electric field distribution of sphere-plane...

IN ,DEGREE PROJECT ELECTRIC POWER ENGINEERING 120 CREDITSSECOND CYCLE

, STOCKHOLM SWEDEN 2016

Electric field distribution ofsphere-plane gaps

A SIMULATION APPROACH

MICHAIL MICHELARAKIS

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING

Electric field distribu on ofsphere-plane gaps

Master Thesis project

Michail MichealrakisSeptember 2015

KTH School of Electrical EngineeringSupervisor & Examiner: Associate Professor Hans Edin

Commissioned by ABB AB, HVDCSupervisors: Dr. Dong Wu

Dr. Liliana Arevalo

TRITA-EE 2016:030

AbstractThe continuous increase of the voltage levels in power transmission systems has leadto the occurrence of higher switching transients during their operation. The designof equipment and grid components able to sustain such a stressful operation, requiresan intensive study of the electric field stress generated by these transients, and theirdistribution to the vicinity of each configuration.

Sphere-plane gaps are the most theoretically and practically interesting electrodeconfigurations. So far, the majority of the conducted work is referred to the study ofthe discharge characteristics of this structure. However, a study of the electrostaticelectric field is required. An accurate calculation of the electric field can contributesignificantly to an even better understanding of the discharge characteristics and theprinciples behind them.

In this project, is presented a simulation approach for the calculation of the electro-static field of a sphere-plane configuration, varying the dimensions of the sphere andthe gap distance. For this purpose, a Finite Element Method (FEM) solver was used, inwhich the configuration was designed and the numerical solution of the problem wasimplemented. After that, an attempt was performed to specify the breakdown voltagebased on the electric field calculation and distribution.

Useful results were recorded from both the simulation of the electrostatic modeland the calculation of the breakdown voltage. One of the most important findings, wasthe specification of an approximate relation between the diameters of the sphere and thetube where this is mounted. As a consequence, the study of the electric field distributionbecame easier, while at the same time an accurate calculation of the breakdown voltagewas achieved.

A series of validations were performed, through the comparison with the already ex-isting, published and unpublished, experimental tests and a number of conclusions werelisted. One of the most significant, was the specification of the correlation between theelectrostatic model and test measurements and how these different approaches can belinked to each other in a practically efficient way. At the end, there is a proposal forfurther work on the subject, and possible improvements of the already conducted work.

Keywords: sphere-plane electrode configuration, electric field, electrostatics, gap dis-tance, breakdown voltage, Finite Element Method (FEM), high voltage testing

i

Sammanfa ningÖkningen av spänningsnivåerna i kraftöverföringssystem har lett till högre kop-plingstransienter under drift. Konstruktionen av utrustning och nätverkskomponen-ter som kan motstå en sådan påkänning, kräver en noggrann förstudie av de elektriskafältet som genereras av dessa transienter.

Ett gap av sfär-platta är den mest teoretiskt och praktiskt intressanta elektrodkon-figurationen att studera. Hittills har majoritetet av genomfört arbete avsett att studeraurladdningsegenskaper för denna struktur. Dock krävs studie av elektrostatiska elek-triska fältet. En noggrann beräkning av elektriska fältet kan bidra till en ännu bättreförståelse för urladdningsegenskaper och principerna bakom dem.

I detta projekt presenteras en simuleringsmetod för beräkning av elektrostatiska fäl-tet av en sfär-plan konfiguration, med varierande dimensioner av sfären och gapavstån-det. För detta ändamål har använts en Finite Element Method (FEM) lösning, där kon-figurationen utformades och problemets numeriska lösningen genomfördes. Därefterhar gjorts ett försök för att ange genombrottsspänningen baserad på beräkning ochdistribution av elektriskta fältet.

Resultat registrerades från både simulering av elektrostatiska modellen och beräkn-ing av genomslagsspänningen. Ett av de viktigaste resultaten var specifikationen av ettungefärligt förhållande mellan diametrarna av sfären och röret där sfären är monterad.Som en konsekvens blev studiet av elektriska fältfördelningen lättare, medan en exaktberäkning av genomslagsspänningen uppnåddes.

En valideringsserie har genomförts genom jämförelse med de redan existerande,publicerade och opublicerade, experimentella tester och ett antal slutsatser har noterats.En av de mest meningsfulla, var specifikation av sambandet mellan den elektrostatiskamodellen och provmätningarna samt hur dessa olika tillvägagångssätt kan kopplas tillvarandra på ett praktiskt och effektivt sätt. Slutligen finns det ett förslag för fortsattaarbete samt eventuella förbättringar av redan genomfört arbete.

Nyckelord: sfär-platta elektrodkonfiguration, elektriskt fält, elektrostatik, gapavstånd,genombrottsspänning, Finite Element Method (FEM), högspänningsprovning

iii

AcknowledgementsThe current Thesis project constitutes the final part of my M.Sc degree in ElectricPower Engineering at KTH, Royal Institute of Technology. This Thesis work wascommissioned by ABB AB, HVDC and was conducted at ABB AB, HVDC, Ludvika,Sweden.

First of all, I would like to express my gratitude to my supervisors at ABB AB,HVDC, Dr. Dong Wu and Dr. Liliana Arevalo for their guidance and contribution onthe project during the five and a half months of its implementation. I enjoyed everysingle meeting and discussion we had. I am very grateful to Dr. Raul Montano, Managerof the DC/US/R&D Studies Department for his continuous support and for makingme feel comfortable within the company from the very first day. I would like also tothank Dr. Joan Hernandez for his assistance during the design of the simulation modelused for the current project.

Furthermore, my deepest thanks to my supervisor and examiner at KTH, AssociateProfessor Hans Edin, for his support and advice.

Finally, I would like to thank my family for being always next to me, and my long-time friends Efi Stragali, Georgios Karmiris, Ioannis Chatzis, Konstantina Pantagakiand Sotirios Missas.

Michail MichelarakisStockholm, September 2015

v

Contents

Abstract i

Sammanfattning iii

Acknowledgements v

Contents vii

List of Figures xi

List of Tables xv

List of Symbols xvii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem definition & Objectives . . . . . . . . . . . . . . . . . . . . . 21.3 Report overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Overview of sphere-plane gaps in High Voltage testing 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Smooth electrode surface . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Surface irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Different environment conditions . . . . . . . . . . . . . . . . . . . . 8

3 Electric field and potential calculation 113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Electrostatic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 The electric field . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.3 Gauss’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

vii

CONTENTS

3.2.4 Rotation of the electrostatic field . . . . . . . . . . . . . . . . 133.2.5 Electric potential . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.6 Poisson’s & Laplace’s equation . . . . . . . . . . . . . . . . . 143.2.7 Electric displacement . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Electric field distribution in electrode configurations . . . . . . . . . . 153.3.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Non-uniform distribution . . . . . . . . . . . . . . . . . . . . 16

3.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.1 Method of images . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Charge Simulation Method (CSM) . . . . . . . . . . . . . . . 183.4.3 Finite Element Method (FEM) . . . . . . . . . . . . . . . . . 19

4 Implementation of the electrostatic model 214.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Design of configuration geometry . . . . . . . . . . . . . . . . . . . . 224.3 Simulation workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Stability of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.6 Post-processing of the results . . . . . . . . . . . . . . . . . . . . . . 28

5 Case study 315.1 Background description . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.1 Electric field over d gap/Rsphere ratio . . . . . . . . . . . . . . . 325.3.2 The tube diameter issue . . . . . . . . . . . . . . . . . . . . . 365.3.3 Stability of the results . . . . . . . . . . . . . . . . . . . . . . 405.3.4 Electric field over actual gap distance d gap . . . . . . . . . . . 425.3.5 Electric field distribution on the sphere electrode surface . . . 425.3.6 Electric field and voltage distribution along the gap . . . . . . 42

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4.1 Sphere mounted on a tube . . . . . . . . . . . . . . . . . . . 475.4.2 Sphere without any mounting object . . . . . . . . . . . . . . 50

5.5 Study of the breakdown voltage . . . . . . . . . . . . . . . . . . . . . 525.6 Validation of the results . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.6.1 Published measurements . . . . . . . . . . . . . . . . . . . . 565.6.2 ABB internal tests . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Closure 616.1 Summary of the conducted work . . . . . . . . . . . . . . . . . . . . 616.2 Conclusions from case studies . . . . . . . . . . . . . . . . . . . . . . 626.3 Proposal for future work . . . . . . . . . . . . . . . . . . . . . . . . . 64

viii

CONTENTS

References 67

Appendices 71

A Electric field distribution on the sphere electrode surface 73

B Sphere and tube diameter ratio for different cases 79

C Calculated values 85C.1 Maximum electric field . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.2 Breakdown voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

ix

List of Figures

1.1 Typical electrode configurations used for electric strength tests. . . . . 2

2.1 Example of a lightning impulse waveform. . . . . . . . . . . . . . . . 62.2 U50% over gap distance for different sphere electrode radii. . . . . . . . 72.3 U50% over gap distance for different sizes and places of installation of

the protrusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Infinite parallel plates of opposite charge. . . . . . . . . . . . . . . . . 153.2 Electric field distribution in needle-plane configuration. . . . . . . . . 163.3 Calculation of potential between a sphere and an infinite grounded

plane using the method of images. . . . . . . . . . . . . . . . . . . . . 18

4.1 Laboratory sphere-plane configuration set-up. . . . . . . . . . . . . . 224.2 Sphere-plane configuration designed in ComsolrMultiphysics. . . . . . 234.3 Schematic representation of the simulation procedure in Comsolr. . . 244.4 Mesh generated in ComsolrMultiphysics with extra fine element size. . 264.5 Custom mesh set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Geometry parameters that should not affect the calculations. . . . . . . 274.7 Parts on the configuration geometry that the post-processing of the

simulation results will be based on. . . . . . . . . . . . . . . . . . . . 29

5.1 Maximum electric field on the sphere surface, appears at the bottom ofthe sphere electrode, and maximum electric field on the tube surface,over the dgap/Rsphere ratio and for different Dsphere values . . . . . . . . 33

5.2 Maximum electric field on the sphere surface, appears at the bottom ofthe sphere electrode, and maximum electric field on the tube surface,over the dgap/Rsphere ratio and for Dsphere < 1m and Dtube = 0.1m. . . 34

5.3 Maximum electric field on the sphere surface, appears at the bottomof the sphere electrode (continuous lines), and maximum electric fieldon the tube surface (dashed lines), over the dgap/Rsphere ratio and forDsphere > 1m and Dtube = 0.45m. . . . . . . . . . . . . . . . . . . . . 35

xi

LIST OF FIGURES

5.4 Electric field at the bottom of the sphere electrode and the maximumelectric field on the tube surface, over the dgap/Rsphere ratio and fordifferent Dsphere values. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.5 Maximum electric field on the sphere and the tube over Dratio. . . . . . 385.6 Maximum electric field on the sphere surface, appears at the bottom

of the sphere electrode, and maximum electric field on the tube sur-face, over the dgap/Rsphere ratio and for different Dsphere values. HereDratio = 4.36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.7 Emax over size of the domains used for the open boundaries set-up. . . 405.8 Emax over Ltube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.9 Emax over dright. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.10 Maximum electric field Emax over the actual gap distance dgap for dif-

ferent Dsphere studied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.11 Electric field distribution on the sphere surface for Dsphere = 0.25m

and different dgap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.12 Electric field distribution on the sphere surface for Dsphere = 2m and

different dgap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.13 Electric field distribution along the gap spacing, for Dsphere = 0.25m

and various dgap values. . . . . . . . . . . . . . . . . . . . . . . . . . . 455.14 Voltage distribution along the gap spacing, for Dsphere = 0.25m and

various dgap values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.15 Electric field distribution along the gap spacing, forDsphere = 2m and

various dgap values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.16 Voltage distribution along the gap spacing, forDsphere = 2m and var-

ious dgap values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.17 Percentage differences between points of figure 5.6. . . . . . . . . . . 485.18 Typical electrode configurations used for electric strength tests. . . . . 495.19 Electric field lines for two different gap distances. . . . . . . . . . . . 495.20 Emax of a sphere without any mounting object and a sphere mounted

on a tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.21 Emax over Vapplied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.22 Breakdown voltage Vb over gap distance for different sphere electrode

diameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.23 Comparison between calculated breakdown voltage based in (5.15) and

(5.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.24 Comparison between calculated breakdown voltage Vb and U50% from

test results included in [3]. . . . . . . . . . . . . . . . . . . . . . . . . 585.25 Comparison between calculated breakdown voltage Vb and U50% from

test results included in [4]. . . . . . . . . . . . . . . . . . . . . . . . . 585.26 Comparison between calculated breakdown voltage and unpublished

test results provided by ABB AB. . . . . . . . . . . . . . . . . . . . . 60

xii

LIST OF FIGURES

6.1 Emax for different enhancement factor f concepts. . . . . . . . . . . . 64

A.1 Electric field distribution on the sphere electrode surface forDsphere =0.25m and different dgap. . . . . . . . . . . . . . . . . . . . . . . . . 73

A.2 Electric field distribution on the sphere electrode surface forDsphere =0.5m and different dgap. . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.3 Electric field distribution on the sphere electrode surface forDsphere =0.75m and different dgap. . . . . . . . . . . . . . . . . . . . . . . . . 74





