ARWt010 Advanced Reseach Wksh n Tansfmes. 3-6 Octbe 010. Santiag de Cmstela Sain Advanced Cmutatinal Tls f Wund Ce Distibutin Tansfme N Lad Analysis Themistklis D. KEFALAS and Antnis G. KLADAS Schl f Electical and Cmute Engineeing, Natinal Technical Univesity f Athens 9 In Plytechneiu Steet, Athens 15780, Geece Phne: (+30) 10 77 336, fax: (+30) 10 77 336, e mail: thkefala@cental.ntua.g Abstact The ae deals with the accuate eesentatin f laminated wund ces with a lw cmutatinal cst using D and 3D finite element (FE) methd. An invese anisty mdel is develed in de t mdel laminated wund ces. The invese anisty mdel was integated t the D and 3D FE methd. A cmaisn between D and 3D FE techniques was caied ut. FE techniques wee validated by exeimental analysis. In the case f n lad eatin f wund ce tansfmes bth D and 3D FE techniques yields the same esults. Cmuted and exeimental n lad lsses agee within % t 6%. The iginality f the ae cnsists in the develment f an invese anisty mdel, secifically fmulated f laminated wund ces, and in the effective eesentatin f electical steels using a cmsite single valued functin. y using the afementined techniques the FE cmutatinal cst is minimised, the numeical stability and cnvegence f the Newtn Rahsn iteative methd ae imved, and the 3D FE analysis f wund ces is endeed actical. Keywds Electmagnetic analysis, Finite element methds, Magnetic ces, Pwe tansfmes I. INTRODUCTION Nwadays an eve inceasing numbe f eseaches and tansfme manufactues evaluate key eatinal aametes f tansfmes by using advanced numeical mdels, Canganu Cetu [1], Henández [], St [3], Kefalas [4], and Kefalas [5]. The n lad lss analysis f wund ce tansfmes is based n the exact evaluatin f the flux density distibutin f wund ces. In de t cmute accuately the flux density distibutin f laminated wund ces, each individual steel sheet and the ai vanish cmsite between successive sheets must be mdelled. This aach was fllwed ecently f the finite element (FE) analysis f tidal ces cnsisting f a small numbe f steel sheets, Zuek [6]. Nevetheless, FE analysis f wund ces equies substantial cmutatinal ecuses. Tyical laminated wund ces ae cnstucted f hundeds steel sheets. The length and width f laminatins anges fm 0.5 m t 1.5 m and 0.1 m t 0.6 m esectively. On the the hand, steel sheet thickness anges fm 0.5 mm t 0.3 mm and the thickness f the ai vanish cmsite between tw successive steel sheets anges fm 10 μm t 8 μm. Cnsequently, the detailed mdelling f a tyical wund ce equies a mesh size f the de 10 8 t 10 9 elements in the D FE case, and 10 11 t 10 1 elements in the 3D FE case, Kefalas [7]. In the esent ae the afementined blem is addessed by elacing the laminated wund ce by a bulk mateial that accuately mdels the cmlicated laminated ce stuctue, Kefalas [7]. The eties f the bulk mateial ae eesented by an invese anisty mdel that can be integated t magnetic vect tential (MVP) based D and 3D FE fmulatins. The advantage f the invese anisty mdel esented in this ae is its bustness and cmutatinal efficiency since it is secifically fmulated f wund ce tansfmes. In cntast, anisty mdels aeaing in the liteatue ae develed f stack ce tansfmes, Sande [8], and ae nt suitable f wund ce tansfmes since wund ce and stack ce tlgies diffe significantly. Accuate mdelling f the electical steel nnlineaity unde all excitatin cnditins, including heavily satuated cnditins, is taken int cnsideatin by using a cmsite single valued functin. And in this case the auths fmulated the afementined cmsite functin s that it can be integated diectly t MVP based D and 3D FE fmulatins and t the Newtn Rahsn iteative technique. 1
