Minor Loops Calculation with a Modified Jiles-Atherton Hysteresis Model

Transcription

1 49S Minor Loops Calculation with a Modifid Jils-Athrton Hystrsis Modl Jan V. Lit, N. Sadowski, Patrick Kuo-Png, GRUCAD/EEL/UFSC, Po. Box 476, , Florianópolis, SC, Brazil, -mail: jan@grucad.ufsc.br Abdlkadr Bnabou L2EP/USTL, Bat P2, 59655, Villnuv d Ascq, Franc, -mail: abdlkadr.bnabou@univ-lill1.fr Abstract This work proposs a modification in th Jils-Athrton hystrsis modl in ordr to improv th minor loops rprsntation. Th irrvrsibl magntization componnt is slightly modifid kping unchangd th othr modl quations and th modl simplicity. Diffrntly to othr proposd mthodologis found in th litratur, th prviously knowldg of th magntic fild wavform is not nd to assur closd minor loops. Masurd and calculatd hystrsis curvs ar usd in ordr to validat th mthodology. Indx Trms Hystrsis modls, magntic matrials, minor loops. I. INTRODUCTION An incrasing harmonic contnt can b obsrvd in lctrical systms. Odd harmonics ar frquntly found in rotating lctrical machins. Th third-harmonic flux, for instanc, xists du to th saturation of th machins iron cor. Morovr non-sinusoidal lctrical sourcs, in particular PWM (Puls Width Modulation) convrtrs ar now th supply standard option for rotating machins and lctromchanical dvics. Ths convrtrs contribut to th incras of th harmonic contnt in lctrical systms and thn to th incras of iron losss. Th incras of coppr losss can b modld in th windings with th additional harmonic contnt but th magntic cor hystrsis losss undr a PWM voltag is a mor complx procss with a strongly non-linar matrial bhavior [1]. Th harmonic contnt causs a distortion in th flux wavform dviating it from th sinusoidal on. Th distortd flux wavform can add minor loops to th major hystrsis loop which incrass th cor losss. By minor loops on can considr any closd B-H symmtric or asymmtric curv, othr than th saturatd on. Th mpirical losss modls as, for instanc, th Stinmtz modl ar not adquat to tak into account minor loops, so a hystrsis modl must b mployd. Nowadays, rsarch in hystrsis modling is mainly focusd on th dvlopmnt of historydpndnt modls. Ths ons must prsnt closd loops which must b immdiatly stabl [2]. Prisach s hystrsis modl is a mathmatically basd formulation widly mployd for modling th bhavior of magntic matrials [3]. It can to rprsnt with good accuracy th cntrd major loop and also not-cntrd minor loops. In fact, th history of th matrial magntic stat is an intrinsic

2 50S proprty of this modl. Howvr, th numrical implmntation of th Prisach s modl and its paramtrs idntification rquir mor ffort in comparison with som othr modls such as th Jils- Athrton (JA) modl [4]. Among th proposd hystrsis modls in rcnt yars, th JA has bn on of th most invstigatd [5]. Th classical JA modl is basd on physical assumptions. It uss a magntic nrgy balanc in th matrial lading to a first ordr diffrntial quation with fiv paramtrs to b idntifid. Nvrthlss, this diffrntial quation is not abl to rproduc proprly minor loops. In fact, th JA modl prsnts a slow accommodation tim, so th magntization trajctory btwn th turning points of a minor loop will not b closd at th nd of its xcursion. This rmains th major limitation in th us of th JA modl. In ordr to ovrcom this drawback, som works hav bn proposd. In [6] a modification in th modl was proposd but it rquirs a priori th knowldg of th magntic fild volution. Othr works propos th us of scaling factors in th main quation of th modl [7] or th us of diffrnt paramtrs st to rprsnt th major and minor loops [8][9]. In this work, w prsnt a gnralizd JA modl abl to prdict th cor losss in th matrial undr sinusoidal or distortd wavform fluxs. Th rsultant modl is basd in a combination of thortical and mpirical approachs and xtnsiv xprimntal obsrvation. No major modifications in th modl ar ndd. Th proposd tchniqu improvs th JA modl allowing it to rprsnt noncntrd minor loops. Contrarily to othr modifications of th original modl, th prvious knowldg of th magntic fild wavform is not ncssary. Calculatd and masurd rsults will b compard in ordr to validat th proposd mthodology. II. THE SCALAR JILES-ATHERTON HYSTERESIS MODEL In th original JA modl, th magntization M is dcomposd into its rvrsibl M rv and irrvrsibl M irr componnts [5]. Th rlationship linking ths contributions is givn by: rv an irr M c M M (1) Th anhystrtic magntization M an is givn by th Langvin function: with: H Man M s coth a a H (2) dmirr Man Mirr (3) dh k

