and and the the measurement parameters physical aspects aspects parameters of of interest. interest. Here, Here, we we shall shall be be concerned concerned with with the the physical of response to to lightning lightning of testing, testing, which which could could allow allow prediction prediction of of the the object's object's response current conditions. current in in aa real real situation, situation, generally generally different different from from laboratory laboratory conditions. current. The The Let Let us us begin begin with with aa laboratory laboratory current current simulating simulating lightning lightning current. best reproduce lightlightbest thing thing to to do do would would be be to to make make aa laboratory laboratory generator generator reproduce ning reliability ning current current exactly. exactly. The The high high requirements requirements on on the the protection protection reliability 200 kA, make up to to 200kA, make one one apply apply maximum maximum currents currents with with an an amplitude amplitude up because especially especially for for testing testing aircraft. aircraft. Tests Tests on on the the 1: 1 :1 scale scale are are attractive attractive because to examine examine they they do do not not require require overvoltage overvoltage measurements. measurements. It It is is sufficient sufficient to no damage. However, the the object's object's equipment equipment after after the the tests tests to to see see that that there there is is no damage. However, the has aa long long the generation generation of of aa high high current current creates creates problems problems when when the the object object has length be short. short. For For length or or when when the the current current impulse impulse front front to to be be reproduced reproduced must must be example, - exp( exp( -pt)] -;3t)] example, the the maximum maximum steepness steepness for for the the impulse impulse i(t) i( t ) = =1 Zo0 [[11 with an an exponential exponential front front is is (di/dt)rnax (dildt),,, = = ;31 pZo. To generate generate such such impulses, with impulses, 0 , To the source source must must develop develop the the voltage voltage Urnax U,,, ==DIOL, where L is is the ;3IoL, where the circuit circuit the object of of length length inductance inductance close close to to that that of of the the test test object; object; L;::;:; L z LII L1l for for an an object The maximum maximum voltage voltage is is U U,rnax , ;::;:; z 12MV 12MV for for L1 x l/1H/m, 1 pH/m, II;::;:; x 100m, loom, I.1. The L I ;::;:; = 200 200 kA, kA, and and ;3p ;::;:; x 0.6/1S-1, 0.6 ps-', corresponding corresponding to to the the front front tf;::;:; t f % 5/1s 5 ps average 1Io average 0 = for the the current current of of the the first first negative negative lightning lightning component. component. A generator generator with with for such parameters parameters would would have have an an enormous enormous size size and and great great cost. cost. such The intuitive intuitive desire desire to to elongate elongate the the current current impulse impulse front front rather rather than than to The to reduce its its amplitude amplitude in in the the testing testing of of objects objects with with aa solid solid metallic metallic sheath sheath has has aa reduce reasonable physical physical basis. basis. Due Due to to the the longer longer front front duration duration tf, tf, the the ohmic ohmic reasonable overvoltage in in the the internal internal circuits circuits of of the the object object to to be be designed designed could could overvoltage change only only when when there there is is an an appreciable appreciable current current redistribution redistribution along along its its change cross section section perimeter perimeter during during the the time time tt ;::;:; x tf. tf. This This would would require require the the time time cross tf > >100 100J..ts. ,us. Therefore, Therefore, the the application application of of impulse impulse fronts fronts with with aa duration duration of of tf of microseconds, microseconds, instead instead of of typical typical lightning lightning impulses, impulses, cannot cannot affect affect dozens of dozens the test test results. results. The The same same is is true true of of overvoltages overvoltages in in aa double double wire wire circuit, circuit, the


292


induced by an induced by an internal internal magnetic magnetic flux. flux. Consequently, Consequently, the the increase increase of of the the impulse be unable unable to to affect impulse front front duration duration by by one one order order of of magnitude magnitude will will be affect appreciably appreciably the the overvoltage overvoltage in in internal internal electrical electrical circuits. circuits. This This considerably considerably reduces the laboratory reduces the the requirements requirements on on the laboratory source source of of impulse impulse current, current, because because its its operating operating voltage voltage decreases decreases in in proportion proportion with with the the increase increase in in ff. the costs, because tf. The The decrease decrease in in U by by an an order order of of magnitude magnitude reduces reduces the costs, because the the actual the costs costs of of high-voltage high-voltage technologies technologies rise rise faster faster than than the actual voltage. voltage. Much attention be given the simulation Much attention should should be given to to the simulation of of the the lightning lightning impulse the test impulse duration duration in in laboratory laboratory conditions. conditions. Anyway, Anyway, the test impulse impulse should not not be shorter than real one, one, for for the the overvoltage overvoltage amplitude amplitude may may should be shorter than the the real thus be underestimated underestimated because because of the skin-effect. be unreasonable unreasonable thus be of the skin-effect. It It would would be to reproduce on bed the the lightning to reproduce on the the test test bed the actual actual amplitude amplitude of of the lightning current current impulse no non-linear test object's impulse if if there there are are no non-linear elements elements in in the the test object’s circuit circuit and and the overvoltages can be registered by detectors. Since the electrical and magthe overvoltages can be registered by detectors. Since the electrical and magnetic netic components components of of overvoltage overvoltage are are similar similar in in shape shape and and equally equally depend depend on on the applied current amplitude, one can recalculate the measurements in prothe applied current amplitude, one can recalculate the measurements in proportion with with higher higher currents test impulse terms of portion currents and and select select the the test impulse amplitude amplitude in in terms of the highest possible accuracy and and registration registration convenience. convenience. the highest possible accuracy Quite another another matter matter is is the situation when test object's object’s sheath sheath is is not Quite the situation when the the test not solid but has has slits windows. The magnetic field solid but slits or or technological technological windows. The 'external' ‘external’ magnetic field of of the lightning partly through the the sheath; sheath; the the field field is is proportional the lightning partly penetrates penetrates through proportional to to the current and and the induced overvoltages overvoltages are are proportional current the current the induced proportional to to the the current impulse total overvoltage now depends depends on both the the current impulse steepness. steepness. The The total overvoltage now on both current rise time and amplitude, so the engineer has no chance to select a convenient rise time and amplitude, so the engineer has no chance to select a convenient test impulse shape. shape. In of measured test impulse In principle, principle, the the recalculation recalculation of measured pulses pulses to to real real ones is also possible, but this requires a detailed analysis of the overvoltage ones is also possible, but this requires a detailed analysis of the overvoltage mechanism of the circuits, which does not mechanism and and the the responses responses of the object’s object's circuits, which does not raise raise the the testing reliability. reliability. testing Another Another problem problem is is to to connect connect the the test test object object to to the the laboratory laboratory is obvious obvious that that aa conductor with the the generated generated current current should generator. It It is generator. conductor with should be connected to the the site of aa possible possible lightning lightning stroke. stroke. In In the the case case of of terrestrial terrestrial be connected to site of objects in in natural natural conditions conditions and and on on aa test test bed, bed, the the problem problem of of current current output output objects is by using using aa grounding grounding bus. bus. The The situation situation for for is solved solved in in aa simple simple way way -- by aircraft aircraft and and spacecraft spacecraft is is more more complicated. complicated. In In real real conditions, conditions, the the lightning lightning sheath (say, (say, the the fuselage) fuselage) and and then then current first first flows flows through through aa metallic current metallic sheath enters the ascending leader leader channel, channel, whose whose length length is is much much greater greater than than enters the ascending that of the object. It It is is difficult difficult to reproduce the current path in to reproduce the real real current path in that of the object. laboratory conditions conditions -- this this would would require high voltage voltage to make the the laboratory require aa very very high to make lightning impulse current current run run through through the the long long conductor conductor simulating simulating aa lightning impulse channel. Besides, Besides, the the test object and and the numerous detectors detectors would be channel. test object the numerous would be under very high high potential relative to to the the earth. earth. under aa very potential relative The return current wire wire is is normally normally located located close close to to the the test test object. object. Its Its The return current magnetic field field interacts interacts with with the the object's object’s metallic metallic sheath, sheath, through through which which magnetic forward forward current current flows. flows. As aa result, result, the the current current distribution distribution along along the the sheath sheath



3-

J/Jo

293 293

.1

2

Point number

01 Or----,....-...... ~-_--_1

1

2

3

4

1

Measured angular angular distributions distributions of of the the linear linear current current density density along along the Figure 6.11. Measured the circular pipe at various various locations locations of of the reverse current current conductor. conductor. circular pipe perimeter perimeter at the reverse Marked points (on the scheme) are are presented presented on on the abscissa axis. axis. The The curve curve Marked points (on the pipe pipe scheme) the abscissa A corresponds to to aa single single reverse reverse wire wire A for for aa = 2r, 2r, curve curve A2 is that for a = 4r, 4r, AlI corresponds A z is that for curve B depicts depicts three three reverse reverse wires wires B placed as shown shown in in the the scheme. scheme. Uniform Uniform curve placed as distribution C corresponds corresponds to to the the coaxial coaxial reverse reverse current current cylinder cylinder of of radius radius 2r. 2r. distribution perimeter changes, the the redistribution considerable if if the return current current perimeter changes, redistribution being being considerable the return wire wire is is close close to to the the object. object. Inducing Inducing the the emf emf of of the the opposite opposite sign, sign, the the reverse reverse current nearby parts metallic sheath of the the metallic sheath current increases increases the the current current load load in in the the nearby parts of but decreases remote parts. parts. but decreases it it in in the the remote For aa particular particular geometry, geometry, the the current current distribution distribution should should be found For be found numerically the set (6.33) by by adding, adding, to to each each equation, equation, aa numerically from from the set of of equations equations (6.33) term for the magnetic magnetic flux the reverse wire, -I -ZMMko, where term for the flux from from the reverse current current wire, M M kO , where M kO is the mutual between the return current Mko is the mutual inductance inductance between the return current wire wire and and the the kth kth conductor simulating simulating the the sheath. sheath. Under Under conditions conditions typical typical of of test test beds, beds, the conductor the distortions return current be appreciable. distortions due due to to the the return current path path may may be appreciable. The The results results presented in presented in figure figure 6.11 6.11 have have been been obtained obtained from from the the tests tests of of aa sheath sheath shaped as as aa circular circular pipe. pipe. In In order order to to avoid avoid the the effects effects of of currents currents induced induced shaped in the the conductive conductive soil, soil, the the sheath sheath was was raised above the the earth earth at at aa height in raised above height H = = 7r, 7 r , where where r is is the the pipe radius. The The role of the the return return current current wire wire was was pipe radius. role of performed by pipe at performed by aa thin thin conductor conductor running running parallel parallel to to the the pipe at the the distances distances a == 2r and and a == 4r from from it, it, three three conductors conductors located located at at 120° 120" at at the the same same provides aa distance, distance, and and aa coaxial coaxial cylinder cylinder of of radius radius 2r. 2r. The The latter latter design design provides perfectly uniform perfectly uniform current current distribution distribution along along the the perimeter perimeter of of the the sheath sheath cross cross section. section. The The return return current current of of aa single single wire wire distorts distorts the the current current distribudistribu0.5ja,and and the the tion to the the greatest greatest extent: extent: its its minimum minimum linear linear density density drops drops to to a.Sjav tion to 2.3ja, (jav (jav =1 Z0/27rr). The current distribution maximum density density rises rises to = /27fr). The current distribution maximum to 2.3jav 0 becomes more uniform uniform when when the the number number of of reverse reverse conductors conductors is is increased. increased. becomes more


294 294

lightning effects on modern structures Dangerous lightning

When aa sheath sheath has has aa complex complex geometry, geometry, it it is is hard hard to to predict predict the the reverse reverse When current effect on the test results. The current redistribution in the sheath may current effect on the test results. The current redistribution in the sheath may lead to both the overestimation and underestimation of overvoltages in the lead to both the overestimation and underestimation of overvoltages in the internal circuits. Much depends on the arrangement of the internal conducinternal circuits. Much depends on the arrangement of the internal conductors and and return return current current wire. wire. tors 6.2.8 Overvoltage in a screened screened multilayer multilayer cable

Overvoltages in in screened screened multilayer multilayer cables cables are are due due to to the the skin-effect. skin-effect. As As aa Overvoltages result the the cable cable wire wire screens screens in in the the layers layers are are loaded loaded differently differently by by the the lightlightresult ning current. current. Every Every layer layer is is formed formed by by wires wires arranged in aa circle circle and and having having ning arranged in their own own screens screens (figure (figure 6.12(a)). 6.l2(a)). Depending Depending on on the the reliability reliability requirements, requirements, their cable may may have have an an outer outer metallic metallic sheath sheath or or aa dielectric dielectric coating coating protecting protecting it it aa cable from mechanical mechanical damage. damage. However, However, aa direct direct lightning lightning stroke stroke produces produces aa from breakdown of of dielectric dielectric material, material, and and the the lightning lightning current current is is distributed distributed breakdown among among the the screens. screens. The The adjacent adjacent screens screens in in aa layer layer contact contact each each other other along the the whole whole cable length. It approximation along cable length. It can can be be assumed assumed in in aa first first approximation that they they form form aa solid solid sheath sheath of of circular circular cross cross section section with with resistance resistance that Rk Rk = = R/nk Rink and and inductance inductance Lk, L b where where R is is the the resistance resistance of of an an individual individual wire wire screen screen and and nk is is the the number number of of screened screened wires wires in in the the kth kth layer layer (figure we shall (figure 6.l2(b)). 6.12(b)). For For simplicity, simplicity, we shall consider consider aa double double layer layer cable, cable, markmarking with kk == 11 and the outer ing the the inner inner layer layer with and the outer layer layer with with kk == 2. 2. The The adjacent adjacent screens the adjacent with one screens of of wires wires from from the adjacent layers layers are are also also in in contact contact with one another. another.

