IEEE Transactions on Electrical Insullation Vol. EI-17 No. 3. June 1982

Low FREQUENCY COMPLEX FIELDS IN POLLUTED INSULATORS

E. Asenjo S. and N. Morales 0.

Electrical DepartmentUniversity of Chile

Santiago, Chile

ABSTRACT

A general method for the solution of ac lowfrequency electric fields in insulating systems

is proposed. The method is based on decouplingMaxwell's equations, and their solution can be

implemented analytically or numerically. Thefinite difference numerical method is easilyimplemented to solve the field in pollutedinsulators.

LIST OF SYMBOLS

= complex electric potentialD = complex electric flux density vector

E = complex electric field strength vector

Et

En

= tangential component of E

= normal component of t

H = complex magnetic field strength vector

p = complex volume charge density

qS = total complex surface charge densityqsd = complex displacement surface charge densityqSC = complex conduction surface charge densityX = angular frequency£ = permittivitya = conductivity

P = permeabilityy = a + iwE

rc = resistivity

INTRODUCTION

The electric field in high voltage ac insulatingsystems is normally solved by neglecting the conductioncurrent, because we>>o in all the intervening dielectricmaterials. The problem is reduced to finding a solu-tion to Laplace's equation V24=-, imposing the boundaryconditions, i.e. the known potentials of conductors

with the supplementary boundary conditions. The latterare the conservation of the normal components of vectorD and the tangential components of vector E across thedielectric interfaces. However, there are insulatingsystems in which a is of the same order of magnitudeas w6, at least in some regions. Examples are pollutedinsulators, insulating systems with resistive coatingand, in general, heterogeneous dielectrics. In suchcases it would be convenient to have a general methodof solution without requiring an exact solution toMaxwell 's equations.

In this work an approximation to decoupling theelectric field from the magnetic field is proposed,concluding with a method for solving the low frequencyac complex electric field by procedures similar to thoseused with electrostatic fields. The key to this methodis in the conditions which the normal components ofvector ; must satisfy across the dielectric interfaces.

Reviewing the literature on heterogeneous dielectrics,we find that the same method has been proposed by Wagner[1] who obtained the conditions on the dielectric inter-faces by imposing the net surface complex charge densityin the initerface be zero. In contrast with Wagner'sprocedure, we obtain the condition on dielectric inter-faces from an approximation to Maxwell's equation and,as a consequence, the vanishing of the net surface com-plex charge density on the dielectric interface isobtained.

The method has also been proposed by T. Takashimaet al. [2] who showed that "there exist dual relation-ships between a complex field due to an alternatingcurrent source in a conducting medium and an electro-static field due to a charge in a dielectric medium".Furthermore, they verified the method experimentally.

c18-9 t.67/82/06)OQ-026U2b$OQc'. 75 C 1982 IIEEE

-I-.26--12

Asenjo and Morales: Low--frequency comple- fields in polluted insulators

We apply the method to calculate the complex acfield in polluted insulators. These constitute veryheterogeneous insulating systems, containing caseswith wc>>a as well as systems with o>>wE.

LOW FREQUENCY FIELDS IN INSULATING SYSTEMS

It is worthwhile reviewing the possible methods forsolving the ac low frequency electromagnetic field ininsulating systems. It is well known that the generalsolution can be obtained by solving Maxwell's equa-tions, but this would be a formidable task due mainlyto the coupling between the electric and magneticfields.

Maxwell's fields equations

VxE = ipH (1)

V* -c p (2)

V x =a + iwcE (3)

V ^ , =O (4)

can be decoupled in the following instances

(a) When the frequency is low and o>>wE, it is possibleto approximate the time-varying field by a dc field(w=0). In this case, Maxwell's equations are reducedto the static approximation.