A.8 Electric field distribution on the sphere electrode surface forDsphere =2m and different dgap. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B.1 Dsphere over Dtube ratio for Dsphere = 0.25m. . . . . . . . . . . . . . . 79B.2 Dsphere over Dtube ratio for Dsphere = 0.5m. . . . . . . . . . . . . . . . 80B.3 Dsphere over Dtube ratio for Dsphere = 0.75m. . . . . . . . . . . . . . . 80B.4 Dsphere over Dtube ratio for Dsphere = 1.2m. . . . . . . . . . . . . . . . 81B.5 Dsphere over Dtube ratio for Dsphere = 1.3m. . . . . . . . . . . . . . . . 81B.6 Dsphere over Dtube ratio for Dsphere = 1.5m. . . . . . . . . . . . . . . . 82B.7 Dsphere over Dtube ratio for Dsphere = 1.6m. . . . . . . . . . . . . . . . 82B.8 Dsphere over Dtube ratio for Dsphere = 2m. . . . . . . . . . . . . . . . . 83

xiii

List of Tables

4.1 Typical tube dimensions used in laboratory experiments. . . . . . . . . 23

B.1 Dratio for every Dsphere value studied. . . . . . . . . . . . . . . . . . . . 83

C.1 Simulation results for maximum electric field in [kV/cm]. Each col-umn corresponds to a different sphere diameter value Dsphere, whileeach row to different gap distance over sphere diameter ratio valuesdgap/Dsphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

C.2 Calculation results for breakdown voltage Vb in [kV]. Each col-umn corresponds to a different sphere diameter value Dsphere, whileeach row to different gap distance over sphere diameter ratio valuesdgap/Dsphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xv

List of Symbols

ε0 Permittivity of free space C/ (V ·m)η Field efficiency factorλ Line charge C/mρ Volume charge C/m3

σ Surface charge C/m2

D Electric displacement field C/m2

Dratio Sphere and tube diameter ratioDsphere Sphere electrode diameter mDtube Tube diameter mE Electric field V/mEb Breakdown strength of air V/mEm Electric field magnitude kV/cmEmax Maximum electric field (norm) V/mEmean Mean electric field (norm) V/mF Coulomb force NN e

k Shape function used in FEMOB Open boundary domain widthQenc Enclosed charge CRsphere Sphere electrode radius mRtube Tube radius mVapplied Applied voltage to the high voltage electrode VVb Electrostatic breakdown voltage VU50% 50% breakdown voltage VUB Breakdown voltage from Menemenlis et al. Vdgap Gap distance m

xvii

LIST OF SYMBOLS

dright Distance to the side on the right in the simulation geometry mf Field enhancement factorpij Potential coefficients used in CSMqr Random charge Cqs Source charge Cr Distance between charges mr̂ Position vector

xviii

Chapter 1

Introduc on

1.1 BackgroundOver the years, the continuous population growth and industrial development resultedin a corresponding increase of power demand, not only quantitatively but also qualita-tively. In addition, the search of more environmentally friendly energy sources has ledto an even more extensive study on the modern power systems and how these can beadapted to the existing requirements.

Due to efficiency, control capability, interconnection distance and cost constraints,HVDC (High-Voltage Direct Current) transmission of electricity is considered as thestate of art for modern power systems. Although, the increase of the voltage levelsrequires the design and development of equipment able to sustain such a demandingoperation, including even more stressful switching transients.

Following the continuously increasing voltage levels, electric fields and breakdownphenomena lead to always new and challenging approaches. This makes the under-standing of the electric field characteristics and their distribution to the vicinity of eachstudied configuration, a very interesting and useful field of study. Furthermore, a suffi-cient understanding of the electric field behaviour will contribute to a very good extentto the understanding of the discharge phenomena and the principles behind them.

So far, many studies were dedicated to the discharge characteristics of differentelectrode configurations and under different voltage stresses, environment conditionsand polarities. Most of these studies were based on laboratory tests and numericalapproaches focusing mainly on the breakdown voltage on each case. Additionally, themajor part of this work was conducted many years ago, when the availability in softwareand equipment was negligible compared to nowadays. All these consist of importantmotivations to approach again, after a long period of limited work, this area of study,this time through in a simulation level.

1

1.2. PROBLEM DEFINITION & OBJECTIVES

A well defined model is expected to return a set of results that will match in a verygood extendwith the already conducted and published in previous works [1-6]. Possiblelimitations in the design of the model and the available resources during the calculationsmay lead to partial convergence of the results. A series of evaluations will be performedto confirm the accuracy of the produced results. All the possible divergences, mutatismutandis, will be discussed in detail and the suggested improvements will be proposedas future work.

1.2 Problem defini on & Objec vesFor electric strength tests there are various electrode configurations which give theopportunity to the involved to create electric fields with different degrees of inhomo-geneity. Some of the typical configurations used in this kind of testing, also illustratedin figure 1.1, are known as needle-plane, rod-plane, sphere-sphere and sphere-plane.

gapd

(a) Needle-plane.

gapd

(b) Rod-plane.

gapd

sphere

D

(c) Sphere-sphere.

gapd

sphere

D

(d) Sphere-plane.

Figure 1.1: Typical electrode configurations used for electric strength tests as theseare depicted in the majority of the bibliography [7-10].

All the configurations presented above are of special interest. One of the most usedin High Voltage equipment are the sphere-plane gaps (figure 1.1d) and therefore one ofthe most interesting to study. Regarding the electric field, in this kind of configuration

2

CHAPTER 1. INTRODUCTION

it is possible to meet all the three types of distribution, the uniform, the quasi-uniformand the non-uniform, depending of the distance that separates the charged electrodewith the grounded plane. This is an additional motivation to study and understandthese configurations with respect to the static electric field behaviour and the dischargecharacteristics.

Available studies on the subject are almost entirely based on the specification ofthe breakdown voltage and its variation with different geometry characteristics of theconfiguration. The demanding calculations required [11-13] and the lack in advancedsoftware and tools, made the systematic electric field calculation a very challengingprocedure, back in the years when the majority of these works were conducted.

Through the next pages, will be presented a simulation approach on the electricfield calculation of sphere-plane gaps. The general purpose is to take the advantage ofmodern software packages, as Finite Element Method solvers, and create a simulationmodel which will give an accurate calculation of the electric field distribution. Therewill be also performed an attempt, after the post-processing of the first batch of results,to estimate the breakdown voltage of the electrostatic model for different dimensionsof the electrode configuration.

The aim of the working procedure described above, and the overall project, is tosuggest a new approach to the understanding of sphere-plane gaps, this time throughthe study of the electrostatic field distribution. A possible sufficient calculation of thebreakdown voltage would constitute a very useful contribution, especially for the caseswhere a quick and precise calculation is required, substituting time- and cost-demandingtests.

1.3 Report overviewChapter 1 is an introduction to the studied field while at the same time are presentedthe definition of the problem that is about to be solved together with its objectives.Chapter 2 constitutes a brief overview of the sphere-plane gaps in High Voltage testing,while Chapter 3 deals with the calculation of the electric field and potential, startingfrom the basic principles of electrostatics until more complicated numerical methods.In Chapter 4 is described the implementation of the electrostatic simulation model,with main focus on how the theoretical assumptions can be adapted during the FEMsolver set-up. Chapter 5 is the largest part of the current report. In this, can be foundthe results from the simulation of the electrostatic model and the calculation of thebreakdown voltage. Finally in Chapter 6 are listed all the conclusions of this thesisproject as well as some ideas for future work and further improvement of the existing.

3

Chapter 2

Overview of sphere-planegaps in High Voltage tes ng

2.1 Introduc onIn order to get a better view on the applications of sphere-plane gaps in High Voltagetesting, it is convenient to present a series of models that have been used in laboratorytests and a brief summary of the corresponding results. The purpose behind this is topresent the wide range of applications that this specific electrode configuration can beused and how useful in the recording of numerous conclusions can be. Together withthe presentation of each case, there will be provided an appropriate description of theprinciples and definitions that are considered in each of the studies.

As it was mentioned before, most of the already conducted work about the be-haviour of sphere-plane gaps, consists of laboratory tests under various conditions.Because of the test procedure and the availabilities in measurement equipment, almostin all of the works, the authors base their observations around the influences on theCritical Flashover Voltage (CFO).

In almost all of the studies, the contributors focus on testing under impulse voltages.This kind of voltage, has the characteristic that it reaches its peak value very fast andthen decays to zero after a certain time [14]. There are two kind of impulse voltages[7, 15, 16],

• The lightning impulse voltages, which are characterised by short front duration,ranging between 1 µs up to a few tens of µs, and then with a slower rate theydecrease to zero. As their name implies, the source of lightning overvoltages arelightning discharges. The standard lightning impulse voltage waveform acceptedis 1.2/50 µs.

5

2.2. SMOOTH ELECTRODE SURFACE

• The switching impulse voltages, which represent the transient overvoltages dueto sudden changes in the power systems, switching operations or faults. Thestandard switching surge voltage is designated to 250/2500 µs.

Time in [µs]0 10 20 30 40 50

Voltage

in[p.u.]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

(1)

(2)

(3) (4)

Figure 2.1: Example of a lightning impulse waveform. Curve derived by imple-menting equation (2) included in [17]. (1) Time to half the peak voltage, (2) Timeto peak voltage, (3)− (4) Time above 90%.

During testing procedure when a number of impulses are applied, usually not all ofthem, but a percentage, will lead to electrical breakdown of the gap [18]. When a specificimpulse voltage leads to breakdown in all of its applications then this voltage representsthe U100%. When breakdown occurs for half of the applications of the applied impulsevoltage then the same voltage represents the U50%. Respectively, U0% represents thelargest peak impulse voltage that does not cause any breakdown during any of its ap-plications [14]. More information on general definitions and test requirements in HighVoltage testing, can be found in IEC 60060-1:2010 [18].

2.2 Smooth electrode surfaceDespite the case that a spherical electrode with a smooth surface, is expected to be“easier” to be studied compared to more special configurations, practically they requirequite much attention. The absence of irregularities of any kind on the surface of theelectrode, makes the observation and the record of the results more sensitive to a variety

6

CHAPTER 2. OVERVIEW OF SPHERE-PLANE GAPS IN HIGH VOLTAGE TESTING

of factors, like the set-up of the entire laboratory. Moreover, it is required very specialtreatment in order to manufacture a completely smooth surface, free of even very smallattached particles.

There are numerous references in the bibliography about sphere-plane gaps. In allof them is included a brief description of the electrode configuration set-up and thevoltage stress preferred. A quite recent and very detailed description is given in [3].For the case of a smooth sphere surface, the fact that during the tests all the break-downs were initiating from the surface of the sphere is a detail not clearly mentionedby the authors of different works [1, 2, 19-21]. The place where the maximum stress issupposed to appear, will be one of the major considerations of this work. As it will beseen in a later chapter, this will play an important role in the design of the simulationmodel.

A very common description and depiction of the breakdown voltage in sphere-plane gaps that can be found in many classical textbooks [8,22] is the one of figure 2.2.Here is presented the variation in U50% with the change of the gap distance, and thesphere electrode diameter.

Gap distance0

U50%

0

I II III

D1

D2 > D1

D3 > D2

Figure 2.2: U50% over gap distance for different sphere electrode diameters. Plotswere redrawn based on material included in [8, 22].

As it is described, in Zone I for small gap distances the electric field is consideredas uniform and the U50% depends mainly on that distance. In Zone II, as the gap be-comes longer, the electric field becomes non-uniform and the 50% breakdown voltagedepends on both the sphere radius/diameter and the gap spacing. Finally, in Zone III,

7

2.3. SURFACE IRREGULARITIES

the electric field is considered as strongly non-uniform and the U50% is affected only bythe gap spacing [8,22]. When the electric field turns strongly non-uniformly distributed,breakdown is preceded from predischarge phenomena such as corona.

In all the cases and corresponding sources it is stated the complexity in the studyingof non-uniform electric fields for such electrode configurations and that the researchis still ongoing. This gives an extra point of interest to the study of the behaviour ofU50%, similar to the estimations in figure 2.2. One possible drawback in this approachis that the sphere electrode is assumed to consist only of an ideal sphere without anyextra component, a tube or a rod, where it is supposed to be mounted.

2.3 Surface irregulari esDifferent electrode surface conditions is a quite interesting topic for study, regardingthe sphere-plane electrode structure. Practically, irregularities on the surface of thesphere electrode can appear as a result of insufficient manufacturing where protrusionsof the same electric constant as the electrode material are introduced. Also, externalfactors such as insects or small particles of different electric constant are responsiblefor surface corrosion.

In [3] and [23] are presented very interesting test results for cases where metallicprotrusions and objects of very high electric constant are installed. In [23], is studied thecase of a segmented sphere as part of the bottom shield of an HVDC valve. In all casesis observed the drastic reduction of dielectric strength of the gap by the installation ofeach kind of protrusions and how the place of their installation affects correspondinglythe place where the discharge is initiated. A brief summary of the results is presentedin figure 2.3 [3].

An interesting study is presented also in [1], where among the other cases, is stud-ied the installation of a spark plug in order to achieve an artificial acceleration of thecorona discharge. Such an experiment gives a view on how the breakdown voltage of arelatively long gap is affected if the predischarge phenomena, like corona, are artificiallyinitiated. As it was expected, also this attempt resulted to the reduction of U50%.

2.4 Different environment condi onsEnvironment conditions defer when the intent of the involved in the implementation ofthe experiment is to study an outdoor or indoor HV installation. Different assumptionsshould be performed, for example, for an indoorHV installation where humidity, pollu-tion, presence of insects and temperature, diverge significantly from the correspondingoutdoor.