Figure 6.12. 6.12. Multilayer Multilayer cable cable (a) (a) and and solid solid sheath sheath model model (b). (b).



295 295

For be regarded regarded as as aa solid solid conductor, conductor, For this this reason, reason, aa set set of of circular circular sheaths sheaths can can be and the second second layer layer to to the the first first and the the current current penetration penetration along along its its radius radius (from (from the one) can also also be be treated treated as as aa one) can can be be considered considered as as aa skin skin effect. effect. Such Such aa system system can set set of of discrete discrete circular circular layers. layers. In the layers layers at at the the initial initial In the the latter latter case, case, the the current current distribution distribution among among the moment the condition condition of of magnetic magnetic flux flux moment of of time time tt = =0 0 can can be be found found from from the coupling valid at at tt> and have have the the coupling equality equality (6.33). (6.33). Equations Equations (6.35) (6.35) are are valid > 00 and = i + i = const: following current 1IM = il i2 = const: following solution solution for for two two layers layers at at constant constant current M l 2

+

R2rM i l ( t ) = ____ [l - exp(-Xt)],

R1+ R2

M [R + R exp( -At)] IM ii2(t) I R22exp(-Xt)l 2(t) = = R 1 R [Rl 1 + RI R22 (6.37) (6.37) ~

+

allowed for for the the mutual mutual where X= = (R (R,I + R R2)/(L1 Equations (6.35) (6.35) allowed where A 2 )/(L I -- L 22).) . Equations = L as in the treatment of the screen-wire inductance , inductance of of the the layers, layers, M M12 = L2, as in the treatment of the screen-wire I2 2 law, the the lightning lightning system system in in section section 6.2. 6.2. In In accordance accordance with with the the skin-effect skin-effect law, current gradually penetrates penetrates into into the the current first first loads loads the the outer outer sheath sheath and and then then gradually inner uniformly between between the the individual individual inner sheath. sheath. The The current current is is distributed distributed uniformly screens = i2/n2. i2/n2' The The overvoltage overvoltage screens in in each each circular circular layer, layer, iiSl = idnl il/nl and and iis2 s1 = s2 = across own screen screen (providing (providing that that the the across the the insulation insulation between between aa wire wire and and its its own is similar to the the current current in in the the skin-effect skin-effect in in an an individual individual screen screen is is neglected) neglected) is similar to layer, UI(t) U,(t)= = Rlil(t) Rlil(t) and and U22(t) ( t )= =R R2i2(t), but not to the lightning current. layer, i (t), but not to the lightning current. 22 If of one one layer, layer, there there is is no no overoverIf aa double double wire wire circuit circuit uses uses the the cores cores of voltage because the the potentials potentials of of the the voltage in in the the instruments instruments connected connected to to it, it, because connected to to the the cores cores of of layer layer cores cores are are identical. identical. If If the the instruments instruments are are connected different different layers, layers, the the voltage voltage between between them them is is U12 =

U2 - U1 = I M R eXp(-Xt). ~

(6.38) (6.38)

M = = 1, 1, expression expression (6.38) (6.38) is is aa unit unit step step function function for At 1I M At for the the set set of of equations equations providing the the solution solution for for the the lightning lightning current current impulse of an an arbitrary arbitrary providing impulse of = IIM[exp(-at) exp(-Pt)], we have shape. In In particular, particular, at at i(t) = [ exp( -at) exp( -,8t)J, we have shape. M

U12 = IIMR2[Bexp(-Pt) exp(-at) ( B -- A U -at) -- (B A)) exp(-At)] exp( -At)] 12 = M R 2[Bexp( -,8t) -- A exp( A= = a/(A ./(A -- a), a),

B = ,8/(A -- p). ,8). = p/(X

(6.39) (6.39)

Owing to to the the relatively relatively small small value value of of LL1I -- L2 Owing L2x ;::,; (p0/27r) (/-Lo/27r) In In (r2/r1) (r2/rl) at at close close layer radii radii r2 r2 and and rl, r l , the the layer layer current current ratio ratio is is redistributed redistributed rapidly, rapidly, for for layer = A-I;::,; A-' FZ 10 l oj..lS. p . This This is is the the reason reason for for aa fast fast damping TT = damping of of the the overvoltage overvoltage pulse U U12 (figure 6.13), 6.13), which which may may be be remarkably remarkably shorter shorter than than the the current current pulse I2 (figure U12 reverses the sign; its impulse. It It follows follows from from (6.39) (6.39) that that the the pulse pulse U reverses the sign; its impulse. I2 opposite tail tail is is damped damped approximately approximately at at the the rate rate of of lightning lightning current current reducreducopposite tion. The The overvoltage overvoltage amplitude amplitude in in aa double double wire is close close to to that that in in aa tion. wire cable cable is wire-shell system, system, exactly exactly as as in in aa sheath sheath with sharply non-uniform wire-shell with aa sharply non-uniform current current distribution. If If the the screens screens are are thin thin and and have have aa high high resistance, resistance, the the hazard hazard of of distribution. damaging the the connected connected measuring measuring instruments instruments is damaging is fairly fairly great. great.


296 296

Dangerous lightning effects on modern structures 1.0

0.8

~

-

0.6

""'"' 0.4 ~ N

I

0.2 125

150

-0.2 J -0.2 6.13. Overvoltage Overvoltage pulse on aa two-layer two-layer cable cable for for the the bi-exponential current Figure 6.13. pulse on bi-exponential current impulse with = 0.007llS, 0.007 ps, ,3 ,3 = = 0.6llS 0.6 ps and and the the redistribution time constant constant T = = 50 50 llS. ps. redistribution time impulse with Qcy =

The be solved way. The The problem problem for for aa multilayer multilayer cable cable can can be solved in in aa similar similar way. The overvoltages between the between the overvoltages between the cable cable cores cores grow grow with with distance distance between the respecrespective layers. Other Other conditions conditions being equal, the the overvoltages overvoltages drop drop with the layer layer tive layers. being equal, with the depth depth in in the the cable. cable. The The use use of of cores cores of of one one cable cable layer layer reduces reduces considerably considerably the overvoltage in in aa double double wire system but does not not eliminate eliminate it it entirely. entirely. the overvoltage wire system but does the cable cable is is pressed pressed under its own own There are are no circular cables cables -- the There no perfectly perfectly circular under its weight and weight and becomes becomes deformed deformed during during its its winding winding on on aa drum. drum. The The result result is is that the perimeter that the current current distribution distribution along along the the sheath sheath cross cross section section perimeter becomes non-uniform, producing additional between the the becomes non-uniform, producing additional overvoltages overvoltages between cores of of the same layer. layer. To To minimize minimize these these overvoltages, overvoltages, it it is is desirable desirable to cores the same to connect to the the adjacent precision connect the the equipment equipment to adjacent cores cores of of the the same same layer. layer. High High precision equipment equipment should should be be connected connected to to the the cores cores of of deeper deeper layers. layers. Overvoltages Overvoltages arising in in aa multilayer cable can can be evaluated from from the the same same set set of of arising multilayer cable be evaluated equations (6.35). equations (6.35).

6.3

pipes as a high high potential potential pathway pathway Metallic pipes

Modern constructions constructions have an abundance abundance of of underground underground metallic Modern have an metallic pipes, pipes, and the the lightning lightning protection engineer must must take take them into account account as as aa and protection engineer them into possible possible pathway pathway for for currents currents from from remote remote lightning lightning strokes. strokes. This This actually actually happens pipe lies happens when when aa pipe lies close close to to aa high high lightning lightning rod rod or or another another object object preferable preferable to to lightnings. lightnings. Spreading Spreading through through the the earth earth away away from from the the 6.2.2, some some of of the the current current grounding electrode electrode in in aa way described in in section section 6.2.2, grounding way described enters pipe and runs along pipe is is sometimes sometimes enters aa metallic metallic pipe and runs along its its length. length. A pipe


Metallic pipes as a high potential potential pathway

R

297 291

R

I 6.14. Underground Underground pipe as the pathway for for aa lightning lightning current current and and the design Figure 6.14. pipe as the pathway the design circuit for for aa simple simple evaluation evaluation of of the the object object potential. potential. circuit

connected directly directly to to the the object object grounding grounding electrode. electrode. Figure Figure 6.14 illustrates illustrates connected the when aa metallic pipe line the typical typical situation situation when metallic pipe line connects connects the the grounding grounding of an an object, object, struck struck by by lightning, lightning, electrode (with (with grounding grounding resistance resistance Rgjg )l )of electrode of aa well-protected well-protected object. object. Although to the the grounding grounding (with (with resistance resistance Rg2) to R g2 ) of Although the lightning lightning is is unable unable to to reach reach the the latter latter directly, directly, some some of of the the current current finds finds the its pipe. For the pipe. For applications, applications, it it is is its way way to to its its grounding grounding electrode electrode -- the 1 and important to to know know the the dependence dependence of of this this current current on on the the line line length length land important on on the the soil soil conductivity. conductivity. considered the problem of of current current distribution distribution for for an an underSection 6.2.3 considered Section the problem underground pipe of present case ground pipe of infinite infinite length. length. The The limited limited line line length length in in the the present case is is an an important parameter, especially because it has the the grounding resistances at important parameter, especially because it has grounding resistances at its its ends. be solved using the ends. Generally, Generally, this t h s problem problem can can be solved analytically analytically using the Laplace Laplace transtransformation. formation. But But the the final final result result is is represented represented as as aa functional functional series series too too complex complex for aa treatment, treatment, so so numerical numerical computations computations are are necessary. necessary. It It is, is, therefore, more for therefore, more expedient this problem problem numerically numerically from beginning. Before expedient to to solve solve this from the the very very beginning. Before presenting the results of presenting the results of aa computer computer simulation, simulation, we we shall shall make make aa simple simple evaluaevaluaan underground underground pipe lumped inductance inductance L = = Ljt L 1I and and tion. Let us tion. Let us replace replace an pipe by by the the lumped its R g == (Gjl) -I. The be represented (G1l)-'. The latter latter will will be represented as as its intrinsic intrinsic grounding grounding resistance resistance R, = 2R 2 R g, by by connecting connecting them them to the ends ends of of the the line line in in two identical resistors resistors R = two identical to the and Rg2 of the the objects objects it it connects connects parallel grounding resistances parallel to to the the grounding resistances Rgl R gI and R g2 of we are (figure 6.14). This This rough rough approximation approximation makes makes sense, sense, since since we are interested interested (figure 6.14). at the far end end connected connected to to the the grounding grounding mat, mat, in the the value value of of current current ii22 at in the far rather rather than than in in its its distribution distribution along along the the line. line. In In this this approximation, approximation, we we have have d i2 R' di2 L Cit L= (. (i1-- 12 i')R 2 ) Rel, el, e2 /2 = dt + Re2i2

+

RR gj. R ej. = RR R -2 , j. = 1 1 , 22 . R + R R + R dg). ' } = , .

(640) (6.40) .