(b) In high voltage ac insulating systems, normallywc>>» and the field is solved as in the electrostaticcase. This implies neglecting a relative to wE in (3),and also assuming wpH=O in (1), which results in de-coupled fields. Since the term proportional to w isneglected only in the second member of (1) and not inequation (3), this solution could be called a quasi-static approximation. The assumption wpP=O is justifiedbecause in insulating systems the magnetic field issmall, and because low frequency is being considered.In this work we shall call this solution the "quasi-static lossless" approximation.

(c) In some high voltage heterogeneous dielectricsystems 0%we. Reasoning as in the quasi-static losslessapproximation, the decoupling is obtained by assumingpH=O in (1). This solution will be called the "quasi-static lossy" approximation.

From now on, this work will consider only systems inwhich p/O only in the interfaces oX conductors and di-electrics. In all previous cases E is irrotaXional andtherefore an electric potential X, such that E = -V¢,can be defined. Then, from (2), V24=o is obtained forall points not in an interface.

From the above considerations, the three problemsare reduced to finding a solution to Laplace's equationconstrained by the boundary conditions. The solutionmust also satisfy the supplementary boundary conditionsin the dielectric interfaces. One is obtained from(1) as the condition that assures the conservation ofthe tangential components of E. The other cannot beobtained from (2), because the surface charge densityis not known. But these last conditions can be ob-tained from (3) as follows.

For the static and quasi-static lossless approxima-tions, respectively, the supplementary boundary condi-tions obtained from (3; are the conservation of thenormal components of oa and the conservation of thenormal components of eE in the dielectric interfaces.Eqs. (2) and (3) applied in the dielectric interfacespermit the surface charge density to be calculated inthe static approximation (w=O), as q. = 01 Eln - a2 E2n-And for the quasi-static lossless approximation,(w#0 ; o=O), Eqs. (2) and (3) imply that the surfacecharge density on the dielectric interfaces is zero.

For the quasi-static lossy approximation, from (3),is obtained

V * V x H = V . [yE' + iwcE] = 0 (5)

Then we have that, in the dielectric interfaces, thenormal components of a/I + iwsE are preserved. The dis-placement surface charge density qSD in the dielectricinterfaces may be found using Eq. (2). In Appendix 1it is shown that for this case (O/O ; o#0) the totalcomplex surface charge density q. = 0.

In summary, we can say that the quasi-static lossyproblem is reduced to finding a solution to Laplace'sequation which satisfies the normal boundary conditions(Dirichlet, Neumann or mixed) and the supplementaryboundary conditions, as specified. To solve the lossycase it is possible to use all the methods applicableto the quasi-static lossless (electrostatic) case(analytical or numerical) by just using (a + iws) in-stead of c. The principal difference is that in thequasi-static lossy problem, the potential is a complexscalar instead of a real scalar.

Once the electric field is obtained, the magneticfield can be calculated from (3) and (4) which are ofthe static type. Nevertheless, in practical applica-tions the solution of the magnetic field is notnecessary.

FINITE DIFFERENCE EQUATIONS FORPOLLUTED SURFACES

The procedures of finite difference equations andfinite elements are well suited to solve the field inregions which contain dielectrics polluted with par-tially conducting surfaces. We have implemented thefirst method to solve Laplace's equation in two dimen-sions and three dimensions with at least one symmetryaxis. This consists of superposing a rectangular gridover the whole field, setting up a finite differenceequation for each node, and solving the resultant setof simultaneous equations. For the points outside ofthe dielectric interfaces these equations are exactlythe same as those in the electrostatic problems. Forthe points in non-polluted dielectric interfaces theequations can be obtained from the corresponding elec-trostatic ones [3], using a+iwE instead of c.

Now we will consider the axisymmetric equation fordielectrics coated with partially conducting surfaces.This is derived in Appendix 2, and the symbols are ex-plained in Fig. 1.

The general grid configuration in Fig. 1 shows eightdielectrics and eight partially conducting surfaces(regions of infinitesimal thickness), each one withspecific characteristics. These dielectrics and sur-faces converge to a common point 0, whose potential isgiven by

26 3.

