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Email faizjsofeeceutacir M OJAGHI Zanjan Regional Electric Company Zanjan Iran There are different numerical techniques for computing electric fields These numerical techniques enable the designer to study the problems that are difficult to be solve

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Instructive Review of Computation of Electric Fields using Different Numerical Techniques JAWAD FAIZ Department of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran. E-mail: faiz-j@sofe.ece.ut.ac.ir M. OJAGHI Zanjan Regional Electric Company, Zanjan, Iran There are different numerical techniques for computing electric fields. These numerical techniques enable the designer to study the problems that are difficult to be solved by analytical methods. This paper attempts to give an instructive review of different numerical techniques

in electric field analysis. These techniques involve finite difference, finite element, boundary element, charge simulation, finite element with variable field intensity and Monte Carlo methods. The merits and limits of the various methods are outlined. Some examples are given in which the field computations using different numerical techniques are compared. As an example the most convenient technique applicable to electric field computation within the tank of power transformers is introduced. INTRODUCTION UNTIL NOW electric and electromagnetic fields have been regarded as probably the most

abstract and difficult part of the undergraduate electrical engineering curriculum. This is largely due to the fact that such fields cannot be visualized directly [1]. Electric fields can be computed using various methods with different precision. However, for insulation of electrical equipment a more accurate electric field prediction is required. With the advent of computing power and numerical tech- niques in recent years, it has become practical to use different techniques to compute the electric fields. Such numerical techniques enable the designer to solve problems that are difficult,

and use of analytical approach with many empirical factors is impossible. The aim of this paper is to give a review of the application of different numerical techniques in the electric fields computation that is useful for an undergraduate course. This material should enable the graduate student to use these analyses in graduate research as well as later in the workplace. The authors will also give their personal views on the field evaluation of transformers. In two-dimensional (2D) analysis, if the field is not time varying, the electric potential V in the actual space is satisfied by

Poisson's equation [2]: = where is the electric charge density, and are relative and absolute permittivity of the free space. In order to solve partial differential Eqn. 1, a number of boundary conditions must be imposed. Analytical solutions for practical boundary con- ditions are difficult, if not impossible and therefore, numerical techniques are necessary. Each technique has its own merits and draw- backs and one technique cannot be generally preferred to others. Based on the proposed problem, the most convenient technique must be selected. Different numerical techniques, so far

used for the electric field analysis, are briefly reviewed and the most convenient technique is then suggested for electric field evaluation within the interior space of power transformers. BOUNDARY CONDITIONS There are two following types of boundary conditions that are considered in the electric field evaluation: 1. Boundary between conducting and dielectric materials. 2. Boundary between different dielectric materials. From the electrical point of view, one of the following conditions may be satisfied on the boundaries of the first type: a. The electric potentials of all points on the

boundary are known. This is possible if the available conductor is connected to a fixed potential source (Dirchlet condition). * Accepted 344 Int. J. Engng Ed. Vol. 18, No. 3, pp. 344356, 2002 0949-149X/91 $3.00+0.00 Printed in Great Britain. 2002 TEMPUS Publications.

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b. The whole charge on the bounda ry surfa ce is known whi le the electric potential over different points is unknow n. Thi is the case when the avail able onductor is not connected to fixe potenti al (floati ng). Norm ally in such case the whol ch arge on the surface bounda ry is equal to zero (Neum ann

cond ition). FINITE DIFF ERENCE METH OD Finite difference method (FDM is the oldest techni que in the field co mputations that was intr o- duced by Gauss Then Bol tzman publ ished it in his notes in 1892. How ever, becau se of the large amoun of computa tions requir ed, the extens ive use of the FD dates back onl to the event of the compu ter. Electric potenti al over any region depend on the (x, and z) co ordinates an its deriva tives. Poten tial of any point may be iven versus the electric potenti al of the ad jacent points using Tylor' expansi on. For inst ance, consider Fig. having point