Putting be i = -at) and Putting the the lightning lightning current current to to be = !M ZM exp( exp(-at) and ii2(0) = 0, 0, we we find find 2 (0) =

ReIA!M [ )] Re 1 AIM exp(-At)], )(A) [exp(-at) exp(-at) -- exp( -At, ReI + Re2)(A R e2 -- a ) (Re1

ii2(t) 2(t) == (


Re1 el + Re2 XA == R L L (6.41) (6.41)

298 298

lightning effects on modern structures Dangerous lightning

At At the the beginning, beginning, while while the the effect effect of of self-induction self-induction emf emf is is still still noticeable, noticeable, the the current at the front end of ] Rel at the front end of current largely largely flows flows through through the the equivalent equivalent resistance resistance R e the -I, the A-', the current current gradually gradually penetrates penetrates to to the the far far end end the line. line. After After time time T == A of of pipe. pipe. Some Some of of it, it, ii82 = i22R/(R R / ( R+ R Re]!, finds its its way way to to the the grounding grounding elecelecg2 = e2), finds trode U22 == iig2Rg1 trode of of the the object object of of interest, interest, raising raising its its potential potential to to the the value value U g2 R g2 relative relative to to aa remote remote point point on on the the earth. earth. For For aa longer longer line, line, the the values values of of iig2 g2 and decrease for two reasons. An increase in L = L]I and G = GIl and U U2 decrease for two reasons. An increase in L = L1l and G = G1l 2 raises raises the the time time constant constant T, T , and and by by the the time time the the current current has has reached reached the the far far end end of of the the pipe, pipe, the the initial initial lightning lightning current current is is considerably considerably damped. damped. Besides, Besides, aa smaller of the the current current ii2 that has has reached reached the the far far end end enters enters the the smaller portion portion of 2 that object's the greater object's grounding grounding electrode electrode because because of of the greater pipe pipe leakage. leakage. The The dependependence rather strong, of iig2 and U U2 on I1 proves proves to to be be rather strong, especially especially when when the the dence of g2 and 2 on effective of the the lightning lightning current, current, tt,p :::::: x ex a -I, - ' , is is comparable comparable with with effective duration duration of T = A-]. A-'. Suppose Suppose we we take take tt,p = = 100/ls 100 ps on on the the 0.5 0.5 level level (ex (a = = 0.007/ls-]), 0.007 ps-'), the the grounding resistances 10 fl, R, and and L1 2.5 pH/m. A metallic metallic R g ] = Rg2 R g2 = 10 L] = 2.5/lH/m. grounding resistances Rgl pipe with pipe with aa 10cm lOcm diameter diameter and and 100m lOOm in in length, length, lying lying at at the the surface surface of of the the soil will deliver 200 fl/m R/m (G] (G1 = 2.1 2.1 x 10- 3 (fl/m)-] (a/m)-', , R = 9.7 9.7 fl), R), will deliver soil with with p = 200 the current iigz 0.171ZAvto to the the ground ground of of the the object object located located at at its its far far end. end. the current g2 :::::: 0.17lh! The be raised raised to I M =30kA. x 50kV at at ZM = 30kA. At The object's object's potential potential will will be to U2, ::::::50kV 1= m, we I = 200 200m, we have have iig2 x 0.0861I 0.0861Zjw and, at at the the same same lightning lightning current, current, g2 :::::: M and, U2 z 25 kV. But even this voltage is quite sufficient for a spark to be ignited U :::::: 25 kV. But even this voltage is quite sufficient for a spark to be ignited 2 between closely located located elements elements of of two two metallic metallic structures, structures, provided provided that that between closely one of of them them is is connected connected to the grounding grounding electrode electrode and and the other is is not. not. one to the the other premises. Such Such aa spark spark can can induce induce an an explosion explosion or or fire fire in in explosible explosible premises. In be transported In low low conductivity conductivity soils, soils, current current can can be transported through through metallic metallic pipes for many kilometres. This This refers, refers, to to aa still still greater greater extent, extent, to external pipes for many kilometres. to external pipes and rails mounted on on aa trestle trestle which which are are grounded grounded only only locally, locally, through through pipes and rails mounted the supports by dozens metres. Here, the supports separated separated by dozens of of metres. Here, evaluations evaluations can can also also be be where R L is is an an average average made with expression expression (6.42), (6.42), putting 2R~/n, where R~ made with putting R == 2Ri/n, resistance of the support grounding resistance of the support grounding and and nn is is the the number number of of supports. supports. A comparison comparison of of the the estimates estimates and and computations computations is is shown shown in in figure figure 6.15 6.15 A for the the above above example example with with I = = 200m. 200 m. The The estimates estimates for for the the current current for amplitude at at the the far far end end of of the the pipe pipe and and for for the the moment moment of of maximum maximum amplitude current show show aa satisfactory satisfactory agreement agreement with with the the numerical numerical computations. computations. current The computations computations will will be be unnecessary unnecessary if if one one finds finds it it possible possible to to ignore ignore the the The initial portion portion of of the the pulse pulse front front and and can can put put up up with with aa 20-25% 20-25% error. error. initial Let Let us us calculate calculate the the potential potential at at the the far far end end of of the the pipe pipe unconnected unconnected to to the grounding grounding electrode electrode at at either either end. end. This This may may happen happen due due to to careless careless design design the or or poor poor maintenance maintenance of of communications communications lines. lines. The The soil soil will will be be considered considered to to have aa low low conductivity, conductivity, p = = 1000 fl/m; L1 L1 = = 2.5 2.5/-lH/m. The curves curves in in have lOOOQ/m; pH,". The figure 6.16 6.16 show show the the variation variation in in the the voltage voltage and and current current amplitude amplitude ratio ratio figure UmaX/ZAv 100 ps and Umax / {/o,f for for impulses impulses of of negative negative lightnings lightnings with with tp tp = = 100/ls and for for those those of of 'anomalous' 'anomalous' positive positive lightnings, lightnings, which which are are an an order order of of magnitude magnitude longer. longer. The The pipe pipe is is capable capable of of delivering delivering aa potential potential of of dozens dozens of of kilovolts kilovolts

+


high potential pathway Metallic pipes as a high

299 299

0.10 0.08 , ,, ,,, ,

,,

I

0.04 0.02

,,

,,

-', ......... "............

,!

0.00 +'---.--.--~--r--.--..--~----.

a

100

200 ~s Time, ~ Time,

300

400

Figure 6.15. 6.15. Portion Portion of of aa lightning lightning current current passed passed to object through Figure to the the object through the the communicommunication pipe pipe of of 200 200 m m length. length. Curve Curve l: 1: numerical numerical computation, 2: simple simple evaluation. cation computation, 2: evaluation.

for aa distance distance of of 11 km km to to the the object object even even at at aa moderate for moderate lightning lightning current current of of 30 kA. kA. Damage Damage of of the the contact contact between pipe and and the object's grounding grounding 30 between the the pipe the object's electrode may may be be fatal fatal if if aa spark spark arising arising in in the air gap encounters an electrode the air gap encounters an inflammable substance. substance. inflammable 25

20 20 -

o0E f

15 15-

+J

. 10-

5-

a+--.---r-,...--r---.--r-...--....--------, 0 200 400 600 800 1000 a 200 1, m m Figure 6.16. 6.16. Computed Computed maximum maximum overvoltages overvoltages transferred transferred to to an an object object at at the the far far end end of of the the underground underground pipe pipe of of 10 10 cm cm diameter diameter and and of of length length 1. I. The The pipe pipe is is not not connected connected with of both both an an object object and and aa lightning lightning rod. rod. Computations Computations were were made made for for with the the grounding grounding of the of 100 100 ps IlS duration, duration, and and for for an an 'anomalous' 'anomalous' impulse impulse the usual usual lightning lightning current current impulse impulse of of 1000 ps, Lightning of 10001lS. Lightning stroke stroke to to the the other other end end of of the the pipe. pipe.


300 300

on modern modern structures stsuctuses Dangerous lightning effects on

The The delivery delivery of of high high potential potential can can be be controlled controlled in in aa simple simple way way -- all all communications communications lines lines must must be be connected connected to to the the same same grounding grounding mat. mat. In In that case, case, the the voltage voltage of of all all mat mat components components will will be be raised raised equally equally by by the the that brought brought current current of of aa remote remote lightning lightning stroke. stroke. It It should should be be noted noted that that this this is is aa reliable reliable means means to to cope cope with with the the overvoltage overvoltage of of kilovolt kilovolt values. values. A A simple simple connection connection of of metallic metallic sheaths sheaths to to the the grounding grounding mat mat cannot cannot remove remove pulse pulse noises noises of of tens tens or or hundreds hundreds of of volts volts having having aa short short rise rise time. time. Steep Steep curcurrent rent impulses impulses spreading spreading across across the the buses buses and and components components of of the the grounding grounding mat mat always always create create an an induction induction emf, emf, producing producing abrupt abrupt voltage voltage changes changes even even in in conductors conductors of of about about 1m l m in in length. length. Electrical Electrical circuits circuits must must be be mounted mounted in in such such aa way way as as to to avoid avoid the the appearance appearance of of closed closed contours contours or or joints joints of of the the conductor conductor screens screens to to points points remote remote from from each each other other in in the the grounding grounding mat. mat. This This sometimes sometimes becomes becomes such such aa delicate delicate matter matter that that the the result result depends depends on on the the engineer's engineer’s intuition intuition rather rather than than on on exact exact knowledge. knowledge,

6.4

Direct stroke stroke overvoltage overvoltage Direct

We described described the the manifestations manifestations of of overvoltage overvoltage due due to to aa direct direct lightning lightning We stroke when when discussing discussing the the lightning lightning current current propagation propagation across across aa metallic metallic stroke sheath. The The highest highest current current enters enters the the sheath sheath when when aa lightning lightning discharge discharge sheath. 6.2.1). This This happens, happens, for for example, example, when when strikes an an object object directly directly (section (section 6.2.1). strikes an aircraft aircraft is is affected affected by by the the return return stroke stroke current current recharging recharging the the descending descending an leader which which has has connected connected the the aircraft aircraft to to the the earth. earth. Below, Below, we we discuss discuss aa leader direct lightning lightning stroke stroke at at aa grounded grounded terrestrial terrestrial object. object. Specifically, Specifically, we we direct shall be be interested interested in in the the voltage voltage applied applied to to the the insulation insulation of of the the object object shall relative to to the the earth earth or or another another construction construction located located nearby. nearby. The The classical classical relative situation is is that that aa voltage voltage arises arises between between the the lightning lightning rod rod that that has has situation rough intercepted the the lightning lightning and and the the nearby nearby object object being being protected. protected. A rough intercepted 1.5.1. The The fast fast variation variation of of treatment of of this this situation situation was was made made in in section section 1.5.1. treatment along the the metallic metallic parts parts of of aa construction construction raises raises high lightning lightning current current ii along aa high = Rgi R,i + L Ldi/dt relative to to aa remote remote point point on on the the earth. earth. its potential potential by by U = its di/dt relative R,g and and Much depends depends on on what what is is understood understood by by the the grounding grounding resistance resistance R Much inductance L. L. These These issues issues are are discussed discussed in in much much detail detail in in the the books books on on inductance [6]).Here, Here, we we outline outline the the most most important important direct stroke stroke overvoltages overvoltages (e.g., (e.g., [6]). direct physical physical aspects aspects of of the the problem. problem.

+

6.4.1 The behaviour behaviour of a grounding electrode electrode at high high current impulses impulses 6.4.1 An An important important parameter parameter of of aa grounding grounding electrode electrode is is the the stationary stationary grounding grounding resistance usually usually measured measured during during the the spread spread of of direct direct or or low low frequency frequency resistance Rgo found from from the the alternative current current of of several several amperes. amperes. The The value value of of R alternative go found measurements may may be be several several times times larger larger or or smaller smaller than than R R,g == Uel Ue/ZM measurements 1M corresponding to to aa rapidly rapidly varying varying kiloampere kiloampere lightning lightning current current (here, (here, U Ue, is is corresponding


301 301

Direct stroke overvoltage

the protector). We the potential potential at at the the current current input input into into the the protector). We have have discussed, discussed, at at sevseveral points points in in the the book, the two two physical physical mechanisms mechanisms affecting affecting differently differently the the eral book, the ability of of aa metallic metallic conductor conductor to to tap off the lightning current current to the earth: earth: the the ability tap off the lightning to the self-inductance and and ionization ionization expansion expansion of of the the surface surface contacting contacting the the soil. soil. self-inductance The voltage drop prevents current the The voltage drop across across the the inductance inductance prevents current flow flow into into the conductor. A long be treated conductor. A long conductor conductor has has to to be treated as as aa line line with with distributed distributed parameters. The input input resistance of the the line, line, Ri, U ( 0 ,t)/i(O, t)/i(O:t) t ) varies varies in in parameters. The resistance of R in == U(O, time, since since the the current current diffuses diffuses along along the the line, line, and and it it takes takes some some time time for for time, the by current the whole whole conductor conductor to to be be loaded loaded by current more more or or less less uniformly. uniformly. As As the resistivity pp.. the limiting limiting case, case, consider consider an an infinite infinite conductor conductor in in aa soil soil with with resistivity From From formulae formulae (6.21) (6.21) and and (6.22), (6.22), the the voltage voltage at at the the conductor conductor input input is is I 2 U e ( t )== z U(O, U ( 0 ,t) t ) '" tt - 1// 2 for for the the current current i(O, i(0,t)t ) == const const == /0 lo and and t > 0. 0. At At Ue(t) /0 = be treated unit step Io = 1, 1, formula formula (6.21) (6.21) can can be treated as as aa unit step function function of of the the system, system, yy(t). ( t ) . This This allows allows us us to to follow follow the the input input voltage voltage of of aa horizontal horizontal grounding grounding conductor with aa real real impulse using conductor at at the the lightning lightning current current i(t) i ( t ) with impulse front front by by using the the Duhamel-Carson Duhamel-Carson integral: integral: N

s:,

y(T)i’(t -- T) 7) dT. dr. U(O, y(t)i(O) + J>(T)i'(t U ( 0 ,t) t) = = y(t)i(O)

+

(6.42) (6.42)

For with an have i(r ) == /0[1 Io[ 1 -- exp( exp(-;3t)] - p t ) ] we we have For aa impulse impulse with an exponential exponential front front i(t)

