A Charge Simulation Method for the Calculation of High Voltage Fields

A CHARGE SIMULATION METHOD FOR THE CALCULATION OF HIGH VOLTAGE FIELDS

H. Singer H. Steinbigler P. Weiss

Technical University, Munich, Germany

ABSTRACT satisfactory accuracy, and therefore digital com-putation is necessary. The method is now illus-

A numerical method for the computation of trated with suitable examples chosen for the com-electrostatic fields is described. The basis of putation of electrostatic fields with one or morethe method is the use of fictitious line charges dielectrics.as particular solutions of Laplace's and Poisson'sequations. Details are given of a digital computer BASIC PRINCIPLEprogram developed for field calculations by meansof this method, and its application is illustrated For the calculation of electrostatic fields,by practical examples involving two- and three- the distributed charge on the surface of conduc-dimensional geometries. tors is replaced by n line charges arranged inside

the conductors. In order to determine the magni-INTRODUCTION tude of these charges, n points on the surface of

the conductors (contour points) are chosen, and itThe calculation of electric fields requires is required that at any of these points the poten-

the solution of Laplace's and Poisson's equation tial resulting from the superposition of thewith boundary conditions satisfied. This can be charges is equal to the conductor potential Oc:done either by analytical or numerical methods. nIn many instances, physical systems are so com- E p - ocplex that analytical solutions are difficult or . j iimpossible, and hence numerical methods are com-monly used for engineering applications. The a-vailable numerical methods are normally based on where Qj is the discrete charge and pi the asso-difference or integral concepts. Many papers havebeen published on solution of Laplace's equationsby finite difference techniques2.-9. The other ap- The application of this equation to the nproach to the solution is the use of integrals of contour points leads to a system of n linear equa-Laplace's or Poisson's equation either by using tions for the n charges:discrete charges9-.14 or by dividigg the electrode [p]surfaceinto subsections of chargesl *1 . The meth- cod described in this paper is based on the conceptof discrete charges2*..22. It proved to be suc- [ Then it must be checked whether the calcu-cessful for many high-voltage field problems. It lated set of charges fits the boundary conditions.is very simple and it is applicable to any sys- For instance the potential in a number of checktem that includes one or more homogeneous media. A points located on the boundary can be calculated.special advantage of this method is the good ap- The difference between these potentials and theplicability to three-dimensional fields without given boundary potential is a measure for the ac-axial symmetry and to space charge problems. curacy of the simulation. Further check possibili-

ties will be described in a later section. If theIn this method, the potentials of fictitious coincidence between the actual conductor surface

line charges are taken as particular solutions of and the corresponding equipotential surface isLaplace's and Poisson's equations. Physically the sufficiently accurate, the electric fields at anydistributed surface charges are replaced by dis- point can be calculated analytically by superposi-crete fictitious line charges. These charges are tion.placed outside the space in which the field is tobe computed. The magnitudes of these charges have In many cases the electrostatic field betweento be calculated so that their integrated effect a system of conductors and an infinite plane withsatisfies the boundary conditions exactly at a se- ground potential is of interest. This plane can belected number of points on the boundary. As the taken into account by the introduction of imagepotentials due to these charges satisfy Laplace's charges.or Poisson's equation inside the space under con-sideration, the solution is unique inside that The basic principle described above is wellspace.. known in field theory . Together with suitable

ways of discretisation, this known principle formsBecause of its discrete nature the charge the basis of electric field computation in two-

simulation method requires the selection and and three- dimensional systems as presented in theplacement of a large number of charges to achieve following sections.

TWO-DIMENSIONAL FIELDS

The simulation of the charge on the surfaceof a conductor by line charges of infinite lengthis a known principle for the calculation of the e-lectrostatic fields of circular cylinders. Par-ticularly for- the calculation of bundle conductorssome methods were developed on this basislO- *13.

T 74 085-7, recommended and approved by the IEEE Transmission & Dis-tribution Committee of the IEEE Power Engineering Society for presentationat Tedsrtsto fsraecagsbnthe IEEE PES Winter Meeting, New York, N.Y., January 27-February 1, 1974.TedcrtstoofufaehrgsbinManuscript submitted August 28, 1973; made available for printing December finite line charges also can be applied to the4, 1973. calculation of the two-dimensional field of arbi-

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trarily shaped conductors. As an example, themaximum gradient of the electrostatic field be-tween a rounded strip conductor and an earthed 5plane is calculated by the use of infinite line 4- 4charges perpendicular to the x-y plane (Fig. 1).The arrangement of the charges and the contourpoints in the rounded part of the conductor isshown in Fig. lb. Aspects for the proper arrange-ment of contour points and charges are given in a 2later section.