An extended research on the how rain affects long air-gaps, including the sphere-

8

CHAPTER 2. OVERVIEW OF SPHERE-PLANE GAPS IN HIGH VOLTAGE TESTING

Gap distance in [m]3 4 5 6 7 8 9

U50%in

[kV]

800

1000

1200

1400

1600

1800

2000

2200

2400

Without protrusion

10mm protrusion - side

50mm protrusion - side

10mm protrusion - underside

50mm protrusion - underside

Figure 2.3: U50% over gap distance for different sizes and places of installationof the protrusion. Here the diameter of the sphere electrode is 1.6m. Plots wereredrawn from figure 5 in [3] and are based on the test results presented in the samepublication.

plane configuration, is described in [21]. It is observed that the presence of rain reducesconsiderably the breakdown voltage in long sphere-plane gaps for switching impulsesof any polarity while at the same time, the affection of rain decreases with the increaseof the gap spacing. Also the probabilistic nature of the U50% seems to be affected fromrain as for long sphere-plane gaps the standard deviation increases significantly.

A useful study for indoor installations is presented in [24], for several differentelectrode configurations. The correction procedure from the test conditions to thestandard conditions, and vice versa, according to IEC 60060-1:2010 [18] is put underobservation in order to verify its accuracy. The understanding of this bidirectionalprocedure is a significant contribution to the understanding of the differences betweenthe two installation types. According to the conclusions included in [24] the correctionprocedure for sphere-plane gaps produces quite accurate results.

9

Chapter 3

Electric field and poten alcalcula on

3.1 Introduc onElectromagnetism constitutes one of the fundamental principles for the understandingof power systems. On one hand, the conversion of mechanical energy into kinetic inrotatingmachines, and the increase and decrease of voltage levels in power transformersare two very obvious cases were the magnetism is the basis of the analysis. On the otherhand, electric fields are crucial in electrical insulation, the description of the energytransfer from charged particles to the insulating materials and the occurrence of severalphenomena in transmission lines and substations [25].

Over time, electric and magnetic fields, regarding their role in power systems men-tioned before, used to introduce limitations in their understanding mostly because ofthe challenging required calculations. Because of this, there are some effective ap-proaches where under some circumstances an electromagnetic problem can be solvedif it is reduced to an equivalent circuit level. Another available option is the use ofrecently developed software packages which give the opportunity to design the config-uration that is under study and provide a solution based on the already known numericalmethods used in electromagnetism.

In all the cases, in order to produce an accurate solution so that someone will beable to interpret this, it is required a sufficient knowledge of the background theory.For the current work, the analysis will be limited in the principles behind electrostaticfields. A brief description of the most popular numerical techniques for the calculationof this kind of fields, will be also presented.

11

3.2. ELECTROSTATIC FIELDS

3.2 Electrosta c fields

3.2.1 Coulomb's lawClassical electromagnetism is also referred in theoretical physics as classical electrody-namics. The very basic principle behind this, is the calculation of the force betweenan electric charge, qs, which is considered as the source, and another random electriccharge qr. Both of the charges are assumed to be in a motion as a function of time. Ifthe source charge is considered as stationary, though the rest of the charges are mov-ing, then this is considered as a case of electrostatics [26]. In that case, the force Fbetween the stationary source charge and a random charge in motion can be calculatedby Coulomb’s law

F =1

4πε0

qsqrr2

r̂ [N] (3.1)

where, ε0 is the permittivity of free space, r is the distance between the two chargesand r̂ is known as the position vector.

In the case where the source charge consists of multiple point charges thenCoulomb’s law is adapted by the principle of superposition,

F = F1 + F2 + F3 + . . .+ Fn (3.2)

3.2.2 The electric fieldIn electrostatics, the electric field is defined in two ways, the one induced by pointcharges and the other induced by continuous distributions, such as line, surface andvolume charges. The electric field from several point charges is given by,

E (r) =1

4πε0

n∑i=1

qir2ir̂ [N · C−1] (3.3)

where this time ri is the distance between the point charge qi and the point wherethe electric field is needed to be calculated. The general equation that expresses thecontinuous charge distribution is

E (r) =1

4πε0

∫dq

r2r̂ (3.4)

where dq is the amount of the distributed charge [26]

12

CHAPTER 3. ELECTRIC FIELD AND POTENTIAL CALCULATION

dq → λdl ∼ σdS ∼ ρdV (3.5)

The parameters λ, σ and ρ correspond to the line, surface and volume charge den-sities with SI units C/m, C/m2 and C/m3 respectively.

3.2.3 Gauss's lawOne of the fundamental laws that characterize the electrostatic fields is the Gauss’s law,and which can be expressed in two forms. The first one, which is easily derived sincethe electric field is also adapted according to the superposition principle, is the integralform ∮

S

E · da =Qenc

ε0(3.6)

The right part of (3.6) denotes the electric flux and the definition behind this equationis that the total electric flux out of a closed surface equals to the enclosed charge overthe permittivity of free space. By using the divergence theorem and the definition ofvolume charge density of (3.5) is derived the differential form of Gauss’s law

∇ ·E =ρ

ε0(3.7)

Gauss’s law is one of the very useful and flexible theorems used in electrostatics,although its application is easier when there is a symmetry of the shapes used in thestudied geometry. The most characteristic of them are the plane, spherical and cylindri-cal but even in these kind of symmetries there are restrictions. For example, cylindricalsymmetry can be studied accurately if someone considers an infinitely or extremely longcylinder.

3.2.4 Rota on of the electrosta c fieldAnother basic characteristic of the electric fields in electrostatics is that they are con-sidered as irrotational

∇×E = 0 (3.8)

This also implies that the electrostatic forces are conservative [27].

13

3.2. ELECTROSTATIC FIELDS

3.2.5 Electric poten alThe equation that links the electric fieldE, which is defined as a vector, with the electricpotential V , which is correspondingly defined as a scalar, is

E = −∇V (3.9)

As it was mentioned before, the electric field is irrotational, so it is possible toexpress it as a gradient of a scalar potential. The minus sign in (3.9) has a conventionalrole and it implies that by moving against the electric field, the potential increases. Itcan also be expressed as

V = −∫

E · dl (3.10)

which is a very useful equation to calculate V when E is already known. With respectto (3.10), (3.4) and (3.5), the potential for a continuous charge distribution will be,

V =1

4πε0

∫dq

r(3.11)

3.2.6 Poisson's & Laplace's equa onCombining (3.9) with (3.7) results to

∇2V = − ρ

ε0(3.12)

Equation (3.12) is known as Poisson’s equations, and in places where there is no charge(ρ = 0) the same equation is reduced to

∇2V = 0 (3.13)

which represents Laplace’s equation. Both of them are partial differential equations ofelliptic type.

3.2.7 Electric displacementMultiplying (3.7) by the constant ε0, leads to,

∇ ·D = ρ (3.14)

where

14


D = ε0E [C/m2] (3.15)

is known as the electric displacement1. As it is shown from (3.8) and (3.14), a staticfield is irrotational but not source-free/divergence-free.

3.3 Electric field distribu on in electrode con-figura ons

3.3.1 Uniform distribu onThe principles behind the uniform distribution of the electrostatic field are quite simpleand can be explained properly through a very common example which describes thefield between two infinite, parallel and opposite charged plates.

z

yx

d

V

V

Figure 3.1: Infinite parallel plates of opposite charge.

In figure 3.1 the electric field directs from the positively charged plate to the nega-tively. By implementing (3.9),

E = −dVdz

= −V+ − V−

d(3.16)

shows that, since the voltage decreases along the gap d from V+ to V− with a stable rate,then the gradient of that voltage will be constant and equal to the potential differencebetween the plates over the distance between them.

Such a behaviour can be found, under circumstances, in some of the common elec-trode configurations such as the Rogowski electrode, the sphere-plane and the sphere-sphere if the gap distance is narrow. Especially for the sphere-plane, for a given sphereelectrode diameter, there is a range in the gap distance where the electric field is dis-tributed uniformly.1Electric displacement or charge density or electric flux.

15

3.4. NUMERICAL METHODS

3.3.2 Non-uniform distribu onIn almost all the electrode configurations, for the majority of the studied cases, thedistribution of the electric field is non-uniform and actually this is the most realisticapproach.

V

Figure 3.2: Electric field distribution in needle-plane configuration.

An example of a non-uniform field gap is shown in figure 3.2. The straight linerepresents the critical field line, which corresponds to the point with the shortest dis-tance to the grounded plane. The maximum electric field in that kind of configurationoccurs always at the tip of the needle. Because of its sharpness, this point, shows anextremely big field “concentration”, and at the same time a steep fall in the vicinity ofit [28].

For the case of sphere-plane gaps, for gap distances relatively bigger than the sphereelectrode diameter, the distribution of the electric field varies from weakly to stronglynon-uniform for very long gaps. A general depiction of such a configuration was shownpreviously in figure 1.1d. Here also the critical field line corresponds to the point in thesphere surface closest to the grounded plane, and this is where normally the maximumelectric field occurs. Through the next chapters will be given a more extensive analysison this specific electrode configuration.

3.4 Numerical methodsSummarizing the principles briefly described above, the purpose of an electrostaticproblem is to properly define a geometry in which will be described and then will besolved the partial differential equations of (3.12) and (3.13). The procedure for this

16


kind of calculations is quite challenging. Several methods are developed, which can beadapted to all the available space dimensions, such as 1D, 2D and more recently 3D,and with a variety of boundary conditions. Each method has its own advantages anddisadvantages over the rest. Usually, what defines the best method are the requirementsof the problem under solution.

The next lines constitute a brief introduction to some of these methods. The pur-pose is to provide an overview of which one and why is used in the current study andfor the model that will be described in the next chapters. References regarding moredetailed theoretical background of each method and examples of their application willbe provided.

3.4.1 Method of imagesAssume a point charge +q placed above a grounded plane. The potential due to thischarge, everywhere in the space, is the sum of two potentials, the one due to +q itselfand the other one due to its image charge −q, placed at the mirror point with respectto the plane.

Vtot = V+ + V− =1

4πε0

(q

r+− q

r−

)(3.17)

This method is based on the uniqueness theorem of Poisson’s equation (3.12), whichimplies that, if there is a gradient that satisfies a specific set of boundary conditions, then this gradientis the only and correct one [26].

Assume the charged sphere of radius R in a distance d above a grounded plane asthis is shown in figure 3.3. The potential at a distance x from the upper sphere+Q, willbe the result of the contribution of both +Q and the image charge −Q and accordingto (3.17) will be given by,

V (x) =+Q

4πε0r++

−Q

4πε0r−

=+Q

4πε0x+

−Q

4πε0 (2d− x)

(3.18)

After specifying the potential at a specific point is easy to calculate also the electricfield by using (3.9). In the same way, can be specified the potential at any point on the2D plane, if just considering that

r2 = x2 + y2 ⇒

r =√x2 + y2

(3.19)

17


d

Q

Q

R

R

d

x

Figure 3.3: Calculation of potential between a sphere and an infinite groundedplane using the method of images.

A popular use of the method of images is for the calculation of the parameters foroverhead transmission lines [29].

3.4.2 Charge Simula on Method (CSM)Another numerical method for the calculation of the electric field is the known asCharge Simulation Method (CSM). Because of numerous different electrode config-urations, the electric field calculation can become extremely demanding. A numericalmethod like CSM, combined with some computational tools, can deliver a sufficientresult for that purpose.

The basic concept behind this method is to replace, in the region where the fieldsolution is needed, the distributed charge with fictitious discrete ones. The magnitudesof these charges have to be calculated so that their induced voltage or electric fieldsatisfies the boundary conditions exactly as they are setted [30]. Also CSM complies interms of the uniqueness theorem of electromagnetic fields as the purpose again is thesolution of Poisson’s or Laplace’s equation.

Taking the advantage of the superposition theorem, the potential of a distributedcharge is the sum of the different potentials resulting from the individual fictional dis-crete charges,

18


Vi =n∑

j=1

pijQj (3.20)

Here Qj is an individual charge, pij is a potential coefficient between the boundary iand the charge j, and n is the total number of simulated charges [30]. For n unknowncharges, according to (3.20) there will be n linear equations

V1

V2

...Vn

=

p11 p12 . . . p1n

p21 p22 . . . p2n...

... . . . ...pn1 pn2 . . . pnn

·

Q1

Q2

...Qn

(3.21)

Equation (3.21) should be able to return results for the simulated charges Q1,Q2, . . . , Qn. It should be confirmed if these charges can produce an accurate cal-culation of potential, preferably on a boundary where this potential is already known.The difference between the calculated and the actual value on the boundary is a factorto determine the accuracy of the solution. In case the returned result is sufficient thenthe potential and the electric field can be calculated.

Further details on this method, improvements and optimizations, can be foundplenty in the bibliography [7, 25, 31] and in publications [11-13].

3.4.3 Finite Element Method (FEM)The basic principle behind the Finite Element Method (FEM) is the division of thedomain of interest into a number of sub-regions, known as elements. The number andthe size of the elements can vary so they can fit within the boundaries of the desiredgeometry. The solution values of the partial differential equation, that is about to besolved, are represented by the nodal values of the elements. The great advantage ofthe FEM is its flexibility and that it can be adapted to very complicated geometries anddifferent media representations [30, 31].

The procedure followed during this method can be divided in the following steps[30]:

• Division of the initial domain into elements.

• If the elements are for example triangles, then the function ue within the elementsthat is about to be solved is approximated by the nodal values

ue =∑k

N ekuk (k = i, j,m) (3.22)

19


where ui, uj, um are the vertices of each triangle andN ek are the shape functions.

• Specification of the element matrix equation, using weighted residual or varia-tional principle [30, 31]

• Form a matrix equation

Ku = B (3.23)

where u is a column matrix of order equal to the number of elementsN , K is theN ×N system matrix and matrix B includes the source from Poisson’s equationand the boundary conditions.

• The solution of (3.23) returns the calculated values at the nodes.