3 51S whr H H M is th ffctiv fild. M S, a,, c and k ar th modl paramtrs which can b obtaind from masurd hystrsis loops; is a dirctional paramtr assuming th valu +1 if dh dt 0 and -1 if dh dt 0. Combining th quations abov th main quation of th original JA modl can b writtn as: dmirr dm an 1 c c dm dh dh dh dmirr dm an 1 1 c c dh dh (4) Th diffrntial quation (4) allows calculating th magntization M with rspct to H variations. Howvr in som applications th induction B is known prior to th fild. Th finit lmnt mthod with magntic potntial formulation is an xampl of such applications. In [10] an invrs modl whr th indpndnt variabl is th induction B was prsntd. Th main quation of this invrs modl is: whr: dmirr c dm an 1 c dm db o dh db dm 1 o db irr c c and B 0H is th ffctiv magntic flux dnsity. 0 dm an dh (5) dmirr Man Mirr (6) db k III. MINOR LOOP REPRESENTATION WITH JILES-ATHERTON MODEL Although th original JA modl is abl to rprsnt a wid rang of major hystrsis loops, in particular thos of soft magntic matrials, it can producs non-physical minor loops with its classical quations. An xampl of such non consistnt curv is th minor loop shown in Fig. 1. Th minor loop starts in th rvrsal point 1, rachs th point 2 but dos not rturn to th initial point mmory, instad, gos to a nw point 3. Bing th magntization, in th JA modl, only dpndnt of its prvious tim stp valu and th solution of th diffrntial quation (4) or (5), thr is no condition or rul that guarants th rturn to th initial point mmory.

4 52S Fig. 1 Non-closd minor loop calculatd with th original JA modl. In fact, th inconsistncy rsids in th slow accommodation tim in th diffrntial quation, spcially in th high slop curvs. With a symmtrical high amplitud xcitation th accommodation tim is nough to produc a closd major loop. On th othr hand, aftr a prturbation in th magntization xcursion (a turning point), it spnds som calculation tim to rcovr a stabl trajctory. IV. PROPOSED METHODOLOGY On can obsrv in th classical modl that th closur of minor loops is not guarantd du a strong variation rat of th global magntization. Whn running through a minor loop, w obsrv a transitory in th trajctory of magntization that rquirs a tim to rcovr its stability. As th minor loop finishs whn th magntization passs by its first turning point (point 3 is far from point 1 in Fig. 1) th accommodation tim is not rachd. If on considrs that running through minor loops yilds to a small prturbation of th magntization around a givn matrial magntic stat, it would b natural to limit th variation rats of th magntization in ordr to allow th stability to b achivd. In th proposd mthodology th first stp consists of rcognizing th magntization running through a minor loop. With th JA modl, on can idntify th minor loop turning points just obsrving th dirctional paramtr. It indicats if th magntization is on its ascndant 1 or dscndnt 1 hystrsis branch. A chang in valu indicats that a turning point occurrd. Storing th B and H valus in th two conscutivs turning points w can stimat if th magntization is running through an asymmtric minor loop or a cntrd curv. Obsrving numrical rsults on can s that th first portion of th calculatd minor loop has a convnint bhavior so th procdur hr proposd is applid aftr th scond turning point. As th global magntization is mainly irrvrsibl in th JA modl on can choos to prform a modification on th invrs magntization componnt. Th variation rat of (3) or (6) is wightd by a continuous function. Th quation (6) in th invrs modl now will b calculatd as:

5 53S dmirr Man Mirr wh. (7) db k 0 with, for xampl, a sigmoid function: wh 1 1 H (8) whr and ar paramtrs to b obtaind from xprimntal data. Th wight factor is applid locally and bcoms 1 onc th minor loop is closd, i.., th magntization is clos nough to that of first turning point. This rul can approximat th modl bhavior to th xprimntal obsrvation. Howvr th rturn to th initial turning point is rathr a tndncy than a rigorous mathmatical rul. Additionally, som adjustmnts can b prformd on th modl paramtrs to improv th minor loop rprsntation. V. RESULTS A voltag fd Epstin fram was chosn to vrify xprimntally th proposd mthodology. A 0.5 mm, non-orintd, F-Si 3% sht was usd. Th flux was controlld imposing th voltag in th scondary winding and th currnt was dirctly masurd in th primary. Th Labviw TM systm was usd to control th induction and to supply svral distortd wavforms. Th invrs modl was usd to prform th calculations. Th fiv modl paramtrs (Tabl I) wr obtaind with a fitting procdur basd on gntic algorithms [11]. TABLE I. PARAMETERS OF JA MODEL. Paramtrs M S x 10 6 [A/m] k x 10 [A/m] c x 10-1 a x 10 [A/m] x 10-3 Fig. 2 shows a masurd hystrsis loop obtaind with a fundamntal frquncy of 1 Hz plus its 3 rd and 5 th harmonics.