The potential coefficients of infinite linecharges are defined by the expression23 o s

1 /(y + yj)2 + ( x - xj)2 0 5 10 15 20cmp - .ln,

i 271E 1( y) x -x )2Fig. 2. Maximum gradient Emax related to the aver-

with the permittivity e and the notation of age gradient U/s of tht field between aFig. lb. This expression also includes the parts strip conductor and a plane as a functionof image charges for the representation of the of the distance s (Fig. 1).earthed plane. Since the line charges are of in-finite length, the quantities to be determined are FIELDS WITH AXIAL SYMMETRYcharges per unit length Aj. After the check of theboundary conditions, the x- and y-components of For fields with axial symmetry, the applica-the field strength at any point (x,y) can be cal- tion of toroidal line charges (ring charges) cen-culated by means of the following relations: tred on the axis of symetry is a very effective

way of discretisation24. Straight line charges offinite length located along the axis of symmetry

n are also used. Both types of charges have a con-A. x-x x-x. stant charge density. This charge simulation tech-

Ex ' i[ a nique is illustrated by the arrangement of ring.1 22e ( ) t 2 2 2J and straight line charges shown in Fig. 3 for the

(y-yxj+vx-x (y+y)2+(x-s ) calculation of a sphere with a cylindrical shank.

n r zAJ[ y-y y+y.Z2

Fig. - 2 ( . I!he1axim gStraight Line charge.' 2jrs Yy)2(_ 222-tweenhe(xonutoraxdthplan (yoynt x-x Zj I

P (r,z)The result of the calculation is shown in

Fig. 2. The maximum gradient Ema in the field be- ring chargetween the conductor and the plane(point A, Fig.1a)is plotted as a function of the distance s. It is Zirelated to the average gradient U/s, where U isthe voltage between the conductor and ground. r

Fig. 3. Arrangement of charges for the calculationty of a sphere with a cylindrical shank.

el -t- T Y /: (xi . yj ) Using the notation of Fig. 3, the potentialP (P1xy) coefficients p; and the components of the field

strength Er ani E. can be calculated with the fol-- 2 -- 10 X *.t 1*@ / lowing expressions20'23; image charges are again

used to represent the earthed infinite plane.A

Ring charges:a) b)b

] .x 1 2 [ ~~~~~~~~~~~~~~~K(kj)K(k)i* ine charge E a1 Of 2x contour point

Fig. 1. Dimensions and arrangement of charges for Er - R 1 %r2-r2+(z-zj)2].E(ki)-P1.K(ki)the calculation of the field between a r - 2strip conductor and a plane; dimensions in 3in1a1 1cm. [r~-r2+(z+z )2] .E(k2)-jt3*K(k2)

a 2.P2 }

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n

E _Q; 2 { ( (I) ( iz+) E(12)1 4= r ot1.131 a2.p2 1002

ar1 - (r+rj)2+(z-.zj)2 a2 - W(r+rjX)2+(z+zj)2 I 30*4000

-*/(r-rj)2+(z-z.)2 ' P2 - (rr)2+(+)2 , 1650

2000

A 100

k1 a c1 k 2 02 610I

with the complete elliptic integrals of the first _ _ i _ _ _kind K(k) and second kind E(k). d

Straight line charges:

Fig. 4. Sphere-gap enclosed by an earthed cage;1 (z32-z+71).(z.1+z+72) dimensions in cm.

lii -z;l(zi ++2p; ln -

47M(ZJ2-zjl) (zj_z+( ).(z +Z+6 ) 'Fig.5 shows an electrode arrangement used forJ 1j2 2) the shielding of a high voltage apparatus. The po-

tentials of the grading rings are fixed at 75, 50n

- z z and 25% of the potential Xbc -U of the top elec-E_. _____Q _ j2 _ji + trode. The result of the calculation is shown in4,r(-(z-) r*7 r*J Fig. 5 for a voltage U X 1 NV. The length of theiI 32 J1 '1 1 arrows is equivalent to the magnitude of the field

z +z -z +z ~ strength, the dash-dotted circle indicates a fieldJA _ J2 1 strength of 5 kV/cm. The maximum field strength on

r.7 r. j the top electrode amounts to 5.6 kV/cm. Only ringr*;r2 r-3 2 charges were used for this example.n