and six ad jacent points The coor- dinate of these points are: For simp licity, the diffe rence betw een the origi and the adjacent points is taken to be h. Poten tial of an arbitrary poi nt may be calcul ated versus pot ential of point as foll ows: ' 1! ' xx xy xz yy yz zz 2! where =@ ii =@ ij =@ If tends to very small value, the terms contain ing the third- and higher-o rder deriva tives may be ignored an Eqn. can be rewritten as foll ows: hV 5h zz hV 5h yy hV 5h zz hV 5h yy hV 5h xx hV 5h zz 6V xx yy zz Solution of Eqn. and substi tuting from Eqns 12 yields: = where is the elect ric

charge den sity at poin 0. As shown in Eqn. 5, there is linea relationshi between the poten tial of point and the potenti als of the adjacent points For the ases where diff er- ence be tween the point an the adjacent poin ts is not the same or these points are wi thin differen insulati ng mate rials, an also for rotating fields in electrica machi nes, sim ilar equati ons can be derived [3, 4]. In the FDM the prop osed region is discr etized using the eq uations sim ilar to Eqn. 5. Dimensions of the meshes must be such that the ap proxima tion is accepta ble. The verte xes of the meshes are

nodes on the bounda ry of the region nd their potenti als are known or they corresp ond to point in Fig. 1, enclosed by the other nodes. For the latt er note (not on the bounda ry), equatio ns simila to Eqn. can be wri tten versus the potentials of the ad jacent nodes. If the number of such node is equal to n, linear algebr aic equati ons with unknown values of the node potenti als are obtaine d. Potentials of the points insi de the meshes may be de termined using different interpo lation techni ques. The FD is not capable of alculati ng elect ric field direct ly at different poin ts on the

proposed region. hen the poten tials of the nodes are obtaine d, num erical derivativ evaluat ion tech- nique is used to calcul ate the elect ric field intensit y: r V. Fig. 1. Point and six adjacent points. Inst ructive Review of Com putation of Electric Field using Differ ent Num erica Techniqu es 345

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FINITE ELEM ENT METH ODS Among the various numeri cal techni ques, the finite elem ent method (FE M) has dominant posit ion because it is versat ile, having strong interch angeabil ity and can be incorpo rated into standar program [5, 6]. FEM is based on this fact that

the physica systems stabi lizes at the minimum level of energy. The gen eral equati on of en ergy in an electric fiel is: dv vd vds wher is the volume of the propo sed region is the elect ric charge de nsity within vo lume is the bounda ry surfa ce with Numm an conditi ons is the surfa ce electric ch arge den sity wi thin Since Eqn. describes the electric poten tial dis- tribu tion in the real syst ems, based on the mini- mum en ergy level theorem it is concluded that Eqn. minimiz es the energy presented by Eqn. 6. In the FEM, the volume of the proposed region is divide into smal polyhed ron

elemen ts wher their sides form grid wi th node s. The pote ntial functi on is then ap proximated by: wher is any point on the pro posed region. is called the shap functi on having the followi ng featu res: a) is eq ual to zero anywhe re, except on the subregi on The sub-re gion consis ts of the elem ents wher node is one of their vertices. b) is co ntinuous on the bounda ries due to and polyhedron insi de eac elem ent. c) any is equal to unit at the locat ion of node and zero at the other node s: for for 6 in Eqn. is equal to the potenti al of node i. Substi tuting Eqn. into Eqn. the approxim

ate energy is pre sented by whi ch is mini mized unde the followin con ditions: =@ 0; Since is qua dratic function of applyin cond itions leads to the foll owing linear algebr aic equati ons: GV wher is the known vector with elem ents is the known vector obtaine from the volume charge density in the pr oposed region and bound- ary con ditions; is the non- singular square symmetrica matrix. Solu tion of this system of equati ons gives the values of and hence an approxim ate dist ribu- tion of the poten tial can be determined ba sed on Eqn. 7. Electr ic field intensit within each elemen is

