Electrical Degradation and Breakdown in ... - IET Digital Library

Materials and Devices Series 9Electrical Degradationand Breakdownin PolymersL.A. Dissadoand J.C. FothergillEdited by G.C. Stevens

IET Materials and Devices Series 9Series Editors: Professor D.V. MorganDr N. ParkmanProfessor K. OvershottElectrical Degradationand Breakdownin Polymers

Other volumes in this series:Volume 4Volume 5Volume 6Volume 8Volume 9Volume 10Volume 11Volume 12Semiconductor lasers for long-wavelength optical fibre communicationssystems M.J. Adams, A.G. Steventon, W.J. Devlin and I.D. HenningSemiconductor device modelling C. M. SnowdenOptical fibre C.K. KaoPhysics and technology of hetrojunction devices D.V. Morgan andR.H. Williams (Editors)Electrical degradation and breakdown in polymers L.A. Dissado andJ.C. FothergillElectrical resistivity handbook G.T. Dyos and T. Farrell (Editors)III-V quantum system research K. Ploog (Editor)Handbook of microlithography, micromachining and microfabrication,2 volumes P. Rai-Choudhury (Editor)

Electrical Degradationand Breakdownin PolymersL.A. Dissadoand J.C. FothergillEdited by G.C. StevensThe Institution of Engineering and Technology

Published by The Institution of Engineering and Technology, London, United KingdomFirst edition © 1992 Peter Peregrinus LtdReprint with new cover © 2008 The Institution of Engineering and TechnologyFirst published 1992Reprinted 2008This publication is copyright under the Berne Convention and the Universal CopyrightConvention. All rights reserved. Apart from any fair dealing for the purposes of researchor private study, or criticism or review, as permitted under the Copyright, Designs andPatents Act, 1988, this publication may be reproduced, stored or transmitted, in anyform or by any means, only with the prior permission in writing of the publishers, or inthe case of reprographic reproduction in accordance with the terms of licences issuedby the Copyright Licensing Agency. Inquiries concerning reproduction outside thoseterms should be sent to the publishers at the undermentioned address:The Institution of Engineering and TechnologyMichael Faraday HouseSix Hills Way, StevenageHerts, SG1 2AY, United Kingdomwww.theiet.orgWhile the authors, editor and the publishers believe that the information and guidancegiven in this work are correct, all parties must rely upon their own skill and judgementwhen making use of them. Neither the authors nor the publishers assume any liabilityto anyone for any loss or damage caused by any error or omission in the work,whether such error or omission is the result of negligence or any other cause. Any andall such liability is disclaimed.The moral right of the authors to be identified as authors of this work have beenasserted by them in accordance with the Copyright, Designs and Patents Act 1988.British Library Cataloguing in Publication DataA CIP catalogue record for this book is available from the British LibraryISBN (10 digit) 0 86341 196 7ISBN (13 digit) 978-0-86341-196-0Printed in the UK by The Redwood Press, WitlshireReprinted in the UK by Lightning Source UK Ltd, Milton Keynes

To our families:Enid and MorwennaandBarbara, Peter and Rosie

PrefaceThe study of electrical breakdown is not a trivial pursuit. Whilst the twentiethcentury has seen great advances in the science of breakdown and prebreakdownprocesses there is still much to be understood. F. W. Peek'sbook, 'Dielectric Phenomena in High-Voltage Engineering' (McGraw HillInc.) published in 1920 described engineering observations of breakdownin commercially-used materials and systems but very little was understoodof the breakdown mechanisms. S. Whitehead's two books on dielectricbreakdown published in 1932 and 1951* show the considerable advancesmade during this period. He noted that it was not until engineers andtheoretical scientists cooperated closely that the various, fundamentallydifferent, types of breakdown could be classified. J. J. O'Dwyer's two bookstpublished in 1964 and 1973 took the analytical models of both conductionand breakdown in solid dielectrics considerably further.Since the publication of these books, the scientific engineering understandingof breakdown has moved on considerably. The 'classical' modelsof breakdown were purely deterministic relying on a chain of causes andeffects to result in breakdown through the system being unable to sustainan energy balance above a critical stress. These models usually assumedhomogeneous insulation and resulted in a single-valued breakdown strengthat which the whole of the insulation would breakdown. Generally howevermeasured breakdown strengths are found to be different every time theyare measured (at least in non-crystalline solids) and may be described by astatistical distribution. Furthermore breakdown usually results in a channelof destruction through the insulation rather than its complete destructionacross a broad front. Also since the publication of O'Dwyer's books extensivestudies have been made on the effects of long-term aging such as electricaltreeand water-tree degradation. These have been found to be particularlydetrimental in synthetic polymeric insulation which is becoming increasinglyused in low and high voltage stress applications. Such degradation resultsin a distribution of times-to-breakdown and the 'tree-type' formations areelegantly described by the new mathematical formalism of fractals. Theseadvances in understanding the stochastic nature of breakdown and thescience and mathematics that describe it have important implications forengineers responsible for designing and maintaining insulating systems. Inview of these advances we decided there was a requirement for a new,comprehensive review of the field.* 'Dielectric phenomena, Vol. 3: breakdown of solid dielectrics', ed. WEDMORE, E. B. (ErnestBern Ltd., 1932) and 'Dielectric breakdown of solids' (Clarendon Press, Oxford, 1951)t 'The theory of dielectric breakdown of solids' and 'The theory of electrical conduction andbreakdown in solid dielectrics' (Clarendon Press, Oxford, 1964 and 1973 respectively)

W'IIPrefaceOur book is divided into five parts. Part 1 serves as an introduction tocater for a broad range of backgrounds of readership. It introduces thechemical and physical structure of polymers, solid-state physics pertinentto charge movement through polymers, and categories of aging and breakdownmechanisms. Part 2 is a comprehensive review of electrical degradationin polymers and, in particular, electrical-tree and water-tree degradation.Part 3 is an introduction and review of conduction and deterministic breakdownin solids as applied to polymers. This part is not intended to be asin-depth as O'Dwyer's 1973 book but assumes less initial knowledge of thesubject. It includes advances made since his book such as filamentary (channel)breakdown, a section on partial discharges, and also examples of thebreakdown mechanisms as they relate to polymers. Part 4 discusses thestochastic nature of breakdown both from the empirical and from thephysical-modelling viewpoints. Part 5 indicates the implications forengineers of the increased scientific understanding presented in earlierparts.In the book we have chosen to concentrate on polymers as they arecommonly used, however much of the discussion applies to non-crystallinematerials generally. We have attempted to make the book both an introductionand a comprehensive review and we appreciate that this has, in places,made some of our introductory arguments rather concise but hopefullycomplete. We envisage that readers using this book as an introductory textwill find the complementary material cited in the references useful and willbe able to also use this book as a guide for further study of the subject.During the writing of the book we have noticed the large amount of researchwhich has been carried out in Japan in the last twenty years and we hopethat this book will serve as a review of this as well as other internationalresearch effort.The style we have adopted reflects the nature of breakdown and degradationin polymers. Not only are such phenomena distributed widely on abasis which is not predictable for an individual sample, but materials giventhe same name often differ from one another in terms of composition andmorphology. Since breakdown is always an extreme condition for a particularsample such differences can have a significant effect on the resultsreported in the literature particularly when undisclosed experimental testfactors such as laboratory temperature, relative humidity, and the formingmethod and prior history of the sample are taken into account. On top ofall this it is a sad fact that very often different features of the same process,such as tree inception time and number density, are often discussed asthough they were measures of the same thing. It is not surprising thereforethat significantly different values are reported for reputedly the samefeatures, and sometimes the results are even more contradictory. We havetherefore adopted an approach in which an identikit' picture of the processesunder consideration is built up by means of mutually-complementarypieces of information. Wherever possible only results and data corroboratedby a number of studies, preferably from different laboratories, have beenused. The picture deduced therefore does not rely on single 'definitive'experiments which it might be justifiably argued do not exist in breakdown