6 54S Fig. 2 Masurd hystrsis loop. Th masurd B curv was usd as th modl ntry. Fig. 3 shows th hystrsis loop calculatd with th original modl quations. Fig. 3 Calculatd hystrsis loops with th classical quations. Fig. 3 illustrats th limitation of th JA modl in prsnc of minor loops. This non-physical bhaviour limits th modl mploymnt in th magntic calculations and circuit analysis undr a harmonic xcitation. Furthrmor, th magntic loss pr cycl in th masurd and calculatd curvs abov prsnts an rror clos to 20%. Th calculation with th proposd tchniqu is shown in Fig. 4. Th usd wighting function is a sigmoid (8) with and Also, th paramtr a, connctd with th slop of Langvin function, was adjustd to improv th rsultant minor loop. If th wighting function is diffrnt from 1, paramtr a is considrd qual to x 10 1 [A/m] and rcovrs its original valu onc th minor loop is closd. Th minor loops in this calculation hav a good agrmnt with th masurd ons. Th diffrnc btwn masurd and calculatd losss is now about 3%.

7 55S Fig. 4 Calculatd hystrsis loops with th modifid modl. VI. CONCLUSION In ordr to ovrcom th drawback of non closur of minor loops, a modification in th Jils- Athrton modl was proposd. Th limitation in th irrvrsibl magntization rats with a sigmoid function, associatd with som paramtr adjustmnts, allowd a mor adquat modl bhavior whn running through minor loops. Th possibility of using othr wight functions is in th offing. Th proposd tchniqu maintains th JA modl simplicity bing an altrnativ to th Prisach modl implmntation, for instanc in a finit lmnt cod. Th agrmnt btwn simulatd and masurd rsults dmonstrats th mthodology fficincy in th cas of F-Si iron stl. Th modl can b suitably incorporatd into a transint circuit simulator taking into account frromagntic matrials. This work was supportd in part by th CNPq. ACKNOWLEDGMENT REFERENCES [1] P. J. Lonard, P. Marktos, A. J. Moss and M. Lu, Iron losss undr PWM xcitation using dynamic hystrsis modl and Finit Elmnts, IEEE Trans. on Magn., 42(4), pp , [2] S. E. Zirka, Y. I. Moroz and E. D. Torr, Combination hystrsis modl for accommodation magntization, IEEE Trans. on Magn., 41(9), pp , [3] F. Prisach, Übr di magntisch Nachwirkung, Zitschrift für Physik, 94, pp , [4] A. Bnabou, S. Clént and F. Piriou, "Comparison of th Prisach and Jils-Athrton modls to tak hystrsis phnomnon into account in Finit Elmnt Analysis", COMPEL, 23(3), pp , [5] D. C. Jils and D.L. Athrton, Thory of frromagntic hystrsis, J. Magn. Magn. Matr., vol. 61, pp ,1986. [6] D. C. Jils, A slf consistnt gnralizd modl for th calculation of minor loops xcursions in th thory of hystrsis. IEEE Trans. Magn. v. 28, n. 5, p , [7] K. H. Carpntr, A diffrntial quation approach to minor loops in th Jils Athrton hystrsis modl, IEEE Trans. on Magn., v. 27, n 6, pp (1991). [8] D. Ldrr, H. Igarashi, A. Kost and T. Honma. On th paramtr idntification and application of th Jils-Athrton hystrsis modl for numrical modlling of masurd charactristics, IEEE Trans. Magn. vol.35, , [9] F. R. Fulgini and A. Salvini, Softcomputing for th idntification of th Jils Athrton Modl Paramtrs. IEEE Trans. Magn., v. 41. n. 3, pp , [10] N. Sadowski, N.J. Batistla, J.P.A. Bastos, and M. Lajoi-Maznc, An Invrs Jils Athrton Modl to Tak Into Account Hystrsis in Tim-Stpping Finit-Elmnt, Trans. on Magn., vol 38, n 2, pp , [11] J.V. Lit, S. L. Avila, N.J. Batistla, W.P. Carps Jr., N. Sadowski, P. Kuo-Png, and J.P.A. Bastos, Ral codd gntic algorithm for Jils-Athrton modl paramtrs idntification, IEEE Trans. on Magn., vol 40, n 2, pp , 2004.