The above description is quite simplified and in practise the whole procedure isquite demanding in calculations and in time, which make it really challenging. All thiscan be replaced by the use of modern sophisticated software packages, which are de-veloped for this exact purpose, to solve partial differential equations with the use ofFinite Element Method. For the current work, among a variety of FEM solvers, waspreferred ComsolrMultiphysics 5.0.

20

Chapter 4

Implementa on of theelectrosta c model

4.1 Introduc onIn the previous chapters the intention was to give a sufficient explanation, to whatextent the understanding of electrode configurations, and especially sphere-plane gaps,is useful. Additionally, the majority of the principles described previously indicate thechallenges, regarding the calculation procedure, scientists had to cope with in the pastyears when the resources were limited.

Nowadays, the development of advanced software packages has given the oppor-tunity to approach this field of study again, this time through electric field calculations.A well designed simulation model, together with an effective calculation procedure,can give a quite accurate result which can be validated from the already existing orupcoming laboratory test measurements.

In this chapter will be presented a simulation model implemented using a modernFinite Element Method (FEM) solver, ComsolrMultiphysics 5.0. The most importantparts of the model design, the theoretical background behind them, and the assump-tionsmade, will be attached here. The purpose is to provide all the required informationto the interested in order to be able to reproduce the simulation model with same orsimilar results.

It is convenient to mention at this point that the model was designed in 2D - Ax-isymmetric dimension, focusing in electrostatics physics and in a stationary study mode.Further details and workflow can be found within the next sections of the current chap-ter.

21

4.2. DESIGN OF CONFIGURATION GEOMETRY

4.2 Design of configura on geometryThe subject of the simulation is the calculation of the electric field distribution withinthe sphere-plane gaps. There are many examples of configurations that have beenpresented in different works from different authors. Some of these configurationscould be considered as complicated in structure [1], while some others are related withsome constraints regarding the environment where the experiment was conducted, forexample indoors or outdoors [19], available space in the laboratory, etc.

A configuration that can be properly transferred to a CAD design and through thisto be simulated using ComsolrMultiphysics, is the one presented in [3] and which isreproduced in figure 4.1.

Figure 4.1: Laboratory sphere-plane configuration set-up with a sphere mountedto a tube, and a double toroid used for the termination of it. Photo taken from [3].

Themetallic sphere electrode, of a specific diameterDsphere, is mounted to ametallictube, of also a specific diameterDtube, which in its turn is terminated through a doubletoroid. The presence of the double toroid eliminates the potential gradient due to thecreation of a sharp edge at the end of the tube. The installation of the test objectsis vertical with a considerable distance from the nearest grounded object inside thelaboratory. Some of the typical values of the tube diameter and length, depending onthe diameter of the sphere, electrode are presented in table 4.1.

Having the laboratory set-up in [3] as a reference, it should be ensured that nothingin the vicinity of the electrode configuration, wall or grounded object, can influence thecalculation of the result. This can be achieved by assuming open boundaries aroundthe configuration. Additionally the double toroid and its role in the elimination of the

22

CHAPTER 4. IMPLEMENTATION OF THE ELECTROSTATIC MODEL

Dsphere [m] Dtube [m] Ltube [m]

< 1 0.10 1

> 1 0.45 3

Table 4.1: Typical tube dimensions used in laboratory experiments according tothe sphere diameter.

sharp edge, can be simplified by assuming a tube of infinite length.An initial version of the simulation model as this is designed in

ComsolrMultiphysics environment is presented in figure 4.2

Figure 4.2: Sphere-plane configuration as this is designed inComsolrMultiphysics.

Domain 1 corresponds to the main part of the design, while domains 2, 3 & 4 havea supporting role in the set-up of the open boundaries mentioned before. This is theone half of a 2D design since it is supposed to be axisymmetric.

4.3 Simula on workflowThe most efficient way to describe the workflow of the simulation, is through a simpleand comprehensive flow-chart. The sequence of steps someone should follow in orderto design, configure and finally run a simulation model using ComsolrMultiphysics, isshown in figure 4.3.

23

4.3. SIMULATION WORKFLOW

Space dimension⊕

Physics interface⊕

Study type

Definition of parameters

Design of the geometry

Materialproperties

Boundaryconditions

CreateMesh

Studyset-up

&

Run the simulation

Post-processing of the results

Figure 4.3: Schematic representation of the simulation procedure inComsolrMultiphysics.

24


Through the model wizard is easy in simple steps to specify the space dimension ofthe geometry, the physics interface and the study type of the model. The definition ofthe model parameters should be done in a practical basis so that the final design will beeasily adjusted when one of the basic parameters is modified. Such parameters are, forexample, those referred to the geometry (Dsphere, Dtube, etc.) and parameters that willbe used for the set-up of the mesh. On the geometry panel there is variety of availableshapes, operations and transforms which make possible the design from simple to evenmore complicated geometries.

After the design of the geometry is completed, there is another batch of settingsthat should be performed before running the simulation. The definition of the materialproperties, the boundary conditions, the creation of the mesh and the study set-upconsist the next step of the procedure. There is no particular order in the set-up of theabove but they have to be done after the entire geometry is well defined and, beforethe final simulation of the model.

In the material library, there are available several built-in materials which can beadded directly. For the purposes of the model of figure 4.2, for all the created domainsthe material used is the Air. The specification of the boundary conditions can be doneby choosing, through a variety of available options, the best fitting to the physical andgeometrical assumptions made during the design of the model. The set-up of the meshis one the most important, if not the most, of the simulation procedure and will be dis-cussed in more detail in section 4.4. The creation of the study gives the opportunity toperform different kind of parametric sweeps using, a single parameter or combinationsof two or more. This requires a proper definition of the model parameters as describedpreviously.

After all the steps of the workflow are well treated then it is possible to run thesimulation and actually solve the “problem” that was defined previously. After thesimulation is finished then comes the post-processing of the results.

4.4 Mesh genera onOne of the most important parts during the implementation of the simulation modelis the generation of the mesh. This procedure is based on the division of the designedgeometry into small units of simple shapes, known as mesh elements [32].

In 2D geometries the domains are discretized into triangular or quadrilateral ele-ments. When curves are included in the geometry, especially on the boundaries, thereis a risk that the surrounding area will not be properly divided and the resulted solution,even if it will be acceptable, it will not be the optimal. A fix to this is to try to use meshstructures that smoothly change in size and resolution near these boundaries, where thesolution is exposed to more steep changes and certainly is an area of higher interest.

25

4.5. STABILITY OF THE RESULTS

Figure 4.4: Mesh generated in ComsolrMultiphysics with extra fine element size.

In figure 4.4 is shown an example of a generatedmesh for the geometry of figure 4.2.Here, for all the four different domains, a free triangular mesh is created with an extrafine element size. The division of the domains seems quite uniform, with a small changeof the distribution of the elements in the vicinity of the electrode curve. From one pointof view this mesh can be considered as sufficient, in case someone is interested in a fastand approximate solution. If the purpose is to produce, an as much as possible, precisesolution, then it is strongly suggested to generate an even finer mesh, at least on theboundaries where the solution is of major importance.

In figure 4.5 is presented the acceptable version for the purposes of the currentstudy. In that case, the mesh is quite denser for the entire geometry compared tothe one in figure 4.4. Furthermore, around the curve and the line that represents themounting tube, the mesh is even finer which means that the returned solutions will bemore precise along these areas. For the additional domains used for the specificationof the open boundaries, is used a quadrilateral mesh structure with predefined size anddistribution of the elements.

4.5 Stability of the resultsWhen a simulation model is designed and a simulation is performed, it is convenient toensure that the produced result remains stable under the variation of parameters whoare supposed indeed to leave it unaffected. The parameters that will be studied for thispurpose are presented in figure 4.6.

26


Figure 4.5: Custom mesh set-up used for the cases studied in the current report.

tubeL

OB

rightd

Figure 4.6: Geometry parameters that should not affect the calculations.

27

4.6. POST-PROCESSING OF THE RESULTS

The parameterOB1 corresponds to the width of the open boundary domains, Ltubeis the tube length for the part that belongs to the main domain of the designed geom-etry (Domain 1 in figure 4.2) and dright is the distance between the edge of the sphereelectrode to the right boundary of the main domain. An acceptable level of variationin the results, is less than or equal to 2%.

4.6 Post-processing of the resultsThe very last step of the simulation procedure is the post-processing of the results.There are several available options through the ComsolrMultiphysics interface butalso the results can be exported, in files of appropriate format, and be used for furtherstudy and calculations through other software packages

(ex. Matlabr

).

For the current study during the post-processing should be included the illustrationsand possible calculations for the below,

(1). The value of the maximum electric fieldEmax and the place in the entire geometrywhere this appears

(2). Electric field and voltage distribution along the sphere-plane gap

(3). Electric field distribution on the sphere electrode surface

(4). Electric field distribution on the surface of the mounting tube

The list above is also depicted in figure 4.7. On these parts of the geometry will be paidmore attention in the next chapter.1OB stands for Open Boundary. An alternative name is IED for Infinite Element Domain.

28


4

3

2

1

Figure 4.7: Parts on the configuration geometry that the post-processing of thesimulation results will be based on. (1) refers to the entire designed geometry,(2) minimum distance between the sphere electrode and the grounded plane, (3)surface of the sphere electrode and (4) surface of the mounting tube.

29

Chapter 5

Case study

5.1 Background descrip onIn this chapter will be studied how the electric field is distributed within the sphere-plane configuration for different values of the sphere electrode diameterDsphere and thegap distance dgap. So far has been given an overview of the model that will be used andthe assumptions have been made about its design. Of major importance is to specifythe value of the maximum electric fieldEmax and the exact point in the entire geometrythis appears.

In one of the most well known electrode configurations, the rod-plane, the max-imum electric field usually appears at the tip of the rod facing the grounded plane.Regarding the sphere-plane gaps, as these were defined in figure 1.1d and for the sim-plicity of the calculations, the maximum electric field should also appear at the bottomof the sphere electrode. Such a behaviour can be achieved by specifying the mountingtube diameter in such a way so it is ensured that the maximum electric field will notappear on the tube surface. The proper way to do this, is to expressDtube as a functionof Dsphere. As it will be shown, this will also give the opportunity for an easier look onthe discharge characteristics of this electrode configuration and will lead to very usefulconclusions.

5.2 InputsThe inputs of the model are related with the geometry parameters of the electrodeconfiguration. The values of the sphere electrode diameter that will be studied are,

Dsphere = [0.25, 0.5, 0.75, 1.2, 1.3, 1.5, 1.6, 2] m (5.1)

31

5.3. SIMULATION RESULTS

while the gap distance will be expressed as a function of this diameter,

dgap = [1, 2, 3 . . . , 15]×Dsphere [m] (5.2)

It is obvious from (5.2) that the ratio between the gap distance and the spherediameter is,

dgapDsphere

= [1, 2, 3 . . . , 15] (5.3)

while the gap spacing over the sphere radius will be for each case,

dgapRsphere

= [2, 4, 6 . . . , 30] (5.4)

The initial values for the diameter of the mounting tube will be based on thosepresented in table 4.1. Additionally, since the scope of the current work is the calcu-lation of the electric field distribution under the application of a DC voltage, then theoptimal physics interface for the simulation model will be the electrostatics. In thatcase, it is required as an input the applied voltage to the high voltage electrode of theconfiguration, and which is set to Vapplied = 1000 kV. Depending on the situation andthe requirements, additional cases for (5.1) and (5.2) will be studied.

5.3 Simula on results

5.3.1 Electric field over dgap/Rsphere ra oIt is essential to have a look on the behaviour of the electric field over the dgap/Rsphereratio. This way of recording can give a clear view how the sphere electrode dimensionsaffect the electric field strength since also the gap distance dgap is expressed as a functionof it. In figures 5.1 to 5.3 are presented the corresponding results of the electric fieldmagnitude Em for the different Dsphere and dgap/Rsphere values mentioned in (5.1) and(5.4) respectively. The curves in these plots refer to the maximum electric field on thesphere and the tube surfaces, as these are depicted in figure 5.4.

For all the differentDsphere and on both the sphere and the tube surfaces, the maxi-mum electric field magnitudes decrease with the increase of the dgap/Rsphere ratio. Fur-thermore, the maximum electric field on the sphere surface is not always higher thanthe corresponding on the tube. As it was mentioned before, such a behaviour requiresspecial treatment.

32

CHAPTER 5. CASE STUDY

dgap/Rsphere ratio0 5 10 15 20 25 30

Electricfieldin

[kV/cm

]

0

20

40

60

80

100

120Sphere - Dsphere = 0.25mSphere - Dsphere = 0.50mSphere - Dsphere = 0.75mSphere - Dsphere = 1.20mSphere - Dsphere = 1.30mSphere - Dsphere = 1.50mSphere - Dsphere = 1.60mSphere - Dsphere = 2.00mTube - Dsphere = 0.25mTube - Dsphere = 0.50mTube - Dsphere = 0.75mTube - Dsphere = 1.20mTube - Dsphere = 1.30mTube - Dsphere = 1.50mTube - Dsphere = 1.60mTube - Dsphere = 2.00m

Figure 5.1: Maximum electric field on the sphere surface, appearing at the bottomof the sphere electrode (continuous lines), and maximum electric field on the tubesurface (dashed lines), over the dgap/Rsphere ratio and for different Dsphere values.