Preface ixstudies. Of necessity such an approach requires a large number of references,and we hope that the reader will bear with us in this, and if necessary forgiveus if the reference list, though substantial, is not exhaustive.We would like to thank STC Technology Ltd. for acting as the originalstimulus for our interest in this field and, together with our Universitydepartments, for actively encouraging that interest over several years.Thanks are also due to our many colleagues including Sue Wolfe, SimonRowland, Robert Hill, John Houlgreave, and Annabel Eccles who havehelped through their discussions. We would particularly like to thank GaryStevens for his careful and constructive criticism of our manuscriptthroughout its preparation and rescind our occasional threats of murder asunfounded reactions to his useful comments. We would like to thank NoreenBerridge, Joyce Meredith and Doug Pratt in the Leicester UniversityEngineering Department Drawing Office for their patient preparation ofthe diagrams. We would like to thank our families, to whom this book isdedicated, for their understanding and patience with our long periods ofinattention to family life during the preparation of the manuscript. We alsoacknowledge the many teachers who sowed the seeds of scientific enquiryin us, among them, Mr. Brace, Mr. Coates, Professor Craig, Mr. Day, Mr.Smith, Mr. Dickinson, Professor Pethig, and Professor Lewis. Finally oneof us (LAD) would like to thank the typist for a valiant effort in the face ofmuch trial and tribulation.Canterbury, April 1991L. A. DissadoJ. C. FothergillEditors prefaceThis book began life as a single chapter in a collection of reviews dedicatedto examining advances in understanding the relationship between thechemical, morphological and defect structure of polymers and their physicalproperties and performance. As with many publishing ventures, that bookwas never completed. However, that early chapter demonstrated that manysignificant advances had occurred in our understanding and description ofelectrical aging and breakdown in polymers which could barely be exploredor appreciated within the constraints of a single review article, and thenotion of a book solely dedicated to this purpose was born.The breadth of this undertaking appeared at first to be staggering inview of the many disciplines involved and the distance travelled since theappearance of the authoritative books of J. J. O'Dwyer in 1964 and 1973.Since that time a more complete view of the structure of semicrystallineand amorphous polymers had been gained and progress made in understandingthe nature of charge injection, electronic excitations and transportin polymers. Advances in band theory and the appearance of molecularelectronics also contributed to and continues to feed this development.Perhaps the most radical influence has been the move away from a purelydeterministic view of electrical breakdown to one centred on the stochasticnature of failure processes. This leads directly to the concepts of spatially

x Prefacedistributed, temporally progressive and cumulative aging processes whichmanifest themselves in specific statistical descriptions of failure. Fortunatecoincidences have also contributed; including the development of fractaldescriptions of damage formation processes, the development of reliablemethods to assess the statistical characteristics of breakdown observationsand the appearance of powerful numerical modelling methods and thecomputing power to implement them.For the first time it now appears that it may be possible to connect themicroscopic physics and chemistry of electrical aging with failure occurringin the bulk and to extend that to real electrical insulation systems wherethe statistical nature of failure has been recognised for many years but notunderstood. Hence the benefits are not confined to the satisfaction felt bythe physicist when finding a model that truly seems to fit the facts but alsohelp the engineer who should be able to define appropriate insulationassessment methods and acceptance criteria which suit his application andhave a sound physical basis.This journey is far from complete and much remains to be done beforeelectrical and water treeing and partial discharge degradation and failurein polymers are fully understood, mathematically described and translatedto better insulating materials and practical systems. What is encouraging isthat the journey has begun. It is in this context that this book plays a keyrole in setting a background against which further advances can be comparedand in directing the novice and the practitioner alike to new avenues ofthought and research.I am grateful to the authors for their endurance and also their patiencein meeting this demanding task. I am confident that the rewards of theirefforts will be seen for many years as the international community benefitfrom the wealth of information in this book and the lead that they have set.G. C. StevensMarch 1992

ContentsPrefaceEditor's prefaceviiixPart 1INTRODUCTION TO POLYMERS AND ELECTRICAL BREAKDOWNIntroduction 11 Polymer structure and morphology 31.1 Structure of polymers 31.1.1 Chemical structure 31.1.2 Physical structure 91.2 Factors affecting crystallinity 101.2.1 Chemical structure 121.2.21.2.3Crystallisation conditionsMechanical flow13141.3 Morphology and factors affecting it1.3.1 Crystallisation from solution14141.3.21.3.3Crystallisation from the meltCrystallisation during flow15171.3.4 Crosslinking 181.4 Bulk defects and free volume 181.5 Techniques for characterising crystallinity and morphology 191.5.1 Calculating crystallinity from density measurements 191.5.2 Permanganic etching 201.5.3 Other techniques 211.6 Additives 22Polymers as wide band-gap insulators 242.1 Conductivity 242.2 Energy band theory: basic concepts 272.2.1 Bonding 282.2.2 Electron energy bands 292.2.3 Band transport 332.3 Band theory applied to polymers 352.3.1 Theoretical considerations 352.3.2 Correlation of calculated and measured electronicenergy bands in polymers 402.3.3 Summary 422.4 Ionic conduction 432.5 Electronic conduction 46