obtaine using the grad ient express ion as follows r 10 Often the first deriva tive of is non-con tinuous Therefor e, reductio of the maxi mum size of the elements and tending to zero, leads Eqn. to the real dist ribut ion of the poten tial. In spit of this no con tinuity of the field intensit on the bound- aries of the elem ents remai ns in force. If functi on is con sidered as complete n-order polynomi al, better resul ts can be obtaine d. If present the maximum size of the elements, red uction of can reduce the potential error with ratio and electric field is continuou an its error

is reduced by rati [7]. Fig. shows the meshin an equipotent ial lines determined using the FEM. The FEM could be also used where the permi ttivity of the pr oposed region is not constant In such case, it is necessa ry to replac wi th which shows the pos ition depend ency. ethods have been introdu ced to consider the floating elect rodes with unknow potenti al or different insul ating materials [7]. Ther are severa reports for automa tic meshing of the prop osed region [911] BOUNDA RY ELEM ENT METH OD If dist ribut ion of electric ch arge for every region (including bounda ry surfaces)

is known, elect ric potential and field intens ity for each poin can be computed using Coulo mb's law of Gauss 's law [2]. In practice, Laplaci an quation is normally used This means that elect ric charge is enclosed only inside the bounda ries of the propo sed region and volume charge den sity insi de the region is equa to zero or ne gligible. The electric pot ential and field intensit are: ds 11 ds 12 where is the prop osed point, is the summ ary of all bounda ry surfaces, is the surfac charge density over surfa ce s, is the dist ance of the different ial elem ents from point and is uni

vector along directed from different ial elemen ds to point I. In practi ce is unknown and it seems that solving Eqns 1112 is impos sibl e. How ever, to Jawad Faiz and M. Ojaghi 346

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overcome this diff iculty, the BEM may be employ ed. In this method, the bounda ry surfa ce is divide into elements. Figure shows typic al bounda ry elements due to flat bounda ry su rface on xy-pl ane. Then taking into acco unt the smal dimens ions of the bounda ry elements, general form may be consid- ered for the surface charge density on elem ent sj Thi gene ral form is often polynomi

al with unknown coeff icients. For instance, if it is taken to be quadrati polyno mial as follows sj 1j 2j 3j 4j xy 13 substitut ing sj in Eqns. 11 an 12 gives sj ds sj 1j 2j 3j 4j xy ds 14 sj ds Fig. 2. Computatio of electric field using FEM: a) meshing, b) equipoten tial lines [4]. Inst ructive Review of Com putation of Electric Field using Differ ent Num erica Techniqu es 347

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sj 1j 2j 3j 4j xy ds 15 It is clear that V(I) and E(I) are linea function of the co efficients of the polyno mials. By us ing the followi ng definitio ns: 1j sj ds 2j sj xd 3j sj yd 4j sj xyds 1j sj ds

2j sj xds 3j sj yds 4j sj xyds Eqns. 14 and 15 become: kj kj 16 kj 17 In the next stage, the num ber of unknowns fun c- tion sj is selec ted on each bounda ry elem ent. Then, based on the bounda ry type wher sj is part of it, the bounda ry co ndition equati on for any selec ted point is form ed using Eqns 16 and 17. Therefor e, syst em of algebra ic linea equati ons is obtaine which final ly pro duces the coefficie nts of the polynomi al. kj an kj are numericall or analytical ly obtaine by integ ration. Hence the surface charge de nsity distribut ion on all bounda ry surfaces is known. Fin

ally the elect ric potenti al and field intens ity ca be determ ined using Eqns 16 and 17. Fig. 3. flat quadrangu lar boundary element. Fig. 4. Electric field calculation using BEM: a) actual electrical system, b) electric field density vectors on the sphere, c) equipotentia countors [12]. Jawad Faiz and M. Ojaghi 348