IEEE Transactions on Electrical Insulation Vol. EI-17 Nol3ilc. June 1982

(6)i=8 8=

Vi=l ii) Vi-El i

where the coefficients Ci, i = 1,2,...,8 have thefollowing expressions:

h1Cl = { 2yCi1 + yl(l + h2/4R) h2 + Y8(l - 4/4R) h 4

h2C2 = {2yC262 + Y2h1 + y3h3} (1 + h2/2R)

h3c3 = {2yC363 + y4(1 + h2/4R) h2 + Y5(1 - h4/4R) 4)

h4C = 12yC464 4- y6h3 + y7h1} (1 - h4/2R)

(7)

h5c5 = {2yC565 (1 + h2/2R)}

h6C6 = {2yC666 (1 + h /2R)}

h7C7 = {2yC767 (1 - h4/2R)}

h8C8 = {2 C868 (1 - h4/12R)}

The point 0 indicated in Fig. 1 must belong to thedielectric interface within which lies the contamina-tion layer of partially conducting dielectric. More-over the point 0 is located at the interface of twolayers, which in general are of width 6i«<hi and6<<hj. The configuration in Fig. 1 permits, with twoline segments at a time taken from hl to h8, the ad-justment of any profile of an axisymmetric insulator.

In all practical applications, at each breakingpoint, only two of the eight partially conducting re-gions considered in Fig. 1 are necessary. In suchcases either some of the coefficients Ci are missing(i = 5,6,7,8) or their expressions are of a simplerform than that shown in (7) (i = 1, 2, 3, 4).

I A

7 4 ~ ~ ~ ~~~~ ~ ~~~~~~~ ; 8Rh 7 Y74

7 - '7 J4 8 8

Fig. 1: Grid configuration for generaZ probZem

VERIFICATION OF THE METHOD AND APPLICATIONTO POLLUTED INSULATORS

The set of simultaneous complex equations, which re-sult from the application of finite difference rela-tions to each point in the grid, are solved by themethod of successive overrelaxation (sor) [4]. Theoptimum accelerating factor is found by Carre's method[5] during the process. The iterations are interruptedwhen the difference between two successive potentialvalues is less than 0.1% for each point in the grid.In all cases, this optimum accelerating factor is foundto be complex and to have a very small imaginary part(positive or negative) compared with the real part [6].

The numerical method was tested for accuracy by eval-uating simple examples which can be solved by circuittheory procedures. These are plane and cylindricalcapacitors with several layers of heterogeneous dielec-tric materials. At the edge of these capacitors a non-uniform field will occur as a consequence of the edgeeffect. Away from the edges the field can be verifiedby lumped parameter circuit methods.

One example corresponds to the cylindrical capacitorshown in Fig. 2. A pronounced edge effect was pur-posely included in order to have a nonuniform field inthat region. But the ratio between the length I andthe radius a is large (10) so that the edge effect isnegligible in the middle part of the capacitor. Inthis region the field can be verified assuming thatthe cylindrical capacitor is infinitely long.

¢ =100 °/.

m= 0 0/0

Fig. 2: CyZindricaZ capacitor of the exampZe

= O+i; Yi 1+

The resulting equipotential lines in the region nearthe edge are shown in Fig. 3. Since the potentialpP+iv is complex, it becomes necessary to plot the

real and imaginary parts independently. We call theplot of i = constant the "real equipotential" and theplot of v = constant the "imaginary equipotential".These equipotentials are shown by solid and brokenlines, respectively, in Fig. 3.

At some distance away from the edge these equipoten-tials tend to have the same shape as those in themiddle part of the capacitor. The numerical values ob-tained through this method have been compared withthose obtained by circuit theory methods, yieldingerrors of less than 1% in both real and imaginary parts.This error includes those due to convergence and dis-cretization. Similar results were obtained with planecapacitors.