xivContents9.2 Charge injection from electrodes 2179.2.1 Schottky injection 2189.2.2 Fowler-Nordheim injection 2239.3 High-field conduction mechanisms 2289.3.1 Space-charge limited conduction 228(a) Trap-free dielectric 228(b) Dielectric with traps 2319.3.2 Hopping conduction 2329.3.3 The Poole-Frenkel mechanism 2349.3.4 The field-limiting space-charge model 23710 Thermal breakdown 24210.1 Steady-state thermal breakdown 24710.1.1 The maximum thermal voltage (the thick-slab, lowfieldapproximation) 247(a) The general form of the maximum thermal voltage(uniform field conditions) 247(b) The maximum thermal voltage for ionicallyconductinginsulators 249(c) The maximum thermal voltage for Klein's conductivity10.1.2 dependence 250The thin-slab low-field approximation 250(a) General solution for thin slabs 250(b) Thermal breakdown in thin slabs of ionicallyconductinginsulators 250(c) Thermal breakdown in thin insulating slabs usingKlein's a(T) dependence 25110.1.3 The low-field thickness-dependent solution 252(a) General solution 252(b) Solution for ionically-conducting insulators 25210.1.4 High-field (field-dependent) conductivity 253(a) The high-field Klein model 253(b) Schottky and Fowler-Nordheim Emission 254(c) Space-charge limited current 255(d) Poole-Frenkel Conductivity 25510.2 Impulse breakdown 25510.2.1 Low-field impulse breakdown 256(a) Klein conductivity relation 256(b) Arrhenius conductivity 25710.2.2 High-field impulse breakdown 257(a) Klein conductivity relation 257(b) Poole-Frenkel conductivity 25710.2.3 Comparison of steady-state and impulse breakdown 25710.3 Filamentary thermal breakdown 25810.3.1 Prebreakdown currents 25810.3.2 Direct observations of localised heating 26111 Electromechanical breakdown 26311.1 The Stark and Garton mechanism 26311.2 Filamentary theories of electromechanical breakdown 26411.3 Discussion 270

Contents12 Electronic breakdown12.1 'Intrinsic' breakdown12.1.1 The von Hippel, single electron, model (the lowenergycriterion)12.1.2 Frohlich's high-energy criterion12.1.3 The Frohlich amorphous solid model12.2 Avalanche breakdown12.2.1 The critical number of ionising generations12.2.2 The avalanche breakdown strength12.2.3 The statistical time lag12.3 Critique of intrinsic and avalanche breakdown13 Partial discharge and free volume breakdown13.1 The nature of partial discharges13. 1.113. 1.213. 1.313. 1.413.11.513.11.613.1.7The statistical time lagEquivalent circuit of voidThe Townsend dischargePaschen's lawDischarge magnitudeA criterion for Townsend discharge extinction andthe transition to streamer formationThe nature of the streamer dischargeSwarming micro-partial discharge13.11.813.2 Partial discharge degradation13.2.1 Ion bombardment13.2.2 Chemical attack13.2.3 Electrical tree initiation by partial discharges13.3 Free volume breakdownxv271272273274275280280282284285287287288289292296298300301303303304304306311Part 4THE STOCHASTIC NATURE OF BREAKDOWNIntroduction 31714 Statistical features of breakdown 31914.1 Statistical description of breakdown 31914.1.1 The nomenclature of lifetime distributions 31914.1.2 The Weibull distribution 323(a) The two-parameter Weibull distribution 323(i) Definition of the distribution 324(ii)The effect of the shape parameter V onthe distribution 324(iii) Obtaining the Weibull parameters 328(b) Comments on the applicability of the twoparameterWeibull distribution 329(c) The three-parameter Weibull distribution 32914.1.3 Other statistical distributions 330(a) The first asymptotic extreme-value distribution 330(b) The log-normal distribution 331(c) Mixed distributions 331

xviContents14.2 The effect of voltage and time on the failure statistics 33214.2.1 The effect of voltage on the characteristic time tofailure 332(a) The inverse power law 332(b) The exponential law 333(c) Comparison of the inverse power and exponentiallaws 333(d) The flat-z characteristic 33514.2.2 The probability of breakdown as a function of timeand voltage 336(a) Incorporating voltage into the Weibull equation 336(b) Estimating Weibull shape parameters 33814.3 Laboratory studies 33914.3.1 Constant-stress tests 339(a) Results in support of the Weibull distributions 339(b) Results in support of other distributions 34414.3.2 Progressive-stress tests 34414.3.3 Effect of voltage on lifetime 34714.4 Service and field studies 35115 Stochastic models of breakdown 35615.1 Statistical and physical connections 35615.1.1 Physical origins of statistical behaviour 35615.1.2 Extreme value statistics 35715.2 The fluctuation model 36115.2.1 Homogeneous breakdown 36415.2.2 Aging or tree assisted breakdown 36515.2.3 Filamentary conduction paths and tree initiatedbreakdowns 36715.2.4 Comparison with experiment 369(a) Predicted Weibull exponents for different typesof fluctuation 369(b) Effect of changes in morphology 372(c) In-service failures of power cables 374(d) Summary 37515.3 Fractal description of breakdown 37615.3.1 What is a fractal? 37615.3.2 Branched filamentary breakdowns 37815.3.3 Computer simulations of breakdown 38115.3.4 Fractal systems and breakdown statistics 38315.3.5 The utility of the fractal model 38815.4 Cumulative defect models of breakdown 38915.4.1 Percolation theory of breakdown 391(a) Construction of the percolation model 392(b) Characteristic breakdown strength 393(i) Theoretical expression 393(ii) Experimental data for metal-loaded 394materials(c) Breakdown statistics 395(i) Theoretical behaviour 395(ii) Experimental observations 396

Contentsxvii15.4.2 The relationship between cumulative defect and percolationmodels 399(a) Spatially random defect accumulation 399(b) Defect extension 400(c) Cumulative free-volume breakdown 40115.5 Some special cases15.5.1 Electrical tree inception40440415.5.2 Partial discharge inception 41115.5.3 Influence of vented water trees on breakdownstatistics 41515.6 Differences and similarities in the model statistics 419Part 5ENGINEERING CONSIDERATIONS FOR BREAKDOWN TESTINGAND DEGRADATION ASSESSMENTIntroduction 42416 Breakdown testing and analysis 42516.1 Breakdown test methods 42616.1.1 Standards 42616.1.2 Manufacturers' qualification tests 43016.1.3 A suggested test for measuring threshold voltage 43116.2 Statistical analysis 43316.2.1 Graphical techniques 435(a) Use of probability graph paper 435(b) Estimation of parameters of distribution 435(c) Confidence limits 43916.2.2 Numerical techniques 440(a) Maximum likelihood estimation 441(b) Computer program 443(c) Monte Carlo estimation 444(d) Incomplete beta function 445(e) Least-squares calculation for threshold value 44616.3 Temperature and frequency acceleration 44716.3.1 Frequency acceleration 44716.3.2 Temperature and multivariate tests 44716.4 Size scaling 44817 Comparison of AC and DC breakdown behaviour 45217.1 Introduction to statistical differences 45217.2 The relationship to space charge and local currents 45317.2.1 Homogeneous dielectrics in AC fields 45317.2.2 Dielectrics in DC fields 45417.2.3 Polymers as heterogeneous materials 45517.3 'Constant* stress conditions and space charge build-up 45617.3.1 Space charge measurement 45617.3.2 Homocharge 45717.3.3 Heterocharge 45817.3.4 Space charge aging 45917.3.5 Weak spots 460