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Figure pr esents typical problem solved by the BEM, wher the electric field between the tw spheres having the same radius is obtaine d. Electr ic field intens ity has be en shown with suit- able vector In additio n, the equipotent ials betw een two spheres have

been calculated by computa tion of the electric potential in different points method has been given in [13] for the curved- shape bounda ry elem ents in or der to mo del prac- tical surfa ces with the de sirable accuracy Bound- ary elem ents with full axial symm etry have bee present ed in the lit erature [1416 ]. In put and outpu da ta process ing methods in the BEM have been intro duced in [12, 17, 18]. CH ARGE SI MULA TION METH OD This method is sim ilar to the BEM. The differ- ence betw een charge simu lation method (CS M) and the BEM is the simulat ion of the surfa ce charge

existing on the bounda ry surfa ces. In the BEM the surfa ce charge de nsity functi on on the different surface bounda ries are estimat ed, whi le in the CSM the su rface ch arge density is sub stitute by set of discr etized linear ch arge distribut ion. The substitut ed linea charge dist ribution is su ch that the electric potenti al and field intensit versus their charges are analytic known functions. Char ge distribut ion on an infi nite length line wi th constant density, on finite line, on circle etc., are exampl es of the ch arge distribut ion. Differ ent distribut ion types and their

equ ations have be en given in [4] and [19] Electric poten tial and field intens ity equatio ns due to the above menti oned charge distribut ion, for the poin ts on the charges have singul arity. To overcome this singul arity, the position of the replac ed charge simulat ion is co nsider ed outsi de the prop osed space and normal ly insi de the elec- trodes The exact position and the replac ed charge distribut ion type are arbitrari ly selected based on the experie nce. Char ge value or their linea charge densit is compu ted such that the bounda ry con- dition are satisfi ed on some su rface

bounda ries, as descri bed in the followin part. Electric potenti al and field intens ity due to the replac ed charge sim ulation in different poin ts are linea function of the charge value or harge densit y: 18 19 wher is the potenti al factor, is the fiel intens ity factor, is the relat ive posit ion of the propo sed point and is the jth replac ed charge distribut ion. and are different for differ- ent charge dist ribut ions. Ther is the followin relationshi between them: r 20 Since there is linear relat ionship between the electric potential (and fiel intens ity) due to the replaced

ch arge dist ribution in different poin ts and charge value or charge densit y, for set of such charge dist ribution, the superpo sition theorem can be applie to calcul ate the electric poten tial and field: 21 22 In the CSM, the num ber of points selec ted on the surface bounda ry is equa to the replac ed sim ula- tion ch arges. Depend ing on the selec ted points Eqns. 21 and 22 are used and the bounda ry condition for individ ual points are con sidered. These eq uations are linea functi ons of Since the exact pos ition of the charges and the selected points are known and in Eqns 21 and 22

are exactly calcula ted. Then only are the unknown values of the ab ove linea eq uations Therefor e, comp utations of the elect ric potenti al and field intens ity are pos sible using Eqns 21 and 22. It is clear that in the CSM, bounda ry con di- tions are satisfied only in the poi nts selected to write the eq uations Befo re using due to the solution of the equ ations, it is necessa ry to study the bounda ry con ditions on other points of the bounda ry surfa ces. When in the CSM is calculated, the bounda ry conditio ns on differen surface points must be de termined. If the accu racy is not

enough num ber, position and type of the replac ed simu lation charges and also position of the selec ted poi nts for writing the equ ations must be varie in ord er to obtain suff icient accuracy Figu re shows typical pro blem solved using the CSM. The prob lem was co mputation of the elect ric field betw een two spheres shown in Fig. 5a. Figu re 5b present the equipotent ials lines by substitut ing the surfa ce ch arge of each sphere with tw point- charges Figure 5c indicates the corres pondin result when three point-charges are su bstitute for each sphere, in which the accuracy is higher. In

ad dition, two point-charges have been used in Fig. 5d, but the position of ch arges is diff erent wi th that of Fi g. 5b Accuracy of the latter case is bette than the other two. FI NITE ELEMEN METH OD WIT VAR IABLE FIELD DENSI TY In the FEM, the main varia ble is the elect ric potential V, while the elect ric field intens ity is normally the requir ed quan tity. For exampl e, in the design of high volta ge device insul ation, it is Inst ructive Review of Com putation of Electric Field using Differ ent Num erica Techniqu es 349

