COMPEL 28,4. Ioan R. Ciric Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, Canada, and

Transcription

1 The current ssue and full text archve of ths journal s avalable at COMPEL 28,4 964 Modellng eddy currents n thn shelds Florea I. Hantla Department of Electrcal Engneerng, Poltehnca Unversty of Bucharest, Bucharest, Romana Ioan R. Crc Department of Electrcal and Computer Engneerng, The Unversty of Mantoba, Wnnpeg, Canada, and Augustn Moraru and Mha Marcaru Department of Electrcal Engneerng, Poltehnca Unversty of Bucharest, Bucharest, Romana Abstract Purpose The purpose of ths paper s to present a smplfed rgorous mathematcal formulaton of the problem of electrc currents nduced n thn shelds wth holes yeldng more effcent numercal computatons wth respect to avalable methods. Desgn/methodology/approach A surface ntegral equaton satsfed by the current densty was constructed, whch s, subsequently, represented at any pont by lnear combnatons of novel vector bass functons only assocated wth the nteror nodes of the dscretzaton mesh, such that the current contnuty s everywhere nsured. The exstence of the holes n the sheld s taken nto account by assocatng only one surface vector functon wth each hole. A method of moments s then appled to compute the scalar coeffcents of the vector functons employed. Fndngs It was found that the nduced current dstrbuton for shelds wth holes havng the complexty of real world structures can be determned wth a satsfactory accuracy utlzng a moderate sze processor notebook n a tme of the order of mnutes. Orgnalty/value The orgnalty of the proposed method conssts n usng specalzed surface vector functons only assocated wth ndvdual nteror nodes of the sheld, ts multply connected structure beng effcently accounted for by ntroducng one unknown for each hole, nstead of unknowns for every node along the hole contours. The method presented s straghtforward and hghly effcent for mathematcal analyss of thn shelds wth holes, and of other physcal felds n the presence of multply connected surface structures. Keywords Eddy currents, heldng, Integral equatons, Method of moments Paper type Research paper COMPEL: The Internatonal Journal for Computaton and Mathematcs n Electrcal and Electronc Engneerng Vol. 28 No. 4, 2009 pp q Emerald Group Publshng Lmted DOI / Introducton There are numerous methods developed specfcally for the analyss of the quasstatonary steady-state and transent magnetc felds assocated wth nduced metallc shelds. These nclude varous fnte dfference and fnte element technques when takng nto account the fnte thckness of the sheld and, also, surface ntegral equaton formulatons for thn shelds. When the sheld thckness s much smaller than the rest of ts lnear dmensons, applcaton of the usual fnte element method s not adequate. To avod dscretzng the regon outsde the sheld, a hybrd fnte element boundary element method can be employed (Felzan and Marade, 1997), whch can also easly be adapted for nonlnear ferromagnetc shelds. However, the fne

2 dscretzaton mesh nsde the sheld yelds large system matrces that are not so well-condtoned. Dscretzng the regon nsde the conductng materal can be avoded by modellng the shelds usng a surface mpedance that takes nto account the relatonshp between the electromagnetc feld quanttes on ther two sdes (Tugulea and Fluerasu, 1974; Felzan and Marade, 1999; Buccella et al., 2005). For conductng shelds wth a thckness much smaller than the depth of penetraton of the electromagnetc feld, the densty of the nduced electrc current can be consdered to be constant along the perpendcular between the sheld sdes and dstrbuted n the form of a current sheet whose densty vector s tangentally orented. Ths densty can be expressed n terms of a surface scalar functon such that the current contnuty condtons are fulflled, as shown n Kameary (1981), where a Lagrangan equaton formulaton usng as unknowns the nodal values of ths scalar functon has been presented for shelds wth no holes. Recently, Alotto et al. (2007), a planar electrc network was constructed to model the nduced sheld whose surface had been approxmated by plane trangular surface elements, the unknowns beng loop currents assocated wth the crcut nodes. In the present work, the current sheet densty nduced n thn shelds s determned as the soluton of a surface ntegral equaton. We employ a polyhedral mesh wth trangular surface elements for the conductng part of the sheld and use surface vector functons assocated wth the mesh nodes to express the current densty. The current contnuty condtons are satsfed over the surface elements, as well as for the edges between adjacent surface elements and for the edges along the sheld boundary ncludng the contours of the holes present n multply connected sheld structures. The mportant features of the proposed method consst n a small number of unknowns, namely, equal to the sum of the number of nteror nodes and of the number of holes, and n an effcent modellng n the case of multply connected shelds. Modellng eddy currents n thn shelds Current sheet ntegral equaton As n the case of 3D structures n an unbounded free space, the tme-harmonc ntegral equaton satsfed by the surface densty J s of the current nduced over the surface of a nonmagnetc thn sheld s derved n the form: J s ðr 0 Þ r s J s ðrþþjl d 0 ¼ 2j2pf A 0 ðrþ 2 7VðrÞ ð1þ R p where r s ¼ r=d s the surface resstvty, wth D beng the sheld thckness, j ; ffffffffffff 21, l ; f m 0 =2 wth f the frequency and m 0 the permeablty of free space, R ¼jr 2 r 0 j wth r and r 0, respectvely, the poston vectors of the observaton pont and of the source pont, A 0 s the magnetc vector potental due to external sources, and 27V s the electrc scalar potental component of the electrc feld ntensty. The boundary condton on s n J s ¼ 0, where n s the unt vector of the local normal drecton. As shown n the next secton, n the proposed formulaton the presence of 27V n equaton (1) can be gnored. 3. Numercal soluton method The current sheet on s modelled by employng a dscretzaton mesh and the surface current densty s represented n terms of surface vector functons assocated to the mesh nodes. In ths paper, s approxmated to be a polyhedral surface wth

