Numerical simulation of the conducting surface of high-voltage insulating systems in 3D

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Numerical simulation of the conducting surface of high-voltage insulating systems in 3D

Philippe Auriol, Qi Huang, Laurent Krähenbühl

To cite this version:

Philippe Auriol, Qi Huang, Laurent Krähenbühl. Numerical simulation of the conducting surface of

high-voltage insulating systems in 3D. IEEE Transactions on Magnetics, Institute of Electrical and

Electronics Engineers, 1988, 24 (1), pp.43-46. �10.1109/20.43852�. �hal-00359866�

IEEE TRANSACTIONS ON MAGNETICS. VOL. 24, NO. 1. JANUARY 1988

NUMERICAL SIMULFITION OF THE CONDUCTING SURFACE OF HIGH-VOLTfGE INSUL9TING SYSTEMS I N 3D

P h . liuria!, 0.S. t!u-.ng, L . Srshsribuhl Depart enen? d ' Elec t r o t e c hnique I?ri+C Associle j u C . N . R . S . No e 2 S

Ecole Centr??r de L.yon EP 153

-

69131 ECVLLY CEDE:<

-

FRAFlCE

43

M S T R A C T as shown in the previcus publrcdticns L o ] , where' T h e numerical simulation described in this paper

is based upon the boundary element method. The effects of the surface conductinp film on the potential distribution of insulating systems are considered. A particular interface condition is used for the resistive surface o f insulators. This approach allows fast subsequent analysis for different positions and resistances of the surface layers.

1. INTRODUCTION

f i s the boundary of V

r i s tbe distance between the point P and t h r

Cp is the solid angle viswing from the pzirt F

4

^{1 5} the potential

E, is the relati**e permitivit;.

B y choosing a set cf pcints P on arid U--!?,

the classical finite element technique, the equaticr

( 1 ) c a n b e wr:tten as:

1 8 the normal vector of current p c i n t of integrsticn

'palp' =

1

( a i l

All

-

&YJ,, EL, ) For the design of high-voltage insulating

systems, i t is very important to andlyse the

conductinp f i l m effects over the surface o f Aharc:

insulators (poilution. semi-conducting glaze for

1. I

anti-pollution).

The presence of the surface current tnrouyhout the insulating system distorts the capacitive potential distribution. There are many published papers presenting a numerical analysis of this effect. in which t h e finite difference method, the finite element method and the charge simulation method are used t 1 1 , C Z I .

These methods, however, suffer from some inconveniencas: the charge simulation method i s very sensitive to the number of charges chosen for a particular problem; the suggestion that the solution quality increases with the number of charge 1 3 not necessarily true [31. For the finite element or' finite difference method, the entire volume o f the problem must be subdivided, making i t difficult t 2 -

treat arbitrarily thin conducting film.

Since the phenomenon can be considered a n interfacial o n e , it is lopica1 t o use the boundary element method (EEM) for the simulation o f this problem. In this paper we present a numerical method based o n RFH for t h e simulation of conducting f i l m

k is the suberript of the boundary e l e m e ~ t

I is the subscript of the diszrete ixds c c g n

$k, y,,

ALi,3nd &de;i.nd only v n t h e ;ecme?r, 2nd - 3 -

e 1 eme? t

:s the value o f

$

^{at the} d i s c r p t e n c d r i o the value cf Y at the discrete PO&

cslculated.

To sol\.? ( Z ! three types o f b o u n d a r y -or;d:t.rsn:

+ "richlet: @ =

v

tn?wn f o r B c.;.nduct.or

* 3r.d

Y

w e 0 C t . k f;r:tncur! f.:r .?fi i.-!,?r

l n e equation i 2 i : 5 estaolianed sn edch cJi~,c:.e:e point of the baundary with the first two types s.P boundary conditions.

On interfaces. equation ( 2 ) is established twice on each discrete point for the two concerned regions.

So we c a n establish equation ( 2 ) a5 many as the number of unlinowns for a problem with several have to be considered:

cftects. different dielectric mediums. After resolutron, G n c

obtains the distributions o f

4

^and

^y

^{o n L a n d il}

, 7 n ~ ~ h e r e by rsusing ( 2 ) .

The field 1 5 obtained in the same mdnrler n,!

T F : , : ~ ~ ~ 9 - ~ ~ r ~ : . I F r j l e ~ t ~ i c t i e l < = - e q ~ i * - e s i r e usin:! the gradients of the two weiyhtinp fLflciAc-t- sclution o f the Laplace's equatro? in one o r savers! I G , o 6 / b n ) .

region's' 0 with the boundary c a d i t r c n s . The , ^is continuous through the interface o f tbc

eguivslert f c r w l a cc th.r Laplace's equation 1:. dia!ectrics, o n condition that there i s no conouctinq f i l m on this interface, otherwise a particular 2. BOUNDHRV INTEGRAL EQUATION

c p @ l P j = - & I ? = bG

-

Z y G I d s I ^{( 1 '} interface condition should be used.