xviiiContents17.4 Progressively increasing stresses (ramps)17.4.1 Statistical features46146117.4.2 Analysis of experimental data 46217.4.3 Do ramps accelerate constant-stress breakdown? 46417.5 Space charge as a critical parameter 46617.6 A model for ramp and constant stress statistics 46817.6.1 Constant stress {E a ) DC tests 46817.6.2 Progressive DC stress tests 46817.6.3 AC 'constant' stress tests: half-cycle space-chargevariation 46917.6.4 AC progressive stress tests: half-cycle space-chargevariation 47017.6.5 Full-cycle space-charge variation 47117.7 The increase of DC breakdown strengths at low ramp rates 47217.7.1 Charge sweeping by DC fields 47317.7.2 Spatial distribution of space charge 47518 Cable assessment procedures 47718.1 Cable validation tests 47718.1.1 Voltage withstand tests 47718.1.2 Partial discharge tests 478(a) Partial discharge detection 478(b) Routine partial discharge testing 48018.2 Lifetime prediction of cables subject to water tree degradation 48118.2.1 Considerations for predicting water treeing 48118.2.2 Accelerated testing techniques 48418.2.3 Modelling water tree growth 48818.2.4 Discussion 48819 Detecting electrical degradation non-destructively 49019.1 Water trees 49019.2 Partial discharges and electrical trees 49219.3 General types of degradation 500Concluding remarks and future directions 5021 Polymer structure and electrical behaviour 5022 Treeing degradation and failure 5033 Deterministic mechanisms of breakdown 5064 Stochastic modelling and breakdown statistics 5075 Engineering aspects 5086 Summary 510Appendix 1: Computer prog* am for calculating Weibull parameters 513Appendix 2: Calculating the Threshold value of a 3-parameterWeibull Distribution using MATHCAD 826 518

ContentsxixAppendix 3: Mathematical proof 520List of Symbols 523References 529Alphabetical list of authors 563Index 588

PART 1INTRODUCTION TO POLYMERSAND ELECTRICAL BREAKDOWNIntroduction'Electrical breakdown is a subject to which it is difficultto apply our usual scientific rigour. A well-designed insulator(in the laboratory) breaks down in service if thewind changes direction or if a fog descends.'Solymar and Walsh (1984) 1Copyright © 1984, Oxford University PressLightning, the electrical breakdown of air, was probably the first electricalphenomenon to be observed. However, despite the tremendous effort whichhas gone into understanding electrical breakdown since the end of theeighteenth century 2 , the subject is still one of the least understood. Polymersare often the best and most economic electrically insulating constructionmaterials. They generally exhibit very high breakdown strengths (typicallyup to ~10 9 V • m" 1 ), they have low dielectric losses (typically tan 8 < 10~ 3 )and high DC resistivities (typically > 10 16 ft • m). These electrical characteristicscoupled with the wide range of mechanical strengths and stiffnessesavailable from plastics, their high corrosion resistance, ease of forming andmanufacture, and their reasonably low cost 3 has led to their increasing useas electrical insulators following the development of synthetic polymersystems in the 1930s.However this widespread use is not reflected by a good physical understandingof their mechanisms of electrical breakdown or the way in whichan applied voltage may cause eventual degradation after a period of perhapsyears. Nor has such breakdown and degradation been well characterised asa function of parameters such as voltage, time of voltage application, thicknessand temperature to name but a few. Recently there have been somemajor advances in both the physical understanding of these processes bythe scientist and their characterisation as a function of parameters associatedwith their operating conditions by the design engineer. Such advances inunderstanding should continue so that the behaviour of polymers can bepredicted in untried environments and critical applications. Better understandingof failure mechanisms will also lead to better component designsto prevent failure.Some of the problems encountered, whilst being frustrating to theengineer, may be fascinating to the scientist! For example one of the reasons

2 Introduction to polymers and electrical breakdownfor choosing polyethylene as an insulator for power cables in the early 1960swas that it was hydrophobic and it was therefore assumed that there wouldbe no problems associated with water ingress into the insulation material.However, after a few years in service many of these early cables started tobreakdown due to insulation failure as a result of a phenomenon knownas 'water treeing'. This degradation mechanism, which is still a major formof insulation degradation in power cables, was not predicted by scientistsat the time. Whilst this phenomenon is still not completely understood, itsinvestigation has led scientists to a much greater awareness of the complexphysical, electrical, and chemical interactions taking place under such conditions.The response of engineers was to introduce triple extrusion techniquesto prevent moisture ingress and improve the dielectric-electrode interface,and to use crosslinked polyethylene for the manufacture of such cables 4 .This has considerably reduced the water treeing problem although it isdifficult to accurately predict the long-term performance of these cables.Throughout this book we hope to consistently consider electrical degradationand breakdown from both the scientific and engineering point ofview. Inevitably many of our examples are often drawn from the world ofpower cables since they fall directly within our experience. This is not toorestrictive because power cables are one of the largest users of polymers asinsulators and they display a wide variety of electrical breakdown anddegradation phenomena which can be carried over to linear and networkpolymers and other component applications. Since the readership of thisbook is likely to be drawn from diverse fields this first part is designed togive introductory overviews of: the nature of polymers; solid-state physicspertinent to polymers; and electrical degradation and breakdown.

PART 2TREEING DEGRADATIONIN POLYMERSIntroductionElectrical treeing degradation has a long history 121 ' 128185 , which goes backto the early development of paper/oil insulation systems. Since then it hasbeen observed in inorganic as well as polymeric materials and must thereforebe a process whose mechanism is independent of the chemical nature ofthe insulation. Water trees however were not recognised until reports inthe early 1970s 185 identified their presence in the polyethylene cable insulationintroduced in the 1960s, and associated them with premature failures.Since then water trees have been shown to occur in a wide variety ofpolymers 119 as well as polyethylene and its lightly crosslinked derivative(XLPE). These include rubbery copolymers (e.g. EPR), side-group (chain)polymers (e.g. PVC and polycarbonate), aromatic derivatives (polystyrenes),and network polymers (epoxy resins) both above and below their glasstransition. As yet water trees have not been reported in inorganic insulationalthough there is evidence that ZnSe crystals can suffer local damage whenattacked by vented water trees grown in coating films of polyethylene 186 . Itis possible therefore that chemical factors may be involved in their formation.Over the years considerable evidence has been accumulated relating breakdownin cable insulation to the occurrence of trees 119 ' 187 . Although thelargest proportion of cable failures can be traced to mechanical or connectorfaults (90%), treeing is now acknowledged to be the major cause of potentiallyavoidable electrical failures 121 .Examination of cables removed from service showed that nearly all electricaltrees were found together with water trees. Where this was not thecase the origin of the electrical tree could often be traced to asperities onthe conductor or minute metallic particles embedded in the polymer duringextrusion. The use of semiconducting tapes to protect the insulation layerdid not substantially improve the situation 188 and in fact introduced newinitiation sites at embedded contaminant particles 189 such as amber 190 silicagel and ferric sulphate 191 . However after the adoption of a triple extrusiontechnique 192 in which semiconducting layers are simultaneously laid downover the inner conductor and outer insulation, Fig. II. 1, the density ofelectrical trees initiated in this way was substantially reduced, leaving onlywater trees as a major problem 187 .In early publications water trees were commonly characterised by theirshape, e.g. broccoli, bow-tie, streamer, micro, dendritic etc. 193 , and termedeither water or electrochemical trees depending upon their chemical composition.Water trees which were deemed to consist mainly of water, were