33



Electricfieldin

[kV/cm

]

20

30

40

50

60

70

80

90

100

110

Sphere - Dsphere = 0.25m



Tube - Dsphere = 0.25m



Figure 5.2: Maximum electric field on the sphere surface, appearing at the bottomof the sphere electrode (continuous lines), and maximum electric field on the tubesurface (dashed lines), over the dgap/Rsphere ratio and forDsphere < 1m andDtube =0.1m.34



Electricfieldin

[kV/cm

]

6

8

10

12

14

16

18

20

22











Figure 5.3: Maximum electric field on the sphere surface, appearing at the bottomof the sphere electrode (continuous lines), and maximum electric field on the tubesurface (dashed lines), over the dgap/Rsphere ratio and forDsphere > 1m andDtube =0.45m. 35


Bottom of

the sphere

Tube

surface

Figure 5.4: Bottom of the sphere electrode and the tube surface as these are definedin the simulation model geometry.

5.3.2 The tube diameter issueBefore proceeding to any comments or any further work regarding the behaviour of theelectric field, it is worth to make a pause and focus on the issue plotted on figures 5.1and 5.2.

In some works have been listed difficulties because of flashovers initiating from thesupporting objects and reaching the grounded electrodes [2, 19]. In other works, it isstated clearly that the observed breakdowns were coming from the sphere surface [3],while in most of them is not included any special reference.

For every electrode configuration of a specific geometry, under standard referenceatmosphere1, the surrounding air will show a maximum dielectric strength expressedin V/m, MV/m, kV/cm, etc. When these levels of dielectric strength are reached,electrical breakdown of the air gap or discharge and pre-discharge phenomena willoccur, depending on the configuration, no matter if the place of occurrence will beon the sphere or the tube surface. As it will be shown in the next sections, if the aimof the study is the behaviour of the electric field in sphere-plane gaps, with a possibleextension to the calculation of the breakdown voltage, the presence of the maximumelectric field at the bottom of the sphere can facilitate the required calculations. Asa reminder, the bottom of the sphere is the point with the shortest distance to the1According to IEC 60060-1:2010, the standard reference atmosphere corresponds to: temperature t0 =20 ◦C, absolute pressure p0 = 1013mbar and absolute humidity h0 = 11 g/m3 [18].

36


grounded plane. Thus, it is imposed that the maximum voltage gradient should appearon the sphere.

Focusing on figure 5.2, for all the three different cases there are useful observations.For Dsphere = 0.25m the maximum electric field on the sphere is always higher thanthat on the tube. For Dsphere = 0.5m, and for longer gap distances, the maximumelectric field appears on the tube surface, while for Dsphere = 0.75m the field on thetube dominates over the corresponding on the sphere for all the gap distances studied.Hence, it would be useful to specify a relationship between Dsphere and Dtube with thepurpose that the maximum electric field of the entire geometry will always appear atthe bottom of the sphere. The ratio between those two parameters can be expressedas,

Dratio =Dsphere

Dtube(5.5)

or,

Dtube =Dsphere

Dratio(5.6)

The smaller the sphere diameter the stronger the electric field will be at its surface.This means that, in order to achieve an electric field magnitude on the tube, closeenough to the one on the sphere, it is required a lower Dratio compared to the cases ofbigger sphere diameters. For a given Dsphere, lower Dratio will result to a bigger Dtube.Consequently, the Dratio which will be specified for the smaller Dsphere studied, will beapplicable also for the bigger ones.

It is possible through ComsolrMultiphysics to perform a parametric sweep withvarying parameter the Dratio. This will give the opportunity to illustrate the maximumgradients on both the sphere and the tube surfaces and specify the appropriate value forthe diameters ratio, and consequently forDtube. As it can be understood from figure 5.2,this procedure is worth to be followed for the maximum assumed gap distance, dgap =15×Dsphere. The implementation of the above is summarized in figure 5.5.

The intersection between the curve of the maximum electric field on the tube andthe maximum electric field on the sphere returns Dratio = 4.37. A value of Dratio ≃4.36 is chosen, which covers, from a simulation point of view, all the different casesof Dsphere studied. The divergence between the two values is around 0.23%, whichpractically corresponds to a few millimetres difference in the diameter of the tube,for the biggest sphere considered2. It is very important to mention that the electricfield at the bottom of the sphere remains almost stable with the variation ofDratio, andconsequently theDtube. There is a divergence of less than 2%between the two extreme2For Dsphere = 2m, when Dratio = 4.37 then Dtube = 0.457m, while when Dratio = 4.36, Dtube =0.458m.

37


Dratio

3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2

Electricfieldin

[kV/cm]

55

60

65

70

75

4.37

Maximum electric field on sphere

Maximum electric field on tube

Figure 5.5: Maximum electric field on the sphere and the tube over Dratio. HereDsphere = 0.25m and dgap = 15× 0.25 = 3.75m.

values of the maximum electric field on the sphere as these are shown in figure 5.5. Fordgap = 1×Dsphere this variation is specified around 0.29%.

After the assumption that Dratio = 4.36, the updated content of figure 5.1 is pre-sented in figure 5.6. Again the maximum electric field on both surfaces decreases withthe increase of dgap/Rsphere for all the different Dsphere. This time the maximum elec-tric field on the sphere surface is always higher than the corresponding on the tubefor all the different cases. Even for dgap/Rsphere = 15, despite they are very close, themaximum electric field of the entire geometry does not appear on the tube surface.

38



Electricfieldin

[kV/cm

]

0

20

40

60

80

100

120Sphere - Dsphere = 0.25m
















Figure 5.6: Maximum electric field on the sphere surface, appearing at the bottomof the sphere electrode (continuous lines), and maximum electric field on the tubesurface (dashed lines), over the dgap/Rsphere ratio and for different Dsphere values.Here Dratio = 4.36. 39


5.3.3 Stability of the resultsAs it was mentioned in section 4.5, it is important to confirm that the produced resultsremain stable within a range of ±2% during the variation of some of the geometryparameters. The checked parameters are, the width of the domains used for the openboundary set-up, the length of the tube Ltube that belongs to the main part of the de-sign, and the distance dright from the edge of the sphere to the right boundary of themain domain. These parameters are also shown in figure 4.6 and the related results arepresented in figures 5.7 to 5.9.

Times bigger to the already designed0 2 4 6 8 10

Emaxin

[kV/cm

]

52.15

52.16

52.17

52.18

52.19

52.2

52.21

52.220.1229%Emax over OB size

Reference point

Maximum difference

Figure 5.7: Emax over size of the domains used for the open boundaries set-up.Here Dsphere = 0.5m and dgap = 1× 0.5 = 0.5m.

In figure 5.7 is shown that with the increase of the width of the Open Boundarydomains the maximum electric field Emax is not affected considerably. Even if thedomain is ten times bigger than the one used in the simulation model, the divergence inEmax is less than 0.2%. The same occurs in figure 5.8 for the case whereLtube increases.The maximum variation in Emax, is less than 0.3%. Even smaller is the differencewhen dright increases, as this is illustrated in figure 5.9. The maximum variation doesnot exceed the negligible 0.0011%.

There is no significant divergence, less than 0.3%, in the calculation of Emax, withthe variation of those three parameters. This implies that the model is properly de-signed and the assumptions made in the model and mentioned in chapter 4 seem to besufficiently adapted during the design and the simulation.

40


Ltube in [m]0 2 4 6 8 10

Emaxin

[kV/cm]

52.14

52.16

52.18

52.2

52.22

52.24

52.26

52.280.2289%Emax over Ltube

Reference point

Maximum difference

Figure 5.8: Emax over Ltube. Here Dsphere = 0.5m and dgap = 1× 0.5 = 0.5m.

dright in [m]0 1 2 3 4 5 6

Emaxin

[kV/cm

]

52.1512

52.1514

52.1516

52.1518

52.152

52.1522

52.1524

52.1526

0.0011%Emax over drightReference point

Maximum difference

Figure 5.9: Emax over dright. Here Dsphere = 0.5m and dgap = 1× 0.5 = 0.5m.

41


5.3.4 Electric field over actual gap distance dgapIn figure 5.10 is illustrated the behaviour of the maximum electric field Emax over theactual gap distance dgap. For all of the cases presented, Emax appears at the bottom ofthe sphere electrode.

The approach here is different from the one presented in section 5.3.1. For all thedifferentDsphere cases, it is visible the reduction in the maximum electric field with theincrease of the actual gap distance. The decrease in Emax with the per meter increaseof dgap is more abrupt for smaller sphere diameters. Additionally, the decrease in theelectric field levels between Dsphere = 0.25m and Dsphere = 0.5m is much larger thanthe corresponding one between Dsphere = 1.2m and Dsphere = 2m.

5.3.5 Electric field distribu on on the sphere electrodesurface

In figures 5.11 and 5.12 is shown the distribution of the electric field on the spheresurface for the two extreme values ofDsphere studied, 0.25m and 2m respectively. Thefifteen different curves correspond to the fifteen different dgap values, from dgap =1 ×Dsphere to dgap = 15 ×Dsphere. The horizontal axis refers to the length of the arcsegment that corresponds to the curve of the sphere electrode as this is depicted infigure 4.7. The vertical grey dashed-dotted line represents the axis of symmetry of thesimulation model and divides the arc length into two equal parts.

For all the different Dsphere values there is a similar trend in the field distributionand how this varies with the increase of the gap distance. This is the reason why theplots for all the differentDsphere values are not attached here. The point for maximumelectric field for each curve always appears at the intersection with the vertical grey linewhich in its turn corresponds to the bottom of the sphere. The closer the high voltageelectrode is to the grounded plane the stronger is the electric field at the bottom. Withthe increase of the gap distance, the more smoothly distributed on the sphere surfacethe electric field becomes. In the appendix A can be found all the plots for all thedifferent cases, of the different Dsphere values studied.

5.3.6 Electric field and voltage distribu on along the gapIn figures 5.13 and 5.14 are shown the distribution of the electric field and the voltagealong the gap for Dsphere = 0.25m and for all the gap distances studied. The samekind of plots, but this time for Dsphere = 2m are shown in figures 5.15 and 5.16. Thevertical axes in figures 5.13 and 5.15 are set to a logarithmic scale. Same here, as theprevious section, the plots for each different case of Dsphere are similar so are attachedonly the two extreme cases.

42


Actual gap distance dgap in [m]0 5 10 15 20 25 30

Maxim

um

electric

fieldE

maxin

[kV/cm

]

0

20

40

60

80

100

120

Dsphere = 0.25m

Dsphere = 0.50m

Dsphere = 0.75m

Dsphere = 1.20m

Dsphere = 1.30m

Dsphere = 1.50m

Dsphere = 1.60m

Dsphere = 2.00m

Figure 5.10: Maximum electric field Emax over the actual gap distance dgap fordifferent Dsphere studied.

43


Arc length in [m]0.4 0.2 0 0.2 0.4

Electricfieldin

[kV/cm]

0

20

40

60

80

100

120dgap = 1 × Dsphere

dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere

dgap = 10 × Dsphere






Figure 5.11: Electric field distribution on the sphere electrode surface forDsphere =0.25m and different dgap. The vertical dashed-dotted line corresponds to the axisof symmetry of the simulation model.

Arc length in [m]4 2 0 2 4

Electricfieldin

[kV/cm]

0

2

4

6

8

10

12


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere







Figure 5.12: Electric field distribution on the sphere electrode surface forDsphere =2m and different dgap. The vertical dashed-dotted line corresponds to the axis ofsymmetry of the simulation model.

44


For all cases, the zero value in the horizontal axis corresponds to the bottom of thesphere where both the electric field and the voltage are maximum. Moving towards tothe grounded plane, the electric field decreases. Depending on how long the gap is, theelectric field will be more, or less, close to zero. The voltage starts with its maximumvalue at the electrode and it becomes zero when the grounded plane is reached.

Gap distance in [m]0 1 2 3 4

Max

imum

electric

fieldin

[kV/cm]

10-1

100

101

102


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere







Figure 5.13: Electric field distribution along the gap spacing, forDsphere = 0.25mand various dgap values. Zero on the horizontal axis corresponds to the bottom ofthe sphere electrode.

45


Gap distance in [m]0 1 2 3 4

Voltage

in[kV]

0

200

400

600

800

1000


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere







Figure 5.14: Voltage distribution along the gap spacing, for Dsphere = 0.25m andvarious dgap values. Zero on the horizontal axis corresponds to the bottom of thesphere electrode.

Gap distance in [m]0 10 20 30

Max

imum

electric

fieldin

[kV/cm]

10-2

10-1

100

101


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere







Figure 5.15: Electric field distribution along the gap spacing, for Dsphere = 2mand various dgap values. Zero on the horizontal axis corresponds to the bottom ofthe sphere electrode.

46


Gap distance in [m]0 10 20 30

Voltage

in[kV]

0

200

400

600

800

1000


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere







Figure 5.16: Voltage distribution along the gap spacing, for Dsphere = 2m andvarious dgap values. Zero on the horizontal axis corresponds to the bottom of thesphere electrode.

5.4 Discussion

5.4.1 Sphere mounted on a tubeComparing the different ways of depiction of the results in figures 5.6 and 5.10, it couldbe said that the variation in Emax can be discussed on two different bases, one is thedgap/Rsphere ratio and the other the actual gap distance dgap.

From figure 5.6 it can be observed that for every differentDsphere, as the dgap/Rsphereratio increases, the Emax that appears at the bottom of the sphere decreases. The rateof decrease is faster for the lower values of the ratio, while for the higher seems that adecrease, of lower rate, still occurs. Someone would expect Emax to saturate for longergaps, which does not happen.