L t t P .

od18-9464/88/0loooo43Wl .WO1988 IEEE

44

3. INTERFACE CONDITION FOR THE CONDUCTING SURFACE L e t S be a p i l l b o n - s h a p e d c l c s e d s u r f a c e ( f i g u r e 1

'

c.f end a r e a $ F . o i e r t h e s w - * 3 c e , whose h i g h + c!! i c e q u a l t c t h e c o n d u c t r q a film's t h i c k n e s s .

Figure 1: The p i l l b o x c o n s t r u c - t e d f o r d e r i v i n g t h e b o u n d a r y c o n d i t i o n of t h e c o n d u c t i n g s u r f a c e

The c h a r g e c o n s e r v a t i o n g i v e s t h e r e 1 s t i 2 3 :

J is t h e c u r r e n t d e n s i t y P is t h e d i s p l a c e m e n t k? i s t h e f r e q u e n c y

zu

f o l lo w I nS

( 3 )

Assuming d l 1 5 n e p l i g i b l e i n c o m p a r i s o n w i t h t h e A F r a d i u s a n d A F i s a l s o v e r y s m a l l , ( 3 ) c a n b e h r i + t e n a5 f e l l c w s :

jSj..n

= I W C ~ ( Y , + K I A F ( 4 )

u h e r e m d a r c t h e va!uas sr

y

on ~ b u t

d e f L n e d i n G f f e r e n t media.

L e t s 5 = d 1 . f b e t h e s u r f a c e c u r r e n t d e n s i t , , we have:

d 1 - C

L l s i r q t h e Ohm's l a w , one o b t a i n s t h e bcaJ,rdary c n n d i d i o n f o r c o n d u c t i n g s u r f a c e :

whcr.-:

E s i s -. t h e t a n p e n t l a 1 f l e l d o n t h e i n t e r f a c 5 cr 1 5 t h e s u r f a c e c o n d u c t i v i t y .

A s we have d e f i n e d t h e i n t e r p o l a t i o n e - p r e s s i o n c f y c n t h e b o u n d a r y ,

2

can be c a l c u l a t e d .

4 . CFILCULFITION OF v . E s

Or! t h e b o u n d a r y e l e n e n ? ( S O ! , nne h a s :

@ = P ; I U , Y I

ai

^{i ? )}

where P i I C t h e i n t e r p c l a t i n p f u n c ? i c n ; U and 'v a r e t h e l o c a l r u r v i l i n e d r c o o r d i n a t e s .

Ey u s i n g d i f f e r e n t i a ! o p e r a t o r s I P 2d .curvilinear s p a c e , snz Q b t e i n s :

3 i s t h e d e t e r n i n a n t o f t h e c c v a r i s n t m e t r i c i ! 3"' , g u y and gvv a r e t h e e l e m e n t s o f t h e

m a t r i x

c c 2 r t - a v a r i x q t m e t r i c a l m a t r i x :

- -

a , , = e , . e , [ 9U"I = [ g""]"

F o r t h e o r t h o g o n a l l o c a l c o o r d i n a t e s :

i . ! ! ! wbere g u u , gur arid 3," a r e t h e e l e m e n t s o f t h e

; c v a r i a n t m e t r i c a l m a t r i y .

F c r t h e a l i s y m m e t r i c c a s e , t h e e q r e s s : o r ' I C mere k i ~ p l e :

5. CONDUCTING FILM WITH FLOATING POTENTIFlL Wheri t h e c o n d u c t i n g f i l m i s not ccnnecte!! h;

e i e c t r c d e s , t h e r e 1 5 no s u r f j c e c u r r e n t ( f n r e.crmp!e t h e s v r f a c e p o l l I J t i o n f i l m I C i n t e r r u p t e d b y t h e d r y b a n d s ! , i n t h i s c a s e , t h e c o n d u c t i n g f i l m b e h s \ , e 5 i t s e l f liCe z good c o n d u c t o r w i t h f l s s t l n g p o t e , ? i i a ! .

! ? \ c j n De w r i t t e n a s :

Numerically, we double the discrete elements o f conducting part of insulator’s surface (these elements constitute a closed surface), so that ( 1 4 ) can be imposed over these elements. Figure 2 shows an example in the axisymmetric case.

Figure 2: AB-conducting film with floating potential on which the discrete elements are doubled

The integration of equation ( 1 ) established at the point C’ is over the boundary o f the medium I;

t h e integration of this equatidn established at C is over the boundary O P the medium 11.

