A NEW VECTOR BOUNDARY ELEMENTS PROCEDURE FOR INDUCTANCE COMPUTATION

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1 Dedcated to the memory of Prof. Augustn Moraru A NEW VECTO BOUNDAY ELEMENTS POCEDUE FO INDUCTANCE COMPUTATION MIHAI MAICAU PAUL MINCIUNESCU 2 IOAN. CIIC 3 MAIAN VASILESCU Key words: Boundary elements method Magnetc vector potental Multply connected domans Inductance computaton. Ths paper presents a new method for solvng the ntegral equaton of magnetc vector potental on the surfaces of perfect conductor bodes. The defnton of magnetc flux of deal cols mples that these are perfect conductors so the normal component of magnetc flux densty on ther surfaces s equal to zero whch s equvalent wth enforcng a zero normal component of magnetc vector potental A and the crculatons of A on any closed path on ther boundares only those surroundng the holes beng equal to magnetc fluxes. A and curl A are descrbed by a lnear combnaton of specal vector functons assocated wth the surface nodes. The ntegral equaton s projected on these functons as well as on another set of specalzed functons orthogonal on the frst set. The small number of unnowns s a great advantage of the proposed method.. INTODUCTION Compared wth the fnte elements method (FEM) the boundary elements method (BEM) based procedures have some mportant benefts: are appled to unbounded domans avod the spurous forces n ar regons have a small number of unnowns assocated wth a varety of smaller sze that compensates for the fact that the system matrx s not sparse. The man drawbac s the lac of drect applcaton n heterogeneous meda. But n the structures wth pecewse homogeneous subdomans the nterface ntegral equatons can be added [] and n the case of nonlnear meda the hybrd FEM-BEM methods can be used [2 3]. Department of Electrcal Engneerng Poltehnca Unversty of Bucharest Spl. Independente no. 33 Bucharest omana E-mal: mm@elth.pub.ro 2 esearch Insttute for Electrcal Engneerng (ICPE S.A.) Spl. Unr no. 33 Bucharest omana 3 Department of Electrcal and Computer Engneerng The Unversty of Mantoba Wnnpeg MB 3T 5V6 Canada ev. oum. Sc. Techn. Électrotechn. et Énerg p Bucarest 20

2 2 Vector boundary elements procedure for nductance computaton 6 The BEM procedures n scalar potental has the advantage of usng a relatvely small number of unnowns but great dffcultes emerge n the case of multply connected domans where ntegrals sngulartes requre specal treatments [4]. Ths form of the ntegral equaton wth the magnetc vector potental as unnown on boundares s most commonly used [5] ' α Ar ( ) = n ( ' Ar ( ')) ( n' Ar ( ')) ( n' A( r')) d ' + 4πA0 ( r) 3 S where Ω s the boundary of homogeneous doman Ω α s the sold angle under whch a small neghbourhood of Ω s seen from the observaton pont r and r ' are the poston vectors of observaton and source ponts respectvely = r r' = n ' s the normal unt vector and A 0 s the vector potental produced by the gven dstrbuton of sources n Ω. The tangental component A t and the gauge condton A = 0 unquely determne the normal component A n and the tangental component of the magnetc flux densty ( A) t equaton () beng based upon these unnowns. In addton the treatment of the sngularty n the ntegral of the thrd term from the rght hand of () presents some dffcultes [6] (only the sngularty of the second term s convergent). From the physcal pont of vew the enforcng of the tangental component A t of the vector potental s too severe n order to defne the normal component of magnetc flux densty. In [7] the restrcton A n = 0 s proposed whch s consstent wth the gauge condton and the A t condton s "weaened" by mposng natural crculatons of A (actually A t ) on any closed path on the boundary surface Ω whch s equvalent to mposng the normal component of magnetc flux densty. It can be shown that these condtons determne unquely the tangental components A t and ( A) t. Integral equaton () has now the followng form: n' α A ( r) = ( ' A( r')) ( n' A( r')) ds' + 4π A 3 0 (r). (2) It can be observed that n the equaton (2) the rght hand thrd term of () does not appear. By approxmatng Ω wth a polyhedral surface wth trangular facets ()

3 62 Mha Marcaru et al. 3 specal edge elements have been proposed n [7] n order to descrbe A t. The number of unnowns corresponds only to the number of edge elements assocated wth a tree resulted form a tree-co-tree spannng. The edge values of A t on all surface mesh sdes result from the crculatons of A t on all loops defned by the co-tree. The tangental component ( A) t s defned n local coordnates of each trangular facet. By projectng the equaton (2) on a set of lnear ndependent specalzed shape functons obtaned from the gradents of the nodal element functons a dfferent numercal soluton of equaton (2) s proposed n ths paper. Ths procedure has sgnfcant advantages over the method proposed n [7] especally n the case of multply connected regons (.e. perfect conductor cols). 2. SIMPLY CONNECTED EGIONS TEATMENT In the case of perfect conductor domans whose boundares are not crossed by electrc currents the crculatons of A and A on any closed path on Ω are equal to zero. We use the followng representaton of these vectors N' A = αv = = A β V (3) wth = N N beng the number of nodes of the dscretzaton mesh over the surface Ω α β = 2 beng the unnowns and the vector functon V assocated to the node havng on the trangular facet (p) contanng the node the followng form (see Fg. ) n p p l ( ) V = (4) 2S p = n p (p) l Fg. The facet (p) assocated to the node.

4 4 Vector boundary elements procedure for nductance computaton 63 n p and S p beng the normal unt vector and the area of trangular facet (p) and l the length vector of the opposte sde of orented accordngly wth the orentaton of n p. Frstly by projectng (2) on the vector functons V we obtan equatons where N' a ' α + N' = = a' b' β = c' = 2 (5) = V ( r) ( n' V ( r') ) dsds' (6) b' = 2 π V ( r) V ( r) ds V ( r) [ ( n' V ( r') )] dsds' 3 (7) c' = 4π A 0 ( r) V ( r) ds. Secondly the remanng equatons are obtaned by projectng (2) on another set of vector functons U defned by [8] assocated also wth each node and we obtan ( 2S ) p (8) U = l (9) where a " α + b" β = c" = = = 2 (0) a" = U ( r) ( n' V ( r') ) dsds' () b" = U ( r) [ ( n' V ( r'))] dsds' 3 (2)

5 64 Mha Marcaru et al. 5 c" = 4π A 0 ( r) U ( r) ds. (3) The vector functons U and V are orthogonal U ( r) V ( r d S = 0. ) At least one ntegral of the above double ntegrals can be evaluated analytcally. In terms of the unnown A (2) s a Fredholm ntegral equaton of the second nd leadng to a well-condtoned matrx of the equatons n (5) and (0). 3. TEATMENT OF MULTIPLY CONNECTED EGIONS In order to present n the smplest way the case of the multply connected regons we consder a doman wth a shape of a torod. The torod hole can be crossed by the electrc currents and magnetc fluxes. The crculatons of A and A along the closed paths Γ ' and Γ " around the torod and surroundng ts hole (as shown n Fg. 2) can be expressed as Γ' ( A ) d l = µ 0 I' ( A ) d l = µ 0 I" (4) Γ' Γ" A d l = φ' A d l = φ" Γ" where I ' I" and φ ' φ " are the electrc currents and magnetc fluxes through the torod and through hs hole respectvely. (5) Γ" Γ' Fg. 2 Closed paths Γ' and Γ'' around the torod and surroundng ts hole on a dscretzaton mesh.

6 6 Vector boundary elements procedure for nductance computaton 65 We defne the vector functon V { K'} G ' = where {K'} s the set of nodes on Γ ' and p ndcates all the trangular boundary elements on the same sde of Γ ' that contans at least one node of {K'}. Smlarly we defne the functon G " assocated closed path Γ ". The followng relatons replace the form of A and A n (3) for the case of multply connected regons = ( G' I' G" I" ) A = α V + µ + 0 (6) A = β V + G' φ ' + G" φ". (7) For a correct formulaton of the vector potental problem on the torod surface t s necessary to now ether the electrc currents I ' and I " or the magnetc fluxes φ ' and φ " or a current and a flux. Equatons (5) and (0) are completed by projectng equaton (2) on the functons G ' and G ". A gven electrc current through the torod hole means that the value of I " s " nown. If another torod crosses the hole of the frst one then the current I that passes through ts hole s the same wth that passng along the second torod ' " I 2 = I. The magnetc flux φ ' along a perfect conductor torod s equal to zero as well as I ". If we mpose the value of magnetc flux φ " through hs hole the electrc current I ' along the torod results. Then the self nductance of the torod s smply obtaned from L = φ" / I'. = 4. ILLUSTATIVE EXAMPLES In order to prove the accuracy of the proposed method we frstly choose the perfect conductor doman of a sphere wth m radus placed n a unform -2 magnetc feld B 0 = T orented along the Oz axs. The sphere surface s approxmated by a polyhedral surface wth trangular 2808 facets and 406 nodes. In Fg. 3 the magnetc flux densty vectors are shown on each trangular element of the sphere boundary. In Fg. 4 the projecton of flux densty on the (x y z) system of coordnates s presented for a merdan located at 30 degrees dependng on lattude. For the perfect conductor sphere an analytcal soluton can be obtaned [9] and for ths example the weghted mean square error of the computed results s 0.954%.