3 COMPEL 28,4 966 suffcently small faces n the form of plane trangles. To each node that s not on the outer contour of we assocate a surface vector functon U, such that on each face p contanng the node (Fgure 1): U ð pþ ¼ 1 l ð pþ ð2þ 2 p where l ð pþ s the length vector along the edge of the face p that s opposed to the node, orented as shown n Fgure 1, and p s the area of the face p. The functon U s null over all the surface elements whch do not contan the node. It can easly be seen that the flux of U ð pþ through the edge L j s: U ð pþ q p dl ¼ 21 ð3þ L j q p beng the unt vector n the plane of the face p perpendcular to the edge L j n the drecton shown n Fgure 1. The same value of the flux through L j s obtaned from the adjacent face. Therefore, expressng the surface current densty as a lnear combnaton of the functons U defned by equaton (2) ensures the current contnuty through all the nteror edges of the polyhedral surface; moreover, snce the functons U are constant over each face, ther normal components are also contnuous through all the nteror edges and, thus, the zero dvergence condton for the current densty s satsfed everywhere over the dscretzed. Therefore, we represent J s over n the form: J s ðrþ < XN ¼1 a U ðrþþ XN h m¼1 b m W m ðrþ where the functons U are assocated wth the N nteror nodes of, whle the functons W m are assocated wth the contours of the N h holes of the sheld. No functon W m needs to be assocated wth the outer contour of. For each hole m, W m s defned as: ð4þ W m ðrþ ¼ [{m} X U ðrþ ð5þ {m} representng the set of nodes on the contour of the hole m. On each surface element p wth one of ts edges on the contour of the hole m, say the edge between the nodes a and b, W m has the expresson: W ð pþ m ¼ U ð pþ a þ U ð pþ b ¼ 1 l ð pþ ab 2 p ð6þ n p Fgure 1. A node and assocated surface elements L k q p L j (p) k (p) l j

4 where l ð pþ ab s the length vector along the edge consdered. Thus, the flux of W ð pþ m through and ts normal component to the edge on the contour are equal to zero, whch ensures that the condtons for the current densty at all the ponts on the edges along the hole contours are also satsfed. The unknown coeffcents a and b m n equaton (4) are determned by takng the nner products of the two sdes of equaton (1) wth U n, n ¼ 1; 2;...; N, and wth W n 0, n 0 ¼ 1; 2;...; N h, and then ntegratng over the dscretzed. Wth equaton (4) substtuted n equaton (1), one obtans the followng system of N þ N h algebrac equatons n a and b m : Modellng eddy currents n thn shelds 967 wth: X N ¼1 A n a þ XN h m¼1 B nm b m ¼ C n ; n ¼ 1; 2;...; N þ N h ð7þ A n ; a n þ jla 0 n ; B nm ; b nm þ jlb 0 nm ð8þ where: for n ¼ 1; 2;...; N and: a n ¼ b nm ¼ r s ðrþu n ðrþ U ðrþd r s ðrþu n ðrþ W m ðrþd a 0 n ¼ 1 R U nðrþ U ðr 0 Þdd 0 0 b 0 nm ¼ 1 R U nðrþ W m ðr 0 Þdd 0 0 C n ¼ 2j2pf U n ðrþ A 0 ðrþd a n ¼ b nm ¼ r s ðrþw n 0ðrÞ U ðrþd r s ðrþw n 0ðrÞ W m ðrþd a 0 n ¼ 1 R W n 0ðrÞ U ðr 0 Þdd 0 0 b 0 nm ¼ 1 R W n 0ðrÞ W mðr 0 Þdd 0 0 C n ¼ 2j2pf W n 0ðrÞ A 0 ðrþd ð9þ ð10þ ð11þ ð12þ ð13þ ð14þ ð15þ ð16þ ð17þ ð18þ