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Asenjo and Morales: Low-frequency complex fields in polluted insulators;

REAL EQUIPOTENTIAL---IMAGINARY EQUIPOTENTIAL

Fig. 3: EquipotentiaZ Zines of the cyZindricaZcapacitor in Fig. 2

Fig. 4: EZectric potentiaZ in the insuZator, o>>»w

Fig. 5: Electric potential in the insuZator, o<<wc

The proposed method has been applied to obtain thecomplex field in polluted insulators. To illustratethe procedure, a theoretical column insulator with asinusoidal profile, polluted with a uniform, partiallyconducting, surface of depth 6 and resistivity r0,was selected. A relative permittivity of 6.5 wasassumed for the porcelain.

There are two extreme situations depending on thepollution parameters. One is obtained when r-I = 5>>U£;in such a case the static approximation is valid. Theother case is obtained when o<<wc where the quasi-staticlossless approximation is valid. In both situations theelectric potential is real. In intermediate situations,when a and we are of the same order, the quasi-staticlossy approximation is valid and the electric potentialbecomes complex.

The results of the static and the quasi-static loss-less approximation are given in Figs. 4 and 5 respec-tively. Fig. 6 shows the results corresponding to thepolluted insulator with (r./6) = 109 ohms, in whichthe imaginary part of the complex electric potential

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IEEE Transactions on Electrical Insulation Vol. EI-17 No.3. June 1982

rises to its maximum value of 6% near the middle parton the surface. The real part of the complex electricpotential lies between that obtained in the static andthat of the quasi-static lossless situations (althoughit is closer to the static case).

CONCLUSION

The proposed method for solving low frequency com-plex fields in insulating systems is capable of wideapplications. It can be implemented in analytical ornumerical form. The finite difference numerical methodis easily implemented and the resulting equations canbe solved in a very efficient manner by Carre's pro-cedure.

REAL EQUIPOTENTIAL---- IMAGINARY EQUI POTENTIAL

Fig. 6: Electric potentiaZ in the insuZator,r /6 = 109 ohms

To the best of our knowledge, the solution of the com-plex field in polluted insulators has not been pre-viously obtained analytically. Although the method hasbeen illustrated for a particular insulator with uni-form contamination, it can be applied to general casesin which the insulator can have any form, within axi-symmetry, and any known contamination distribution.

It is worth pointing out that the proposed method cantake into account time variations of the contaminationresistivity. This, for example, could be used to gaina deeper insight into the dry band formation phenomenon.

APPENDIX 1

PROOF THAT THE SURFACE CHARGE DENSITY q. = 0

In the quasi-static lossy case (w/0, o/O). applying(2) to a small coin-shaped volume element with its flatsurface parallel to the interface, as shown in Fig. 7,we have qSD = c-El n - £2E2n, where qSD is the interfacecomplex displacement surface charge density.

The conduction current is given by JC = uE, and theaccumulated surface charge density due to this current,in the same volume element considered in Fig. 7, is

tqSC (a E1 E2 )dt - (Cy -o )ClSC 1 ln '22n dt ln- 22n

The total surface charge density in the dielectricinterface is

q=+ qC E+ 1(a E

qs qSD q 1 ln 2 2n iw 1 ln 2 2n

Therefore, from (5), we can see that qs=0.

E

C-

/C2ZE2E

Fig. ?: DieZectric boundary

APPENDIX 2

AXISYMMETRIC DIFFERENCE EQUATION FORDIELECTRICS WITH PARTIALLY CONDUCTING SURFACE

The derivation of the finite-difference equation forthe general case shown in Fig. 1 will be illustrated.If we rotate the rectangle of Fig. 1 in a small angle earound the symmetry axis, a volume V i-s generated, asshown in Fig. 8.