7 66 Mha Marcaru et al. 7 z Fg. 3 B over the surface of a perfect conductor sphere n a unform feld B z B (T) B x B y θ (degrees) θ Fg. 4 Magnetc flux densty components versus θ at ϕ = 30º for the sphere n Fg. 3. Fg. 5 A coarse dscretzaton mesh of the torod surface.

8 8 Vector boundary elements procedure for nductance computaton 67 L (µh) Number of nodes Fg. 6 Dependence of the self nductance of the torod n Fg. 5 on the densty of the dscretzaton grd. A second example presents the computaton of self-nductance of a perfect conductor torod wth average radus of 4 m and m radus secton (see Fg. 5). We choose a magnetc flux equal to unty through the torod hole ( φ " = Wb) and compute I ' along the torod. The torod surface s dscretzed n the manner shown n Fg. 5. The computed self nductance s presented for dfferent denstes of the dscretzaton mesh (number of nodes) n Fg CONCLUSIONS The replacement of the vector potental tangental component boundary condton by mposng only the crculaton of A on any closed paths on Ω corresponds to the normal component of flux densty natural boundary condton on the surface of perfect conductor domans. Ths "weaenng" of restrctons on A t allows us to choose the normal component of vector potental to be zero on the boundary and therefore results a smplfed form of the ntegral equaton () and removes the sngularty contaned n the thrd rght hand sde term. If we use the vector shape functons V assocated wth nodes and the test functons V and U orthogonal to V we obtan a system of algebrac equatons wth the number of unnowns equal to twce the number of nodes. In the case of multply connected domans we add specalzed vector functons assocated to the closed paths around the torods and boardng ther holes. The number of such functons s equal to twce the order of connecton. In the case of perfect conductor domans the number of these functons s equal to the order of connectvty. The method descrbed s partcularly effectve for determnng the nductances of perfect conductor cols. We note that Neumann formula can be used

9 68 Mha Marcaru et al. 9 only to determne the approxmate mutual nductance and can not be appled to obtan the self nductance. ACKNOWLEDGMENTS Ths wor was supported n part by the omanan Natonal Councl of Scentfc esearch (CNCSIS UEFISCSU) project number PNII IDEI 200/2008 and the Sectoral Operatonal Programme Human esources Development of the omanan Mnstry of Labour Famly and Socal Protecton through the Fnancal Agreement POSDU/89/.5/S/ eceved on January 8 20 EFEENCES. P. Mncunescu Contrbutons to ntegral equaton method for 3D magnetostatc problems IEEE Trans. on Magn pp I. F. Hănţlă I. Nemoanu M. Marcaru I. Hănţlă P. Palade An teratve procedure for solvng FEM-BEM equatons Journal of Electrcal and Electroncs Engneerng 2 pp I.. Crc M. Marcaru I.F. Hănţlă S. Marnescu Iteratve FEM-BEM technque for an effcent computaton of magnetc felds n regons wth ferromagnetc bodes Proc. of XIX Internatonal Conference on Electrcal Machnes (ICEM) pp Sept M. Marcaru T. Maghar M. Slagh F. Hănţlă Scalar BEM for Magnetc Feld Computaton n Multply Connected Domans n Proc. of th Internatonal IGTE Symposum Graz Austra pp Sept A.J. Poggo K. Mller Integral equaton solutons of three-dmensonal scatterng problems (Ch. 4) n Computer Technques for Electromagnetcs. Mttra ed. Oxford New Yor Pergamon Press pp T. Onu S. Waao Novel boundary element analyss for 3-D eddy current problems IEEE Trans. on Magn pp F.I. Hănţlă I.. Crc Magnetc vector potental tree edge values for boundary elements IEEE Trans. on Magn pp F.I. Hănţlă I.. Crc A. Moraru M. Marcaru Modellng eddy currents n thn shelds Compel 28 4 pp I.. Crc On the electromagnetc levtaton of sphercal conductors ev. oum. Sc. Techn. Électrotechn. et Énerg. 4 pp