In figure 5.17 is illustrated the percentage decrease in the maximum electric field,as dgap/Dsphere increases. For all the different Dsphere the percentage reduction is thesame as dgap/Dsphere gets higher. A possible interpretation of this, is that the closer tothe grounded plane the high voltage electrode is, the stronger the electric field will be.In such cases, the voltage gradient is higher, and more exposed to variations duringchanges in the gap distance. As the high voltage electrode is moved away from thegrounded plane the lower the variation in Emax will be with any further increase in

47

5.4. DISCUSSION

the gap distance. This is also supported from figure 5.18, where the density of theequipotential lines when the gap distance is short (figure 5.18a) is much higher than thecorresponding when the gap distance is longer (figure 5.18b). For example, when thegap distance changes from dgap = 1×Dsphere to dgap = 2×Dsphere, the change in thedensity of the equipotential lines close to the sphere bottom will be much bigger thanthe case where it changes from dgap = 14×Dsphere to dgap = 15×Dsphere.

dgap/Dsphere ratio1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15

Percentagedifferen

ce[%

]

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Figure 5.17: Percentage decrease of maximum electric field with the increase ofdgap/Dsphere. Here are included eight different curves, one for each Dspehre value,but they are not all of them visible since they coincide with each other.

Considering figure 5.10, for the same gap distance and applied voltage to the elec-trode, the smaller the sphere diameter is the stronger the electric field on its surfacewill be. From the definition of electrostatic potential (eq. (3.11)), with an increasingsphere surface, the charge density will decrease. Consequently, the concentration ofthe charge at the bottom point of the sphere electrode will be lower for biggerDsphere.For instance, for Dsphere = 0.25m in the vicinity of the sphere, it is expected the de-crease in the potential to be steeper compared to Dsphere = 0.5m or higher. Such abehaviour will lead to a higher gradient at the same point of the sphere surface. Thiscan be observed in all the figures 5.11 to 5.16.

The results in figures 5.13 and 5.15 are well expected, after the study of the previousplots. Here it seems that for short gap distances, even close to the grounded plane aweak electric field will still exist. This is more understandable after having a look onthe field lines as these are depicted in figure 5.19. For a short gap distance these linesreach the grounded plane while for longer they do not3. The closer the high voltage3Actually there is a very small electric field in most of the cases< 0.5 kV/cm. Its depiction depends onhow many levels are considered in the contour line plot.

48


(a) dgap = 1×Dsphere (b) dgap = 15×Dsphere

Figure 5.18: Equipotential lines for two different gap distances. Here Dsphere =0.25m.

electrode is to the grounded plane the closer to perpendicular to the same plane theelectric field lines will tend to become. On the other hand when the equipotential linesare close to a grounded plane, or any other grounded surface, they will tend to becomeparallel to it.

(a) dgap = 1×Dsphere (b) dgap = 15×Dsphere

Figure 5.19: Electric field lines for two different gap distances. Here Dsphere =0.25m.

49

5.4. DISCUSSION

The shortest the gap distance the closer the electric field will be to the uniformdistribution, and this is observable in both figures 5.14 and 5.16. The voltage beginsfrom Vapplied = 1000 kV at the electrode and for the short gap distances, for instancedgap = 1 × Dsphere, the decrease in the voltage is almost linear. As the gap distanceincreases the voltage distribution comes closer to an exponential decaying behaviour.For all cases in figures 5.13 to 5.16 the distribution is studied along the so-called criticalline of the sphere-plane gap. This line represents the distance between the bottom ofthe sphere and the plane, where the shortest dgap occurs.

5.4.2 Sphere without any moun ng objectIn [19] the authors reproduce an approximate equation which was first included in [33]many years ago, and is referred to the calculation of the maximum electric field insphere-plane configurations. According to this, Emax is expressed as,

Emax =Vapplieddgap

(

4dgapDsphere

+ 1)+

√(4dgapDsphere

+ 1)2

+ 8

4

(5.7)

where Vapplied is the applied voltage on the high voltage electrode. The authors statethat the results from (5.7) differ from the exact values less than 1% for a range ofdgap/Dsphere ratio between 0 and 25. By subtracting the tube geometry from the modelused in the current study (figure 4.2), the obtained results match with those from (5.7)with a maximum divergence of 0.34%. The difference in Emax between the case of asphere without any mounting object and the model of figure 4.2, is presented in fig-ure 5.20. The percentage difference between the two different cases of configurations isthe same for all the differentDsphere values studied. For short gap distances the two dif-ferent configurations can be considered as quite close in their results. The disadvantageis that so short gap distances, are not realistic and after a value of dgap/Rsphere ≃ 3− 4their difference becomes quite big. After this, it is obvious the importance of an accu-rate representation of the electrode geometry with well defined assumptions regardingits design. Approximations as this of (5.7), can result to over- or under-estimations inthe electrode dimensioning, especially when calculating gradients, and possibly insuf-ficient conclusions regarding the discharge characteristics of large electrode gaps likethose considered in the current study.

50



Electricfieldin

[kV/cm

]

0

20

40

60

80

100

120

Dsphere = 0.25mDsphere = 0.50mDsphere = 0.75mDsphere = 1.20mDsphere = 1.30mDsphere = 1.50mDsphere = 1.60mDsphere = 2.00m

Figure 5.20: Emax of a sphere without any mounting object (dashed lines) and asphere mounted on a tube (continuous lines). The results for the first case are basedon (5.7), and for the second are exported from the simulation model. Because ofthe limited free space on the plot, in the legend are included only the continuouslines while the sequence of the colours is the same also for the dashed. 51

5.5. STUDY OF THE BREAKDOWN VOLTAGE

5.5 Study of the breakdown voltageThere are notmany available test results listed regarding the electric field in sphere-planeconfigurations. A proper way to validate the results produced from the electrostaticsimulation model, is to make an attempt to estimate the corresponding breakdownvoltage Vb. The way that the link between the electrostatic model and the breakdownvoltage will be achieved constitutes a quite interesting process.

In [7] is described the “field efficiency factor” η which is defined as,

η =Emean

Emax(5.8)

where Emax is the maximum electric field and Emean is the mean value of the electricfield,

Emean =Vapplieddgap

(5.9)

Based in (5.8), for a uniformly distributed electric field η = 1, while for a stronglynon-uniform electric field η → 0. By inserting (5.9) to (5.8) results to,

η =Vapplied

dgap · Emax(5.10)

Solving (5.10) for Vapplied,

Vapplied = η · dgap · Emax (5.11)

As it is already known from the electrostatic simulation model, the maximum elec-tric field Emax finally appears at the bottom of the sphere electrode. When the break-down strength of air Eb is reached then this will lead to an electrical breakdown of thegap or other discharge or pre-discharge phenomena depending on the geometry char-acteristics of the configuration. In such a case, the applied voltage on the electrode canbe considered as the breakdown voltage Vb. An equivalent expression of (5.11) is

Vb = η · dgap · Eb (5.12)

A similar approach to η is that of the “field enhancement factor” f which is metalso in the bibliography [8] and publications specific on the subject [1], and which ismost commonly defined as,

f =Emax

Emean= 0.9

Rsphere + dgapRsphere

(5.13)

52


The relationship between η and f is obvious,

η =1

f=

Rsphere

0.9 ·(Rsphere + dgap

) (5.14)

Equation (5.14) shows that f , and hence η, depends only of the geometry charac-teristics of the configuration. This gives the opportunity to substitute (5.10) into (5.12)which will result to,

Vb = VappliedEb

Emax(5.15)

Equation (5.15) is very helpful when an electrostatic model is available, throughwhich is defined the maximum electric field Emax for a specific applied voltage Vappliedon the high voltage electrode. Knowing in advance the breakdown strength of air forthis specific geometry, it returns the corresponding breakdown voltage Vb. Anothervery useful observation is that (5.15) represents a linear dependence of the maximumelectric field to the applied voltage Vapplied,

VbEb

=VappliedEmax

(5.16)

In a laboratory test, when the breakdown strength of air Eb is known, and through thetest is specified the breakdown voltage, then it is possible with the use of (5.16) to switchthe study from a laboratory configuration to an electrostatic simulation model, similarto the one studied in the current work. In figure 5.21 is presented the variation of Emaxover Vapplied as this is exported from the simulation model in ComsolrMultiphysics.

The line of figure 5.21 passes through the origin of the axes, which implies that themathematical expression that represents it, is of the form,

Emax = aVapplied orVappliedEmax

=1

a= const. (5.17)

which actually confirms (5.15) and (5.16).In order to be able to apply (5.15) to the exported results of figure 5.10, it is required

to know the breakdown strength of air Eb. A useful approximation for the calculationof this is given in [1], where the authors summarized published and unpublished testresults in order to conclude to an expression for the field strength of air,

Eb =1 +

(dgap/Rsphere

)0.42 + 0.30

(dgap/Rsphere

) [MV/m] (5.18)

The application of (5.18) is limited to the quasi-uniform distribution of the electricfield, which according to their presented results ranges between,

53

5.5. STUDY OF THE BREAKDOWN VOLTAGE

Applied voltage Vapplied in [kV]0 200 400 600 800 1000

Max

imum

electric

fieldE

maxin

[kV/cm]

0

20

40

60

80

100

120

Figure 5.21: Emax over Vapplied. Here Dsphere = 0.25m and dgap = 1×Dsphere.

dgap/Rsphere ∈ [0.1, 4.5] (5.19)

There is not any relationship available in the bibliography, nor any of the publications,that could give a quick calculation of the breakdown strength of air for the cases whendgap/Rsphere > 4.5, where the electric field is considered as strongly non-uniform. In-stead, in order to attempt an extended re-calculation of Eb, it is required a completelydifferent model which will be based in principles out of the scope of the current work.Consequently, the analysis regarding the breakdown voltage, will be limited in the quasi-uniform electric field distribution. The combination of the data of figure 5.10 with(5.15) and (5.18), gives the results of figure 5.22.

The breakdown voltage has an opposite behaviour to that of the electric field anddepends on both the sphere diameter and the gap distance. The bigger theDsphere or thedgap or both of them, the higher the required voltage to lead to an electrical breakdownof the gap. In the range of quasi-uniform electric field, as it described in [1], breakdowncoincides with corona inception. This differs from the cases where dgap/Rsphere > 4.5,during which corona discharge phenomena usually precede the breakdown [6].

54


Gap distance dgap in [m]0 1 2 3 4 5

Breakdow

nvoltageVbin

[kV]

0

500

1000

1500

2000

2500

3000

Dsphere = 0.25m

Dsphere = 0.50m

Dsphere = 0.75m

Dsphere = 1.20m

Dsphere = 1.30m

Dsphere = 1.50m

Dsphere = 1.60m

Dsphere = 2.00m

Figure 5.22: Breakdown voltage Vb over gap distance for different sphere electrodediameters. Here dgap = [0.1, . . . , 4.5]×Rsphere, based on (5.19).

55

5.6. VALIDATION OF THE RESULTS

5.6 Valida on of the results

5.6.1 Published measurementsCompared to the electric field, for the breakdown voltage there are more recorded andpublished measurements which can be used for the validation of the calculations ofsection 5.5.

In [1], the authors summarize the results from multiple works [19-21], with thehelp of which they derived an equation which gives the 50% breakdown voltage forthe quasi-uniform filed distribution of sphere-plane gaps in dry conditions,

UB =1

0.38/dgap + 0.27/Rsphere[MV] (5.20)

where dgap and Rsphere are substituted in meters. The comparison between the calcu-lated results of figure 5.22 and (5.20) are summarized in figure 5.23. Additionally, it isincluded a curve that corresponds to the discharge voltage of the rod-plane configu-ration. This curve is usually used as a reference, mostly to compare with the sphere-plane gaps for the strongly non-uniform electric field distribution

(dgap/Rsphere > 4.5

).

Equation (5.21) is known as the Paris formula and was firstly included in [5].

U50% = 500 · d0.6 [kV] (5.21)

The variation of the UB has a similar behaviour to the calculated Vb. For everyDsphere, a very sharp increase is observed for short gap distances and a more mild asdgap approaches 4.5×Rsphere. Despite the obtained values from both cases seem to bevery close, there is a significant difference between them. The maximum percentagedifference is specified around 6.76%, with the results from (5.20) to be always lowerfrom those in figure 5.22.

In figure 5.24 are compared a batch of U50% measurements included in [3] with thecalculated Vb. The test results refer to dgap/Rsphere values greater than 4.5. In order tobe able to compare them with the corresponding Vb, (5.18) is used “inappropriately”for the calculation of the breakdown strength of air. This is because, the limits within(5.18) is considered as accurate, are those specified in (5.19). However the results seemto match to a very satisfactorily level, a feature that introduces some concerns about theexact limits of the dgap/Rsphere that correspond to the quasi-uniform field distribution.

In the same spirit, in figure 5.25 are compared the results for the case of Dsphere =1.3mwith test results included in [4]. The final result is slightly controversial as for theshorter gap distances the percentage difference is higher than for the longer. Due tothe probabilistic nature of U50% and the fact that the calculation of Vb is limited along

56



Breakdow

nvoltagein

[kV]

0

500

1000

1500

2000

2500

3000

Dsphere = 0.25m

Dsphere = 0.50m

Dsphere = 0.75m

Dsphere = 1.20m

Dsphere = 1.30m

Dsphere = 1.50m

Dsphere = 1.60m

Dsphere = 2.00m

Rod-plane

Figure 5.23: Comparison between calculated breakdown voltage based in (5.15)(continuous lines) and (5.20) (dashed lines). Because of the limited free space onthe plot, in the legend are included only the continuous lines while the sequence ofthe colours is the same also for the dashed. 57



Break

dow

nvoltagein

[kV]

0

500

1000

1500

2000

2500

3000

3500

1.30%

0.36%0.56%

Dsphere = 1.3m

Dsphere = 1.6m

U50% - Test results

dgap/R0.65 ∈ [0.1, 4.5]

dgap/R0.80 ∈ [0.1, 4.5]

Figure 5.24: Comparison between calculated breakdown voltage Vb andU50% fromtest results included in [3].