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necessa ry to have the amp litude and direction

of the electric fiel on the surfa ces of the elect rodes and insulati on. In study of the dischar ge phe nom- enon the path of the force lines is requ ired whi ch can be determined if the elect ric field distribut ion in the propo sed space is know n. In the FEM numerical integ ration techni que is used in order to calculate that normal ly has error. Ther is more accurat FEM techni que in which the main variab le has been taken instead of [20]. If the volume charge density in the proposed space is zero and Fig. 5. Calculation of electric field using CSM: a) electric system, b) equipo tential

when two charge points used for each sphere, c) equipoten tial when three charge points used for each sphere, d) equipoten tial when two charge points in different positions used for each sphere. Jawad Faiz and M. Ojaghi 350

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permi ttivity is con stant and isot ropic, then the Maxw ell's equatio ns for elect rostatic fields are as follows: div 23 Curl 24 wher is the electric charge densit and is the electric field intensit and 25 Apply ing Eqn. 24 and using Eqn. 23 an 25 leads to: 26 which is the Laplaci an equati on. For 2D fields: =@ =@ =@ =@ 27 But vector wi ll be zero if

all its componen ts are zero: =@ =@ =@ =@ 28 For each componen of the field intens ity, the FEM is used onc and finally the electric fiel intens ity over the whol pr oposed region is obtaine d. How ever, with de fined bounda ry cond itions, the eq uation will have uniqu solut ion. In practice, the bounda ry condition on the first- type bounda ry versus elect ric poten- tial re known but there is no idea con cerning their electric field intensit y. In [20], the BEM has been used in ord er to overcome this difficul t; it means that the BEM is employ ed on the mentio ned bounda ries then fini te

elem ent method with varia ble field densit (FEMVFD is applied Analysis has been arried out based on the FEM and FEMV FD and the resul ts have be en present ed in Fi g. 6. The problem was co mputation of the electric field in the region between the two cylinde rs having potenti als of 100 and 200 V. Bec ause of the axial symm etry and its bounda ries an infin ite lengt h, the field has only radial compo nent varyi ng in radial direct ion. Therefor e, analys is of the field is possibl in two dimens ions. Due to the symm etry only one-qu arter of the cyli nder is used for analy- sis. Figure 6b

shows the meshi ng for both tech- niques and Fig. present the equ i-field intensit lines. As seen in spite of the low er numb er of meshes in the FEMV FD, the equi-fiel intensit lines have better co ntinuity an their accuracy is also high er. In this ex ample, the maxi mum error in solut ion by the FEMVF is abou 2% while this error is 9.3% when the FEM is used. MON TE CAR LO ETHOD In the FDM, the potenti al of every node is equ al to the mean alue of the poten tials of the ad jacent nodes. When the dist ance between the pr oposed node an adjacent node re the same, same weigh potentials are

involved in the computa tion of the mean value. Othe rwise, the weigh ts wi ll not be equal [4]. But the closer nodes wi ll be heavier. However, the sum of all wei ghts is unity. Genera lly in the FDM potential of each node (V versus potential of adjacent node is as follows 29 where 1, is the potential of the node adjacent to i-th node and is its weigh t. Val ue of depend on the relat ive dist ance of the i-th node from the pro posed node. The basic equa tion of Monte Car lo Method (MCM) is simila to Eqn. 29. Therefor e, these two methods are sim ilar, except that determ ination of the

adjacent node and method of calcul ation of diff ers. In the FDM, all nodes are de fined after meshing process and at the same time ad jacent nodes of every node are de termined. In co m- putation, analytical relation ships are used But in the MCM, ad jacent nodes are alw ays on the bounda ries and their exact pos itions are deter- mined using rando pro cess and is obtaine using prob ability techni ques. In Fig. 7, show bounda ry surfa ces of the problem Thes bounda ry surfaces are Drich let type with potenti al Calculat ion of potenti al at any poin such as is requir ed using equ ations

similar to Eqn. 29. In the MCM, simu lation of random movem ent is use in order to determine each node Any movem ent begins from and after success ive jumps with varia ble lengt hs nd in random direct ions, leads to Condi tions governi ng on every random mo ve- ment are as follows a) All movem en ts begin from b) Length of an jump is eq ual to the minimum distance of the beginni ng point with bounda ry surfaces (S s). c) Directio of very jump is rando m. d) The end of any rando mo vement wi ll reach when the minimum distance mentioned in item is smaller than that of the predefi ned value such