J-1 4=(zj2-zj1) 71 31 72 2 1u1m 0

't1 w / r+(j2_Z)2 , 72 " ( 2U~0 20000

- r2+(z 2-z)2 , 52 ' r2+(zjj+z)2 . 200

500 --E- - 75%The application of ring charges and straight 5 75%

line charges is demonstrated by two examples. In -201the first example, the influence of an earthedcage on the field of a sphere-gap is investigated. - - 50%Fig. 4 shows a 2 m sphere-gap surrounded by a 750 r k-°oo&-- -I 1000closed cylindrical cage of 16.5 m height. The di- l ltributed charge on the sphere and on the cage is 500replaced by ring charges, the charge on the shanks 25%by straight line charges. In Table I the increasea E of the maximum field strength at a gap spacing 25011 zof 1 m is shown as a function of the diameter d ofthe cage. 4 E is related to the maximum field t itstrength without a cage at this gap spacing.

Fig. 5. Electrode arrangement for the shielding ofa high voltage apparatus; dimensions in cm(The cylindrical dielectric is used forJE ( 3)3.5 1.0 0.6 O calculations in a later section).

d (in) 10 20 50 O_________________________ HREE DIMENSIONAL FIELDS WITHOUT AXIAL SYMMETRY

The charge simulation method can be appliedTable I. Increase of the maximum field strength in with great advantage to the calculation of three-

a sphere-gap by an earthed cage (Fig. 4). dimensional fields without axial symmetry. Theprinciple of this method is explained by a simple

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example, a rod-rod gap with a trigger electrode with D (z-z )2+r+r2 and as the Legendre(Fig.6).

function of the second kind of t-e order )m -1/2, a

The original axial symmetry -of the field of so-called torus function.The symbols r. and z arethe two rods disappears because of the presence of the sae ones as in the case of constint cha4ges.the trigger electrode. Within the half spheres ofthe rods and the conical part of the trigger elec- In a similar way as it is described before, atrode ring type charges are arranged in a similar system of linear equotions is established andway to that used for geometries with axial symme- solved for Atry. The rings, however, have a variable chargedensity. For the cylindrical parts of the rods and The field strength components, obtained bythe trigger electrode, straight line charges are differentiation of the potential function withsituated parallel to the axis of symmetry of the respect to r,v and z, areelectrodes on a circle on the axis. To consider Athe asymmetry with reasonable accuracy it is suf- E . _ -_ L. cos;t,)ficient to arrange 3 or 4 lines around the circum- ur 5r 2nEference in the cross section of each electrode.

Wo _ { 2r;AL z D(2== 2_ 2r: (2r2-D2).

D2I ~+(-)2

\ @ yr a~~~~~~~~rr--2rr' D4 >z2ru @@~~~~~~~~~1 ,&-z-. - *~ -Dos)2

8-G~ ~ ~ ~ ~ LLZ2rr 2r 2?D D ~ ~ ~ ~

j *4rJr(Z-Z~~~~~~Yj )[r +(j ) 2rr; .-t(r,

2

arrngeentcros sctin tQ feldstrngt ofan Qiecio siscalcu

-I1/ of r 2e r 2rr i

6fiaqiW' 1

E V p3COS(jVI~)./PL IsZ 27rr r iA4r2r2

2 2 2

orrangement co 3e tion 2rr P -r r

known at first. For this reason the distribution Tofris divided into a constant part and several cc-sinusoidal (or sinusoidal) harmonics with unknown In principle it is also possible to arrangepeak values Aa similary to a Fourier analysis. several rings with constant charge distributionThisa charge strifution is a function of the ro- instead of a single ring with variable chhrgetatiQn angle a shown in Fig. 6 and is given by distribution. Fig. 7 shcws an example of this way

e rof discretisation, where the centers of the ringknonabcharges are placed on a circle centered on the

A(a)i a A * cospa) * ais of the electrode. Such an arrangement cannotinuoida rsinuoidlarmnicswit konIbeused in the case of toroidal electrodes.p0

nh denotes the nusber of ha onics.

The ale ofsh is not calculated by fulfill- f ting an orthogonalitwy condition as in the case of a d w theJ r

Fourier analysis, but by giving several contour /points with known potential values around theelectrodes.The total number (nH+1) of the constantcharge (p 0) and the harmonics (p- 1..nH) must Fig.7. Ring coharges of constant charge density forbe equal to the nubber of contour points 'around the simulation of three-dimensional fields.the circumference.