Gap distance dgap in [m]0 5 10 15 20

Break

dow

nvoltagein

[kV]

0

500

1000

1500

2000

2500

3000

12.82%

2.64%0.73%

Dsphere = 1.3m

U50% - Test results

dgap/Rsphere = 4.5

Figure 5.25: Comparison between calculated breakdown voltage Vb andU50% fromtest results included in [4].

58


the critical field line, it could be said that it does not occur any surprisingly abnormalsituation.

5.6.2 ABB internal testsA series of measurements were provided from ABB AB for validation purposes of thesimulation model. In this set of data, are included test measurements for sphere radiiDsphere = 0.125m and Dsphere = 1m which are not presented so far in the previoussections. Additionally to the cases of (5.1), the two new Dsphere values are simulated.Through (5.15) is calculated the breakdown voltage Vb and the results are summarizedin figure 5.26.

The test results seem to match quite well with the calculations based on the electro-static simulation model. The fit is even better for lower dgap values, however this doesnot mean that they lack of precision for the higher. Details on the exact set-up of theexperimental configuration are not available and possibly there are factors which affectthe measurements when the gap spacing becomes longer.

The curvatures shaped from the test results are slightly lesser than those from thecalculated. A possible explanation is similar to the one discussed in section 5.4.1. As theDsphere values tested are relatively small (≤ 1m), the occurred electric fields, and there-fore the U50%, are more sensitive to the presence of surrounding objects or variationsin the test configuration, especially when the high voltage electrode moves increasinglyhigher from the grounded plane.

Additionally to the above, taking also into account restrictions, such as the prob-abilistic nature of U50% and the fact that the Vb calculations are limited to the criticalfield line, the results of figure 5.26 can be considered as surprisingly convergent. Afterall the series of validations, it could be said that the simulation model used, togetherwith the suggested calculation procedure of Vb, can return results sufficiently close tothe test measurements.

59


Gap distance dgap in [m]0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

Breakdow

nvoltagein

[kV]

0

150

300

450

600

750

900

1050

1200

1350

1500

Dsphere = 0.125m - Simulation




Dsphere = 0.125m - Test meas.




Figure 5.26: Comparison between calculated breakdown voltage Vb and unpub-lished test results for U50% provided by ABB AB. Here dgap = [0.1, . . . , 4.5] ×Rsphere which, according to [1], corresponds to the quasi-uniform electric field dis-tribution.60

Chapter 6

Closure

6.1 Summary of the conducted workIn the current work, it was attempted to give a clear view on the electric field distributionof sphere-plane gaps through the implementation of an electrostatic simulation model.An overview on the already conducted research work regarding this field of study wasrequired in order to list properly all the assumptions made in order to built the model.Additionally, a brief description of the numerical methods used for the calculation ofthe electric field and potential was presented. The intention behind this was to give anoverview to the reader of the complexity behind the calculations, and the constraintsintroduced because of them in the past studies.

After the introductory parts, it was paid more attention on the used model. First, itwas described the implementation of the electrostatic simulationmodel, using ComsolrMultiphysics, the theoretical background, the assumptions made and the expectations.The initial objective of the current work, the specification of the electric field distribu-tion for different sphere-plane gap geometry characteristics, was fulfilled and to someextent validated despite the very limited availability in electric field calculations in thepreviously conducted works. Obstacles, like the tube diameter issue described in sec-tion 5.3.2, were treated with respect to the theoretical and practical principles coveredby the model, and after that useful conclusions were derived. Finally, a series of valida-tions were performed by studying the breakdown voltage of the model and comparingthe results with the already existing from previous test measurements.

The next and final step is to list all the conclusions that can be exported fromthe comparison of the simulation model and the already existing published material.This will be part of an attempt to describe the contribution of the current work tothe corresponding field of study. Then is added a list of ideas for future work andsuggestions for improvement of the already existing.

61

6.2. CONCLUSIONS FROM CASE STUDIES

6.2 Conclusions from case studies• The breakdown voltage Vb was specified in two ways. One was based on thetheoretical principles and background regarding the electric field distribution be-tween the electrodes (eq. (5.16)), and the other through the simulation model(figure 5.21 and eq. (5.17)). This validates the use of the derived equation andconfirms that it can work as a “bridge” for the calculation between Emax and Vbwhen an electrostatic simulation model is available.

• The linear relationship between the maximum electric field and the applied volt-age on the high voltage electrode (eq. (5.16)), could be said that it constitutesthe link between a test model and an electrostatic simulation similar to the oneused in the current study. From a test model is derived U50%, and by knowingEb for the specific electrode geometry used, it is possible to switch the study toan electrostatic simulation model. The opposite path, and which was one of theobjectives of this study, is that having Vapplied andEmax from the electrostatic sim-ulationmodel and knowing againEb can lead to the calculation of the breakdownvoltage Vb.

Electrostaticmodel

VbEb

=VappliedEmax

Testmodel

• During the design and the simulation of the used model, it was observed howthe mounting tube affect the maximum electric field. A tube of relatively smalldiameter compared to the sphere diameter, can lead Emax to appear on the tubesurface. In the current work it was produced an approximate relationship be-tween those two diameters (eq. (5.6)). Knowing the diameter of the sphere it iseasy to calculate the appropriate diameter for the tube, so that it will be ensuredthe maximum electric field will always appear at the bottom of the sphere.

• The relationship between the sphere and the tube diameters can be a very usefulapproach regarding laboratory testing. The specification of the tube diameterduring the initial set-up of the electrode configuration can be time and cost ef-fective for the whole test procedure.

• The calculated breakdown voltage Vb was validated through the comparison withU50% test results (figures 5.24 to 5.26). This was performed for gap distanceswhich correspond to the quasi-uniform electric field distribution (eq. (5.19)).

62

CHAPTER 6. CLOSURE

This implies that the results from the simulation model regarding Emax are suffi-ciently accurate for that range. Since Emax values for the uniform and the quasi-uniform electric field distributions are reliable, then also the values that corre-spond to longer gap distances can be considered as accurate.

• During the validation procedure, it was observed a sufficient convergence be-tween Vb and U50%, also for gap distances longer than those which correspondto the quasi-uniform field distribution (figures 5.24 and 5.25). These results con-tribute to the growing doubts about how wide is that range and if it is properlydefined in previous works.

• Menemenlis et al. in their calculations use the “field enhancement factor” f(eq. (5.13)) for the calculation of the 50% breakdown voltage. The analyticalexpression of this factor differs, to a small extent, through different resources.This makes slightly offish the use of the mathematical definition of this factorsince there is a possibility to introduce an amount of error in the calculation. Inthe current study, the calculation of the breakdown voltage Vb (eq. (5.15)) is freeof this enhancement factor, which is an extra advantage during the calculationprocedure, with a predefined Emax.

• A more reliable approach for the calculation of the “field enhancement factor”f is given from Schneider et al. (eq. (5.7)). According to the authors the pro-duced results are sufficiently validated from measurements. These results arealso confirmed from the simulation model of the current work if only a highvoltage sphere, without any mounting tube or rod, is considered together witha grounded plane. A drawback of this approach is the absence of a mountingobject to the high voltage sphere.

The comparison regarding Emax between different concepts and models is sum-marized in figure 6.1. The simulation model without a mounting tube and theSchneider et al. approach match almost perfectly. The differences from Men-emenlis et al. and the configuration with a tube are significant. These resultsshown the importance of the proper definition of f and the incorporation of themounting tube in the simulation model and the calculations.

• For the calculation of the breakdown strength of air for the quasi-uniform electricfield, it was used the Eb as this was determined by Menemenlis et al1 (eq. (5.18)).This expression constitutes a common reference between the calculation of Vb,performed for the purposes of the current study and the UB suggested by Mene-menlis et al (eq. (5.20)). The divergence between the two methods of calculationis introduced probably because of differences in the calculation of Emax.

1In [1] it is denoted as Emax, but here is used Eb to avoid possible inconvenience with the surfacemaximum electric field.

63

6.3. PROPOSAL FOR FUTURE WORK

Gap distance dgap in [m]0 1 2 3 4 5 6 7 8

Max

imum

electric

fieldin

[kV/cm]

30

35

40

45

50

55

Simulation - without tubeSchneider et al.Menemenlis et al.Simulation - with tubedgap = 4.5 · 0.25 = 1.125m

Figure 6.1: Emax for different enhancement factor f concepts. Here Dsphere =0.5m and dgap = [1, . . . , 15]×Dsphere.

6.3 Proposal for future workThe current master thesis project has achieved its purpose but in engineering problemsthere are no complete solutions and there are always many alternative versions andideas for further improvement.

Regarding the current subject, there is no doubt that an accurate calculation ofthe breakdown strength of air is the missing piece of the puzzle, and which would beworthy of further study. In this work it was followed an opposite path of calculationcompared to what had been used so far. This gave the opportunity to obtain a clearview, from a different perspective this time, to what happens regarding the distributionof the electric field in sphere-plane configurations.

In this type of electrode structures, for uniform distribution of the electric field,even the theoretical approaches that can be found in the bibliography are enough tocover the practical part. For the quasi-uniform, despite there are many significant con-tributions, conducted mainly many years ago, an update regarding the calculation ofEb would be welcomed, so it would be possible to compare again with the already ex-isting material. Regarding the strongly non-uniform field distribution for longer gaps,despite it is the most practically useful and interesting, the availability of information isrelated primarily with test results and the specification of the 50% breakdown voltage.

64

CHAPTER 6. CLOSURE

In parallel with the calculation of the breakdown strength of air, it would be useful toreconsider the exact limits of the three different distribution types of the electric field.It is important during all this procedure, to pay extra attention to the existence of amounting tube and not just an ideally hanging sphere.

A suggested working procedure on the above could be that, firstly each subject isstudied at an electrostatic simulation level. In that way, it is possible to adapt efficientlyalmost all the theoretical assumptions. After that, the results produced from the sim-ulation model can be validated through a well designed laboratory experiment. Apartfrom the financial advantages, in that way it would be easier also to get a view on how toperform an appropriate set-up of the laboratory in order to avoid obstacles that couldaffect the accuracy and the quality of the measurements.

It would be very interesting also to see a simulation model where a double toroidis included for the termination of the tube, similar to the one of figure 4.1 [3]. It isexpected that the produced results will not differ considerably from those of the con-figuration studied in this project. However, the appropriate design and the adjustmentof the geometry, seem challenging since there are some additional parameters relatedwith the double toroid dimensions and which should be taken into account. An evenmore extensive study could be performed by redesigning the current simulation model,but this time in a 3D space dimension. In that way, it would be possible to examine howthe presence of protrusions on the surface of the sphere electrode affects the electricfield distribution.

After the implementation of an appropriate numerical or simulation model for thespecification of Eb, even for longer gaps, it will be possible to use these values with theEmax calculated in this project. This approach will give a complete calculation of Vb.After this, it would be wise to examine to what extent is achievable to adapt the newapproaches to different electrode configurations.

65

References

[1] C. Menemenlis, G. Harbec, and J. F. Grenon, “Switching-impulse corona incep-tion and breakdown of large high-voltage electordes in air,” IEEE Transactions onPower Apparatus and Systems, vol. PAS-97, no. 6, pp. 2367–2374, 1978.

[2] J. K. Hepworth, R. C. Klewe, and B. A. Tozer, “Impulse breakdown of largesphere-plane gaps,” Proceedings of the Institution of Electrical Engineers, vol. 119, no. 12,pp. 1751–1753, 1972.

[3] D. Wu, L. Arevalo, L. Ming, and M. Larsson, “Switching impulse test of largesphere-plane air-gaps with protrusion on large spheres,” in 18th International Sym-posium on High Engineering, Seoul, Korea, 2013.

[4] L. Arevalo, “Numerical simulations of long spark and lightning attachment,”PhD thesis, Uppsala University, 2011. [Online]. Available: http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-149171

[5] L. Paris and R. Cortina, “Switching and lightning impulse discharge characteristicsof large air gaps and long insulator strings,” IEEE Transactions on Power Apparatusand Systems, vol. PAS-87, no. 4, pp. 947–957, 1968.

[6] K. Feser, “Mechanism to explain the switching impulse phenomena,” SchweizerischeTechnische Zeitschrift, vol. 46, 1971.

[7] E. Kuffel, W. Zaengl, and J. Kuffel, High Voltage Engineering: Fundamentals.Butterworth-Heinemann, 2000. [Online]. Available: http://www.sciencedirect.com/science/book/9780750636346

[8] S. Naidu and V. Kamaraju, High Voltage Engineering. McGraw-Hill Education(India) Pte Limited, 2013.

[9] H. Ryan, High-Voltage Engineering and Testing (3rd Edition). Institution ofEngineering and Technology, 2013. [Online]. Available: http://digital-library.theiet.org/content/books/po/pbpo066e

67

http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-149171

http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-149171

http://www.sciencedirect.com/science/book/9780750636346


http://digital-library.theiet.org/content/books/po/pbpo066e


REFERENCES

[10] V. Ushakov and V. Ushakov, Insulation of High-Voltage Equipment. Springer, 2004.[Online]. Available: http://www.springer.com/us/book/9783540207290

[11] H. Singer, H. Steinbigler, and P. Weiss, “A charge simulation method for thecalculation of high voltage fields,” IEEETransactions on Power Apparatus and Systems,vol. PAS-93, no. 5, pp. 1660–1668, 1974.

[12] A. Yializis, E. Kuffel, and P. H. Alexander, “An optimized charge simulationmethod for the calculation of high voltage fields,” IEEE Transactions on Power Ap-paratus and Systems, vol. PAS-97, no. 6, pp. 2434–2440, 1978.