70 Treeing degradation in polymersmetal shield insulation conductorjacketFig. II. 1Cut-away section of a power cable with the components as labelledopaque and disappeared on drying out. In contrast electrochemical treeswere often coloured and still visible after drying, as for example the dendriticsulphide trees 194 . However it has now been recognised that these aredifferences in degree only depending upon the ion content and type ratherthan the treeing mechanism. Thus the term 'water tree' has been adoptedto apply to all trees with an opaque electrolyte content.It is now usual to distinguish just two sub-categories, vented trees andbow-tie trees. Vented trees have a stem joining them to the surface of theinsulation, and are therefore in direct contact with a reservoir of aqueouselectrolyte. In contrast bow-tie trees, Fig. II.2, originate from contaminants195 , boundary surfaces 196 , or water-filled voids 1 , within the insulationand thus have only limited access to an aqueous reservoir. Althoughthis categorisation was based purely on visual grounds, there is evidence toshow that there are significant differences particularly in their influenceupon breakdown 175 . A similar nomenclature has been applied to electricaltrees, with bow-tie trees originating within the bulk insulation at metallicimpurities and vented trees from surface asperities and imperfections. Inthis case however there is no evidence for a difference in their effect upondielectric breakdown.Because the initiation and growth of both electrical and water trees ininsulation systems is a long time process under service conditions, theirlaboratory investigation requires some form of acceleration. This is mostconveniently supplied by the use of needle electrodes to enhance appliedfields of kV/mm to nearly MV/mm. Thus metal electrodes in a doubleneedle 198 or point-plane 129 geometry are used to initiate electrical trees,whilst needle-shaped depressions filled with aqueous electrolyte 199 performthe same role for water trees. These geometries fulfil the dual role of highreproducibility combined with field intensification but have the drawbackthat the polymeric material investigated does not necessarily possess eitherthe physical properties or the morphology that exists in engineered insulationsystems . For this reason water tree research in cables in particular

Treeing degradation in polymers 71(a)(b)(c)Fig. IL2 Bow tie water trees found in an XLPE cable removed from service(Rowland, unpublished data), (a) In a group with no obvious initiating defect, (b)initiated by a void, (c) initiated by a contaminant

72 Treeing degradation in polymershas adopted a number of alternative test geometries, Fig. II.3. The mostprominent among these uses miniature cables which are bent into a U shapeand immersed in a bath of electrolyte. In this case the inner conductor mayeither be removed to allow conditions at the inner surface to be controlled 115or retained to simulate service conditions more closely 200 . This latter aimis now closer to realisation through the recent development of procedures201 * 202 which allow sections of production line power cable to betested in the laboratory, Fig. II.3. One other geometry of interest makesuse of Rogowski profiled 203 plaques which are circular parallel-plate specimenswith a specially-profiled increasing thickness around the edge toeliminate the influence of divergence in the applied field. Here the polymeris compression moulded to the required shape with the electrolyte formingthe profiled electrode 204 .All of these water tree investigations involve the need to slice the materialin order to locate the trees and measure their lengths. The problem offading as the trees dry out has been overcome in polyethylene and crosslinkedpolyethylene by boiling with methylene blue dye which renders thempermanently visible. Until recently the opacity of EPR has made it impossibleto see water trees clearly in this material, however a new staining techniquecombined with an oblique illumination of the microscope has now removedthis limitation. Both methods are given in detail by Shaw and Shaw 119 , andby their aid it is possible to determine the number density and lengthdistribution of water trees produced during the time of application of thevoltage 205 .Such features and especially the characteristic tree length, when obtainedin a fixed time under standard conditions, are often used for a comparativeassessment of the treeing resistance of different materials. In additiongrowth curves can be constructed by examining materials that have beenunder test for different periods of time 115 ' 206 . Such plots however do notnecessarily give the growth behaviour of an individual tree or even anensemble of trees, because of the existence of sample-to-sample variationwhich is present even in the standard needle geometry 199 . The best thatcan be hoped for is that standard production techniques will reduce thevariation to a minimum, thereby allowing the constructed plot to be areasonable approximation to the time development of an ensemble of trees.A recent advance in technique now allows these results to be checked bycomparison with the direct observation of the growth of individual trees 20 .Here a television camera is used to view growing trees in a needle pointgeometry, with accurate measurement being made on the image transmittedto a television screen. A simpler version used first by Fournie el a/. 208 is toremove the system from test transiently while the tree is photographed.These latest developments place the study of water trees on a level withthat of electrical trees whose growth has been directly observed both in thinpolyethylene films 209 and transformer oil (see Forster 210 and referencestherein) for some time.This wide range of procedures has been developed because the aim ofthe experiments is not always the same. For example non-destructive directobservations are designed to answer fundamental questions concerning the

Treeing degradation in polymers 73(a)-ElectrolyteSpecimenContainer(b)ContainerElectrolyteSpecimenSemiconductorContainer(c)Miniature cable-Electrolyte(d)ElectrolyteSpecimenElectrolyte fills needleshaped tubeSemiconductorCopper wireSpecimen(e)ElectrolyteStripped cable corePE box for electrolyteMetal conductorBrass earth electrodeEpoxy sealsFig. ILS Standard geometries used for water treeing measurements in the laboratory,(a) Ashcraft geometry; (b) Rogowski plaque; (c) Miniature cables; (d) Needleplane;(e) Disc; (/) Lengths cut from production line cables*

74 Treeing degradation in polymersmechanism of the treeing phenomena, whereas tree formation in uniformfield plaques gives an assessment of the material quality. The testing ofcomplete insulation systems requires the use of cable sections and additionallya statistical analysis. A complete research programme would thusideally embrace each type of technique, using direct observation to investigatethe growth pattern of individual trees, together with destructivemeasurements to determine the influence that the material morphologyand the insulation system has on initiation and growth rates.