(

;3LI )1/2 U ( 0 ,t) t ) == 2/ 2100 ( g (3t) U(O, 71"Gy 2 1 1h(3t)

(6.43) (6.43)

1

where h(;3t) by the the last h ( @ )is is aa function function given given by last integral integral in in (6.29) (6.29) and and figure figure 6.5. 6.5. Its Its where maximum h at ;3t ::::: 0.9 permits the calculation of the maximum voltage maximum h,,max at pt,m M 0.9 permits the calculation of the maximum voltage drop drop across across the the grounding grounding electrode: electrode:

(6.44) (6.44) The of an an extended extended horizontal horizontal grounding grounding electrode, electrode, The effective effective input input resistance resistance of corresponding to to U U,,,,max' is is expressed expressed as as corresponding

R

g

,tT

=

Umax i( tmax )

:::::

1.82 (;3L 1)1/2

71"G 1

(6.45) (6.45)

In with R, R g ::::: p,, the the input Mp input resistance resistance In contrast contrast to to aa lumped lumped grounding grounding electrode electrode with of with the resistivity, R,,, Rg,tT '" ppl/2. of an an extended extended one one varies varies much much less less with the soil soil resistivity, li2. Extended because only Extended grounding grounding electrodes electrodes are are ineffective, ineffective, because only aa short short initial initial the portion of portion of their their length, length , leff::::: leffM (Rg,tTG1)-I, (RgerG1)-’, is is actually actually operative operative during during the impulse resistance is Rg,tT ::::: nand M 13 13 R and impulse front front time. time. For For example, example, the the effective effective resistance is R,,, the pipe with with L the the effective effective length length of of aa long long grounding grounding pipe L1 = 2.511H/m 2.5 pH/m at at the 1 = earth's m in leff::::: x 22 22m in the the case case of of the the first first component component current current of of aa earth’s surface surface is is leff negative with the negative lightning lightning with the rise rise time time tf:::::511s tf M 5ps (;3:::::0.611S( p M 0 . 6 ~ ~I )- I and and ) n . m. with an p == 100 100 R m. In In aa soil soil with an order order of of magnitude magnitude lower lower conductivity, conductivity, the the R&tT ::::: respective values are respective values are R,, M 42 42 nand R and leff leff::::: M 75 75 m. m. N

-


302 302


Extending the grounding bus beyond beyond the the limit we are unable to to Extending the grounding bus limit lerr, leff,we are still still unable reduce appreciably appreciably the the maximum maximum voltage voltage drop drop across across the the bus. bus. For For this this reduce reason, it it is is better better to to introduce introduce current current at at the the centre centre of of aa long long bus bus rather rather reason, than at at its its end, end, such such that that two two current current waves waves would would run run in in opposite opposite directions directions than along the half-length half-length conductors. conductors. Still more effective effective are three conductors conductors along the Still more are three arranged at angle of and so so on. on. When When aa grounding grounding mat mat with with the the arranged at an an angle of 120°, 120”, and lowest desired, it of R,, Rg. ff is is desired, it is is preferable preferable to to load, load, more more or or less less lowest possible possible value value of uniformly, the the whole of the the adjacent adjacent soil soil volume. volume. For For this aim, aa set set of of uniformly, whole of this aim, horizontal conductors conductors or or aa conductor conductor network network is is combined combined with with vertical vertical horizontal rod electrodes. electrodes. To To avoid avoid the the interaction interaction effect effect of of the the grounding grounding elements elements rod and and to to achieve achieve the the maximum maximum loading loading of of them them by by current, current, the the distance distance between between the the elements elements should should be be made made comparable comparable with with their their length length (or (or with the the height, height, for for vertical vertical rods). rods). But But even even in in that that case, case, only only part part of of the the with grounding mat, mat, within within the the radius radius of of leIT leEfrom from the the current current input, input, will will operate operate grounding effectively effectively at at the the impulse impulse front. front. Thus, the the resistance resistance of of aa grounding grounding electrode electrode for for rapidly rapidly varying varying curcurThus, rents is is much much higher higher than than for for direct direct current. current. A A grounding grounding mat mat network network rents with numerous numerous horizontal horizontal buses buses and and vertical vertical rods rods is is able able to to reduce reduce the the effeceffecwith tive resistance resistance to to the the value value of of R R,,, x 1 0. But when a large number of objects tive ~ 1 n. But when a large number of objects geff is being being constructed, constructed, for for example, example, the the towers towers of of aa power power transmission transmission line, line, is one has has to to deal deal with with resistances resistances as as high high as as Rg. R,, ff ~ x 10 10 n R and and more. more. one Laboratory experiments experiments show show that that the the grounding grounding resistance resistance of of an an Laboratory electrode delivering delivering to to the the earth earth very very high high currents currents is is lower lower than than for for low low electrode currents. The The grounding grounding resistance resistance decreases decreases with with the the current current rise. rise. The The currents. grounding resistance resistance ratio ratio of of aa high high impulsed impulsed current current and and low low direct direct current, current, grounding cui = =R R,/Rpo, is often often called called the the impulse impulse coefficient coefficient of of aa grounding. grounding. The The Ctj g / RgO' is coefficients Ctj ai used used in in the the literature literature are are sometimes sometimes as as small small as as Ctj ai~ x 0.1. 0.1. To To coefficients illustrate, we we shall shall cite cite the the generalized generalized function function Ctj cui = =f(plM) which has has f(pIM ) which illustrate, been suggested suggested for for aa vertical vertical rod rod of of 2.5 2.5m in length length from from the the results results of of been m in small-scale laboratory laboratory experiments experiments [7] [7] (figure (figure 6.17). 6.17). The The grounding grounding resistance resistance small-scale is is reduced reduced by by aa factor factor of of four four at at p == 1000 1000 n. R m m and and 1IM = 30 30 kA. kA. M = In principle, principle, this this reduction reduction in in resistance resistance might might be be due due to to aa larger larger effective effective In radius of of the the grounding grounding electrode electrode because because of of the the soil soil air air ionization. ionization. In In section section radius 6.2.2, 6.2.2, we we gave gave formula formula (6.15) (6.15) for for the the linear linear conductivity conductivity of of aa long long rod rod lying lying on on If the the rod rod is is fixed fixed in in the earth earth with with one one half half of of its its surface surface contacting contacting the the soil. soil. If the the vertical vertical position, position, the the whole whole of of its its surface surface contacts contacts the the soil soil but but its its leakage leakage the conductivity is is lower lower by by aa little little less less than than aa factor factor of of 22 at at the the same same length length (due (due conductivity to the the poorer poorer operation operation of of the the upper upper end end of of the the rod rod located located at at the the earth's earth’s to surface, because because current current cannot cannot flow flow upward upward into into the the air). air). The The linear linear conducconducsurface, tivity G GI1 and and the the grounding grounding resistance resistance R R,g of of aa rod rod of of radius radius ro, ro, fixed fixed vertically vertically tivity into the the earth earth for for aa length length I,1, are are into

-

_ G1 -

27r . pin (21/ro)·


Rg =

pin (21/ro) 27r1 .

(6.46) (6.46)

Direct stroke overvoltage overvoltage Direct stroke

U aij

303 303

I

1.0 1.0’ 0.8 0.8-

2.5 m

0.60.6

0.4 0.4. 0.2

-

0,

pI PI,M MVrn MVm to

I

1

20

30

40

Impulse coefficient coefficient for for the the grounding grounding rod rod of of 2.5 m m length. length. Figure 6.17. Impulse

To R g by by aa factor To reduce reduce R, factor of of 44 at at the the initial initial rod rod radius radius ro ro = = I1 em cm and and 1= 1 = 2.5 2.5 m, m, the must be be increased the radius radius must increased to to rl r1 == 105 105 em. cm. The The field field at at the the ionized ionized volume volume boundary threshold in boundary must must exceed exceed the the ionization ionization threshold in the the soil, soil, E Eig 10 kV/cm, jg :x: : : IOkV/cm, 2 p= = 1000n/m IOOOR/m must must be be j == Ejg{p:::::::: Eig p = I1 kA/m kA/m2. and the the current current density density at at P • and 2 , l For the the ionized + 7rrl : : : : 24 m the For the surface surface area area of of the ionized volume volume S:::::::: S M 27rr 27rrll = 24m2, the l total be ZI = total leakage leakage current current would would be = jjS S = 24 24 kA, kA, corresponding corresponding to to the the current current of moderate lightning power. of aa moderate lightning power. However, uniform radial However, the the uniform radial ionization ionization expansion expansion of of the the initial initial groundground5 ing rate rl/tf:::::::: m/s (this be completed ing volume volume at at aa rate r l / t f = 22 x 10 105m/s (this process process must must be completed within rise time )..ls) can hardly occur within the the rise time of of the the current current impulse, impulse, tf:::::::: tf = 5 5ps) can hardly occur in in reality. Anyway, no experimental reality. Anyway, there there is is no experimental indication indication for for this. this. More More probable probable would rod 'elongation' would be be the the rod ‘elongation’ owing owing to to the the leader leader development development into into the the soil, soil, because the because the current current density density and and the the field field at at the the rod rod end end are are higher higher than than at at its its lateral more effective lateral surface. surface. The The elongation elongation of of aa grounding grounding electrode electrode is is aa more effective means of resistance R,, R g , because because of R g r-v 1//, means of reducing reducing the the grounding grounding resistance of R, 1/1, since since the with increasing the resistance resistance decreases decreases only only logarithmically logarithmically with increasing radius radius (but (but only no only at at r« r

. 0.5 0.0 0 .40 t . 0

1

2

.

,

4

q.

, 6

.

, 8

Time, IJS ks Time,

,

1

I

10 10

6.19. Computed Computed overvoltage overvoltage of of the the direct direct stroke stroke at at the the transmission transmission line line tower tower Figure 6.19. with the grounding resistance resistance R, 10 s2 for for the the lightning lightning current current with with tf tf = = 51ls 5 ps and and with the grounding R g == 1012 amplitude of 100 100kA; kA; VU,r == 0.3c. 0.3~. amplitude of lightning rise and lightning current current impulse impulse because because of of an an abrupt abrupt rise and an an equally equally abrupt abrupt fall fall of UM, with with time (figure 6.19). 6.19). of time (figure 6.4.3 6.4.3 Voltage between between the affected affected and neighbouring neighbouring objects It many applications to know the voltage It is is important important for for many applications to know the voltage affecting affecting the the insulation height h, by aa direct h, affected affected by direct lightning lightning insulation gap gap between between an an object object of of height this, one stroke, hl < h, h, located located nearby. nearby. For For this, one stroke, and and another another object object of of height height hi should of the the direct direct should find find the the difference difference between between the the evaluated evaluated overvoltage overvoltage of , induced stroke, , and Ud,,,, and the the maximum maximum overvoltage overvoltage U Uinmax, induced on on the the stroke, U d in neighbouri~g dep~~ds on neighbouring object. object. The The latter latter value value strongly strongly depends on the the object's object’s construction. us analyse two extremal So, let let us analyse two extrema1 situations. situations. construction. So, Suppose Suppose aa lightning lightning strikes strikes aa lightning lightning rod rod located located near near the the mast mast it it protects (figure 6.20). 6.20). The The magnetic magnetic components components of of the the overvoltages overvoltages are are protects (figure determined by the EM from from formula formula (6.49). (6.49). For For the the determined by the vortex vortex field field strengths strengths EM yo can can be be taken taken to to be be equal equal to to its its average average radius radius r1' rl. For For the the mast, yo rod, ro mast, ro rod, can be equal the distance between the can be be assumed assumed to to be equal to to the distance d d between the rod rod and and the the mast. between the magnetic mast. The The maximum maximum time-dependent time-dependent difference difference between the magnetic components voltage at between the where the the distance distance between the components of of the the voltage at the the height height h jl,, where constructions is is minimal, minimal, is is equal equal to to constructions

t:J.U

, Mma

~ /LoAihj In ~. 2Jr

r1

(6.50) (6.50)

Expression the value moment this v,t » >> dd at at the the moment this maximum maximum Expression (6.50) (6.50) allows allows for for the value vrt occurs. the insulation occurs. The The magnetic magnetic component component of of the the overvoltage overvoltage across across the insulation