0. At the end of rand om movem en t, poin on one of the bounda ries with the closest distance to the end point is selected as an ad jacent poin t. Finall y, potential of point is calculated as foll ows: 30 where is the total number of the random Inst ructive Review of Com putation of Electric Field using Differ ent Num erica Techniqu es 351

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movem ents and is the adjacent poi nt due to the j-th random movem ent. In order to eliminat the stat istical errors, it is necessa ry that becomes large enough and often severa thousand s. In Eqn. 30 all wei ghts are apparent ly the same

and equal to 1/N. In fact it is not so, because taking into acc ount the large num ber of random movem ents (N), prob ability of severa reputations of one point exists an for relat ively sho rter Fig. 6. Comparison of computed electric field using FEM and FEMVFD a) electric system, b) meshing based on the two techniques, c) equi-fie ld intensity lines using both methods. Jawad Faiz and M. Ojaghi 352

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distan ce from this probabil ity is larger. Suppos constant pot ential for any bounda ry surfa ce, Eqn. 30 can be writt en as follo ws: 31 where is the number of bounda ry surfa

ces and is the number of random movem ents ended to point at Alth ough the CM was descri bed using the FDM, it is necessa ry to note that the MCM is itself an independ ent method having specia Fig. 7. Representatio of the possible random movement paths for reaching from the proposed point (r to one of the boundary surfaces (S ). Fig. 8. Geometry of laboratory high voltage electrodes [7]. Inst ructive Review of Com putation of Electric Field using Differ ent Num erica Techniqu es 353

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fundame ntals [21]. Compu tation of electric poten- tial us ing MCM is sim ilar to the micro

scopic analys is of gas pressur wher the random move- ment of the molec ules is simu lated. techni que for compu tation of the elect ric field over the spaces consisting of different isolati ng mate rials has been presented in [22]. In [7, 22], num ber of techni ques have be en introdu ced to speed up the calculati on using the MCM. Figure shows the geo metry of the labo ratory high vo ltage elect rodes in which the Fig. 9. Comparison of the computed electric fields of the system shown in Fig. using three techniques a) along path AB, b) along path CD and c) along path EF [7]. Jawad Faiz and M.

Ojaghi 354

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electric pot ential and field along line AB, CD an EF have bee estimat ed using the three methods MCM, CSM and FEM. The resul ts are present ed in Fig. 9. It is lear that the accuracy of the MCM is sim ilar to the other techni ques. Ther are abou 24000 random movem en ts in this simu lation. CO MPAR ISON OF METH ODS AN CONC LUSION Any numeri cal techni que for elect ric fiel evaluat ion ha its own merits and drawba cks an it is not general ly possibl to prefer one techn ique over the others Structu re of the power trans former indica tes that normally 3D computa

tion of the electric field is requir ed beca use there is no symm etry in order to ignore one dimens ion. In addition, numer- ical techn ique must be capable of dealin with the narrow insul ating or condu cting layer s. FDM an FEM in the elect ric field computa tion have two draw backs: a) In determinat ion of potenti al dist ribution using two methods num erical deriva tive techniq ues must be used in order to obtain the elect ric field intens ity. Thi has co nsiderab le error that leads to large error in the elect ric field compu tation. In many cases, the fiel is required for the design of

