Powrd by TCPDF (www.tcpdf.org) This is an lctronic rprint of th original articl. This rprint may diffr from th original in pagination and typographic dtail. Upadhaya, Brijsh; Martin, Floran; Rasilo, Paavo; andgrubr, Paul; Blahcn, Anouar; Arkkio, Antro Modlling anisotropy in non-orintd lctrical stl sht using vctor Jils-Athrton modl Publishd in: Compl: th intrnational journal for computation and mathmatics in lctrical and lctronic nginring DOI: 0.0/-0-0-0 Publishd: 0/0/0 Documnt Vrsion Pr rviwd vrsion Plas cit th original vrsion: Upadhaya, B., Martin, F., Rasilo, P., andgrubr, P., Blahcn, A., & Arkkio, A. (0). Modlling anisotropy in non-orintd lctrical stl sht using vctor Jils-Athrton modl. Compl: th intrnational journal for computation and mathmatics in lctrical and lctronic nginring, (), -. https://doi.org/0.0/-0-0-0 This matrial is protctd by copyright and othr intllctual proprty rights, and duplication or sal of all or part of any of th rpository collctions is not prmittd, xcpt that matrial may b duplicatd by you for your rsarch us or ducational purposs in lctronic or print form. You must obtain prmission for any othr us. Elctronic or print copis may not b offrd, whthr for sal or othrwis to anyon who is not an authorisd usr.
Modlling anisotropy in non-orintd lctrical stl sht using vctor Jils-Athrton modl Journal: : Th Intrnational Journal for Computation and Mathmatics in Elctrical and Elctronic Enginring Manuscript ID Draft Manuscript Typ: Original Articl Kywords: Magntic anisotropy, Rotational magntic fild, Vctor hystrsis modl
Pag of 0 0 0 0 0 0 Modlling anisotropy in non-orintd lctrical stl sht using vctor Jils-Athrton modl Abstract Purpos Non-orintd lctrical stl prsnts anisotropic bhaviour. Modlling such anisotropic bhaviour has bcom a ncssity for accurat dsign of lctrical machins. Th main aim of th study is to modl th magntic anisotropy in th non-orintd lctrical stl sht of grad M00-0A using a phnomnological hystrsis modl. Dsign/mthodology/approach Th wll-known phnomnological vctor Jils-Athton hystrsis modl is modifid in ordr to corrctly modl th typical anisotropic bhaviour of th non-orintd stl sht, which is not dscribd corrctly by th original vctor Jils-Athrton modl. Th modification to th vctor modl is implmntd through th anhystrtic magntization. Instad of th commonly usd classical Langvin function, w introducd -D bi-cubic splin to rprsnt th anhystrtic magntization for modlling th magntic anisotropy. Findings Th proposd modl is found to yild good agrmnt with th masurmnt data. Comparisons ar don btwn th original vctor modl and th proposd modl. Anothr comparison is also mad btwn th rsults obtaind considring two diffrnt modification to th anhystrtic magntization. Originality/valu Th papr prsnts an original mthod to modl th anhystrtic magntization basd on projctions of th anhystrtic magntization in principal axis, and apply such modification to th vctor Jils-Athrton modl to account for th magntic anisotropy. Th rplacmnt of th classical Langvin function with th splin rsultd in bttr fitting. Th proposd modl could b usd in th numrical analysis of magntic fild in an lctrical application. Kywords Magntic anisotropy, Rotational magntic fild, Vctor hystrsis modl Papr typ Rsarch papr. Introduction Non-orintd (NO) lctrical stl shts hav bn found to b magntically anisotropic. Although thir anisotropic bhaviour is mitigatd compard with th grain orintd (GO) lctrical stl shts, accurat modl for th local magntic quantitis improv th prdiction of th global faturs for lctrical applications. Among divrs mthods for prdicting locally th isotropic hystrsis, th mathmatical Prisach modl and th phnomnological Jils-Athrton (J-A) modl rcivd a growing attntion ths last dcads. In ordr to account for th anisotropic bhaviour of lctrical stl shts, th improvmnt of th Prisach modl can b prformd ithr with a phnomnological bistroid modl (Vrnscu-Spornic t al., 000) or with a gnralizd Mayrgoyz modl (Mayrgoyz, 00; Dlala, 0; andgrubr t al., 0). Although th formr prsnts som discrpncis at low amplitud of th flux dnsity undr rotational fild, th divrs improvmnts of th lattr can accuratly rprsnt both altrnating and rotating dissipativ bhaviour, including th xcss fild, of th frromagntic matrials. In comparison with th Prisach modl, th J-A modl (Jils and Athrton, ; Jils and Athrton, ) sms intrsting in trms of simplicity and computational fficincy (Bnabou t al., 00). Morovr, both th altrnating and rotating aspcts could b modlld (Ivanyi, 000). Th implmntation of th J-A modl in finit lmnt mthod (Gyslinck t al., 00; Bnabou t al., 00) can also account for th