PART 3DETERMINISTIC MECHANISMSOF BREAKDOWNIntroduction'Breakdown theory faces a dilemma. The relatively generalmodels can be stated exactly with specific parametersand solutions computed in a straightforward manner.However, the results lend themselves only to order ofmagnitude estimates on real substances, and do notreflect the complexity of experimental data. Alternatively,one could propose a different model for everydifferent dielectric (and possibly for different thicknessesand methods of preparation of the same dielectric)leading to complex computing that offers little insightinto the physical processes involved.'O'Dwyer 501© IEEE 1984Even good insulators such as polymers conduct to some extent (see Chapter2) and the useful range of materials is restricted to those whose conductivitiesare so low that their impedance is dominated by their capacitive capabilitiesunder service conditions. Thus at working stresses the DC leakage currentis negligible and is expected to remain stable as the stress is raised eventhough it may exhibit a non-linear field dependence 502 ' 503 . At some highvalue of the field however an unstable situation will develop and the currentwill rise by many orders of magnitude without any further increase of thefield. During this process the power dissipated in the insulator will meltand probably vapourise it 150 forming a narrow conducting channel betweenthe electrodes unless the power supply is limited in some way, as for examplein reverse-biased zener diodes. In this latter case the situation is reversibleand the system will revert back to an insulating state on reducing the appliedfield. Whilst a similar situation could occur in polymers, perhaps undertrap-filled space-charge limited current conditions 147 , or due to theSchottky 148 or Poole-Frenkel 149 mechanisms (see Chapter 9), the termbreakdown is here reserved for the destructive generation of a conductingpath. Under such circumstances the insulator will have been irreversiblyconverted to a conducting state.In the above description we have reduced breakdown to its fundamentals,namely a cross-over in the current from stability to instability at some field,with consequent material modification. In order for the current to behave

200 Deterministic mechanisms of breakdownin this manner it is essential that a positive feedback mechanism exists, i.e.an increase in the current changes some material property in such a waythat the current is enhanced. The means by which this occurs depends uponthe particular breakdown mechanism, the most important of which forpolymers are discussed in this part of the book. Typical feedback processesare local heating 162 , and impact ionisation 378146 . The current reinforcingeffect of these processes will be opposed by a 'dissipative* mechanism suchas thermal conduction 162 , or the counter field of the ions 378146 (in the caseof avalanches), and at fields below breakdown the current will in time reacha stable equilibrium value. The breakdown field for a particular model isthen defined as the maximum field for which a stable equilibrium can exist.A schematic representation of the current-time characteristics in both stableand unstable regimes is given in Fig. III.la. An interesting comparison canbe made with Figs. 4.3 and 5.2 from which it can be seen that water treegrowth usually corresponds to a nearly stable change, whereas the propagationof most electrical trees represents an unstable process. These considerationsclearly reflect the different degree of degradation that is representedby the two processes.CurrentTime(t)(a)1.0 - 1.0 --Field (E)Time(t)-(b)Fig. III.l (a) Schematic representation of current-time characteristics for a deterministicbreakdown model, where t is the time of application of the field. Examplesare shown for fields below the breakdown value E B and fields above E B . (b) Thecumulative failure probability, P F , for a deterministic model shown both as a functionof the applied field, and as a function of the time to breakdown, t Bt at fields inexcess of E B

Deterministic mechanisms of breakdown 201Models of breakdown which attempt to define balance equations for thecurrent (or energy density) in terms of inter-linked causal mechanisms aredenoted here as deterministic 504 . In this class of model each step in thecurrent response to an applied field is completely determined by previousconditions and itself completely determines future events. A model-dependentbreakdown field, E by can be determined for which the system responseto the applied field just becomes unstable. At fields below E b the system isstable and breakdown will not occur. Above E h a dynamic process respondingto the applied field becomes unstable, thereby 'switching' from being selflimitingto self-enhancing, leading to breakdown. This process may be asingle event or a sequence of 'cause and effects' which finally establish theinstability rather like an electrical tree. In either case however the causalchain of events uniquely determines the time to breakdown for a givenapplied field. These results are summarised in Fig. III. Ib. In practice breakdownis a statistically-distributed process (see Chapter 14) and thus to begenuinely applicable to real systems the deterministic models must bemodified to allow for the possibility of stochastic features in the developmentof the instability (see Chapter 15). (By stochastic we mean that there is achoice of alternative possibilities for development at each step in the process.)Because of this limitation a comparison between predicted breakdown fields(or times) and experimental values can only be made for measurementsaccurately defined as the 'mean', 'minimum', or another characteristic valueof the observed distribution. Thus unless the whole distribution follows apredicted dependence a given deterministic model can only be demonstratedto be the underlying breakdown mechanism for a small fraction ofthe observed breakdowns. In spite of these limitations and problems ofevaluation the deterministic models represent a useful basic framework inwhich to examine the types of process that may lead to breakdown.The catastrophic process whereby a conducting path across the system isfinally formed will be electrically power driven and ultimately thermal inthe sense that the discharge track in solids involves at least the melting andprobably the vapourisation of the insulator 150 (see Section 6.2). Manyfeatures of this final stage are likely to be common to all solids and allmechanisms 161 , and hence deterministic models are usually categorisedaccording to the process(es) that are postulated to lead up to it. Workersoften subdivide breakdown models into various groups dependent onexperimental parameters rather than physical models. We have found ituseful to consider four basic categories for polymeric insulating systems:thermal (E b ~~ 10 5 —10 9 V • nT 1 ), in which the applied power causes (local)heating and a concurrent increase in conductivity; electro-mechanical (E b >—10 8 V- m~ ! ), in which the combination of electrostatic attraction of theelectrodes and (local) softening (or cracking) causes the dielectric to collapse(or fracture); electronic (E b ~l0 7 —10 9 V-m~ 1 ), in which breakdowninitiation is caused by (local) high electric fields; and partial discharge breakdown(E b >~~ 10 6 V« m" 1 perhaps), in which voids in the dielectric ioniseand thereby cause progressive deterioration.It is unlikely that any single mechanism can be responsible for the manydiverse phenomena observed in the breakdown of polymers, and hence inthe following chapters the principles of each breakdown mechanism will be

202 Deterministic mechanisms of breakdownqualitatively described; this will be followed by a more quantitative treatmentand examples of polymer breakdown attributed to this mechanism will begiven. An overview of these mechanisms has already been given in Section3.2. The reader is also referred to the many works on breakdown in solidsthat already exist (e.g. References 82, 133, 150, 159, 161, 163, 165, 378,503, 505, 506, 507). Much of the experimental work using highly viscousliquids is also capable of aiding a better understanding of breakdown insolids and will be of interest (e.g. References 210, 330, 427, 426).Breakdown processes (including degradation processes which ultimatelylead to breakdown) are generally thought of as being in competition suchthat the 'winning' process is that which leads to breakdown quickest underthe service conditions of the insulating system. For example see Vermeer'sresults for different times and temperatures reported by Klein 161 in asystematic way in which it was clearly shown that thermal and electricalbreakdown mechanisms were both operative but which one dominateddepended on both these factors. The results of Park et a/. 508 on PVDF atlow thicknesses and different temperatures indicate a similar electricalbreakdown mechanism dominating at low temperatures and a thermalbreakdown mechanism operating at high temperatures. Most deterministicmodels are based on ideal insulating systems and the instability criterion ischosen such that breakdown quickly ensues after this criterion is satisfied.Thus breakdowns attributed to deterministic breakdown processes usuallyoccur in a very small time scale (typically 10~ 9 to 1 second) at a well-definedhigh field. Under service conditions the fields would be much lower (typicallyby a factor of at least 100) to avoid such breakdown. However degradationprocesses may occur at these fields leading to a weakening of the insulationover a relatively-long period of time and ultimately culminating in breakdown.The 'winning' breakdown mechanism is therefore highly dependentupon field and operating conditions (e.g. temperature) and history and onemechanism may take over from another as time progresses.As well as competing, breakdown mechanisms may also cooperate oraccumulate. These 'cumulative* models of breakdown/degradation aredescribed in Section 15.4 and take account of the variety of breakdownmechanisms by considering the breakdown process to occur in severalseparate stages during which the dielectric is progressively degraded, eachstage establishing the conditions in the dielectric necessary for the initiationof the subsequent stage. In this case breakdown generally ensues at fieldslower than that predicted by an individual breakdown mechanism. Whilstthe model appears to be essentially deterministic, the state of deteriorationis subject to considerable local and temporal variation because of the cooperationrequired between local sites to produce the necessary and sufficientconditions required for the subsequent stage in the overall breakdownprocess. Potentially therefore it embodies considerable stochasticpossibilities, and hence is included in Chapter 15.It is unclear which category of breakdown/degradation partial dischargebreakdown should fall into. The onset of ionisation in voids can be welldefined(and is well characterised) in terms of field and other parametersso it could be thought of as a deterministic mechanism. However the