The effectiveness of the method for three-di-The potential coefficient of a periodically mensional electric fields is shown in two practi-

variable ring charge due to the s-th harmonic in cal field problems of high voltage engineering:any point (r,y,z) of a cylindrical coordinate sys- the influence of adjacent conductors on the fieldtem is (without image charges)21r25 of a sphere-gap, and the field dstribution near

the bundle of a three-phase overhead-line under1h D2 the influence of the tower.

i/S 2rnE 0r QPI(2rrj3 * cosWt)The first example is shown in Fig. 8, namely

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2m- diameter spheres at gap spacing of s - O.5... As the second example, the influence of the1.5 m. The computed field strengths were compared tower window on the field near the middle phase ofwith those of two isolated spheres, which can be a 735-kV three-phase overhead-line is calculated.calculated analytically. In this way the influence Fig.9 shows the dimensions of the tower and the 4-of connecting tubes was investigated. The results bundle conductor27. For the computation, the towerof these computations with the field strengths at was slightly simplified, assuming particular conethe upper and lower sphere have been represented contours, each of them with an axial symmetry asin Table II. For both arrangements the maxinmum shown in Fig. 9. The cross arm was approximated byfield strength occurs at the high voltage sphere; a cylinder. These simplifications are allowable,this field strength is reduced by the influence of as a good accuracy of results is needed only nearthe connecting tube, whereas the field at the low- the bundle conductors.er sphere increases. The greater the distance be-tween the spheres the greater is the influence of The calculations show that the maximum fieldthe connecting tube. While the variation amounts strength occurs on the upper conductors of theto 0.4 % for 0.5 m gap spacing, for 1.5 m gap middle bundle at the moment when this phasespacing the difference is around 3 %. Indeed for a reaches the maximum voltage. The maximum fieldsphere diameter of 2 m this distance is just strength is about 8 % higher than in a distance ofbeyond the range, which is foreseen by IEC for a more than 20 m from the tower (Fig. 10).measuring sphere-gap. For each distance the influ-ence on the field is greater on the earthed spherethan on the high voltage sphere. More detailedcalculations concerning the influence of ad'cent Zconductors have been done for rod-plane gaps 426J--A