308 308

Dangerous lightning effects on modern structures structures

- - -

-

-,-----,-

h

R 6.20. Estimation Estimation the the voltage voltage between lightning rod struck and and an an object. object. Figure 6.20. between aa lightning rod struck

gap D. AUMm,, UMrOd U,wob, increases with distance d, d, because UMrOd is gap UMmax == UM d -- U with distance because UM d is MOJb· increases ro ro d and and V\10bJ drops drops as as the the distance distance between the object object and and independent of of d independent between the the current has practically practically no no effect effect the current increases. increases. The The return return stroke stroke velocity velocity Vv,r has Its upper limit is is the the value value of of U M for for the the affected affected lightning lightning on AUMm,,. D.UM . Its upper limit on ~ UMmaxl2 . max rod (D.U::" (AUMm,,M UMm,,/2at at dirt d/rl ~ RZ 100). 100). rod The The situation situation with with the the electrical electrical component component of of overvoltage overvoltage is is less less definite. definite. The overvoltage overvoltage is is also also determined determined by the difference difference between two values, values, by the between the the two The AU, Uerd Ueobl, Uerd = -Rg,IM is a definite quantity and Ueobl -- U D.Ue == U ' but but U = -RgJM is a definite quantity and U erod erod eObi eObJ varies with the be varies with the design design of of the the object's object’s grounding grounding mat. mat. The The latter latter may may be common common with with the the lightning lightning rod rod grounding grounding grid grid and and quite quite compact; compact; in in that that case, we have have A D.U bases of the rod the object U e, == 00 because because the the bases of the rod and and the object are are case, we interconnected. be another the grounding interconnected. There There may may be another extreme extreme situation: situation: the grounding mat mat of the the object object may may be be so so far far from from that that of of the the lightning lightning rod that it it may of rod that may be be unaffected by by the unaffected the electric electric field field of of the the lightning lightning current current spreading spreading through through the the UeOb, = 00 and and AU,” = Rg,IM, soil. In In that that case, case, we we shall shall not not have have U == D.Uemax = RgJM' because because soil. eObi this possible only only in in the the absence absence of of current current through the object's object’s this would would be be possible through the grounding mat. mat. In In reality, reality, there there is is an an electric electric charge charge induced induced on on the object’s grounding the object's surface so aa current current flows flows surface due due to to the the electrostatic electrostatic induction induction (section (section 6.1.1), 6.1. l), so across the the object, object, creating creating the the electrical electrical component component of of the the overvoltage. overvoltage. Its Its across value can be be found found from from formula formula (6.5) (6.5) and and the the maximum maximum value from (6.6), (6.6), value can value from provided that the the return return stroke stroke is is simulated simulated by by aa rectangular current wave wave provided that rectangular current in aa vertical vertical lightning lightning channel. channel. Let Let us us evaluate evaluate the the possible voltage from from in possible voltage formula formula (6.6). (6.6). To go go beyond beyond the the zone zone of of the the current current spread spread away away from from the the lighting lighting rod rod To move away grounding, grounding, it it is is necessary necessary to to move away at at aa distance distance ",20 -20mm from from it. it. The The radius of of the grounding grid, grid, within the electrodes electrodes are are located, located, is is radius the grounding within which which the so that that the the distance distance between and the the hardly larger larger than than 55m, hardly m, so between the the rod rod and


Direct stroke overvoltage overvoltage

309 309

object in in formula formula (6.6) (6.6) can can be to be = 25 25m. Assuming that that the the height height object be taken taken to be r ::::; m. Assuming ::::; of m, its 10 pF 1m, of aa typical typical object object is is h = = 30 30m, its linear linear capacitance capacitance C C1 z 10pF/m, j . C= Cjh = 300pF, and R = lOn, we shall have U ::::; 30kV at the light= Clh = 300pF, and g, = l o a 2 ,we shall have UeDb, 30kV at the lighteob) Z,v = = 30 30 kA. Uerod and U Ueobi have different different signs signs and and ning current current 1'>1 ning kA. Although Although U and e e have iAU,I > IUeWdl, IUerodI,the the additional additional value is not not essential essential because it is is an an order order IDoUel> value is because it in the the above above example. example. The The situation situation when when aa of magnitude magnitude less less than than R,ZM of RgIM in lightning rod is put up at at aa distance distance sufficient sufficient for for the the separation separation of of its its own own lightning rod is put up grounding grid grid and and that of the object is is quite quite realistic. realistic. This This is is done done for for the that of the object the grounding protection of of especially especially important important constructions constructions to to avoid avoid pulse noises or or protection pulse noises sparking due due to the induction induction emf, emf, when some of of the the current current finds finds its its way way sparking to the when some to to the the object's object’s grounding grounding through through the the soil. soil. Another extreme Another extreme case, case, in in which which the the electrical electrical component component of of the the object object overvoltage is is dominant, dominant, is is aa lightning lightning stroke stroke at at aa metallic metallic grounded grounded tower overvoltage tower of aa power power transmission transmission line. line. The The direct direct stroke stroke overvoltage overvoltage affects affects an an of insulator to which power wire wire is insulator string, string, to which aa power is suspended. suspended. Consider Consider first first aa simple and and frequent frequent variant variant (in (in lines lines with with an an operation operation voltage voltage below simple below 1lOkV) when the the line line has no protecting wire. In In that that case, case, we we do do not not have have 110 kV) when has no protecting wire. to solve solve the difficult problem of lightning lightning current current distribution distribution between to the difficult problem of between the the affected the wire, affected tower tower and and the wire, repeatedly repeatedly grounded grounded by by the the adjacent adjacent towers. towers. Nor should bother about the protecting Nor should we we bother about the the electromagnetic electromagnetic effect effect of of the protecting wire previous situation, the in the the previous situation, the wire on on the the power power wire wire (section (section 6.4.4). 6.4.4). As in insulator A U equal equal to to the the potential potential insulator string string is is affected affected by by the the overvoltage overvoltage Do difference the tower the string power difference of of the tower at at the the point point of of the string suspension suspension and and the the power wire. The wire. The calculation calculation of of the the tower tower overvoltage overvoltage is is similar similar to to that that for for aa lightlightning just described. A specific ning rod, rod, just described. A specific feature feature of of this this problem problem is is the the existence existence of horizontally, it not respond magof the the wire. wire. Being Being suspended suspended horizontally, it does does not respond to to the the magnetic field power wire wire netic field of of the the current current in in the the lightning lightning vertical vertical channel. channel. The The power is tower grounding the insulator is well well insulated insulated from from the the tower grounding by by the insulator string. string. Owing Owing to being grounded, would be be able potential, to its its far far end end being grounded, it it would able to to maintain maintain zero zero potential, but by the redistribution of but for for the the current current created created by the redistribution of the the charge charge induced induced on on the wire. The the wire. The induced induced charge charge is is very very high high because because some some of of the the wire wire length length is the total total capacitance is located located close close to to the the lightning lightning channel, channel, and and the capacitance of of aa long wire is Naturally, the the small wire and long wire is very very large. large. Naturally, small distance distance between between the the wire and the the lightning lightning channel channel does does not not mean mean the the existence existence of of aa direct direct contact contact between between them, we can wire. so we can speak speak only only of of the the effect effect of of electrical electrical induction induction on on the the wire. them, so Even the wire wire is resistance, the Even though though the is connected connected to to the the earth earth at at zero zero resistance, the induced induced charge charge cannot cannot respond respond immediately immediately to to the the lightning lightning charge charge variation variation and wire potential point is at zero. zero. The The grounding grounding point is located located and the the wire potential cannot cannot remain remain at far by the so the the charge charge liberated liberated by the induction induction far away, away, at at the the end end of of the the wire, wire, so cannot than with with light c. For For the the induced induced cannot be be delivered delivered to to it it faster faster than light velocity velocity c. charge point x, x, aa current q,n to to appear appear at at the the point current wave wave must must be be excited excited at at this this charge qin point, which will eventually point, which will eventually transport transport the the charge charge -qin -q,n out out of of the the wire wire to to the will propagate propagate at the earth. earth. This This wave wave will at light light velocity. velocity. During During its its motion, motion, the potential at rise due the potential at the the wave wave front front will will rise due the the voltage voltage drop drop on on the the wave wave ~


~

o~

310 310


resistance parameters, i.e., i.e., along along aa long long wire. wire. resistance of of the the line line with with distributed distributed parameters, Elementary at any any point point on on the the wire, wire, Elementary current current and and potential potential waves waves arise arise at the lightning lightning field. field. Propagating Propagating where where the the induced induced charge charge is is changed changed by by the the origin, origin, the the currents currents of of with right of with light light velocity velocity to to the the left left and and to to the the right of the elementary the voltage voltage between between the the wire wire and and elementary waves waves are are summed, summed, raising raising the the this voltage, voltage, naturally, naturally, drops drops to to the earth. earth. After After the the waves waves are are damped, damped, this response of of aa long long line line to to the the external external zero, is grounded. grounded. The The response zero, because because the the wire wire is field wire is is described described by by the the field Eox(x, Eo,(x, t)t ) acting acting along along aa horizontally horizontally suspended suspended wire equations equations di at

di dx

-aue = R l i + L l - - E o , ( x , t ) ,

dx

--=

C aue 1 T

(6.51) (6.51)

where to the the line line response response to to the the where the the potential potential Ue(x, U e ( x ,t)t ) is is due due exclusively exclusively to field Eox(x, t). The total potential of the wire relative to the earth, Uge(x, t) field Eo,(x, t ) . The total potential of the wire relative to the earth, U p e ( xt,) = U o ( x ,t)t ) + Ue(x, U e ( x ,t), t ) , contains contains another another component, component, Uo(x, Uo(x,tt),) , defined Uo(x, defined by by the the Neglecting the the ohmic ohmic voltage voltage drop drop charges charges of of the the lightning lightning return return stroke. stroke. Neglecting = -Eox -Eox into into account, account, we we relative relative to to the the induction induction term term and and taking taking aUo/ax b’Uo/b’x = arrive driving force force and and containing containing arrive at at the the wave wave equation equation with with aa distributed distributed driving no damping damping term: term: no

+

2

a Uge ax 2

2

1 a Uge -

c2

-----aT =

1 a

2

[1,0

- c 2 at2

'

C=

1/2

(L 1C1) - .

(6.52)

The solution solution to to this this equation equation represents represents aa general general solution The solution to to aa homogeneous homogeneous equation and a particular solution to an inhomogeneous one, equation and a particular solution to an inhomogeneous one, corresponding corresponding to the the two two identical identical waves waves propagating propagating in in opposite opposite directions to directions along along the the line: line:

zIo

Uge(x,t) = 1 ‘ da 8O U o ( X l , O ) d O +

Uge(x, t) =

a UO(X2 , e) de 2lJI0 ae UO(X1, e) de + 2lJI0 ae

XI = x - C ( t - 0 ) . X1=x-c(t-e),

(6.53) (6.53)

x, = x + c(t - 0 ) . X 2 =x+c(t-e).

The integrals integrals give give the the sum sum of of the the above above elementary elementary waves The waves moving moving at at light light velocity. The The waves waves are are excited excited by by the the time of the the external external field field velocity. time variation variation of U,. For For the the elementary elementary wave wave to potential Uo. potential to arrive arrive at at the the point point x at at the the moment of of time time t,t , the the causative causative variation variation in in U U0o must must occur occur at at the the points points moment f Ax earlier, earlier, by by the the time time e 0= = .6.x/c. A x / c . If If the xx ±.6.x the time time is is counted counted from from the the moment of of the the lightning lightning contact contact with with the the line line tower, tower, the the lower, lower, zero zero limit limit moment of the the integrals integrals of of (6.53) (6.53) should should be be replaced replaced by of by the the time-of-flight time-of-flight of of light light for aa minimum minimum distance distance between between the the lightning lightning channel for channel and and the the wire, wire. In the the general general case, case, the the difficulties difficulties that arise in In that arise in the the calculation calculation of of the the integrals depend depend on on how how one one approximates approximates the integrals the lightning lightning current current related related to to the linear linear charge charge in in the the return return stroke stroke wave wave inducing inducing the field Eo, Eo, as as well well as as the the field on the the lightning lightning channel channel position position relative relative to on to the the wire. wire. Of Of significance significance are are the following following factors: factors: what what object object the the lightning lightning strikes the strikes (the (the earth earth or or an an element of of aa power power transmission transmission line line raised raised above above the the ground), ground), the the channel channel element