Deterministic mechanisms of breakdown 203degradation it causes is cumulative, usually changing over to an electricaltreeing mechanism, and stochastic depending on the shape of the void andthe local morphology. We have included partial discharges in this part ofthe book as their description follows on naturally from electrical avalanchebreakdown.In practice breakdown fields may be lowered (and the degradation processesenhanced) by imperfections in the insulating system. In fact breakdownis usually initiated at an inhomogeneity occurring either in thebulk of the insulator or at the insulator/electrode interface. Typicalinhomogeneities include:• electrode aberrations such as 'splinters' of conductor penetrating theinsulation or surface scratches, asperities and depressions;• regions of free volume in the polymer due to morphological irregularitieswith sizes typically less than ~10nm (the so-called Matsuoka voids 168 ),small (sub-micron) voids due to movement of additives etc. which mayonly affect very-long term degradation processes, and larger (supermicron)voids generally due to imperfections in the manufacturingtechnique such as insufficient pressure during high-temperature crosslinking;• impurities in the insulation, including impurity particles (inclusions),moisture and sites of imperfect mixing and coagulation of fillers.All such imperfections will give rise to local changes in electrode geometryor changes in dielectric, mechanical, and/or thermal properties resulting inlocally non-uniform electrical and mechanical stresses and temperaturegradients thereby leading to an increased probability of breakdown eitherin the inhomogeneity or in the surrounding insulation. Such 'physical'inhomogeneities may also give rise to space charge clouds which may beimmobile relative to the charge carriers; these then form a type of electricalinhomogeneity as they cause field distortion. Furthermore the thermallyactivatedmovement of polymer chains within the free volume gives rise totemporal fluctuationsof mechanical and electrical stresses at any point withinthe bulk (Section 15.2). This is one of the factors contributing to the stochasticbehaviour of breakdown (Part 4 of the book).Much of the recent work on breakdown mechanisms has been on 'filamentary*variations of established models. In such cases breakdown is precipitatedfrom a weak point or develops in a filamentarymanner. In order toavoid the effect of weak points it is possible to use self-healing electrodesin which the heat produced by a localised breakdown, usually in a thin film,leads to evaporation of the thin electrodes and isolates the region of breakdownfrom the remaining intact insulation. Klein has advocated this techniquefor investigating deterministic breakdown in which the samples wereconditioned by an electric field until all the weak spots had been isolated.Using this technique breakdown could be produced on a broad front acrossthe sample rather than simply at an unrepresentative weak spot. Self-healingelectrodes are also commonly used to increase the reliability of capacitorsfollowing the suggestion of Strab 509 . However the self-healing process isnot feasible in most insulation systems either because they do not have thin

204 Deterministic mechanisms of breakdownelectrodes or because the breakdown would be too extensive; the firstbreakdown therefore usually leads to catastrophic failure.Whilst terms such as breakdown field/stress/strength' are widely usedand usually defined as the applied voltage at breakdown divided by thedielectric thickness, it should be noted that:(i) the electric field is likely to vary considerably through the thickness ofthe dielectric because of electrical and physical inhomogeneities whichmay also be fluctuating in time and space;(ii) in imperfect materials there is always a distribution of breakdownstrengths, not a single breakdown strength.There is usually very little direct evidence to confirm that a particularbreakdown mechanism is operative in a particular situation. For examplealthough there have been some observations of the spatial evolution of atemperature rise prior to thermal breakdown, these have been generallylacking. Indeed O'Dwyer 82 notes that whilst the best currently availableconcept of purely electrical breakdown has, as its precursor, collision ionisationthere is in fact no direct and convincing evidence of collision ionisation;it merely seems reasonable as it is known to happen in semiconductors andreasonable alternatives are difficult to visualise. Coupled with the problemsof measuring *a specific breakdown strength' and the difficulty of estimatingabsolute values of breakdown strength from theoretical considerations, theapplicability of most theories has been tested by comparing the theoreticallyand experimentally determined dependence of breakdown strength onthickness, time, and temperature.In spite of these problems of evaluation, definition, and attribution, thereare some general features of breakdown in polymers which are worthmentioning in an introductory manner. It is generally found that breakdownfield decreases with increasing time of application of the electric field andwith increasing sample thickness 161 . This is shown schematically in Fig. III.2. The limiting value of electric field at very short times and very thinsamples is now known as the 'intrinsic' breakdown strength and is typicallyabout 10 9 V • m~ l for polymers 158159160 . This term was originally used toimply a specific breakdown mechanism which was independent of specimenshape or volume (e.g. Whitehead 510 ) rather than the maximum strengthavailable. This usage is now generally out of favour (e.g. Cooper in Bradwell511 ) since it seems likely that under such conditions breakdown will beelectrode dominated and is not therefore intrinsically related to the material.O'Dwyer 82 however defines the intrinsic critical field strength as 'the moreor less abrupt transition from circumstances in which collision ionisation isnegligible to those in which it is not'. Under the technologically-importantconditions of working lifetimes and thick dielectrics, the breakdown fieldmay be two or three orders of magnitude lower than this intrinsic value.The effect of temperature on the breakdown strength of polymers hasbeen reviewed by Cooper 512 and Ieda 163 (most of their results are based onthe original work of Ball 513 and Oakes 514 ). It is usually found that thebreakdown strength is reasonably constant (±—20%) from very low temperaturesup to a critical temperature, often around room temperature, at

Deterministic mechanisms of breakdown 205k Log breakdown stress"Intrinsic" breakdownFig. III.2 A schematic representation of the relationship between the breakdownfield, the time to breakdown and the thickness of the insulation. The shaded areamarked 'intrinsic breakdown' corresponds to thin samples and short times andrepresents the highest breakdown strength attainable. At the technologically importantlonger times and greater thicknesses the breakdown strength may be two orthree orders of magnitude lower than this intrinsic valueCopyright © 1984 The IEEEwhich it suddenly starts to decrease, Fig. III.3. This can usually be associatedwith the softening of the polymer although not necessarily with its glasstransitiontemperature 133 ' 5 2 . This effect may also be dependent on the typeof test used, in particular the sample thickness and the electrode thermalconductivity and rigidity 161 .At low temperatures breakdown in polymers is likely to be of the avalanchetype which may be exacerbated by carrier injection from electrodes. Justabove the softening point various mechanisms could be responsible such aselectromechanical breakdown, scattering of hot electrons with consequentlattice damage (the Frohlich amorphous-solid model), or thermal breakdown.Thermal breakdown is likely to dominate over electromechanicalbreakdown at higher temperatures especially in lower-resistivity anddielectrically lossy materials. Notice the 'competitive' nature of the breakdownmechanisms are exemplified here as different mechanisms dominate(rather than operate) at different temperatures. In thicker insulation overa service lifetime the breakdown is likely to be due to the progressivedeleterious effects of thermal aging, partial discharges and electrical (andpossibly water) trees.

206 Deterministic mechanisms of breakdown15-200 -100Temperature /°CFig. III.3 Breakdown field as a function of temperature for various polymers. Thesolid lines are taken from Ball 513 who reported work by W. G. Oakes and theElectrical Research Association. These tests were carried out on recessed specimensusing impulses at the higher temperatures and progressive-stress tests at low temperaturesto prevent thermal breakdown occurring and give the highest possibleresults. The broken lines are reported by Ieda 163 for DC breakdown in films andwill tend to be lower than the other data by as much as 50%. Notice that for mostcases, and especially the non-polar polymers, the breakdown strength drops rapidlyat a critical temperature. (PVA = polyvinyl alcohol; PMMA = polymethyl methacrylate;P VC-AC = polyvinyl chloride-acetate; PS = polystyrene; LDPE = low-densitypolyethylene; PIB = polyisobutylene; iPP/aPP = isoactic/atactic polypropylene; E-P =ethylene-propylene copolymer, PVC = polyvinyl chloride, PA6 = polyamide 6)Many of the breakdown mechanisms to be described require an understandingof the way in which carriers may be injected into insulators fromelectrodes and subsequently move through the bulk of the material underhigh-field conditions. In particular it is important to realise how externalfactors such as electric field and temperature influence charge carrierinjection and movement. A detailed discussion of this subject is worthy ofa book in its own right and there have been many reviews to which thereader is referred (60, 82, 515, 516, 517). For the sake of completenesshowever a brief overview of the factors affecting charge carrier injectionand transport is given in chapter nine.The references quoted in this part of the book represent only a smallselection of those published on the subject. For example between 1970-1990there were over ten thousand papers published on electrical breakdownover a thousand of which related to polymers. We hope to have includedmajor works but the selection of other papers is to some extent arbitrarydepending largely on the authors' experiences.

PART 4THE STOCHASTIC NATUREOF BREAKDOWNFrom the point of view of the individual life is asequence of choices made for the best of reasons. Toan external observer of the whole population the basisfor each choice may be impossible to ascertain, and theindividual's progress appears to be stochastic; i.e. itcan only be described by a probability function.IntroductionMany theoretical treatments of electrical breakdown in polymeric insulatingsystems are based on the deterministic models described in the previouspart of the book. As well as being applied to 'ideal' insulating systems, theyare also applied to systems with defects (such as voids in polymers) and toinsulating systems which have been aged or degraded in ways such as thosedescribed in Part 2. Although these applications of deterministic models tonon-ideal insulating systems give values of breakdown strength which arelower than the 'intrinsic' value otherwise obtained, they generally predicta specific breakdown strength (e.g. O'Dwyer 82 ) which is not usually found.Such models suggest then that the insulating system will always breakdownwhen, and only when, a critical breakdown voltage is exceeded; if this criticalvalue is not exceeded the system will last forever. When coupled with adeterministic degradation mechanism a specific time to breakdown may, atleast in principle, be predicted given known voltage, environmental factors,and insulator history (e.g. Klein 146 ).Whilst such specific values of breakdown voltage may be obtained in somecompletely crystalline insulating systems and in some very well made amorphoussystems, such as the capacitors in some MOS devices 699 , this is notfound to be the case in polymeric insulating systems. If apparently identicalpolymeric insulating systems are exposed to identical tests in which thevoltage is linearly increased with time, a different breakdown voltage willbe observed for each insulating specimen. The breakdown voltages soobserved may be characterised by a statistical distribution which describesthe probability of breakdown as a function of voltage; the parameters of thisdistribution are dependent upon the test conditions and the specimenconstruction. If a set of similar tests are carried out at a constant voltage(AC or DC) then the times from the start of the test to breakdown may beobserved. In all types of insulating system no specific time to breakdown

318 The stochastic nature of breakdownwould be observed; rather each specimen would be found to breakdown ata different time. These times would be found to fit a similar statisticaldistribution giving the probability of failure as a function of time; theparameters of the distribution would be found to depend upon the valueof voltage used and other test conditions including the specimen construction.For this reason the presentation of such data in most publications isusually in the graphical form 'cumulative probability of failure' or 'fractionof samples failed' as a function of 'time to breakdown* or 'breakdownvoltage'.This part introduces the use of the 'Weibull function' 700 in characterisingthe distribution of times and voltages at breakdown. The physical originsof this distribution are then discussed in Chapter 15 together with otherso-called 'extreme value' distributions. In the final part of the book engineeringapplications of these distributions are considered and lifetime predictionin particular is investigated.

PART 5ENGINEERING CONSIDERATIONSFOR BREAKDOWN TESTING ANDDEGRADATION ASSESSMENTIntroductionIn earlier parts of this book we have reviewed the scientific understandingof electrical degradation and breakdown in polymers. In this part we wishto review and suggest techniques for assessing polymeric insulation systems.The first chapter in this section reviews current breakdown testing techniquesand the procedures which are used for analysing the data. The basisof these tests and procedures are examined and complementary techniquesare also suggested. We have noticed in the course of our experience withtesting cables under AC and DC conditions that there are significantdifferences in the statistics describing the results. Chapter 17 analyses thesetwo cases and indicates the differences in the techniques necessary forunderstanding the data. It also explains commonly observed, empiricalrelationships between AC and DC breakdown strengths and suggests thatthe build up of space charge may be the critical factor linking AC and DCbreakdown. Since extruded cables are a major user of polymers for insulationpurposes, Chapter 18 is devoted specifically to cable assessment procedures.Since these tests are currently being reviewed by many bodies we have notattempted to specify tests in any detail in this chapter but, rather, weattempted to identify the major considerations in carrying out such tests.Finally we have outlined some techniques for the non-destructive diagnosisof electrical polymeric degradation.