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Asenjo and Morales: Low-frequlency cnmplex fields in polluted -insulators

42Yci ( ) R061 y1+ hi )

+ YC5( )(+

+ Yc2 (2 )(R +

C h6)h2

-267

(R +2 \ 24 2 +

h2 '2SO) h2) h12065 + Y2(\h) 2,R 2)0

h+062

~)666+

Y3( )(h +

Y4()(R + 4

_3___ Rpq0\ h(R4h h4+YC3'h7 R063+ (5 h R -- 023/ 3 Y5 3/ 4

h32

+

+

h2e 2 +

+

Fig. 8: VoZume generated by rotation of generaZgrid configuration

+ YC7( h)( 2)7 6(h 4)(

+ YC4( h)(4 2 064 + Y7( h4) (R 2

Similarly, the rectangle formed by the mid-points1', 5', 2', ... 8' generate a volume v = V/4. Apply-ing Eq. (5) to this last volume we obtain

f V - [(a+iw)JE]l dv = f (a+iweJE * ds = 0 (8)v s

in which s is the surface surrounding the volume.

The following symbols, shown in Fig. 1, are defined:

y. = O.+iue. : parameter of dielectric medium i.2- 2-

YCi = ac +iWECi : parameters of partially conductingt'CiCi surface between point 0 and point i.

h. = length of segment Oi2-

6. = width of partially conducting surface between2 point 0 and point i.

The right hand member of (8) can be developed infinite difference form, assuming 6i«<<i. Consideringthe elementary surface generated by rotation of seg-ment 1'5' in Fig. 8, the following c6ntributiQn to thesurface integral is obtained:

YCl R)06+Y1 ( i1o) (R+ 2) 2 +

+ YC5 ( 5 ) (R+ h2) 06

Similarly, considering the contributions of all theelementary surfaces which conform s, the followingexpression is obtained:

+ YC8 ()(h 2) 068 + Y8( h)

(9)

Multiplying (9) by 2/Re and rearranging terms, wefinally obtain:

h- 12 yC161 + y1 (1 +h2

h2 + Y8 (1- )h4h +

~j2 y22yh yhl ( h~ ++ h2 2| YC26 2 f Y2(h + )3h3 + (1+ 2R )+

+ 23 C + +h2 h (i 4h

+4T 12 YC464 f Y6h3 + Y7h, |1 R)

+ h '2 C565(1 + 2 + h6 2 YC666 (1 + 2R +

h | 2YC767( 2R

2 ycldl° -h

h32

+

(R- 24 2-4 = °

h8 Y2YC868 (1 -2R

2 yC363+ h3 +

5

IEEE Transactions on Electrical Insulation Vol. EI-17 No.3. JLne 1982

h 2 yC262 2 Y 2 yy0h1 + y3h3\+ + 2R h2 + h5 h6 h

+ h4) 2 yC464+ 2 YC767 2 YC868 Y6h3 +Y7h1)R) h4 h7 h8 4

hi h2 h

+ (1 + j)( h +Y4 h1)+( h4 h4 h4)1

4R) 8 hj+ y5 h~~

(10)

REFERENCES[1] K. W. Wagner, Archiv fur Elektrotechnik, Vol. 2,

pp. 371, 1914.

[2] T. Takashima, T. Nakae and R. Ishibashi, "Calcu-lation of complex fields in conducting media",IEEE Trans. Elect. Insul., Vol. EI-15, No. 1,pp. 1-7, Feb. 1980.

[3] J. H. Wensley and F. W. Parker, "The solution ofelectric field problems using a digital computer",Elect. Energy No. 1, pp. 12-16, 1956.

[4] D. Young, "Iterative methods for solving partialdifferential equations of elliptic type", Trans.Amer. Math. Soc., Vol. 76, pp. 99-111, 1954.

[5] B. A. Carre, "The determination of the optimumaccelerating factor for successive overrelaxation",Computer J., No. 4, pp. 73-78, 1961.

[6] E. Asenjo and N. Morales, "Determinaci6n delfactor acelerador del m6todo de sobrerrelajacionsucesiva con elementos complejos", Revista Sigma,Universidad de Chile, Vol. 3, No. 3, pp. 29-37,1977.

Manuscript was received 25 September 1980, in finaZform 17 October 1981.

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