-tOS9 4 ^-1074e

300 1200 -

300 7~~~~~00600~~~~~~~~~~1

; t Fig. 9. 735-kV single-circuit towe; dimensions in

- ~~~~~~~~~~~~~cm( y is the in-line distance from the

Fig. 8 Sphere-gap with connecting tubes8 and tower).earthed plane; dimensions in cm.

|| DeviBtiondof thetietdstrength

arrangement i SI°i.|8 W

4~~~.08 0--

150 1.909 -2.7 % 2 -100 1.517 -1.7 % 0 - , .350 1.199 -0.4 % 51052m

Distancy fromthetower'Earthed sphere

150 1.209 +3.4 %100 1.201 +1.9 % Fig. 10. Maximum deviation of the field strengths50 1.146 +0.4 % at the middle bundle near the tower

___________ from the field strengths without towerTb IIFe faos'=. (arrangement of Fig. 9).

Table(c) Field factorsE'vUsofasphere-gati

High volta esphere ~ ~ ~ ~ 166


ELECTRIC FIELDS WITH SPACE CHARGES the conditions for the potential gradients alongthe channel together with the charges within the

By means of the charge simulation method it electrode. Thus the matrix of the potential coef-is also possible to calculate electric fields ficients [p] is enlarged.with space charges. These calculations are mainlyused to investigate the physical principles of CALCULATION OF TWO-DIELECTRIC ARRANGEMENTSbreakdown mechanism. There are two possibilities:known or unknown space charge distribution. In a dielectric, dipoles are re-aligned by

the electric field. In the interior, they compen-The first case may be illustrated by an ex- sate each other; but, on the surface of the die-

ample28involving a cloud of ions travelling in a lectric they have the effect of a net surfacefield . The ion cloud has known charge density charge30"1 .Therefore, in the digital computationand can be approximated by point or ring charges. of electrodes, a dielectric boundary can be simu-Ring charges can be used with greater advantage lated by discrete charges. There are only two im-as they cover a greater area than a point charge, portant differences from the previously consideredand therefore fewer ring charges than point situations:charges are needed for the simulation of an ion (, v~~~~~~~~~~~~1)In general the dielectric boundary does notcloud. correspond to an equipotential surface.

Because of the field-induced motion of the (2) It must be possible to calculate the electricions, the calculation is time-dependent, and the field on b o t h sides of the dielectricion motion must be considered by a step-by-step boundary; this is necessary for the formationprocedure. After each step the field strengths at of the system of equations.the places of the space charges are computed anew,and all the space charges are shifted according to As shown in Fig. 11, a simple example with athe amplitude and the direction of the field small number of discrete charges is chosen to ex-strengths. This calculation procedure is continued plain the method. At the electrode, there are nEuntil the ions reach the opposing electrode. As contour points and charges, nED of them are onthe space charges are known, it is only necessary the side of the dielectric (No.1) and nE-nED areto calculate the charges required for simulating on the side of the air (No.2,3). These nE chargesthe electrodes. The potentials of the contour are valid for the calculation of potentials andpoints result from field strengths in both media, i.e. for the die-

lectric and for the air. At the dielectric bounda-[P] [Q] + [PBI[JQ. ] [1PcI ry there are nB contour points (No.4,5) with nB

charges in the air (No.4,5) - valid for the die-with [p.] as matrix of the potential coefficients lectric - and nB charges in the dielectric (No.6,of the space charges and [Q.] as vector of the 7) - valid for the air. In total there are nc =space charges. As [p5] and [Qs1 are known they can n B+ nB (=5) contour points and nq nE + 2 nBbe multiplied and brought to the right side of the (=7) charges.equation system and subtracted from the potentials[0c] . In this way the right side becomes a vec-tor again.

[p ] Q[] [cl -1[P] [Q] x contour point .3

*charge ivector 3he

Thus it is not necessary to enlarge the matrix of *4 2the potential coefficients and the calculation ofthe charges within the electrodes is done in the *5 1same way as without space charges. The potentialof any point in the field is then calculated by electrode

n n5 07Pk * Qk + 2 Psk sk -AIR- -ODIELECTRIC-

k=1 k-1

no denotes the number of the space charges, n the Fig. 11. Discrete charges at an electrode and at anumber of contour points and charges which simula- dielectric boundary.te the electrodes. The field strength is calcu-lated in a corresponding way. The system of equations required for determi-

nation of the simulation charges is formed by de-If on the other hand the space charge distri- fining the boundary conditions which must be sat-