311 311

deviation from from the the normal, normal, as as well well as as its its bendings and branching. branching. It It is is imposimposdeviation bendings and sible to to solve solve this this problem problem without without numerical numerical computations. computations. The The question question sible then arises arises as as to the stage stage in in the study, at at which computer simulation simulation is is then to the the study, which aa computer most helpful. One should should not not ignore ignore aa numerical numerical integration integration of of initial initial most helpful. One equations (6.51), (6.51), allowing allowing the the control control of of the the effect effect of of active active resistance resistance R equations R,I which sometimes sometimes has has aa large large value. value. The The effective effective value value of of R 1Imay may be be much which much higher than than the the resistance resistance of of the the line line wire to direct direct current current because because of of the the higher wire to skin-effect, the the soil soil resistance, resistance, used used by by the wave as as aa 'return ‘return wire', wire’, and and due due skin-effect, the wave to the the energy energy consumed consumed by by the the impulse impulse corona. corona. The The corona corona is is excited excited in in to the wire wire by by overvoltages overvoltages and and absorbs absorbs some some of of the propagating wave wave the the propagating energy, contributing contributing to to its its damping. damping. The The impulse impulse corona corona also also increases increases the the energy, effective linear linear capacitance capacitance of of the the wire, wire, since since the the electrical electrical charge charge is is localized localized effective not only on on the the wire surface but but in in the the adjacent adjacent air. air. The The charge charge is is delivered delivered not only wire surface there by streamers of of metre lengths. The The capacitance capacitance ClefT CleRdepending depending on on the the there by streamers metre lengths. local wire wire overvoltage overvoltage varies varies together together with with the the velocity velocity of of perturbations perturbations in in local the wire, wire, vU = = (C'efTL, (C1,,L1)-1/2. This greatly greatly distorts distorts the the wave wave front, front, since since different different the )-'/2. This sections of of the the wave wave front front have have different different velocities. velocities. The The problem problem becomes becomes sections greatly greatly non-linear non-linear and and definitely definitely requires requires aa numerical numerical solution. solution. The The calculation calculation formulas formulas given given below below describe describe simple simple situations situations neglecting neglecting the the wave wave damping damping in in the the wire. wire. They They have have been been derived derived by by direct direct integration integration of of (6.53) (6.53) and and borrowed borrowed from from [3]. [3]. The The lightning lightning channel channel is is considered considered to to be be vertical; vertical; the the return return stroke stroke wave wave moves moves towards towards the the cloud cloud at at constant constant velocity velocity VU,. For aa rectangular rectangular charge charge wave wave in in the the channel channel of of r . For lightning lightning that that has has struck struck the the earth earth (but (but not not aa line line tower) tower) at at aa horizontal horizontal distance distance r from from the the wire, wire, we we obtain obtain for for the the wire wire point point nearest nearest to to the the lightning lightning channel channel IMh I [ (3] Uge(O, t) = - -27r-s-oc-r -",2-+-1 '" - -(",-2-+-1-_-(3-2)--:-'"""""/2 '

vrt

'" = -

r

( 6.54 )

p= = vrl ur/c, is the the return return stroke stroke velocity, velocity, and and h is is the the wire wire height height above above where (3 c, VU,r is where the of the the lightning lightning the earth. earth. The The time time in in (6.54) (6.54) is is counted counted from from the the moment moment of channel channel contact contact with with the the earth. earth. This This formula formula can can be be used used at at t > ric, r / c , i.e., i.e., after the the electromagnetic electromagnetic signal signal has has covered covered the the distance distance between between the the channel channel after and and the the wire. wire. The The overvoltage overvoltage is is still still active active at at aa large large distance distance from from the the stroke stroke point (x (x -+ + 00), CQ), where where the the lightning lightning field field effect effect is is negligible, negligible, E Eox x 0. The point ~ O. The ox wave reaches reaches that that point point through through the the wire, wire, as as in in the the case case of of aa communications communications wave R I= = 0). 0). Such Such overvoltage overvoltage waves waves are are line (this (this occurs occurs without without damping damping at at R, line known known as as wandering wandering waves. waves. For For these these waves, waves, we we have have _ _ Uge (00, t ) -

IMhvrt 2 2 2 27rsoC( V r t + r )

(6.55) (6.55)

where the the time time is is counted counted from from the the moment moment of of the the wave wave front front arrival arrival at at the the where ‘infinitely’remote remote point point of of interest. interest. At At t m , == rIV r/wr, the function function - UUgeCoo, g e ( m ,t)t ) has has 'infinitely' n the


312 312

Dangerous lightning effects on modern structures 0.5

:2

0.4

~

~

~o OJ t::

N

'-'

;:J~ 0.2

0.1

0.0 -+WL....I----r--....--r-........---r-~_,__-.-__,

o

2

t/ 3

Vr r

4

5

Evolution of of an an overvoltage overvoltage at at the the wire wire point point nearest nearest to to the the lightning lightning Figure 6.21. Evolution stroke stroke point point (solid (solid curves) curves) and and aa wandering wandering wave wave voltage voltage (dashed (dashed curve). curve). For For aa return stroke, stroke, aa rectangular rectangular current current wave wave model model is is used. used. return

maximum: aa maximum: IMh -UgeeXJ , t m ) = -47T" corc

(6.56) (6.56)

which At t» t >> tt,,m , the the overvoltage overvoltage is is damped damped as as tf-’.1• which is is independent independent of of Vq.. r . At Typically, the the amplitude amplitude of of aa wandering wandering wave wave is is somewhat somewhat higher higher than than Typically, the voltage voltage amplitude amplitude relative relative to to the the earth earth at at the the site site where where the the wire wire is is closest closest the to U.o to the the stroke stroke point point (figure (figure 6.21). 6.21). The The reason reason for for this this is is the the opposite opposite signs signs of of U and U U,, causing aa reduction reduction in in the the value value of of U Uge U,e + U U,o in in the the close close and ge == U e, causing of the the wire, wire, where where U U.o -I# O.0. Far Far from from the the stroke stroke point, point, we we have have vicinity of vicinity M 0, 0, and and the the overvoltage overvoltage is is totally totally defined defined by by the the wire wire response. response. Although Although Uo, >:::: this the overvoltage overvoltage maximum maximum at at the the closest closest point point does does vary vary with with Vwr, the n this variation is is not not appreciable. appreciable. This This is is good good because because there there are are few few measurements measurements variation of the the return return stroke stroke velocity velocity and and practically practically no no synchronized synchronized measurements measurements of of the the lightning lightning current. current. of If the the lightning lightning current current is is supposed supposed to to rise rise at at the the impulse impulse front front as as If i(t) == Ait A,t with with Ai A , == const, const, the the overvoltage overvoltage Uge(O, U,,(O. t), t ) ,for for the the same same conditions conditions and and designations designations as as in in (6.54), (6.54), is is

+

+

2 1 - 32)”2 (2 3K U (0 ) Aih In (K + I _32)1/2 -- .3K Uge(0. t ) = -In ge , t = - 27rcocvr 1 _ .32 27r&oCWu, 1 - 32

+

(6.57) (6.57)

+

Formula (6.57) (6.57) has has aa sense sense at at ric:;:::; r / c 6 t:;:::; t 6 t[t f + ric r / c (3:;:::; ( 3 < KK :;:::; < (vrtr/r) ( t + t f / r )+.3). 3).After After Formula > t[, tf, the the calculation calculation can can be be made made using using this this the current current impulse impulse maximum, maximum, tt > the formula and and aa superposition, superpositior. by by representing representing aa real real current current wave wave as as two two formula



313 313

waves of of different different signs signs shifted shifted in in time, time, as as was was done done in in the the comments comments on on waves formula (6.49). (6.49). As As the the time time increases increases within within the the lightning lightning current current rise rise time, time, formula the value value of of Uge(O, Ug,(O,t) t ) rises rises monotonically. monotonically. The The calculated calculated pulses V,,(O, t) the pulses Uge(O, for aa rectangular rectangular current current wave wave and and for for aa wave linearly rising rising current current for wave with with aa linearly have something in in common common at at r/ r/v, tf, which which is is valid valid for for remote remote lightning lightning have something Vr x ~ tf, strokes with 100m. Both approximations approximations lead lead to to an an increase increase in in the the overover~ 100 m. Both strokes with r 2 voltage during during the the current current front. front. But But for for close close strokes, strokes, especially especially for for aa direct direct voltage stroke at at aa line line tower, tower, the discrepancy between the calculated calculated pulses stroke the discrepancy between the pulses becomes becomes remarkable. This This is is the the reason for the sceptical attitude attitude to analytical solusoluremarkable. reason for the sceptical to analytical tions, which which we showed at at the the beginning of the the discussion. discussion. No doubt, aa tions, we showed beginning of No doubt, linearly rising rising current current is is closer closer to to reality reality than than aa rectangular rectangular impulse, impulse, but but it it linearly cannot simulate simulate the the actual actual current current rise rise accurately. accurately. The The same same is is true true of of the the cannot impulse within the impulse amplitude. amplitude. The The discrepancy discrepancy in in the the calculations calculations made made within the models models considered considered grows grows with with decreasing decreasing distance distance r.r . Nevertheless, another another analytical analytical solution solution [3] [3] may may be be useful useful for for the the estimaestimaNevertheless, tions. tions. It It concerns concerns the the case case of of aa direct direct lightning lightning stroke stroke at at aa transmission transmission line line tower, tower, when when the the shortest shortest distance distance between between the the lightning lightning channel channel and and the the and wire is is determined determined only only by by the the height height difference difference between between the the tower, tower, hh,,s , and wire the wire, wire, h. h. If If aa charge charge wave wave corresponding corresponding to to the the lightning lightning current current front front the i(t) = = Ait Air moves moves up up along along the the vertical vertical lightning lightning channel channel from from the the tower tower top top to we shall have at the point of the wire to the the cloud cloud with with constant constant velocity velocity Vw,, we shall have at the point of the wire n suspension suspension (6.58)

Let Let us us calculate calculate the the overvoltage overvoltage due due to to the the first first component component of of aa negative negative lightning with with the the average average parameters parameters 1IM = 30 kA, tf = 5 ps, = 6 kA/~s, kA/ps, lightning = 30 kA, tf = 5 ~s, Ai = 6 M and = 0.3c 0 . 3= = ~ 90m/~s. 90m/ps. We We shall shall have have Uge(O, U,,(O, tr) tf) ~ = 320kV 320kV for for hhoo = = 30m 30m and Vw,r = and m at and h = = 20 20m at t = = tf. tf. Similar, Similar, but but of of the the opposite opposite sign, sign, is is the the potential potential rise rise on on the the tower tower grounding grounding resistance resistance R R,g ~ x 10 10n R due due to to the the lightning lightning current, current, R l M =300 kV. This doubles the electrical component of the overvoltage RgIM =300 kV. This doubles the electrical component of the overvoltage g across across the the insulator insulator string. string. It It is is worth worth noting noting the the specific specific effect effect of of overvoltage overvoltage on on the the line line insulation. insulation. Overvoltage Overvoltage is is not not strictly strictly related related to to any any point point on on the the line, line, as as is is the the case case with with the the voltage voltage drop drop across across the the tower tower grounding. grounding. It It has has been been mentioned mentioned that that the the charge liberated liberated by by electrical electrical induction induction moves moves along along the the wire, wire, creating creating aa charge wandering wandering overvoltage overvoltage wave. wave. With With aa negligible negligible damping, damping, itit can can cover cover aa distance of of several several kilometres, kilometres, affecting, affecting, on on its its way, way, all all the the insulator insulator strings strings it it distance encounters. An An insulation insulation breakdown breakdown may may occur occur even even far far from from the the lightning lightning encounters. stroke, where where the the line line insulation insulation is is poor poor for for some some reason. reason. Really Really hazardous hazardous is is stroke, the encounter encounter of of the the wandering wandering wave wave with with aa high-voltage hgh-voltage substation, substation, because because the the the overvoltage overvoltage wave wave penetrates penetrates to to the the internal internal insulation insulation of of its its transformers transformers and and generators, generators, which which is is always always poorer poorer than than the the external external insulation. insulation.