xcss fild (Sadowski t al., 00). In ordr to account for th anisotropic bhaviour, Brgqvist () suggsts to rplac th scalar paramtrs of th J- A modl by tnsor paramtrs in th diffrntial quation rprsnting th displacmnt of th walls. In spit of improving th prdiction in th rolling and th transvrs dirction of th lamination, discrpncis ar obsrvd for th rotational loci. Lit t al. (00b) xtnd this improvmnt by considring anisotropic suscptibility for th anhystrtic magntization dfind by two Langvin functions. In spit of nglcting th cross-coupling trms in th diffrntial anhystrtic suscptibility tnsor, this improvd vctor J-A modl can corrctly rproduc both th altrnating loops and th rotating loci. Furthrmor, this modl will b dnotd th original anisotropic J-A modl. Albit Li t al. (0) xtnd this improvmnt by including th off-diagonal trms in th diffrntial anhystrtic suscptibility tnsor, vry paramtr of th modl is tabulatd with rspct to th amplitud and th orintation of th magntic flux dnsity which significantly affcts th simplicity of this anisotropic hystrtic modl. Although, th Langvin function holds a physical maning for rprsnting th anhystrtic bhaviour of an idalizd frromagntic matrial, th cavitis and impuritis within th magntic domains can affct its accuracy spcially if th alloy contains som porosity (Nél, ). For GO matrial, Ramsh and Jils () accounts for th uniaxial anisotropy with a Boltzmann distribution. Thir modl rquirs a numrical intgration which significantly affct th computational ffort. Modlling th anisotropic bhaviour for th anhystrtic magntic curv could b prformd with th conrgy or th nrgy dnsity. Dpnding on th formulation of th magntic problm, th conrgy dnsity could b mployd with magntic scalar formulations (Péra t al., ) whras th nrgy dnsity is mor adaptd for magntic vctor formulations (Bíró t al., 00; Chwastk, 0; Martin t al., 0). Th contours of
Pag of 0 0 0 0 0 0 th conrgy/nrgy dnsity can b modlld with a modifid llips for improving th accuracy of th anisotropic anhystrtic modl. Although, this approach can b asy to implmnt, th intrpolation of vry componnts of th fild with rspct to all th componnts of th magntization can significantly improv th accuracy of th anhystrtic curv (Enokizono and Soda, ; Martin t al., 0). Th aim of this papr is to prsnt a furthr improvmnt of th original anisotropic vctor J-A modl for modlling NO lctrical stl shts. Th modification consists in rplacing th Langvin anhystrtic curvs by th intrpolation of vry componnts of th anhystrtic magntization with rspct to all th componnts of th ffctiv fild. nc, only thr diagonal tnsors ar idntifid from altrnating flux dnsity masurmnts at 0 z in principal dirctions (Goričan t al., 000; andgrubr t al., 0). Th proposd improvmnt is thn compard with both th anisotropic J-A modl and th rotating flux dnsity masurmnts.. Mthodology. Vctor Jils-Athrton Modl Th vctor gnralization of th original scalar J-A modl has bn introducd by Brgqvist (). Th chang in magntization (M) following any chang in th magntic flux dnsity (B) is givn by th diffrntial quation (Lit t al., 00b): i) If ( χ f d ) > 0 ii) If ( χ f d ) 0 whr = k ( M M) f an dm = I+ f f f ( α) + c ξ ( α) f f f + c ξ d µ χ χ χ I I B χ χ χ 0 χ χ χ χ χ χ () dm = I+ c ξ ( I α) c ξ db () µ 0 χ is an auxiliary vctor quantity. c, k and α ar modl paramtrs, ξ is th diffrntial anhystrtic suscptibility, I rprsnts th unit matrix, and M an rprsnts th anhystrtic magntization. In th vctor J-A modl th anhystrtic magntization vctor is a function of th ffctiv fild, ( = + α ) and is assumd to b paralll with it (Gyslinck t al., 00; Lit t al., 00b). M = M i+ M j= M an anx any an ( ) M, anx anx x y ξ = () any any x y Givn any sigmoid function to xprss th anhystrtic magntization, th lmnts of () can b xprssd as: anx x = M& x x anx( ) + M anx( ) () ()
Pag of 0 0 0 0 0 0 any y = M = M & y y any( ) + M any( ) x y ( ) M anx( ) anx x y anx y = M & x y ( ) M any( ) any x y any x Th output of ()-() can b changd to th diffrntial rluctivity using following xprssion: B = ν 0 I B whr, I is th unit matrix, ν 0 is th air rluctivity. Equation () is solvd using ()-(), and hnc th nxt tim stp valu of th magntic fild can b computd as: whr t is th tim stp valu. ( t+ t) = ( t) + B( t+ t) B( t) B [ ]. Modification of th anhystrtic magntization Th mthod proposd in this study introducs two modifications to th anhystrtic magntization. Th first modification considrs only th amplitud of th anhystrtic magntization for th intrpolation, and in th scond modification, both th componnts of anhystrtic magntization ar takn into account... Modification- Th curv fitting don with th classical Langvin function in th vctor J-A modl dos not always produc good rsults for crtain lctrical stl having stp kn in th masurd B- loop (Kokornaczyk and Gutowski, 0). Jils and Athrton () providd a frdom for th fr choic of th sigmoid function in plac of th classical Langvin function, to modl th anhystrtic magntization. Kokornaczyk and Gutowski (0) compard diffrnt sigmoid functions in a scalar J-A modl. On of th possibl choic is to us cubic splins for rprsnting th anhystristic magntization. Th -D splin is constructd considring th avrag valus of th magntic fild strngth for th sam valu of th magntic flux dnsity in ascnding and dscnding B- loop (Chwastk, 0). Th -D splin for th anhystristic magntization is thn obtaind by using th amplitud of th ffctiv fild, and th dirction of th masurmnt (Figur and Figur ). Th formulations prsntd in () rmain th sam xcpt for th us of th Langvin function, and its drivativ, that ar rplacd by th bi-cubic splin and its drivativ. Th modifid anhystrtic magntization function can b xprssd as: M = M an an (, θ ) () () () () (0) () whr θ is th angl of th ffctiv fild with rspct to th rolling dirction. In (), th assumption of th collinarity btwn th anhystrtic magntization and th ffctiv fild is still rtaind. Thrfor, partial diffrntiation of () is takn only with rspct to th componnts of th ffctiv fild but not with th angl of it.
Pag of 0 0 0 0 0 0.. Modification- Th anhystrtic magntization is now considrd for ach of th projction of th vctors B and in th principal axis (rolling and transvrs). This givs both th componnts of th anystrtic magntization vctor as a function of th amplitud of th ffctiv fild and th argumnt of th ffctiv fild as follows: (, θ ), (, θ ) M = M M = M anx anx any any Car must b takn whn valuating lmnts of th diffrntial anhystrtic suscptibility tnsor using () in (). Thus, a small chang in on of th componnt of th ffctiv fild brings simultanously a chang in argumnt of th ffctiv fild, and in th partial drivativ trms associatd with th lmnts in (). Whil building th -D bi-cubic splin, M an obtaind in th rang π/ to +π/ (positiv half plan, s Figur ) is rotatd ( st quadrant by π and th quadrant by +π) to covr th full rang of th rotation. This assumption of rotation in basd on th fact that th ffct of applid magntization is π priodic.. Masurmnt Th masurmnt of th magntic fild strngth is don using a rotational singl sht tstr (RSST). Figur dpicts th schmatic viw of th circular silcon stl sampl of grad M00-0A and 0. mm thicknss. Th sampl is magntizd using th stator of an induction machin. Th dtails of th masurmnt stup is xplaind in Goričan t al. (000), and andgrubr t al. (0). Th masurmnt systm is a B controlld and prformd at 0 z. On full cycl of th B- curv consists of 000 masurmnt points (000 ach in ascnding and dscnding curvs). In th cas of unidirctional altrnating magntic fild, masurmnts ar don for vry o ranging from -0 o to +0 o (s Figur ). Th unidirction altrnating magntic fild masurmnts ar shown in Figur, and Figur dpict th rotational masurmnts. Thus, th silcon stl sampl of grad M00-0A rvals wak magntic anisotropy (Figur ) as compard to th masurmnt don on a NO lctrical stl sampl at z by Martin t al. (0). θ -θ Figur. M00-0A stl sampl and arrow rprsnting th dirction of th masurmnts (lft), and anhystrtic curvs obtaind from th unidirctional altrnating masurmnts (right) ()
Pag of 0 0 0 0 0 0 by [T] Figur. Masurd and simulatd b-h curvs in th transvrs dirction (lft), and in th rolling dirction (right) using splins Figur. Masurd and simulatd b-h curv using Langvin function in th rolling dirction. Paramtr idntification Th J-A modl paramtr idntification has bn a topic of dbat vn in th prsnt day rsarch. Diffrnt optimization algorithms hav bn proposd in last fw dcads (Jils and Athrton, ; Ldrr t al., ; Lit t al., 00a; Marion t al., 00). Gntic algorithm and particl swarm optimization ar th two most widly usd optimization tchniqu for th J-A paramtr optimization (Lit t al., 00a; Marion t al., 00). Th fitting don using cubic splin for th anhystrtic magntization function xcluds th us of two modl paramtrs (M s and a). Th rmaining modl paramtrs ar dtrmind with th pattrn sarch optimization mthod (Vasghi t al., 0) from th unidirctional altrnating masurmnts in rolling and transvrs dirctions at. T. On mor considration is to valuat th original anisotropic J-A modl using (), and with th tnsor modl paramtrs, whos componnts in th rolling (k x =.; α x =. 0 - ; c x = 0.0) and in th transvrs (k y =.; α y =. 0 - ; c y = 0.) dirctions ar obtaind from th masurmnts using th abov mntiond optimization tchniqu.
Pag of 0 0 0 0 0 0. Rsults and Discussion Th comparison btwn th rsults obtaind from th modification- and th original vctor J-A modl is prsntd in Figur. Th original vctor J-A modl rsults in an lliptical wavform, and clarly, th loci match masurmnt rsults only in th xtrms (rolling and transvrs dirctions) but fails to do so in all othr intrmdiat dirctions. On th othr hand, th modification- brings immdiat improvmnts by following th curvatur in intrmdiat dirctions (Figur ). owvr, th discrpancis btwn th masurmnt and th simulation rsult is quit visibl. In th nxt approach, modification- is applid, that uss projctions in principal dirctions of th unidirctional altrnating magntic fild masurmnts. Th rsults ar shown in Figur and Figur. Th simulation rsult obtaind using modification- prsnt som improvmnt in th symmtry of th curvatur. Figur. Anhystrtic magntization from modification- (lft), and th simulatd and masurd hystrtic hx-hy loci for circular B input ( B =. T) (right) Figur. Componnts of th anhystrtic magntization from modification-: in th rolling dirction M anx (lft), and in th transvrs dirction M any (right)
Pag of 0 0 0 0 0 0 Figur. Comparison btwn th simulatd rsults from two modifications (lft), and th simulatd and masurd hx-hy loci obtaind using modification- only for highr magntization lvl ( B =. T,. T, and T) (right). Conclusion Th papr prsnts a phnomnological modl to prdict th anisotropic hystrtic bhaviour of th M00-0A NO silicon stl sht. Instad of rprsnting th anhystrtic magntization with th classical Langvin function in th J-A modl, th authors suggst to dvlop two bi-cubic splins to account for th imprfction of th frromagntic matrial. Both componnts and th cross-coupling trms in th diffrntial anhystrtic suscptibility tnsor ar invstigatd and implmntd in th anhystrtic J-A modl. Th anisotropic paramtric tnsor for rprsnting th domain intraction, th rvrsibility of th wall displacmnt and th corciv fild ffct on th losss ar idntifid from th altrnating masurmnts at rolling and transvrs dirctions. Th suggstd improvmnt of th anisotropic J- A modl is compard with th rotational flux dnsity masurmnts. Th original anisotropic J-A modl prsnts som noticabl discrpancis which bcom mitigatd with th prsntd modification for modlling th anhystrtic charactristic. Morovr, th considration of only th amplitud of th magntization instad of both its componnts significantly dcras th vracity of th anisotropic modl. nc, th cross-coupling trms in th diffrntial anhystrtic suscptibility tnsor affct th accurat prdiction of th anisotropic hystrtic fatur of non-orintd stl matrial. Th prsntd modl givs bttr rsult for th highr amplitud of th rotating magntic fild xcitations. Nvrthlss, largr discrpancis ar sn at low xcitation lvls. Furthr invstigation must b don to undrstand th influnc of ach of th modl paramtrs in such dviations. A mor accurat analysis of th modl at low frquncy masurmnts should hlp in bttr modlling of th magntic anisotropy. In futur, th study of anisotropic paramtrs and its influnc in th rotational loss, modl rspons to th lliptical flux dnsity, and implmntation in th finit lmnt subroutin will b considrd. Rfrncs Brgqvist, A. (), A simpl vctor gnralization of th Jils-Athrton modl of hystrsis, IEEE Transactions on Magntics, Vol., No., pp. -. Bnabou, A., Clént, S. and Piriou, F. (00), Comparison of Prisach and Jils-Athrton modls to tak into account hystrsis phnomnon for finit lmnt analysis, Journal of Magntism and Magntic Matrials, Vol., pp. 0-0. Bnabou, A., Lit, J.V., Clént, S., Simão, C. and Sadowski, N. (00), Minor loops modlling with a modifid Jils-Athrton modl and comparison with th Prisach modl, Journal of Magntism and Magntic Matrials, Vol. 0, pp. 0-0. Bíró, O., Außrhofr, S., Pris, K. and Chn, Y. (00), A modifid lliptic modl of anisotropy in nonlinar magntic matrials,, Vol., No., pp. -. Chwastk, K., (0), Anisotropic proprtis of non-orintd stl shts, IET Elctric Powr Applications, Vol., No., pp. -.
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