[13] N. H. Malik, “A review of the charge simulation method and its applications,”IEEE Transactions on Electrical Insulation, vol. 24, no. 1, pp. 3–20, 1989.

[14] V. Cooray, The Lightning Flash. Institution of Engineering and Technology,2014, (ed). [Online]. Available: http://digital-library.theiet.org/content/books/po/pbpo069e

[15] S. Okabe, T. Tsuboi, and G. Ueta, “Study on lightning impulse test waveform forUHV-class electric power equipment,” IEEE Transactions on Dielectrics and ElectricalInsulation, vol. 19, no. 3, pp. 803–811, 2012.

[16] S. Okabe, G. Ueta, T. Tsuboi, and J. Takami, “Study on switching impulse testwaveform for UHV-class electric power equipment,” IEEE Transactions on Di-electrics and Electrical Insulation, vol. 19, no. 3, pp. 793–802, 2012.

[17] A. P. Brede, P. Werle, E. Gockenbach, and H. Borsi, “A newmethod of determin-ing the mean curve of lightning impulses according to IEC 60060-1,” in EleventhInternational Symposium on High Voltage Engineering, (Conf. Publ. No. 467) 1999, vol. 1,Conference Proceedings, pp. 74–77 vol.1.

[18] IEC, “High-voltage test techniques - Part 1: General definitions and test require-ments,” p. 149, 2010-09-29 2010.

[19] H. M. Schneider and F. J. Turner, “Switching-surge flashover characteristics oflong sphere-plane gaps for UHV station design,” IEEE Transactions on PowerApparatus and Systems, vol. 94, no. 2, pp. 551–560, 1975. [Online]. Available:http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1601485

[20] G. Carrara and L. Thione, “Switching surge strength of large air gaps: A physicalapproach,” IEEE Transactions on Power Apparatus and Systems, vol. 95, no. 2, pp.512–524, 1976.

[21] F. A. M. Rizk, “Influence of rain on switching impulse sparkover voltage of large-electrode air gaps,” IEEE Transactions on Power Apparatus and Systems, vol. 95, no. 4,pp. 1394–1402, 1976.

68

http://www.springer.com/us/book/9783540207290



http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1601485

REFERENCES

[22] C. Wadhwa, High Voltage Engineering. New Age International (P) Limited, 2007.

[23] L. Arevalo and D. Wu, “Effect of high dielectric protrusions on the breakdownphenomena of large electrodes under positive switching impulses,” in IEEE Con-ference on Electrical Insulation and Dielectric Phenomena (CEIDP), 2014, Conference Pro-ceedings, pp. 51–54.

[24] M. Li, F. Sahlen, D. Wu, G. Asplund, and B. Jacobson, “Humidity effects on di-electric strength of air-gaps for indoor HV installations,” in 2005 Annual ReportConference on Electrical Insulation and Dielectric Phenomena, 2005. CEIDP ’05, Confer-ence Proceedings, pp. 43–46.

[25] F. Rizk and G. Trinh, High Voltage Engineering. Taylor & Francis, 2014.

[26] D. Griffiths, Introduction to Electrodynamics. Pearson Education, 2014.

[27] D. Bromley and W. Greiner, Classical Electrodynamics. Springer New York, 2012.[Online]. Available: http://www.springer.com/us/book/9780387947990

[28] R. Arora and W. Mosch, High Voltage and Electrical Insulation Engineering. Wiley,2011.

[29] H. Saadat, Power System Analysis. PSA Pub., 2010.

[30] P. Zhou, Numerical Analysis of Electromagnetic Fields. Springer Berlin Heidelberg,2012. [Online]. Available: http://www.springer.com/us/book/9783642503214

[31] V. Chari and S. Salon, Numerical Methods in Electromagnetism. AcademicPress, 2000. [Online]. Available: http://www.sciencedirect.com/science/book/9780126157604

[32] Comsol AB, “Comsol Multiphysics Reference Manual, ver. 5.0,” 2014.

[33] G. R. Dean, “The maximum voltage gradient in a spark gap in terms of the radiusof curvature of the electrodes,” General Electric Review, vol. 117, no. 16, pp. 148–150, 1913.

69





Appendices

71

Appendix A

Electric field distribu on onthe sphere electrode

surface


Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere







Figure A.1: Electric field distribution on the sphere electrode surface for Dsphere = 0.25mand different dgap.

73


Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere









Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere








74

APPENDIX A. ELECTRIC FIELD DISTRIBUTION ON THE SPHERE ELECTRODE SURFACE


Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere









Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere








75


Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere









Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere








76

APPENDIX A. ELECTRIC FIELD DISTRIBUTION ON THE SPHERE ELECTRODE SURFACE


Electricfieldin

[kV/cm]

0

20

40

60

80

100


dgap = 2 × Dsphere

dgap = 3 × Dsphere

dgap = 4 × Dsphere

dgap = 5 × Dsphere

dgap = 6 × Dsphere

dgap = 7 × Dsphere

dgap = 8 × Dsphere

dgap = 9 × Dsphere







Figure A.8: Electric field distribution on the sphere electrode surface forDsphere = 2m anddifferent dgap.

77

Appendix B

Sphere and tube diameterra o for different cases

Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm

]

60

65

70

75

80

85

4.3783



Figure B.1: Dsphere over Dtube ratio for Dsphere = 0.25m.

79

Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm]

30

32

34

36

38

40

42

4.363




Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm

]

20

21

22

23

24

25

26

27

28

4.3924




80

APPENDIX B. SPHERE AND TUBE DIAMETER RATIO FOR DIFFERENT CASES

Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm]

12

12.5

13

13.5

14

14.5

15

15.5

16

16.5

17

4.5726




Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm

]

11

11.5

12

12.5

13

13.5

14

14.5

15

15.5

4.6222




81

Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm]

9.5

10

10.5

11

11.5

12

12.5

13

4.7258




Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm

]

9

9.5

10

10.5

11

11.5

12

12.5

4.7793




82

APPENDIX B. SPHERE AND TUBE DIAMETER RATIO FOR DIFFERENT CASES

Dratio

4 4.5 5 5.5 6

Electricfieldin

[kV/cm]

7

7.5

8

8.5

9

9.5

4.9808



Figure B.8: Dsphere over Dtube ratio for Dsphere = 2m.

Dsphere [m] Dratio

0.25 4.378

0.50 4.363

0.75 4.392

1.20 4.573

1.30 4.622

1.50 4.726

1.60 4.779

2.00 4.981

Table B.1: Dratio for every Dsphere value studied.

83

Appendix C

Calculated values

C.1 Maximum electric field

0.125 0.25 0.5 0.75 1 1.2 1.3 1.5 1.6 2

0.05 1707.1 853.59 426.78 284.52 213.39 177.83 164.15 142.25 133.37 106.680.1 909.19 454.62 227.29 151.53 113.65 94.71 87.42 75.76 71.03 56.820.2 512.87 256.4 128.19 85.46 64.1 53.41 49.3 42.73 40.06 32.050.3 382.98 191.41 95.7 63.79 47.85 39.87 36.8 31.9 29.9 23.920.4 319.25 159.54 79.76 53.17 39.88 33.23 30.68 26.59 24.92 19.940.5 281.68 140.77 70.38 46.92 35.19 29.32 27.07 23.46 21.99 17.590.6 257 128.44 64.21 42.81 32.11 26.76 24.7 21.4 20.07 16.050.7 239.59 119.74 59.87 39.91 29.93 24.94 23.03 19.96 18.71 14.970.8 226.64 113.28 56.64 37.76 28.32 23.6 21.78 18.88 17.7 14.160.9 216.64 108.29 54.14 36.09 27.07 22.56 20.82 18.05 16.92 13.541 208.66 104.31 52.15 34.77 26.08 21.73 20.06 17.38 16.3 13.041.1 202.15 101.06 50.53 33.68 25.26 21.05 19.43 16.84 15.79 12.631.2 196.72 98.34 49.17 32.78 24.59 20.49 18.91 16.39 15.37 12.291.3 192.11 96.05 48.02 32.01 24.01 20.01 18.47 16.01 15.01 12.011.4 188.15 94.07 47.03 31.36 23.52 19.6 18.09 15.68 14.7 11.761.5 184.71 92.35 46.17 30.78 23.09 19.24 17.76 15.39 14.43 11.541.6 181.68 90.83 45.42 30.28 22.71 18.92 17.47 15.14 14.19 11.351.7 178.98 89.49 44.74 29.83 22.37 18.64 17.21 14.91 13.98 11.191.8 176.58 88.28 44.14 29.43 22.07 18.39 16.98 14.71 13.79 11.04

85

C.2. BREAKDOWN VOLTAGE

1.9 174.4 87.2 43.6 29.07 21.8 18.17 16.77 14.53 13.62 10.92 172.43 86.21 43.11 28.74 21.55 17.96 16.58 14.37 13.47 10.782.1 170.64 85.32 42.66 28.44 21.33 17.77 16.41 14.22 13.33 10.662.2 168.99 84.49 42.25 28.16 21.12 17.6 16.25 14.08 13.2 10.562.25 168.21 84.11 42.05 28.03 21.03 17.52 16.17 14.02 13.14 10.513 159.4 79.7 39.85 26.57 19.92 16.6 15.33 13.28 12.45 9.964 152.21 76.1 38.05 25.37 19.03 15.85 14.63 12.68 11.89 9.515 147.46 73.73 36.86 24.58 18.43 15.36 14.18 12.29 11.52 9.226 144 72 36 24 18 15 13.85 12 11.25 97 141.32 70.66 35.33 23.55 17.66 14.72 13.59 11.78 11.04 8.838 139.15 69.58 34.79 23.19 17.39 14.5 13.38 11.6 10.87 8.79 137.35 68.68 34.34 22.89 17.17 14.31 13.21 11.45 10.73 8.5810 135.81 67.91 33.95 22.64 16.98 14.15 13.06 11.32 10.61 8.4911 134.48 67.24 33.62 22.41 16.81 14.01 12.93 11.21 10.51 8.4112 133.31 66.65 33.33 22.22 16.67 13.89 12.82 11.11 10.42 8.3313 132.26 66.13 33.07 22.04 16.53 13.78 12.72 11.02 10.33 8.2714 131.32 65.66 32.83 21.89 16.42 13.68 12.63 10.94 10.26 8.2115 130.46 65.23 32.62 21.75 16.31 13.59 12.55 10.87 10.19 8.15

Table C.1: Simulation results for maximum electric field in [kV/cm]. Each columncorresponds to a different sphere diameter valueDsphere, while each row to differentgap distance over sphere diameter ratio values dgap/Dsphere.

C.2 Breakdown voltage

0.125 0.25 0.5 0.75 1 1.2 1.3 1.5 1.6 2

0.05 14.32 28.64 57.28 85.91 114.55 137.46 148.92 171.84 183.28 229.130.1 27.5 54.99 109.99 164.98 219.97 263.97 285.96 329.97 351.98 439.960.2 50.55 101.12 202.24 303.35 404.47 485.39 525.84 606.74 647.18 808.950.3 69.63 139.32 278.65 418.01 557.3 668.81 724.54 835.98 891.72 1114.70.4 85.43 170.94 341.93 512.91 683.88 820.67 889.06 1025.8 1094.2 1367.80.5 98.62 197.33 394.71 592.08 789.44 947.33 1026.3 1184.2 1263.1 1578.90.6 109.75 219.6 439.23 658.86 878.48 1054.2 1142 1317.8 1405.6 17570.7 119.25 238.6 477.24 715.87 954.48 1145.4 1240.8 1431.7 1527.2 19090.8 127.47 255.01 510.06 765.09 1020.1 1224.2 1326.2 1530.2 1632.2 2040.3

86

APPENDIX C. CALCULATED VALUES

0.9 134.63 269.34 538.71 808.06 1077.4 1292.9 1400.6 1616.1 1723.9 2154.81 140.95 281.97 563.96 845.94 1127.9 1353.5 1466.3 1691.9 1804.7 2255.91.1 146.57 293.2 586.41 879.64 1172.8 1407.4 1524.7 1759.2 1876.5 2345.71.2 151.61 303.27 606.55 909.81 1213.1 1455.7 1577 1819.6 1941 2426.21.3 156.16 312.35 624.71 937.06 1249.4 1499.3 1624.3 1874.1 1998.6 2498.91.4 160.29 320.6 641.21 961.83 1282.4 1538.9 1667.2 1923.6 2051.9 2564.91.5 164.06 328.14 656.29 984.43 1312.6 1575.1 1706.4 1968.9 2100.2 2625.21.6 167.52 335.06 670.14 1005.2 1340.3 1608.3 1742.4 2010.4 2144.4 2680.61.7 170.72 341.45 682.91 1024.4 1365.8 1639 1775.6 2048.7 2185.3 2731.61.8 173.67 347.36 694.73 1042.1 1389.5 1667.3 1806.3 2084.2 2223.1 2778.91.9 176.43 352.86 705.74 1058.6 1411.5 1693.8 1834.9 2117.2 2258.3 2822.92 178.99 357.99 715.98 1074 1432 1718.4 1861.6 2148 2291.2 28642.1 181.39 362.8 725.6 1088.4 1451.2 1741.4 1886.6 2176.8 2321.9 2902.42.2 183.65 367.31 734.62 1101.9 1469.2 1763.1 1910 2203.8 2350.8 2938.52.25 184.73 369.46 738.92 1108.4 1477.8 1773.4 1921.2 2216.8 2364.5 2955.7

Table C.2: Calculation results for breakdown voltage Vb in [kV]. Each columncorresponds to a different sphere diameter valueDsphere, while each row to differentgap distance over sphere diameter ratio values dgap/Dsphere.

87

TRITA TRITA-EE 2016:030

www.kth.se

electric field distribution of sphere-plane...

Documents