Three-dimensional eddy current analysis by the boundary element method using vector potential

Transcription

1 Physcs Electrcty & Magnetsm felds Okayama Unversty Year 1990 Three-dmensonal eddy current analyss by the boundary element method usng vector potental H. Tsubo M. Tanaka Okayama Unversty Okayama Unversty Ths paper s posted at escholarshp@oudir : Okayama Unversty Dgtal Informaton Repostory. and magnetsm/121

2 ~ 4x 454 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 2, MARCH 1990 THREE-DIMENSIONAL EDDY CURRENT ANALYSIS BY THE BOUNDARY ELEMENT METHOD USING VECTOR POTENTIAL H. Tsubo and M. Tanaka Department of Electrcal and Electronc Engneerng, Okayama Unversty, Tsushma, Okayama 700, Japan Abstmct: A boundary element method usng a magnetc vector potental for eddy current analyss s descrbed. For three-dmensonal problems, tangental and normal components of the vector potental, tangental components of the magnetc flux densty and an electrc scalar potental on conductor surfaces are chosen as unknown varables. When the approxmaton that the conductvty of conductor s very large n comparson wth the conductvty of ar s ntroduced, unknowns can be reduced. Futhermore, for axsymmetrc models the scalar potental can be elmnated from unknown varables. Formulaton of the boundary element method usng the vector potental and the computaton results by the proposed method are presented. INTRODUCTION For lnear problem n three-dmensonal eddy current analyss, boundary element methods are attractve n term of pre-processng and computer requrements. Several formulatons of boundary ntegral equatons were reported[ The boundary element method usng unknown electrc feld vector and magnetc flux densty vector has been developed for three-dmensonal eddy current analyss[5], and can be appled to conductng, magnetc and delectrc regons. In ths paper, we propose a boundary element method usng a magnetc vector potental as an unknown vector varable. For the conductng and magnetc regons n whch felds are'. excted by current source, the eddy current analyss can be performed by usng the magnetc vector potental and an electrc scalar potental. When the approxmaton that the conductvty of conductor s ve'y large n comparson wth the conductvty of ar IS ntroduced, the normal component of the gradent of the scalar potental on the boundares can be related to the normal component of the vector potental and unknown varables are reduced. Formulaton of the proposed boundary element method and ts computaton results are descrbed. FORMULATION Boundary element methods usng vector varables are formulated by the vector Green's theorem[6]. In the proposed method, a magnetc vector potental s ntroduced as an unknown varable and the boundary ntegral equatons are formulated. A magnetc vector potental A and an electrc scalar potental cp are generally ntroduced to solve the Maxwell's equatons. The electrc feld E and the magnetc flux densty B wth snusodal tme dependence are defned by usng the potentals as follows: E = -joa - Vcp (1) appears on the boundary surfaces between conductor and ar. From Eqs. (1) and (2). the equatons for A and cp to solve eddy current dstrbutons whch are governed by the Maxwell's equatons are gven by VxVxA-o2pc*A+jwpE*Vcp=pJo (3) V2cp + j0v.a = -pole* (4) where Jo and po are the source current densty and the source charge densty, respectvely. Furthermore the Lorentz gauge s ntroduced as a gauge condton. That s gven by V.A + jwp&*cp = 0 where E* s the generalzed complex permttvty. In the case of. ntegral equaton methods usng the vector potental and the scalar potental, we can prove that the Lorentz gauge s equvalent to the followng contnuty equaton. (5) V.J + jop = 0 (6) Usng the vector Green's theorem for the vector potental A and the Green's theorem for the scalar potental cp, followng boundary ntegral equatons at computaton pont are obtaned from Eqs. (l)-(s). Q - 4x A=I Q cp = (-(A.n')V'@+(Axn')xV't$-(V'xA)xn'@ -jope*cp@n') ds + p J&dv + Ao (7) Iv (@V'cpm'-cpV'@.n') 1 ds + - Iv po@dv +W (8) where Q s the sold angle subtended by S at, Ao and Cpo are potentals whch are nduced by the external sources, s the fundamental soluton gven =- e-jkr 4xr k = o w The boundary condtons between the regon 1 and the regon 2 are gven by the contnuty condtons of the potentals and the tangental components of the magnetc feld and the normal component of the electrc flux densty on the boundares as follows: A=Az (9) Cp 1 'W (10) B=VxA (2) The sources of the vector potental A and the scalar potental cp n free space are electrc current and electrc charge, respectvely. Here, we consder the regon ncludng conductors and magnetc materals n whch the felds me excted by electrc currents and no dsplacement current arses. Therefore the source of the scalar potental (V 'xa) 1 x n '/p 1 =( V 'x A )2x n '/p 2 (11) EI* /90/ $ IEEE (-joal-v'cpl).n'=~~* (-joa2-v1cp2).n' (12) After applyng Eqs. (7) and (8) to the both sde of the boundary surfaces, we can obtan the boundary ntegral equatons whch s expressed by the unknown varables n regon 1 as follows:

3 ds 455 trangular elements s 64 on one eghth part of the sphere surface and unknown varables are defned to be constant on each element. Fgure 2 shows the dstrbutons of the vector potental along x-axs for 50(Hz). Computaton results agree wth theoretcal values[4]. Fgure 3 and 4 show the dstrbutons of the vector potental and the tangental component of magnetc flux densty whch are unknowns on the boundary surface, respectvely. In the conductng sphere model, the external potentals Ao and cpo were defned by 1?l= [-@(jw (?*- El* l)(am')l+ -(V'cp.n')l) E2* E2* When the approxmaton that the conductvty of conductor s very large n comparson wth the conductvty of ar can be ntroduced, the term, V'cp.n', can be related to A.n' by Eq. (12) because the normal component of the electrc feld s equal to zero. The relaton between V'cp.n' and A.n' s gven by V 'cp. n'=-j o (A.n ') (17) Therefore the nteracton between the smultaneous equatons for A and that for cp can be removed, and the fnal smultaneous equatons whch consst of only Eqs. (13) and (14) are solved for A, (V'xA)xn and cp. In ths case, V'pn' s removed from the unknowns. In axsymmetrc models, the normal component of the vector potental and the scalar potental become zero because the eddy current has no normal component and the dvergence of the vector potental s zero n all places. As the result, the number of unknowns can be reduced. The magnetc flux densty B and the electrc feld E at the feld pont n the regon to be analyzed are gven by usng the rotaton of Eq. (7) and the gradent of Eq. (8) as follows: qo = 0 (22) Eqs. (21) and (22) gve the external magnetc flux densty Bo= k and the external electrc feld Eo=jo(y/2-x/2j) whch arse from a large crcular loop current. h E v m VI 8 V z Fg. 1 Conductng sphere model. :o7 B=(VxA) =Is [-[(Axn').V)V'@+k2(Axn')@ + V@x [(V'xA)xn') -jop~*cpv@xn']ds E m 1. 0 E.- 1- _' JaA-(vY') =Is (cpo(a,n')v'@-jo(axn')xv'@ +jw(v'xa)xn'@ +k2cp+n' -V@(V'cp.n')+cp(n'.V)V'@]dS COMPUTATION RESULTS In order to verfy the applcablty of the proposed boundary element method, a conductng sphere model n an unform alternatng magnetc feld as shown n Fg. 1 was chosen as a computaton model whch can be solved theoretcally. The conductng sphere model s an axsymmetrc model but the computaton of rhe model was pcrforrned as a three-dmensonal model. The number of -8-6 t Fg. 2 Dstrbutons of the magnetc flux densty and the electrc feld along x-axs, (a) magnetc flux densty, (b) electrc feld.

4 456 Fgure 5 shows a conductng cube model n unform alternatng magnetc feld whch s a truly threedmensonal model. The external potentals were gven by Eqs. (21) and (22). The number of trangular elements s 216 on one eghth part of the cube surface. Fgure 6 shows the dstrbuton of the eddy current densty on the x-y. y-z. x-z planes for 50(Hz). Fgure 7 shows the dstrbuton of the magnetc flux densty on the y-z plane. The computaton results agree wth those of the boundary element method usng magnetc flux densty and electrc feld as unknowns[5]. Equ-value lnes of the normal component of the vector potental on the boundary between the conductor and ar are shown n Fg. 8. Large values appear at the locatons where the normal components of the external vector potental are large. Therefore the scalar potental s nduced so that the normal component of the electrc feld whch s expressed by -joa.n-vpn becomes zero. Fg. 3 Computaton results of the vector potental on the boundary surface, (a)real part, (b)magnary part. Fg. 5 Conductng cube model. Fg. 4 Computaton results of the tangental component of the magnetc flux densty, (a)real part, (b)magnary part. Fg. 6 Eddy current dstrbutons on the x-y, y-z, x-z planes, (a) real part, (b) magnary part.

5 CONCLUSION I X b - Y --c- 0.2 (T) The boundary element method usng the magnetc vector potental was proposed, and the formulaton and the computaton results were descrbed. The conclusons can be summarzed as follows: 1) For the three-dmensonal problems, the tangental and normal components of the vector potental, the tangental components of the magnetc flux densty whch s gven by the curl of the vector potental and the scalar potental are defned as unknown varables on the boundares. 2) Usng the approxmaton that the conductvty of conductor s very large n comparson wth the conductvty of ar, the normal component of the gradent of the scalar potental on the boundares can be related to the normal component of the vector potental and unknown varables can be reduced. In ths case, the scalar potental s nduced so that the normal component of the electrc feld becomes zero. 3) By usng the computaton results, the applcablty of the proposed method were verfed.! \ I \, I Fg. 7 Magnetc flux densty dstrbutons (a) real part, (b) magnary part. (b) on the y-z plane, REFERENCES [l] A. J. Poggo and E. K. Mller: "Integral Equaton Soluton of Three-Dmensonal Scatterng Problems," Comuuter es for Electromagnetcs, Edtor: R. Mttra, Pergamon Press, [2] W. R. Hodgkns and J. F. Waddngton: "The Soluton of 3- dmensonal Inducton Heatng Problems Usng An Integral Equaton Method," JEEE Transactons on Maenetcs, Vol. Mag-18, pp , March, 1982 [3] I. D. Mayergoyz: "Boundary Integral Equatons of Mnmum Order for the Calculaton of Three- Dmensonal Eddy Current Problems," IEEE Transactons on Mapnetcs, Vol. Mag-18, pp , March, 1982 [41 T. Morsue and M. Fukum: "3-D Eddy Current Calculaton Usng The Magnetc Vector Potental," on Mametcs, Vol. 24, No. 1, pp , January, 1988 [5] H. Tsubo and T. Msak: "Three-Dmensonal Analyss of Eddy Current Dstrbutons by the Boundary Element Method Usng Vector Varables," IEEE Transactm on Magnetcs, Vol. Mag-23, pp , September, 1987 [6] J. A. Stratton: Electromaenetc Theory, MacGraw Hll, New York, 1941 Fg. 8 Equpotental lnes of the normal component of the vector potental on the boundary, (a) real part, (b) magnary part.