Although C and C ’ are peometrically t h e sane point, w e can obtain two different equations because the boundaries o f the two media a r e not the same. The points A , B become the geometrical singular points and

Y

has not a unique value on these points.

The numerical method proposed in 2 4 1 is very convenient for treating these points.

It is obvious that this technique can be u s e d Cor treating metallic filmr.If the metallic f i l m I >

i n a medium. not over the surface of insulator, we

can add an artificial interface i n order that the metallic film becomes a part of t h i s interface

( Pisure 3 ) .

{.

- ‘ \ \ ,

/ ’

conductive film

F i g u r e 3: h-tificial interface

The metallic film can also be treated a s source.

I C :u:b r e g l e n , 1 becomes:

< ! 5 ’

-

^{7 j%rG}

c p + i w =

- & f * z

b G

-

^{GYGlds I}

’

^.

wnere:

r

is the surface CharQe density Sc is the surface of the metallic film.

This treatement can reduce the number \ c f unknowns, but complicates programming.

6 . SONE EXAMPLES

The above developrent is now applied i n PHI32 prograr 1 4 1 for the simulation in 30 space aqd 11-

& H I A X program 151 for a*iayrvetric cases.

. Tbe illustrating results of bone eramples ace Given hers.

Figure 4 shows the potential distributiop 13ver the surfaces of an insulator between tuo elactrcdes.

O f 4 surfsces, only one is conductive. I ? can be $per that the presence o f ths surface currevt distc-ts t h = poteqtia! distributi2n:

/

\

\ ,d

Figure 4: Potential distribu- tion over the surfaces of an insulat or

Figure 5 gives an e.zsmp case. I ? shows t h e effect of f i l m on the potential dis?ribu 10v:

5’

9

I

i i I

I

I

I

I

I

I I i I

I I

I

Figure 5 : Pctential d i s t r i t u -

? i o n for an a ~ i s y m n e t - ~ c ~ R E ~ I -

lator where A6 is the resistance surface and the dark-line segment the floating pctential film

46

7. CONCLUSlON

Conducting f i l m e f f e c t s a r e v e r ) e a s y t 5 b e i m p l e m e n t e d w i t h t h e b o u n d a r y i n t e g r a l e q u a t i o n method u s i n g a p a r t i c u l a r b o u n d a r y condition.

T h i s a p p r o a c h a l l o w s f a s t s u b s e q u e n t a n a l y s i s t o r d l f f e r e r i t p o s i t i o n s a n d r e s i s t d n c e s of t h e L J n d u c t i n g f i l m s . I n a d d i t i o n , t h e i n f l u e n c e o f t h e m,etallic t o i l s w i t h floating p o t e n t i a l c a n b e easily , L m u l a t e d .

REFERENCES

[ ! ! ?!.!.!?i!iing s n d J.T.Stcre: " C o n s i d e r s t i c n zf t h e e f f e c t of po!!uticn on % h a p o t e n t i a l d i s t r i - b u t i o n of i n ~ u l % t i n g s y s t e m s " - I E E ^{P r c c} Vol. 1 1 5 , 1561-1665 - 1968 -

C21 5.M. S a d o ~ i c : " M 2 c r o E l e m e n t s 1'1 !he F i n i t e E l e m i n t M e t h o d - s p p l i c s t i o n ti. t h e h i g h - v c ! t = . p p

i n s u ! ~ t i r ! g a y s t e m " - IEEE ?-MAC- l8!2, 519-523 - M a r c h 1 9 8 2 -

[:I M . J . Fh?n a n d P . H . Pla,?ander: "Charge s i m u l a t l n n r! c d e 1 I ng o f p r a c t 2 c 5 ! 2 n s u 1 a t o r ge cm e t r 1 e E " - lEEE ?-E1 ! 7 / 1 , 325-352 - f!u,~ust 19512 -

[ 4 1 L . FliccIa5 and L. K r a h e n b u h l : "PHI30: 3 g r z p h i c i n ? r r a c ! i v e p a c k a g e f o r 3D fields c a m p u t d t i c n " -

P r c c . of t h e 2 n d n c u n d a r y E!emen? Teihnc!cp;

C o n f e r e n c e I K . 1 . T . V S A ! p p 33-44 - Ccmp:??ticn3!

A e c h s n i c s P u b l i c a t i c n s , June 1986 -

1 5 1 L . Y r a h e n b u h l s n d A . N i c o l a s : " A ' i 4 y w e t r i c S c r m u l * ? i o n for E I E m e t h c d in s c a 1 . r p c t e n ? i e ! pr-cblens" - IEEE T-MFlE 518/8, 2361-2365 -

N o v e m b e r 1935 -