ARWt010 Advanced Reseach Wksh n Tansfmes. 3-6 Octbe 010. Santiag de Cmstela Sain Figue 1. H cuves f vaius angles t the lling diectin Figue. Tajecty f in the q lane Figue 3. Tyical wund ce and, q, z, diectins Figue 4. Ellise in the q lane II. LAMINATED WOUND CORE REPRESENTATION An accuate eesentatin f the wund ce is achieved by cnsideing the in laminated mateial as hmgeneus and anistic media at the level f finite elements. An ellitic anisty mdel is best suited f the wund ce tansfme in cntast with the stack ce tansfme, Sande [8]. The anisty mdel develed in the esent ae is an invese ne i.e., in cntast with the anisty mdel esented by Kefalas in [7], it is based n the assumtin that the flux density has an ellitic tajecty f the mdulus f magnetic field intensity H cnstant. As a esult, it can be integated t MVP based D and 3D FE fmulatins whee in each element f the FE mesh can be diectly evaluated by the cul f the magnetic vect tential A. A simlified gahical inteetatin f this assumtin is given in Figs. 1,. Fig. 1 deicts H chaacteistics f gain iented steel f vaius angles t the lling diectin. F cnstant magnetic field intensity the flux density tends t decease as the angle t the lling diectin inceases. y jecting the values f the flux density f diffeent diectins f the field t the q lane, whee and q is the diectin tangential and nmal t the lling diectin esectively, a cuve is fmed as shwn in Fig. In the case f laminated wund ces, this cuve can be aximated by an ellise and this aximatin leads t an e that aely exceeds a few ecentage ints. This is due t the fact, that the eluctivity f the wund ce alng diectin q i.e., nmal t the lling diectin f the electical steel, is thee t fu des f magnitude lage than the eluctivity alng diectin i.e., tangential t the lling diectin, as shwn in Fig. 3. Theefe, if v is the magnetic eluctivity tangential t the electical steel lling diectin, v q is the eluctivity nmal t the lling diectin, and is the ati f the ellise semi axes t the ellise q semi axes, shwn in Fig 4, then
ARWt010 Advanced Reseach Wksh n Tansfmes. 3-6 Octbe 010. Santiag de Cmstela Sain vq = v, >1. (1) Fig. 4 illustates an ellise in the q lane f an abitay value f magnetic field intensity, whee ( 0 ), ( 90 ) ae the esective values f the flux density tangential and nmal t the lling diectin. The equatin f the ellise f Fig. 4 is given by () and the ati f the ellise semi axes is given als by (3). Substituting (3) int () yields ( 0 ) + ( 90 ) = 1 () q ( 0 ) ( ) = 90 (3) q = ( 0 ) + (4) F an abitay angle θ t the lling diectin and fm Fig. 4, it fllws that the tw cmnents 1, q1 f the flux density ae given by The int ( ), 1 q1 = csθ, 1 sinθ (5) 1 q = satisfies als the equatin f the ellise. S by substituting (5) int (4) it fllws that ( θ ) = ( 0 ) 1+ ( 1) sin θ (6) In de t detemine the H chaacteistics f the hmgeneus anistic media f the laminated wund ce, ne can exess fm magnetic cicuit cncets the magnetic eluctivity tangential and nmal t the lling diectin by (7) and (8) esectively v [ c + ( c ) v ] = v0 v 1 (7) v q sf sf ( csf ) 0 = csf v 0 v + 1 v (8) whee v 0 is the eluctivity f ai, v is the elative eluctivity f the electical steel btained by the nmal magnetisatin cuve f the electical steel and c sf is the stacking fact f the wund ce. The eluctivity v z thgnal t the q lane, shwn in Fig. 3, is taken t be equal t the eluctivity alng the lling diectin f the electical steel v. Even thugh the afementined assumtin is nt tue it des nt affect the flux density distibutin evaluatin f wund ces due t the axial symmety f the FE blem. Finally, due t the gemety f the wund ce, shwn in Fig. 3, the eluctivity tens must be tated t a diffeent cdinate system deending n the lcatin f the aea vlume f the D and 3D FE mdel esectively. III. ELECTRICAL STEEL REPRESENTATION UNDER HEAVILY SATURATED CONDITIONS If the slutin f nnlinea electmagnetic blems by the FE methd is based n the MVP fmulatin, sft magnetic mateials ae exessed by the elative eluctivity vesus squaed flux density cuve v and nnlineaity is tackled by the Newtn Rahsn iteative methd. 3
ARWt010 Advanced Reseach Wksh n Tansfmes. 3-6 Octbe 010. Santiag de Cmstela Sain The evaluatin f v f whee v is the elative eluctivity value f i eluctivity is exeimentally knwn, and n, is caied ut by cubic slines intelatin f tabulated sets f (, i ) i, v i, n is the maximum squaed flux density f which elative 0 i n. In de t cmute v f n the cnventinal aach is t use a linea a quadatic functin t extalate the magnetisatin cuve Fujiwaa [9]. In the case f the linea extalatin functin as, v µ 0. The afementined is f cuse incect. Cnsideing the cnstitutive equatin f femagnetic mateials it is easily veified that the elative eluctivity v f a sft magnetic mateial as v 1. tends t unity ( ) As a esult, it fllws that the cnventinal sft magnetic mateial eesentatin duces eneus esults in cases whee the excitatin level is high. In the esent ae an altenative extalatin functin is sed f (9), whee a, b ae aametes f the extalatin functin. The dimensins f a aamete ae n [ T ] wheeas b is a dimensinless aamete. ( ) = ex[ ( a b) ] v 1 + (9) The functin f (9) as well as its sle is cntinuus. Als, it can be seen fm (9) that as, v tends t unity since the exnent tem tends t ze. Thus, the sed extalatin functin yields cect esults and it can be integated t the Newtn Rahsn iteative technique and t the FE methd. The values f the tw aametes ae btained by satisfying the fllwing tw cnditins. 1. The fist deivative f functin v at. Functin v at must be cntinuus. n n must be cntinuus. Since the sed eesentatin f electical steels is based n a cntinuus single valued cmsite functin with a cntinuus sle, it can be integated diectly t the Newtn Rahsn iteative technique and the FE methd. Als, the sed eesentatin esents advantages ve cntemay aaches using a quadatic functin t extalate the magnetisatin cuve Fujiwaa [9]. The afementined aaches equie the evaluatin f fu thee aametes wheeas the sed extalatin methd invlves nly tw aametes. IV. FE ANALYSIS AND EXPERIMENTAL VERIFICATION The invese ellitic anisty mdel was integated t the MVP based D and 3D nnlinea FE methd. A cnventinal cedue was used since the invese ellitic anisty mdel is secifically fmulated f the MVP based FE fmulatin, Silveste [10]. Als, the sed cmsite single valued functin f Sectin III was integated t the MVP based D and 3D FE methd as well as the Newtn Rahsn methd using the exact same cedue used f the integatin f the cnventinal eesentatin f electical steels, Silveste [10]. The tw aametes f the sed extalatin functin, a and b, ae evaluated nly nce f each electical steel and thee is n need t e evaluate them at evey iteatin ste f the Newtn Rahsn iteative methd. A D and 3D FE nnlinea ackage based n the magnetic vect tential and Newtn Rahsn methd has been develed by the auths. The secific FE ackage cnsists f a e cessing cde, a fist de tiangle and tetahedal mesh geneat, the nnlinea D and 3D FE slves, and a gahics st cess. Figs. 5 t 8 shw the D and 3D eak flux density vect lt that cesnds t magnetisatin levels fm 0.3 T t 1.75 T f a wund ce cnstucted f the Hi gain iented electical steel M OH 0.7 mm. The thickness f the ce is 51.3 mm wheeas the ce windw height, width, and length ae 183 mm, 57 mm, and 190 mm. The simulated magnetisatin f the wund ce t an abitay level is achieved by using magnetstatic FE analysis and a systematic iteative cedue based n the bisectin technique. Tyically 0 4
ARWt010 Advanced Reseach Wksh n Tansfmes. 3-6 Octbe 010. Santiag de Cmstela Sain Figue 5. Detail f D flux density vect lt ( = 0.3 T) Figue 6. Detail f D flux density vect lt ( = 0.6 T) Figue 7. Detail f D flux density vect lt ( = 1.3 T) Figue 8. 3D flux density vect lt ( = 1.75 T) Table I. Cmaisn f Cmuted and Calculated N Lad Lss f Wund Ce Magnetizatin level (T) Cmuted D n lad lss (W) Cmuted 3D n lad lss (W) Measued n lad lss (W) 1.14 18.69444 18.78791 19.3538379 1.5 33.30397 33.43719 34.6901545 1.67 41.8981 4.109 40.933181 1.7 46.53759 46.85404 49.43109859 iteatins ae enugh in de t detemine the cuent density which duces the desied magnetisatin level with an e f the de f 10 4. The evaluatin f the eak flux density distibutin with the FE methd is used in cnjunctin with the exeimentally detemined lcal secific ce lss SCL, f the evaluatin f the wund ce n lad lss. The auths develed tw st cessing algithms f the evaluatin f the n lad lss in the D and 3D case esectively. In bth cases the n lad lss f each element that belngs t the ce dmain i.e., tiangle in the D case and tetahedal in the 3D case, is cmuted based n the lcal secific ce lsses that cesnd t the eak flux density f the element, the ce stacking fact c sf, the electical steel density d ms, and the vlume V f each element, whee in the D case the vlume is equal t the duct f the aea f each element with the length f the ce. The exeimental setu used f the n lad lss evaluatin f wund ces cnsists f a 0 tun excitatin cil which is sulied with a sinusidal vltage wavefm fm a gammable ac we suly, in de t magnetise the wund ce. N lad cuent and vltage ae catued using a cuent be based n the Hall effect, an active high vltage diffeential be, and a Natinal Instuments NI6143 data acquisitin cad. Analysis f the catued data was caied ut using LabVIEW sftwae, Lizs [11], Lizs [1]. Table I summaises the cmuted and measued n lad lss f the tested wund ce f diffeent wking inductin atings. Calculated and measued n lad lsses agee within % t 6%. 5
ARWt010 Advanced Reseach Wksh n Tansfmes. 3-6 Octbe 010. Santiag de Cmstela Sain V. CONCLUSION y using the invese ellitic anisty mdel in de t mdel the laminated wund ce thee is n need t mdel hundeds electical steel sheets and the ai vanish cmsite between them. The esulting FE mdel is vey simle and duces accuate esults. The mesh size in the D case is educed fm 10 8 t 10 9 elements t, 10 3 t 10 4 elements and cnsequently the cmutatinal cst is minimised. Als, the afementined anisty mdel makes actical the 3D FE analysis f laminated wund ces by educing the mesh size fm 10 11 t 10 1 elements t, 10 4 t 10 6 elements. Even thugh, the cmutatinal efft f the 3D analysis is a multile f that f the D analysis, the esults btained by the D and 3D FE analysis agee within 0.3% t 0.7%. This is due t the inheent D symmety f wund ces. As a esult, it fllws that in the case f n lad analysis f wund ce tansfmes the 3D FE analysis is nt necessay. REFERENCES [1]. Canganu Cetu, J. Smajic, J. Ostwski, W. Renhat, and C. Magele, Sftwae Integated Slutin f Design Otimizatin f Industial Devices, IEEE Tansactins n Magnetics, Vl.44, N.6,.11 5, June 008. [] I. Henández, J. C. Olivaes Galván, P. S. Gegilakis, and J. M. Cañed, A nvel ctagnal wund ce f distibutin tansfmes validated by electmagnetic field analysis and cmaisn with cnventinal wund ce, IEEE Tansactins n Magnetics, Vl. 46, N. 5,. 151 58, May 010. [3] A. St, D. Sut, J. Tuwski, X. M. Lez Fenandez, and D. Cut, Sftwae f fast inteactive thee dimensinal mdeling f electmagnetic leakage field and magnetic shunts design in shell tye tansfmes, in Pc. ICEM, 008,. 1 6. [4] T. Kefalas and A. Kladas, FEM ackage f in lss evaluatin and minimizatin f tw gade laminatin wund ces, Junal f Otelectnics and Advanced Mateials, Vl. 10, N. 5,. 1197 10, May 008. [5] T. D. Kefalas and A. G. Kladas, Hamnic imact n distibutin tansfme n lad lss, IEEE Tansactins n Industial Electnics, Vl. 57, N. 1,. 193 00, Januay 010. [6] S. Zuek, F. Al Naemi, and A. J. Mses, Finite element mdeling and measuements f flux and eddy cuent distibutin in tidal ces wund fm electical steel, IEEE Tansactins n Magnetics, Vl. 44, N. 6,. 90 5, June 008. [7] T. D. Kefalas and A. G. Kladas, Rbust numeical analysis f wund ce distibutin tansfmes, in Pc. ICEM, 008,. 1 6. [8] H. V. Sande, T. nen, I. Pdleanu, F. Hentte, and K. Hameye, Simulatin f a thee hase tansfme using an imved anisty mdel, IEEE Tansactins n Magnetics, Vl. 40, N.,. 850 5, Mach 004. [9] K. Fujiwaa, T. Adachi, and N. Takahashi, A sal f finite element analysis cnsideing tw dimensinal magnetic eties, IEEE Tansactins n Magnetics, Vl. 38, N.,. 889 9, Mach 00. [10] P. P. Silveste and R. L. Feai, Finite Elements f Electical Enginees, ISN 0 51 44953 7 1996 Cambidge Univesity Pess. [11] G. Lizs, T. Kefalas, A. Kladas, T. Suflais, and D. Paaigas, Flux distibutin in single hase, Si Fe, wund tansfme ces, Junal f Magnetism and Magnetic Mateials, Vl. 30,. 874 7, 008. [1] G. Lizs, T. D. Kefalas, A. G. Kladas, and A. T. Suflais, Flux distibutin analysis in thee hase Si Fe wund tansfme ces, IEEE Tansactins n Magnetics, Vl. 46, N.,. 594 7, Febuay 010. 6