bution is unknown, the space charges have to be isfied:-calculated from physical conditions. Thio case isillustrated by fields associated with discharge (i) the potential of the contour points of thechanels, e.g. the simulation of a "leader" chan- lectro4e on the side of the dielectricne129 in the breakdown of a long air gap. Experi- (No1)mental data suggest that a reasonable physical m cmodel is given by the assumption that a constantpotential gradient occurs along the channel, forinstance 1 kV/cm. Assuming this, the "leader"' is -nE nE+nBconsidered as a quasi-electrode. At i-ts boundary, .? + sR=some contour points are given (with different po- j=1 i i j-nE+1l P fctentials), and inside as many charges as contvourEpoints are arranged. Straight line charges are (i...3) (4...5)most suitable for the simulation of long channels.These unknown charges are determined by fulfilling

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(2) it must be also 0c on the side of the air(No.2,3), E Et. a

Et ' *

nE C 140% Pi + Q * pj . .1T E-1 Q" P+ +1 % 'a =~cJ /B Cr 2.2

(1.3) ~~(6.-7) 14

(3) the potential of the contour points on the di- Cr 6.0electric boundary is uinkiown, but for eachpoint it must be the same in the air ( OA) andin the dielectric (ID). Thus for the air-die-lectric boundary A - D, and hence it may beshown that22 932.

nE nE+2nB nE nE +nB

-jP + 2 -pJ p q -pi0 Vii%P . 1 %P -~ Il piP3 i~i- 1 .'nE+nB+l i-1

(1 5) (673 (1*~~~~~~~~1-3) ...... ..(4--5)0.6or simplified, Et. tangential fieLd strengthnE+nB nE+2nB U - voltage

a. Length of the dieLectric boundary (1000cm)- sa +1 J Pjn+1 J z- height at the dielectricJ'nE+l LJmn+nB+E~~~~~~~(-7 0.2-

(4) pj has been defined as the coefficient consid-ering the effect of the charge j on the poten- °tial at a given contour point. In the sane way 0 200 40 600 800 1000cmfj is defined as the contribution of thecharge j to that component of the field vec- Fig. 12. Normalised tangential field strength Et'tor, which is vertical to the dielectric along the dielectric boundary of the ar-boundary in a given contour point. Then, at rangement shown in Fig. 5.the contour points of the dielectric boundary,the normal field strength in the air must beEr times greater than in the dielectric, thatis, tC ECr E'

nE nE + n.B nE nE+2nB 1 2.19Cr (EQ% fj + i: Qj.fj)a Q..f.÷+ Q.f.l _ 2 2.62J~~~

j-1 j=nE+1 j=1 Jin +n+l 3 2.86(1 -.3) (4..5) (1 ..3) (6..7) electrod 200 5z

Or, 10 y | 773.31r ~ ~ ~ ~ ~ ~~~~~~~~~<l 3.46

nE nEnB nE+2nIB Mr E

J-1 j-nE+l J-nE+nB+l(1--3) (4.-5) (6..7)

Thus nE + 2 * nE C(- 7 ) linearequations are Fig. 13. Sphere electrode with Table III.diplectric slab. Field factorsgiven for the calculation of the same number of of Fig. 13.unknown charges.

As an example a dielectric cylinder is in- It is also of interest to mention that sys-serted into- the system shown in Fig 5 (broken tematic application of this method led to the dis-lines). Fig.12 shows the tangential field strength covery of a new effect in electrostatic fieldat the dielectric boundary with the dielectric conceRing electrodes partially embedded inconstant cr as a parameter. a dielectric .

Another example is given in Fig. 13- which MPPLICATION ASPECTSshows a sphere electrode with a dielectric slab. o h ot efcieapiaino hThe results of the calculatiozi are presented in metod the questio offaesuitabearrliatingmnofthT!able III. The method described above also was ap- ejhdthehares sndiontourapointsisoiragmportancepliedtthieldn calecultione of Vthesfiengstrengthrma A practical criterion is obtained by the defini-erg3.iedn lcrdso FYtsigtasom tion of an assignment factor fa = a2/a1 with -the

distance a1 between two successive contour points

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and the distance a2 between a contour point and necessary for the calculation of the examplesthe corresponding charge (Fig. 14a). For curved described in the previous sections. For the com-contours the distances between the charges should putations a digital computer TR 440 was used,not be too small, and this necessitates the formu- which is comparable with a digital computer CDClation of a curvature criterion (radius q ) for 6600. The electrode arrangement shown in Fig. 13such charges. Based on the geometric mean of a1 was calculated with 15 charges for the electrodeand a2 the following expression with the notation and 40 charges for the dielectric. Without the di-of Fig.14 b was derived: electric the computation time amounted to 4 s,

with the dielectric to 13 s. The arrangement showna-f1)2 -r a1 in Fig. 4 was calculated with 58 charges, 42 of

i/o a r +a r them were used for the spheres and the shanks and16 for the cage. The computation of this example

qi is valid for convex curvature,qo for concave needed 21 s. The amount of computation for three-curvature. For dielectric boundaries, both cases dimensional fields is higher. For the computationmust be used accordingly. of the three-phase overhead-line shown in Fig.9, a

total of 259 charges was used, 172 for the simu-lation of conductors and the ground wires and 87for the tower. The computation time amounted to

/ 594s.--_. j- { so A comparison with the finite difference method

l - -.4< shows that the charge simulation method leads to

a) >9-t- b) I- -2-a-7_1 shorter computation times in many geometries usedr / - - ~ in high voltage technology. This is a consequence

F . 0*1 _~-\~-t,,r of the fact that in high voltage apparatus curvedsurfaces are generally preferred to sharp edges.As an example, the electric field of a sphere-gap

x contour point . charge similar to that shown in Fig. 4 was calculated withboth methods. The computation time needed for thecharge simulation method was only 25 % of the com-

Fig.14.To the definition of the assignment factor. putation time of that for the finite differencemethod. The error of the field strength between

Experience shows that the assignment factor the spheres was less than 1 % in both cases. Afa should be between 1.0 and 2.0. The accuracy of successive overrelaxation technique495 was usedthe calculation depends on the choice of the asin the finite difference method.signment factor and the density of the contourpoints. In the areas of interest, the accuracy There are further advantages of the chargecan be improved by an increase of the density of simulation method compared with the finite differ-the contour points. The following criteria can ence method, such asbe used to check the accuracy of the simulation:(1) potential of various points on the surface of (i) the field strength can be calculated analyti-

the electrode(2) continuity of the potential on the dielectric (2) it is not necessary that the field region is

boundary limited by a closed boundary;(3) continuity of the tangential component of the (3) the computation of three-dimensional fields

field strength on the dielectric boundary without symmetry is possible with reasonable(4) relation of the normal components of the field amount of computation.

strength on the dielectric boundary.An additional check is the derivative of the po- CONCLUSIONStential gradient perpendicular to the surface of Cthe electrode, especially for areas of the elec- As shown in this paper the charge simulationtrode with a small radius of curvature. This de- met ho wn itale waper the solutionimulatiorivative divided by the gradient must be equal to method is a suitable way for the solution of manythe curvature at this point35. A measure for the electric practical field problems. The presentedaccuracy of the calculation is the "potential examples give an idea of the variety of possibleerrot" in various check points on the surface of applications of this method. Of course furthertheelectrode between two contour points. This developments are inherent. So the detailed compu-the el ro de fin as the di nce be- tation of arrangements with space charges is an"potential error" i's defined as the difference be- inestgapctfr nimovetof hs

tween the known potential of the electrode and the interesting aspect for an improvement of thiscomputed potential. Experience shows that the er- method.ror of the gradient is up to ten times greaterthan the corresponding potential error. Thereforethe potential error should be less than 1 0/00 inan area of the electrode if a field strength accu-racy of 1 % is desired in this area. The practical The authors acknowledge with grateful thankslimit for the accuracy of the simulation of elec- the keen interest shown in this work by Prof. H.trodes is given by the manufacturing tolerance of Prinz, Director of the Institute of High Voltageconductors. In the same way the accuracy of the Technology and Power Plants, Munich.simulation of dielectrics has its practical limitin the accuracy of the determination of dielectric REFERENCESconstants.--

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In this section some data are presented about 2 G. Shortley, R. Weller, P. Darby, E.H. Gamble:sthe nulmber of charges and the computation time Numerical Solution of Axisymetrical Problems,

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3 R. V. Southwell: Relaxation Methods in Engi- of Electric Field Strength. Electra (1972),3 F. V Southell: Rlaxatin Methds ~ ~No.23, pp. 62-69

neering Science. Oxford Univ. Press 194927 J. Aubin, D. T. Mc Gillis, J. Parent: Compos-4 R. S. Varga: Matrix Iterative Analysis. ite Insulation Strength of Hydro-Quebec 735-kV

Prentice - Hall, Englewood Cliffs, N. J. 1962 Towers. Trans. IEEE, pt.3, 85(1966), pp. 633

5 K. J. Binns, P. J. Lawrenson: Analysis and -648Computation of Electric and Magnetic Field 28 K. Feser, H. Singer: fier den Durchschlag ausProblems. Pergamon Press, Oxford 1963 der Glimmentladung. ETZ-A 93(1972), pp. 36-39

6 G. E. Forsythe, W. R. Wasow: Finite Differen-ce Methods for Partial Differential Equations. 29 "LeasRenardi4res Group": Research on Long AirWiley, New York 1965 Gap Discharges at Les Renardidres. Computation

of Electric Field with Leader Channels and7 D. F. Binns: Calculation of field factor for Space Charges. Electra (1972), No.23, pp.120-

a vertical sphere gap, taking account of sur-* 124rounding earthed surfaces. Proc. IEE 112(1965)pp.1575-1 582 30 R. Becker, F. Sauter: Theorie der Elektrizitdt

8 R. H. Galloway, H. Mal. Ryan, M. F. Scott: vol.1, Teubner, Stuttgart 1962Calculation of electric fields by digital 31 A. Roth: Hochspannungstechnik. Springer, Wiencomputer. Proc. IEE 114(1967), pp. 824-829 1959

9 H. Prinzs: Hochspannungsfelder. Oldenbourg, 32 P. WeiB: Berechnung von Zweistoffdielektrika.Miinchen-Wien 1969 ETZ-A 90(1969), pp. 693-694

10 E. Clarke: Three - phase Multiple - conductor 33 J. Moeller, H. Steinbigler, P. WeiB: Feldstar-Circuits. Trans. AIEE 51 (1932), pp. 809- 823 keverlauf auf Abschirmelektroden fur ultrahohe

11 A. S. Timascheff: Field Patterns of Bundle Wechselspannung. Bull. SEV 63(1972), pp.574-Conductors and their Electrostatic Proper- 578ties. Trans. AIEE pt.3, 80(1961), pp.590-597 34 P. WeiBs: Feldstlrkeeffekte bei Zweistoffdi-

12 M. S. Abou-Seada, E. Nasser: Digital Computer elektrika. Bull. SEV 63(1972), pp. 584-588Calculation of the Potential and its Gradi- 35 J. Spielrein: Geometrisches zur elektrischenents of a Twin Cylindrical Conductor. Trans. Festigkeitsrechnung. Arch. Elektrotechn. 4IEEE, pt.3, 88(1969), pp.1802-1814 (1915), pp.78-95 and 5(1917), pp.244-254

13 M. P. Sarma, W. Janischewskyi: ElectrostaticField of a System of Parallel Cylindrical Con-ductors. Trans. IEEE, pt.3, 88(1969), pp.1069-1079

14 M. S. Abou- Seada, E. Nasser: Digital Computer DiscussionCalculation of the Potential and Field of a L. 0. Barthold (Power Technologies, Inc., Schenectady, N. Y. 12301):Rod Gap. Proc. IEEE 56(1968), pp. 813-820 The authors are to be commended for the significant work they have

15 J. Higgins, D. K. Reitan: Calculation of the done and for a fine documentation of that work for the benefit ofCapacitance of a Circular Annulus by the Meth_ otherslEEE members.od of Subareas. Trans. AIEE, pt.1, 70(1951 - It would be interesting to hear the authors' comments on the

9-93 T70(195) relative complexity and time required in setting up a problem usingpp. 9-933 the method they have developed, as compared with more traditional

16 A. Kessler, A. Vlcek, 0. Zinke: Methoden zur electrolytic tank approaches. It would seem that reducing a problemBestimmung von Kapazitaten unter besonderer to a form amenable to digital computation might be as great or greaterBerUcksichtigung der TeilflAchenmethode. Arch. than construction of models where a convenient electrolytic tank waselektr. Ubertrag. 16(1962), pp.365-380 available. Has thought been given to the application of digitizers toelektr. ttbertrag.161962 P365-380simplify the problem descriptions for digital solution?

17 R. F. Harrington: Matrix Methods for FieldProblems. Proc. IEEE 55(1967), pp. 136-149 Manuscript received February 6, 1974.

18 D.Pfluigel: Berechnung von Kapazitaten und Fel-dern zwischen Leitern mit geschichtetem Dielek- H. Singer, H. Steinbigler, and P. Weiss: The authors wish to thank Mr.trikum nach der Teilflichenmethode.Zeitschrift Barthold for his comments and for his discussion of the paper.fiir angewandte Physik 23(1967), pp.86-89 The authors are aware of the fact that not only the computation

time but'also the time necessary to prepare a problem for the comp-19 R. F. Harrington: Field Computation by Moment utation must be regarded in order to judge a computation method with

Methods. Mc Millan, New York 1968 respect to its economy. Therefore emphasis was laid on the develop-

20 H. Steinbigler: Anfangs3feldstarken und Ausnut- ment of an auxiliary program which calculates the coordinates of con-zungsfaktoren Arotationssyf etrischer Elektro- tour points for the case that the contour consists of straight lines ordenanordnungen in Luft. Doctoral Thesis TH parts of circles. The coordinates of the charges are calculated auto-

matically for all parts of the contour. The application of a digitizerMunich 1969 would reduce the time of preparation too, but we did not have the

21 H. Singer: Das Hochspannungsfeld von Gitter- possibility until now to use such a device for the purpose of electricelektroden. Doctoral Thesis, TH Munich 1969 field calculation.

A comparison with the electrolytic tank depends on the kind of22 P.WeiB: Rotationssymmetrische Zweistoffdielek- the field problem. In some cases the analogue method may have ad-

trika. Doctoral Thesis TU Munich 1972 vantages in comparison with the field computation, if an automatic

23 F. Ollendorff: -Potentialfelder der Elektro- tank is available. But in general the computation method needs shortertechniSpriger, erlin1932 preparation times than the construction of a model, especially if the co-tehi*Srne,Bri ordinates of contour points and charges are determined automatically.

24 H. Steinbigler: Digitale Berechnung elektri- For some cases -for instance the tower shown in filg. 9 of the paper -scher Felder. ETZ-A 90(1969), pp. 663-666 the application of the electrolytic tank method may lead to difficulties,

especially with respect to the accuracy.25 H. Singer: Das elektrische Feld vron Polycon-

elektroden. Bull. SEV 63(1972), pp. 579-583 Manuscript received April 22, 1974.

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