5 COMPEL 28,4 968 for n ¼ N þ 1; N þ 2;...; N þ N h ; wth n 0 ; n 2 N. The system (7) has complex coeffcents and only N þ N h unknowns, a and b m, where N s the number of nteror nodes and N h the number of holes. Its soluton allows the calculaton of the surface current densty n equaton (4) at any pont on. It should be notced that the electrc scalar potental component 27V n equaton (1) does not contrbute to the surface ntegral over the dscretzed. Indeed, for each face p one has (Fgure 1): ð pþ U ð pþ ð7vþd ¼ 1 2 p ¼ 1 I ð pþ 2 p ð pþ ¼ 2 1 L j l ð pþ ð7vþd Vðn p l ð pþ Þ dl Vdl þ 1 L k Lk Vdl ð19þ L j where ð pþ s the contour of ( p) andn p s the unt normal to the face p. Thus: X U ð7vþd ¼ U ðpþ ð7vþd ¼ 0 ð20þ p2{} ðpþ snce the contrbutons of the last two ntegrals n equaton (19) are exactly compensated by the correspondng contrbutons from the adjacent faces p belongng to the set {} of faces contanng the node. mlarly: W m ð7vþd ¼ 0 ð21þ 4. Illustratve results Consder frst an mm nonmagnetc conductng plane plate of thckness D ¼ 1 mm wth four mm holes, as shown n Fgure 2. The conductng materal has a resstvty r ¼ 0: Vm. The nducng magnetc feld s produced by a coaxal square turn carryng a snusodal current of 100 A rms wth f ¼ 50 Hz, whose sdes of 120 mm are parallel to those of the plate. The turn plane s at a dstance of 10 mm from the plate. The mesh employed s shown n Fgure 2, and nduced current lnes are shown n Fgure 3. All the current lne sketches presented n ths secton are taken at the moment n tme when the nduced current densty has relatvely hgh values, correspondng to a phase of 2pft ø 176 degrees of the current n the nducng turn. In another computatonal example, a thn sheld wth four holes occupes a porton of the surface of a crcular cone of equaton x 2 þ y 2 2 4z 2 ¼ 0, the orgn of the coordnate system beng at the sheld centre, wth the z-axs along the sheld s axs of symmetry, as seen n Fgure 4, where the generated mesh s also shown. The thckness of the sheld D ¼ 1 mm and ts projecton on a plane z ¼ const has the lnear dmensons of the plate consdered n the frst example. The nducng feld s generated by a coaxal square turn that carres a current of 200 A rms wth f ¼ 50 Hz. It has sdes

6 Modellng eddy currents n thn shelds 969 Fgure 2. Thn conductng plane plate wth holes and trangular mesh Fgure 3. Current lnes on the sheld n Fgure 3 of 40 mm and ts plane s at a dstance of 30 mm from z ¼ 0, as shown n Fgure 5. Current lnes nduced n the sheld are shown n Fgure 5. In a thrd example, the sheld occupes a porton of the surface of a parabolod of revoluton of equaton x 2 þ y 2 þ 0:06z 2 0: ¼ 0 and has seven holes, as

7 COMPEL 28,4 970 Fgure 4. Thn concal sheld wth holes and trangular mesh Fgure 5. Inducng turn and current lnes on the concal sheld shown n Fgure 6, where the mesh employed s also shown. D ¼ 1 mm and the projecton of the sheld on a plane z ¼ const s a dsk of a dameter of 120 mm wth seven dentcal 20 mm dameter crcular holes, one of them beng concentrc wth the dsk and the other sx placed unformly wth ther centres on a 80 mm dameter concentrc crcle. The nducng turn has the same dmensons and carres the same

8 Modellng eddy currents n thn shelds 971 Fgure 6. Thn parabolodal sheld wth holes and trangular mesh current as n the prevous examples. It s placed n the plane z ¼ 30 mm as shown n Fgure 7, where nduced current lnes n the sheld are also sketched. 5. Conclusons The method presented n ths paper s based on the surface ntegral equaton satsfed by the current sheet densty nduced n thn metallc shelds whch s solved numercally by usng a Galerkn method of moments wth a novel choce of the set of vector bass functons such that the current contnuty s ensured everywhere. Fgure 7. Inducng turn and current lnes on the sheld n Fgure 6