insul ation of elect rical equipment Mean while, study of some phe nomena such as elect rical dischar ge is pos sible by elect ric fiel compu tation. b) In order to pr event large lectric field an its draw backs, the sha rp edges on the different surfa ces are avoided. Ther efore, the curved surfa ces are often preferred. FDM, FEM an FEMV FD hav difficul ty in modeli ng such curved surfaces; but CSM and BEM can be easily ad apted for such cases. On the other han d, althoug FDM, FEM and FEMVF can theo retically compu te 3D fiel ds, there are many problem in deali ng with this matt er. One

seriou problem is 3D mesh generat ion and its modificat ion to app roach the requir ed accuracy Genera lly manual calcul ation is cumbers ome an tim co nsumin an also computer program ming is reall compli cated. Anothe difficul ty is the large size of the co effi- cient matr ix of the syst em of equatio ns. In these methods num ber of equatio ns is pro portion al to the memor requ ired for co mputation whi le in other methods (CSM an BEM) this number is propo rtional with area of the bounda ry surfaces. It means that in FDM FEM and FEMVF D, the coeff icient matr ix has one more dimens ion than

the coeff icient matrix due to the other ab ove-m entione method s. Therefor e, more computer memory and longer computa tion time are requir ed. Hence FDM, FEM an FEMV FD may not be con sidered conven ient techni ques for lectric field computa tion. MCM, at least in elect ric field co mputation, is not so common and has no consider able progres in recent years. At the present ap plication of this method in na rrow layer problem is difficul t, if not impossibl e. In spite of the sim plicity of computer program ming and high accuracy of the method in compu tation of 3D electric fields having

narrow layers CSM is confront ed with major difficul ty. In this method it is ne cessary to consider charges within the mentio ned layer suc that they have en ough distance from two sides of the layer But the thickne ss is too na rrow and such an assum ption may not be correct. Fin ally BEM is capable of nalyzing 3D fields and there are some rep orts sho wing its applic ations to narrow layer s. In conclusi on, BEM may be consider ed the most conven ient techn ique for elec- tric field computa tion within the interior of the power trans former tank [23, 24 ]. REFE RENCES 1. N. N. Rao,

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22. M. Krause and K. Muller, Monte Carlo method for two- and three-dimen sional electrostatic field calculation in materials of different permittiv ity, Int. Symp. High Voltage Engineerin Athens, Greece, Sept. 1983, p. 11.04. 23. J. Faiz and M. Ojaghi, fast boundary element method to electric field computa tion within the tank of power transformer s, Int. J. Comput ation and Mathematic in Electrical and Electronic Engineering (COMPE L) 17 (1/2/3), 1998, pp. 6977. 24. J. Faiz and M. Ojaghi, Novel techniques for treating singula rity problems in the boundary element method of evaluation

within the tank of power transform ers, IEEE Trans. Power Delivery 15 (2), April 2000, pp. 592598. Jawad Faiz received the Bachelor and Masters degrees in Electrical Engineering from Tabriz University in Iran in 1974 and 1975 respectively graduating with First Class Honours. He received the Ph.D. in Electrical Engineering from the University of Newcastle-upon-Tyne, England, in 1988. Early in his career, he served as faculty member in Tabriz University for 10 years. After obtaining his Ph.D. degree he rejoined Tabriz University where he held the position of Assistant Professor from 1988

to 1992, Associate Professor from 1992 to 1997, and has been Professor since 1998. Since February 1999 he has been working as Professor at Department of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran. He is the author of 50 publications in international journals and 60 publications in conference proceedings. Dr. Faiz is Senior Member of IEEE and member of Iran Academy of Sciences. His teaching and research interests are switched reluctance and VR motor design; design and modelling of electrical machines and drives. Mansour Ojaghi received the B.Sc. degree in

Electrical Engineering from the Shahid Chamran University, Ahwaz, Iran in 1993 and the M.Sc. degree in Electrical Power Engineering from the University of Tabriz, Iran, in 1997. He is now working as senior engineer of Zanjan Regional Electric Company, Zanjan, Iran. His area of interest is on the modeling and field analysis of transformers, protective relaying and electrical machines modeling. Jawad Faiz and M. Ojaghi 356

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