314

Dangerous lightning effects on modern structures structures

A A wandering wandering wave wave also also arises arises when when aa lightning lightning strikes strikes aa power power wire. wire. The The lightning the wire point in both lightning current current spreads spreads along along the wire from from the the stroke stroke point in both directions, waves, U(x, directions, producing producing very very strong strong overvoltage overvoltage waves, U ( x ,t)t ) == Zi(x, Z i ( x , t)/2. t)/2. Since wave resistance resistance is Since the the wave is Z:::::; 2 x 250-400 250-400Rn (the (the smaller smaller value value is is typical typical of of ultrahigh voltage with split wires of ultrahigh voltage lines lines with split wires of aa large large equivalent equivalent radius), radius), the the current current 1ZM = 30 30 kA kA would would produce produce an an overvoltage overvoltage with with an an amplitude amplitude of of M = 3.5-6 MV. In to the breakdown 3.5-6MV. In reality, reality, the the overvoltage overvoltage is is limited limited to the value value of of breakdown voltage for where voltage for the the tower tower insulator insulator string string closest closest to to the the lightning lightning stroke, stroke, where the the flashover flashover does does occur. occur, A A wave wave with with an an amplitude amplitude equal equal to to the the string string breakbreakdown down voltage voltage is is aa wandering wandering wave wave in in this this case. case. Of Of course, course, the the overvoltage overvoltage may the string may rise rise again, again, after after the string flashover, flashover, due due to to the the self-induction self-induction emf emf of of the tower and to the the voltage the tower and to voltage drop drop across across its its grounding, grounding, to to which which the the lightning lightning current runs after waves are current runs after the the string string flashover. flashover. Wandering Wandering waves are damped damped by by the the same processes that same processes that determine determine the the resistance resistance R RIj in in (6.51). (6.51). It is is important important for for lightning lightning protection to compare compare the overIt protection practice practice to the overvoltages to direct wire. In the former voltages due due to direct strokes strokes at at aa line line tower tower and and aa wire. In the former case, the voltage drop drop across across the the insulator insulator string string is is the sum of of three case, the voltage the sum three components. voltage drop the tower components. The The voltage drop across across the tower grounding grounding and and the the induced induced voltage voltage of of the the wire wire are are approximately approximately the the same same quantitatively quantitatively but but have have opposite ,,-,2Rg I M over magopposite signs. signs. This This totally totally gives gives about about -2R,ZM over the the string. string. The The magnetic netic component component LsA L,Aii has has aa real real effect effect only only on on the the current current impulse impulse front, front, L,Z,w/tf (L (L,s is is the self-inductance). and its its average average value value is is equal equal to to Ls/,ldtf and the tower tower self-inductance). The magnetic magnetic component component for for the the first first leader leader of of aa moderate moderate negative negative lightning lightning The (ZM = 30 30 kA, kA, tft f == 55 Ils) ps) and and for for aa tower tower of of standard standard size size (h (h,o :::::; M 30 30 m) m) does does not (I M = not exceed 200kV 200 kV but 2R,Iy > 600 kV because of R, 2 10 Cl. It appears that but 2R > 600kV because of R ~ IOn. It appears that exceed / g M g overvoltages due due to to aa direct direct moderate moderate stroke stroke at at aa tower tower can can flashover flashover the overvoltages the insulation only only in in lines lines with with voltages less than than 110 llOkV, strings insulation voltages less kV, which which have have strings less than than 1m 1 m in in length. length. For For aa 220 kV transmission line, line, aa hazard hazard may arise less kV transmission may arise when the currents are are twice twice as as high as the the average average value, value, but such lightnings lightnings when the currents high as but such occur only only with 10% frequency. frequency. The The hazard of aa lightning lightning stroke stroke at at aa tower tower occur with aa 10% hazard of is not not high high for for 500-750 500-750 kV transmission lines, lines, since since they they have have long long strings. strings. A is kV transmission A reverse flashover flashover may may arise arise from from aa lightning lightning with with 100 100 kA kA currents currents and and more, more, reverse but number is but their their number is less less than than 1% 1YOof of the the total. total. If If the the lightning lightning current current strikes strikes the the current current spreads spreads in in both directions along along it. it. With With the the wave wave resisresisthe wire, wire, the both directions tance Z > 200 n, 52, we we get get ZZM/2 MV even even for for aa moderate moderate lightning. lightning. This This tance ZIM /2 > 33 MV is sufficient sufficient to to flashover flashover the the insulation insulation of of any any of of the the currently currently operating operating lines. lines. is A lightning lightning stroke stroke at at aa wire wire should should always always be be considered considered to to be be hazardous. hazardous. A

6.4.4 Lines with overhead overhead ground-wires ground-wires 6.4.4

When aa lightning lightning strikes strikes aa tower tower of of aa power power transmission transmission line line protected protected by When by aa grounded wire, wire, the the current current is is split split between the tower tower and and the grounded wire, wire, between the the grounded grounded due to to which which the the current current load load on on the the tower tower is is reduced. reduced. However, However, the the engineer engineer due is then faced with complex problem of calculating calculating the the current current distribution. distribution. is then faced with aa complex problem of



315

Another aspect aspect of of this this problem problem is is the the account account of of the the screening screening effect effect of of aa Another protecting wire. wire. Since Since the the wire wire is is connected connected to to the the tower, tower, it it acquires, acquires, in in aa protecting first approximation, approximation, the the tower tower potential, thus creating creating aa voltage voltage wave wave of of first potential, thus the same sign. sign. Owing electromagnetic induction, similar wave the same Owing to to the the electromagnetic induction, aa similar wave of of aa lower amplitude amplitude is is excited excited in in the the power power wire. wire. As As aa result, result, the the voltage voltage in in the the lower insulator string, string, equal equal to to the the potential potential difference difference between between the the tower tower and and insulator the power power wire, wire, drops. drops. These These additional additional problems problems complicate complicate the the calculation calculation the of direct direct stroke stroke overvoltage overvoltage for for aa line line with with an an overhead overhead ground-wire. ground-wire. The The of problems that that arise arise here here reJate relate to to electric electric circuit circuit theory theory rather rather than than to to problems physics, so we we shall shall discuss discuss them only briefly. physics, so them only briefly. Many engineers try current distribution distribution between Many engineers try to to calculate calculate the the current between aa tower tower and overhead wire and an an overhead wire within within the the model model of of an an equivalent equivalent circuit circuit with with concenconcentrated as aa source source of current trated parameters. parameters. The The lightning lightning channel channel is is regarded regarded as of current i(t). The inductance L, and grounding i(t). The tower tower is is replaced replaced by by its its inductance L s and grounding resistance resistance R,, R g, the grounding wire of the the two two grounding wire branches branches (on (on the the left left and and on on the the right right of the stroke stroke point) are are represented represented by by the the branch branch inductances inductances L L,/2 and their their grounding grounding point) e /2 and resistances in in the adjacent towers, towers, R,/2, connected in in parallel. One also also introintroresistances the adjacent Rg/2, connected parallel. One duces the current in duces the mutual-induction mutual-induction emf emf M Me, dildt, di/dt, induced induced by by the the lightning lightning current in the circuit (figure the wire-towers-earth wire-towers-earth circuit (figure 6.22). 6.22). This This circuit circuit can can be be simplified simplified further on the further by by putting putting R, Rg = = 0, 0, because because the the principal principal interest interest is is focused focused on the current of tf 1-10 ps current front front of tf = ~ 1-10 /-lS and and because because the the cable cable inductance inductance along along the the many of metres as hundreds many hundreds hundreds of metres of of its its length length is is as as high high as hundreds of of microhenries microhenries and the the time time constant constant is is usually usually taken taken to to be be L LsiR /-lS. This This model model circuit circuit and J R ,g > 100 100 ps. then presents presents no no calculation calculation problems, problems, provided provided that that the the mutual mutual inductance inductance of of then is the vertical vertical lightning lightning channel channel and and the the circuit circuit including including the the ground-wire ground-wire is the known. As As the the return return stroke wave moves moves up, up, the the channel channel is is filled filled by by current current known. stroke wave so that that the the value value of of M Me, rises rises in in time. time. The The calculations calculations similar to those those for for so similar to the derivation for the the time time delay yield [3] the derivation of of formula formula (6.49) (6.49) and and allowing allowing for delay yield [3] Moho [ vrt + 2h o ] Ue(t) = ~ In 2(1 + (3)h + 1 . o

(6.59) (6.59)

The design design circuit circuit for for aa current current aa tower tower of of the the line line with with the the protective protective Figure 6.22. The cable. cable.


316 316


The real process process can justified only The considerable considerable simplification simplification of of the the real can be be justified only at at low low the towers, the current wire circuit grounding resistances of grounding resistances of the towers, when when the current in in the the wire circuit is is limited by its neglect the the current branching limited mostly mostly by its inductance, inductance, and and one one can can neglect current branching off to the the grounding resistances of towers except the one off to grounding resistances of all all other other towers except the one nearest nearest to to the affected affected tower. tower. the The R g in real transmission transmission line with low The value value of of R, in aa real line in in areas areas with low conductivity conductivity soils may may be several times times higher than the the normal normal value, value, reaching reaching 100 100 O. 0. Then Then soils be several higher than the must also take into the current current distribution distribution problem problem must also take into account account the the removal removal of of some the lightning to 2-5 some of of the lightning current current to 2-5 towers towers away away from from the the stroke stroke point. point. The The equivalent becomes more more complicated equivalent circuit circuit becomes complicated (chain-like), (chain-like), representing representing aa of link link circuits circuits identical identical to to the the first first one. one. For For aa more more rigorous solution, series of series rigorous solution, the ground-wire ground-wire is is to to be be considered considered as as aa long long line line with wave resistance resistance Zc Z, the with aa wave and many many local local non-uniformities produced where ground-wire contacts contacts aa and non-uniformities produced where the the ground-wire tower. Each Each tower tower is is then then represented as aa chain chain of of L, and R, connected in in represented as L s and R g connected tower. series. Figure Figure 6.23 6.23 illustrates illustrates the the variation of the current impulse impulse in in the the series. variation of the current tower with with the the design design circuit. circuit. For circuit with with lumped lumped parameters, the neglect tower For aa circuit parameters, the neglect of the the tower grounding resistance resistance R, x 10 10 0R does does not affect the the result result much, much, of tower grounding Rg ~ not affect while at at R, 1000, the the tower tower current current impulse impulse shape shape changes changes radically. radically. A while Rg x ~ 1000, circuit with with distributed distributed parameters parameters permits permits one one to to follow follow the the effect effect of of conconcircuit secutive wave wave reflections reflections at at the contacts between the ground-wire ground-wire and and the the secutive the contacts between the towers. The The current current impulse impulse distortion distortion by by the reflected waves waves is is especially especially towers. the reflected

R=100hm

14

~.g

/

,/ ,/

12

10

,,

,

, ,,

,,

,

, ,,

,,

"",

6

4

--------------------------2

o-f--.--,-.....,....--.--.....---,,.----.---, o 5 10 15 20 Time, ps Time, ~s

6.23. Current Current impulse impulse on on the the struck struck tower tower with with (solid (solid curves) curves) and and without without Figure 6.23. (dashed curves) curves) allowance allowance for for aa grounding grounding resistance resistance of of the the nearest nearest tower. The tower. The (dashed is used for return return stroke. stroke. model with linearly raising raising current current front front is model with aa linearly used for



317 317

appreciable short impulse front tftf characteristic subsequent lightappreciable for for aa short impulse front characteristic of of the the subsequent lightning effect of of ning components. components. As As the the linear linear resistance resistance of of the the line, line, R Rj,I ,rises, rises, the the effect the the reflections reflections becomes becomes less less pronounced pronounced because because the the reflected reflected waves waves are are damped more strongly. In any case, case, the some of damped more strongly. In any the overhead overhead wire wire takes takes some of the the lightning current current away, away, thereby thereby unloading unloading the the affected affected tower; tower; this this current current lightning fraction fraction cannot cannot be be less less than than 2RgjZe. 2R,/Z,. Let us us evaluate evaluate the the screening screening effect effect of of the the protecting protecting wire, wire, which which also also Let reduces the the direct direct stroke stroke overvoltage. overvoltage. Engineers Engineers had had become become aware aware of of this this reduces effect long long before before overhead overhead wires wires were were used used as as lightning lightning protectors. protectors. Some Some effect even supposed supposed that wire could could reduce reduce the the voltage of an an insulator insulator string string even that aa wire voltage of to lower than The wire electrical to aa value value lower than the the flashover flashover voltage. voltage. The wire acquires acquires the the electrical potential of the tower, tower, which which has has increased increased by by the the value value of of the the voltage voltage potential of the drop the grounding grounding resistance. resistance. As aa result, result, aa high-voltage high-voltage wave wave runs runs drop across across the along the wire. wire. The The nearby nearby power power wire wire finds finds itself itself in in its its electromagnetic electromagnetic along the field inducing similar wave. If U U,e is in the field inducing aa similar wave. If is the the voltage voltage wave wave amplitude amplitude in the ground-wire, the the voltage voltage produced produced in in the the power power wire wire is is U Ucoup = kcoupUc, ground-wire, = k U eoup e , eoup where kkcoup = ZCw/Z, is a coupling coefficient and Z,, is the wave resistance where = Zew j Ze is a coupling coefficient and Zew is the wave resistance eoup of of the the grounded grounded wire-power wire-power wire wire system system which whch can can also also be be regarded regarded as as aa long long The line. line. We We have have Zew Z,, == (LlewjClcw)I/2 ( L ~ c w / C l ~ )by by 1 ’definition. definition. 2 The linear linear inductance inductance LL1, lcw and the capacitance Clew Clcw between this two two wires wires are are calculated calculated in in aa and the capacitance between this conventional way, allowance for for the conventional way, with with the the allowance the earth’s earth's effect. effect. With With Llcw L lew "-' In[(h + h)j(h h)] and C "-' {In[(h + h)j(h h)]} -I, the coupling ln[(ho h ) / ( h o h ) ] and Clcw {ln[(ho h ) / ( h o h ) ] } ’ , the coupling o lew o o o is coefficient is coefficient

-

+

N

~ In[(h o + h)j(ho - h)]

k eoup

In(2h ojr e )

(6.60) (6.60)

where where rY,e is is the the ground-wire ground-wire radius. radius. For For aa rigorous rigorous calculation, calculation, the the geometrigeometrical cal radius radius in in (6.60) (6.60) should should be be replaced replaced by by an an equivalent equivalent radius radius of of the the space space charge charge region region at at the the wire, wire, (the (the space space charge charge is is incorporated incorporated by by aa impulse impulse high voltage). the corona the action corona under under the action of of high voltage). This This somewhat somewhat increases increases the value of k eoup . Measurements Measurements give k eoup = ~ 0.25, instead of value of kcoup. give approximately approximately kcoup 0.25, instead of the calculated calculated 'geometrical' value of k eoup z ~ 0.2. Therefore, the the electrical electrical the ‘geometrical’ value of kcoup 0.2. Therefore, component power wire wire overvoltage is reduced reduced once once more, more, this this time time by by component of of power overvoltage is the value value U . = k U The total overvoltage reduction owing to the Ucoup = kcoupUc. The total overvoltage reduction owing to the the eoup eoup e ground-wire makes makes up up several dozens percent, percent, decreasing decreasing the the tower-stroke tower-stroke ground-wire several dozens effect effect on on the the transmission transmission line line insulation. insulation. We We should should like like to to mention mention aa certain certain relationship relationship between between the the type type of of lightning lightning action action and and the the transmission transmission line line cut-off. cut-off. Even Even induced induced overvoltages overvoltages are are hazardous hazardous for for low low voltage voltage lines lines (primarily (primarily those those of of 0.4-10 0.4- 10 kV). kV). Induced Induced overvoltages much more the main overvoltages are are much more frequent frequent than than direct direct strokes strokes and and are are the main reason for the line line cut-offs. cut-offs. A protecting protecting wire wire is useless in in this this case, low reason for the is useless case, so so low at all. all. For For aa line line of 35 kV or or voltage transmission transmission lines lines do do not not have have it voltage it at of 35kV more, induced induced overvoltages overvoltages are are practically practically harmless harmless and direct lightning lightning more, and direct strokes dominant. The effect of of an an overhead grounded wire wire strokes are are dominant. The favourable favourable effect overhead grounded


318 318


becomes ~ 110 kV, when when the the lightning lightning becomes apparent apparent at at an an operating operating voltage voltage U 2 110 kV, current after aa stroke stroke at at aa line line tower tower current leading leading to to the the insulation insulation flashover flashover after of cut-offs cut-offs for for 110-220 kV exceeds exceeds an an average average value value ",30 ~ 3 kA. kA. 0 About About 50% 50% of 110-220 kV lines to strokes strokes at at towers towers and and 50% 50% lines equipped equipped with with aa grounded grounded wire wire are are due due to of through to to get get to to the the power power wire. wire. of cut-offs cut-offs occur occur when when lightning lightning breaks breaks through number of are due due to to lightning lightning Beginning Beginning with with 500 500 kV, kV, an an increasing increasing number of cut-offs cut-offs are breakthroughs breakthroughs to to the the power power wire. wire.

6.5

Concluding Concluding remarks remarks

We by describing the lightning lightning effect effect on on We finish finish this this chapter chapter and and the the book book by describing the power unable to to offer offer aa clear clear power transmission transmission lines. lines. Scientists Scientists are are still still unable mechanism. Modern Modern computer computer mathematical mathematical description description of of its its complicated complicated mechanism. simulations refine aa mathematical mathematical model model of of the the simulations can can infinitely infinitely specify specify and and refine field and and the the object’s object's lightning lightning effect, effect, with with respect respect to to both both the the electromagnetic electromagnetic field interesting, useful useful and and makes makes response response to to aa stroke. stroke. This This is, is, to to some some extent, extent, interesting, has no no limit. limit. To To illustrate, illustrate, aa sense. sense. The The process process of of refining refining computations computations has parameters with with the the account account of of detailed detailed analytical analytical treatment treatment of of long long line line parameters the of pages pages in in the the work work by by the earth's earth’s effect effect has has taken taken several several hundreds hundreds of been created created for for the the solution solution of of Sunde Sunde [6]. [6]. Suppose Suppose aa superprogramme superprogramme has has been the lightning lightning protection protection problem; problem; its its application application will the will immediately immediately show show that the the great great efforts efforts it it has has required required can can change change but that but little little the the existing existing low low predictability of of lightning-induced lightning-induced cut-offs. cut-offs. The The key predictability key problem problem today today is is not not aa rigorous mathematical mathematical solution solution of of the the available available equations equations but but an an adequate adequate rigorous physical description description of of the the principal principal physical physical processes processes producing producing aa lightning lightning physical discharge, its its electromagnetic electromagnetic field, field, and and the the object's object’s response For this this discharge, response to to it. it. For reason, we we have have tried tried to to present present simple simple qualitative reason, qualitative models models rather rather than than stringent solutions solutions to to the the equations. equations. On On the of stringent the other other hand, hand, many many aspects aspects of this problem problem have have been been omitted, omitted, partly partly because this because they they are are not not directly directly related related to lightning lightning as as aa physical physical phenomenon phenomenon and and partly of space space or or to partly due due to to the the lack lack of to the the limited limited knowledge knowledge about about the the key key physical to physical phenomena. phenomena. Let us us look look back back at at the the material material presented presented in in this this book book in in order order to to Let emphasize the the points points of of primary primary importance. importance. After After the the numerical numerical value value of of emphasize an overvoltage overvoltage has has been been calculated, calculated, it it is is necessary an necessary to to compare compare the the result result obtained with with the the flashover flashover voltage voltage of of the the insulation insulation in in order order to to identify identify its its obtained possible flashover. flashover. Most Most of of the the voltage-time voltage-time characteristics characteristics of of insulation insulation possible strings have have been been found found from from tests tests by standard 1.2/50 1.2/5Ops IlS impulses impulses (here, (here, strings by standard the first first value value is is the the front front duration duration and and the second is the second is the the impulse impulse duration duration the on the the 0.5 0.5 level). level). Such Such aa refined refined impulse impulse has has little little to to do do with with lightning lightning overoveron voltages, and and this this is is clear clear from from figure figure 6.24. 6.24. A A lightning-induced lightning-induced overvoltage overvoltage voltages, has necessarily necessarily aa short-term short-term overshoot overshoot arising arising not has not only only from from the the current current wave wave reflection by by the the grounding grounding of of the the neighbouring neighbouring towers towers but but also also from from the the reflection magnetic induction induction emf. emf. It It is is not not quite quite clear clear how how this this rapidly rapidly damping damping magnetic


Concluding remarks 30

319 319

lightning current .............................................................

25

20

12

t

15

U

10

~ ::l

5

040

5

10 10 Time, J.!S ps Time,

15

20

Figure 6.24. Current Current impulses impulses in in aa struck struck tower tower (is) ( i s ) and and in in the few neighbouring neighbouring Figure the few towers (il-i3). The wave wave problem problem was was solved solved allowing allowing for for wave wave reflection reflection from from the the towers (i)-i 3 ). The places wire is with the places where where grounded grounded wire is connected connected with the towers. towers.

overshoot overshoot affects affects the the electrical electrical strength strength of of an an insulation insulation string. string. Under Under certain certain conditions, aa powerful powerful corona corona flash flash saturates saturates the the gap gap with large space space conditions, with aa large charge [12]. As charge and and can can 'lock ‘lock up' up’ the the leader leader process, process, increasing increasing the the strength strength [12]. for by positive for 'anomalously' ‘anomalously’ long long overvoltages overvoltages induced induced by positive lightnings, lightnings, the the electrical may, on be several of air air may, on the the contrary, contrary, be several dozens dozens percent percent electrical strength strength of smaller real [5,13]. The The question question of of the the real smaller than than what what standard standard tests tests give give [5,13]. of the the UHV UHV transmission transmission line line insulation insulation is is still still to to be electrical strength strength of electrical be answered. answered. The The return return stroke stroke models models discussed discussed above above ignore ignore the the lightning lightning channel channel branching and whereas an branching and bending, bending, whereas an actual actual discharge discharge channel channel is is far far from from being straight vertical vertical conductor. conductor. The The channel channel can can deviate deviate from from the the being aa straight normal by dozens normal by dozens of of degrees, degrees, especially especially when when it it approaches approaches high high conconstructions. structions. Another Another complicating complicating point point in in aa return return stroke stroke model model is is the the counterleader. The The assumption assumption that the return wave starts starts directly directly from from counterleader. that the return wave of an an affected affected construction, construction, say, say, from from aa line line tower, tower, is is far far from from the the the top top of the height of reality. The reality. The length length of of aa counterleader counterleader is is comparable comparable with with the the height of the the construction with the construction it it starts starts from. from. Together Together with the total total length length of of the the streamer streamer this will zones 1.5-3 zones of of the the descending descending and and ascending ascending leaders, leaders, this will give give aa value value 1.5-3 times greater greater than than the object’s height. height. Such Such aa high high altitude altitude of of the the return return times the object's stroke point of stroke origin origin and and its its propagation propagation in in both both directions directions from from the the point of on contact, not not only only towards towards the the cloud, cloud, may may have considerable effect effect on contact, have aa considerable the electromagnetic electromagnetic field field of of the lightning. The The available available theoretical theoretical models models the lightning. the data on on counterleaders counterleaders do not not take facts into into account. account. There There are are no do take these these facts no data related to the subsequent subsequent lightning lightning components. components. Today, Today, it it is is even even impossible impossible related to the


320 320


to mere existence travelling to meet aa to confirm, confirm, or or to to disprove, disprove, the the mere existence of of aa leader leader travelling to meet dart leader. leader. dart Another weak point point of Another weak of the the models models is is the the set set of of statistical statistical data data on on the the amplitude amplitude and and time time characteristics characteristics of of the the lightning lightning current current impulse. impulse. There There is because these is some some information information on on medium medium current current lightnings, lightnings, because these are are numernumerous, poorly understood. ous, whereas whereas lightnings lightnings of of extremal extremal parameters parameters are are poorly understood. The The consequences of of this this are are quite quite serious. serious. The The choice choice of of protection means and and consequences protection means measures what has measures depends, depends, to to aa large large extent, extent, on on what has actually actually caused caused the the storm storm cut-off reverse flashover cut-off of of aa particular particular transmission transmission line line -- aa reverse flashover of of the the insulation insulation string, when lightning strikes strikes aa tower, tower, or or the lightning breakthrough breakthrough to to string, when the the lightning the lightning the bypassing the Underestimating the power power wire wire bypassing the overhead overhead protecting protecting wire. wire. Underestimating or, on on the the contrary, contrary, overestimating overestimating the the high current probability by ignorignoror, high current probability by ance, one one may may arrive arrive at at the the wrong wrong conclusion conclusion concerning concerning the contribution the contribution ance, of reverse reverse flashover flashover in in UHV UHV transmission transmission lines, lines, which which may may cause cause great great of losses. The The determination determination of of extremal extremal lightning lightning parameters parameters is is one one of of the the losses. key problems in natural investigations. We We should should like like to to emphasize emphasize again again key problems in natural investigations. that the exceptions are are more more important important than than the the rules rules to to lightning lightning protection that the exceptions protection practice. practice.

References Refe rences [1] AlII 75 (Pt [l] Wagner Wagner C CF F 1956 1956 Trans. Trans. AIII (Pt 3) 3) 1233 1233 1958 Power Apparatus and Systems 34 34 121 Lundholm Lundholm R, R, Finn Finn R RB B and and Price Price W W S 1958 [2] 1271 1271 [3] Razevig Razevig D DV V 1959 1959 Thunderstorm Overvoltage on Transmission Transmission Lines (Moscow: (Moscow: [3] 216 (in (in Russian) Russian) Gosrenrgoizdat) p 216 Gosrenrgoizdat) H (ed) (ed) Lightning 1977 1977 vol. vol. 22 (London, (London, New York: Academic Press) [4] Golde Golde R RH [4] New York: Academic Press) [5] Bazelyan Bazelyan E EM M and and Raizer Raizer Yu P 1997 1997 Spark Discharge (Boca (Boca Raton: Raton: CRC CRC Press) Press) [5] Yu P p294 p 294 [6] Sunde Sunde E ED D 1949 1949 Earth Conduction Effects in Transmission Transmission Systems (Toronto: (Toronto: [6] Van Nostrand) Nostrand) p 373 373 Van [7] Ryabkova Ryabkova E E Ya 1978 Grounding Grounding in High-Voltage (Moscow: [7] Ya 1978 High- Voltage Installations (Moscow: Energiya) (in (in Russian) Russian) Energiya) M, Chlapov Chlapov A V and and Shki1ev Shkilev A V V 1992 1992 Elektrichesrvo 9 19 19 [8] Baze1yan Bazelyan E M, [8] AV H 1934 1934 Archivfur 818 191 Kaden Kaden H [9] Archiv fur Electrotechnik Electrotechnik 12 818 [lo] Babinov Babinov M M Band B and Baze1yan Bazelyan E EM M 1983 1983 Elektrichesrvo 6 44 44 [10] EM M and and Goryunov Goryunov A A Yu Yu 1991 1991 Elektrichesrvo 1 29 29 [ l l ] Babinov Babinov M M B, Bazelyan E [11] B, Baze1yan [12] Dokl. Akad. Nauk SSSR 155 784 [12] Bazelyan Bazelyan E EM M and and Steko1nikov Stekolnikov II S 1964 1964 Dokl. 784 V 1982 1982 Elektrot. Ser. Appar. wysokogo napryaz[13] Burmistrov Burmistrov M MV [13] Elektrot. promyshlennost’; promyshlennost'; Ser. 123 heniya 1 123


Lightning Physics and Lightning Protection

Lightning Physics and Lightning Protection

Lightning Physics and Lightning Protection

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Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

Lightning

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