9 COMPEL 28,4 972 One surface vector functon s attached to each nteror mesh node and the exstence of the holes n the sheld s effcently accounted for by assocatng only one surface vector functon wth each hole, correspondng to only one unknown ntroduced nto the resultant matrx equaton nstead of unknowns for ndvdual nodes along the hole contour. Thus, the total number of unknowns s equal to the sum of the number of nteror nodes of the trangular mesh used and the number of holes n the sheld. In the case of planar shelds, the coeffcents (9)-(18) of the system of equatons (7) can smply be determned by an exact analytc ntegraton. For the frst case consdered (Fgure 2), wth a mesh of 1,403 nodes and 2,501 trangular surface elements, the computaton of the nduced current sheet densty dstrbuton requred a tme of only 100 s when usng a dual-core 2.5 GHz processor notebook. For curved shelds, the elements of the matrx of the system n equaton (7) have to be calculated by numercal ntegraton, the computaton tme beng greater and dependng on the mposed accuracy. Most of ths tme s spent on calculatng the double ntegrals n equatons (11), (12) and (16), (17), where the number of ntegraton elements on the mesh plane trangles also depends on ther sze and on the dstance between them. The current dstrbuton shown n Fgure 5 was obtaned n about 10 mn usng the same notebook and a mesh of 1,403 nodes and 2,501 trangular elements, whle that n Fgure 7 requred 8 mn for a mesh of 1,216 nodes and 2,087 trangular elements. Once the current dstrbuton s determned, the magnetc feld ntensty at varous ponts due to the nduced sheld s obtaned by smple superposton of the felds produced by the current densty of the plane trangular surface elements. References Alotto, P., Guarner, M. and Moro, F. (2007), A boundary ntegral formulaton on unstructured dual grds for eddy-current analyss n thn shelds, IEEE Transactons on Magnetcs, Vol. 43, pp Buccella, C., Felzan, M., Marade, F. and Manz, G. (2005), Magnetc feld computaton n a physcally large doman wth thn metallc shelds, IEEE Transactons on Magnetcs, Vol. 36, pp Felzan, M. and Marade, F. (1997), Fnte-dfference tme-doman modelng of thn shelds, Proceedngs of Internatonal ymposum on Electromagnetc Compatblty, Austn, TX, August, pp Felzan, M. and Marade, F. (1999), Feld analyss of penetrable conductve shelds by the fnte-dfference tme-doman method wth mpedance network boundary condtons (INBC s), IEEE Transactons on Electromagnetc Compatblty, Vol. 41, pp Kameary, A. (1981), Transent eddy current analyss on thn conductors wth arbtrary connectons and shapes, Journal of Computatonal Physcs, Vol. 42, pp Tugulea, A. and Fluerasu, C. (1974), The complex surface conductvty and permeablty n the study of a.c. n thn wall conductors, Rev. Roum. c. Techn.-Electrotechn. et Energ., Vol. 27 No. 4, pp About the authors Florea I. Hantla receved Electrcal Engneer degree n 1967 and PhD degree n 1976, both from the Poltehnca Unversty of Bucharest, where he s currently a Professor and the Head of the Department of Electrcal Engneerng. Hs teachng and research nterests are n

10 electromagnetc feld analyss wth non-lnear meda and numercal methods. Florea I. Hantla s the correspondng author and can be contacted at: hantla@elth.pub.ro Ioan R. Crc s a Professor of Electrcal Engneerng at the Unversty of Mantoba, Wnnpeg, MB, Canada. Hs major research nterests are n the mathematcal modellng of electromagnetc felds, feld theory of specal electrcal machnes, dc corona onzed felds, methods for wave scatterng and dffracton problems, transents, and nverse problems. Augustn Moraru, PhD started hs actvtes over 50 years ago, mmedately after graduatng n 1951 wth Mertorous Dploma from the Electrcal Faculty of the Polytechnc Insttute of Bucharest, and s reflected n the publshed 20 books, 163 papers n journals and conference volumes, and 86 reports of scentfc research grants. Mha Marcaru graduated n 2001 from the Faculty of Electrcal Engneerng, Poltehnca Unversty, obtaned PhD n Electrcal Engneerng from the same unversty n 2007, where he s currently workng as an Assstant Professor. Hs research nterests are manly n electromagnetc feld computaton (ntegral and hybrd methods). Modellng eddy currents n thn shelds 973 To purchase reprnts of ths artcle please e-mal: reprnts@emeraldnsght.com Or vst our web ste for further detals: