Fixed-Point Harmonic-Balanced Method for Nonlinear Eddy Current Problems

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1 Intrnational Journal of Enrgy and Powr Enginring 206; 5(-): 37-4 Publishd onlin Octobr 4, 205 ( doi: 0.648/j.ijp.s ISSN: X (Print); ISSN: X (Onlin) Fixd-Point Harmonic-Balancd Mthod for Nonlinar Eddy Currnt Problms Xiaojun Zhao, Yuting Zhong, Dawi Guan, Fanhui Mng, Zhiguang Chng 2 Dpartmnt of Elctrical Enginring, North China Elctric Powr Univrsity, Baoding, China 2 Institut of Powr ransmission and ransformation chnology, Baobian Elctric Co., Ltd, Baoding, China addrss: @63.com (Xiaojun Zhao), yuting35@yah.nt (Yuting Zhong), zhynh23@63.com (Dawi Guan), mngfh990@63.com (Fanhui Mng), mlabzchng@yahoo.com (Zhiguang Chng) o cit this articl: Xiaojun Zhao, Yuting Zhong, Dawi Guan, Fanhui Mng, Zhiguang Chng. Fixd-Point Harmonic-Balancd Mthod for Nonlinar Eddy Currnt Problms. Intrnational Journal of Enrgy and Powr Enginring. Spcial Issu: Numrical Analysis, Matrial Modling and Validation for Magntic Losss in Elctromagntic Dvics. Vol., No., 206, pp doi: 0.648/j.ijp.s Abstract: A nw mthod to optimally dtrmin th fixd-point rluctivity is prsntd to nsur th stabl and fast convrgnc of harmonic solutions. Nonlinar systm matrix is linarizd by using th fixd-point tchniqu, and harmonic solutions can b dcoupld by th diagonal rluctivity matrix. h -D and 2-D non-linar ddy currnt problms undr DC-biasd magntization ar computd by th proposd mthod. h computational prformanc of th nw algorithm provs th validity and fficincy of th nw algorithm. h corrsponding dcomposd mthod is proposd to solv th nonlinar diffrntial quation, in which harmonic solutions of magntic fild and xciting currnt ar dcoupld in harmonic domain. Kywords: Eddy Currnt, Fixd-Point, Harmonic solutions, Rluctivity. Introduction Non-linar ddy currnt problms can b solvd by th tim-stpping mthod [] or th harmonic-balancd mthod [2]. h tim-stpping mthod rquirs many priods to approach th accurat stady-stat solution, whil th harmonic-balancd mthod computs th magntic fild dirctly in th frquncy domain. Compard with th frquncy-domain mthod, th so calld brut forc mthod using tim-stpping tchniqu spnds much mor tim on th transint procss. A rlationship btwn magntic fild intnsity H and th magntic induction B is rprsntd by introducing th fixd-point rluctivity ν [3]. h fixd-point rluctivity ν can b rgardd as a priodic variabl or a constant in harmonic-balancd mthod. h convrgnt spd of th solutions dpnds mainly on th stratgis to dtrmin th valu of ν in non-linar itrations. Diffrnt mthods to optimally dtrmin ν hav bn prsntd and invstigatd in ordr to nsur th stabl and fast convrgnc of solutions [4-6]. Whn th powr transformr works undr DC-biasd magntization, th frromagntic cor will b significantly saturatd. Owing to th nonlinarity of magntic matrial in lctromagntic dvics, thr oftn ar high-ordr harmonics in th xciting currnt and magntic fild. hrfor, lctrical dvics such as powr transformrs and ractors may work abnormally du to th magntic storm and th transmission of high voltag dirct currnt [7]. In that cas, th DC-biasd ddy currnt problm should b solvd fficintly and accuratly. Furthrmor, lctromagntic coupling should always b considrd in numrical computation whn th strand coils or solid conductors ar connctd to th voltag sourc. In this papr an fficint algorithm to dtrmin th fixd-point rluctivity ν is proposd. It is aimd at fficintly computing th non-linar ddy currnt problm undr DC-biasd magntization. h nonlinar magntic fild is computd in harmonic domain. h corrsponding dcomposd algorithm is prsntd to solv th nonlinar diffrntial quation squntially or concurrntly, which dcrass th mmory cost of harmonic-balancd computation of larg scal problms. 2. Fixd-Point Harmonic-Balancd Mthod 2.. Fixd-Point Mthod Maxwll s quations hold in non-linar ddy currnt

2 38 Xiaojun Zhao t al.: Fixd-Point Harmonic-Balancd Mthod for Nonlinar Eddy Currnt Problms problms, H= J + σe () 0 B E = (2) t B= 0 (3) whr E is th lctric fild intnsity, J 0 is th imprssd currnt dnsity and σ is th conductivity. A rlationship btwn magntic fild intnsity H and magntic flux dnsity B is rprsntd by introducing th fixd-point rluctivity ν [8], ν H B = B M B (4) whr M is a magntization-lik quantity which varis nonlinarly with B. hrfor th magntic fild intnsity H is split into two parts: th linar part, which is rlatd to th ν, and th non-linar part, which varis with th magntic induction B [3] Harmonic-Balancd Mthod h priodic variabls in th lctromagntic fild undr DC-biasd xcitation can b approximatd by th Fourir-sris with a finit numbr of harmonics [9], ( W nωt W nωt n n ) Wt () = W + sin + cos (5) n= whr W(t) can b rplacd by currnt dnsity J, vctor potntial A, magntic flux dnsity B, magntic fild intnsity H, and th magntization-lik quantity M. Equation () can b rwrittn in isotropic matrial by mans of harmonic vctor in 2-D problms, in which H = ν B M H = ν B M x x x y y y H = H H H H H H = H H H H H x x,0 x, x,2 x,3 x,4 y y,0 y, y,2 y,3 y,4 Each of harmonic vctors B x, B y, M x and M y has a similar xprssion with (7) D Eddy Currnt Problms Non-linar ddy currnt problms can b solvd dirctly with th prscribd imprssd currnt dnsity. Howvr, th imprssd currnt dnsity is unknown whn th solid conductor or strand coil is connctd to th voltag sourc. hrfor, th coupling btwn th magntic fild and lctric circuits should b invstigatd if th non-linar ddy (6) (7) currnt problm is solvd in th harmonic domain Solid Conductor Connctd to Voltag Sourcs Whn th solid conductor is fd by th voltag sourc, th ddy currnt xists in th solid conductor and th othr conducting matrials. h non-linar problm can b dscribd by th following quation, ν A = J M (8) whr J is th currnt dnsity. h magntic vctor potntial A and th scalar potntial V can b linkd to th currnt dnsity J by th quation as follows, J= σ σ V h non-linar quation, including th 2-D magntic and lctric filds, can b prsntd as follows [0], ν + + = ( A) σ σ( V) M (9) (0) h fixd-point harmonic-balancd quation can b stablishd by applying th finit lmnt mthod on th whol problm domain, m S D + N A = K + P ( j=,2,..., m) () ij ij j j j i= S = N NdΩ (2) ij i j Ωw ij = ωσnndω (3) i j Ωd ( M ) P = - j NdΩ (4) j Ωn K = U j σn dω (5) j Ωc whr Ω w, Ω d and Ω n rprsnt th finit lmnt in th whol problm domain, ddy currnt rgion and th non-linar rgion, rspctivly. N i is th shap function on nod i in ach finit lmnt, and m is th total numbr of nods in on lmnt. D and N ar th squar matrics rlatd to th fixd-point rluctivity and harmonic numbr [2], rspctivly. A j is th harmonic vctor of th magntic vctor potntial on nod j and P j is th harmonic vctor obtaind from M. U is th voltag in harmonic domain pr unit lngth. By intgrating (3) on th solid conductor, w can obtain sinc C A + ZI= U (6) Ωc

3 Intrnational Journal of Enrgy and Powr Enginring 206; 5(-): ω C = N i S R 0 Z = 0 R NdΩ (7) i cd Ωc (8) whr Ω c rprsnts th finit lmnt in th conducting rgion, R is th conductor s rsistanc pr unit lngth, S cd is th cross-sctional ara of th solid conductor Strand Coil Connctd to Voltag Sourcs h strand coil consists of fin wirs whr th ddy currnt is gnrally too small to b considrd for computation. h supplid voltag U and th xciting currnt I in th coil can b linkd by Kirchhoff s Law and Faraday s Law [], N coil Ωc dω+ RI= U (9) whr N coil is th turn numbr of th strand coil. h frquncy-domain systm quation considring lctromagntic coupling can b obtaind according to th harmonic-balancd thory, S A + A+ G I= P (20) Ω Ω Ω Ω w d c n C A + ZI= U (2) Ωc matrix D ar non-zro, which indicats harmonic solutions ar coupld with ach othr. In that cas th mmory dmand will incras significantly in th larg-scal computation, although fast convrgnc is achivd. In fact th ν can b a constant in th harmonic domain, and is dtrmind as follows, ν = H B / B (24) max max whr B max rprsnts th maximum valu of th magntic induction in on priod. h man ( man ) and maximum ( max ) variation of th rluctivity dfind by ν=h/b can b obsrvd to chck th convrgnc of th harmonic solutions. In this papr th stopping critrions ar st to man = 0.% and max = % in th numrical computation of th on-dimnsional and two-dimnsional ddy currnt problms. 3. Computational Rsults and Analysis 3.. Laminatd Stl Sht As shown in Fig., A thin lctrical stl sht carrying ddy currnt is modlld and computd to obsrv variation of th magntic induction whn th lamination oprats undr diffrnt typs of magntization. h 30Q40 orintd stl sht of 0.3 mm thicknss is first tstd undr sinusoidal flux in 50 Hz, and thn th DC flux is providd. h conductivity of th sht is σ = S/m. sinc G = -I N coil i u i Scoil Ωc NdΩ (22) whr I u is a unit matrix of th sam siz with D and N. Consquntly, th harmonic solutions of th magntic fild and magntizing currnt can b computd simultanously by solving (2) whn th solid conductor and strand coil ar both connctd to th voltag sourc, Figur. Elctrical stl sht. S + G A K P + C Z = I U (23) whr G is rlatd to th spatial distribution of th magntizing currnt whn th strand coil is fd by th voltag sourc, whil K appars on th right sid of th quation whn th solid conductor is connctd to th voltag sourc Dtrmination of Fixd-Point Rluctivity h fixd-point rluctivity ν can b rgardd as a priodic variabl whn it is dtrmind in ach tim stp. Consquntly, th harmonic cofficints of ν can b usd to calculat D in th harmonic-balancd mthod [2]. All lmnts in th squar Figur 2. Magntic induction in diffrnt dpths of th stl sht.

4 40 Xiaojun Zhao t al.: Fixd-Point Harmonic-Balancd Mthod for Nonlinar Eddy Currnt Problms h spatial distribution of magntic induction in th thin sht is wll clarifid in Fig. 2. h magntic flux for th boundary condition is B av,ac = and B av,dc = Fig. 3 compars th wavforms of magntic induction undr sinusoidal (indicatd by ac ) and dc-biasd (indicatd by dc ) magntizations. Notic that th ddy currnt in th sht lads to th non-sinusoidal wavform of th magntic induction. Furthrmor, th distribution of th magntic induction varis with th dpth of th sht in th x-dirction. Figur 5. Calculatd currnt in th coppr conductor Coppr Conductor Surroundd by Frromagntic Scrn Figur 3. Comparison of th magntic induction undr sinusoidal and dc-biasd magntizations. h two-dimnsional problm consists of a solid coppr conductor and an iron scrn with an air gap. As shown in Fig. 4, th iron scrn surrounds th conductor. h ddy currnt xists in both th coppr conductor and iron scrn. h conductivitis of th coppr and iron ar σ= S/m and σ = S/m, rspctivly. h coppr conductor is connctd to a voltag sourc of 50 Hz. h B-H curv is dtaild in [3]. 892 scond-ordr lmnts with 278 nods ar usd in th numrical computation. Computational costs of th proposd mthod and th traditional mthod [2] ar compard in abl I. M c and c rprsnt th mmory dmand and computational tim, rspctivly. N h is th truncatd harmonic numbr. Compard to th traditional mthod, th proposd mthod significantly rducs mmory rquirmnts with a slight incras in computational tim du to a fw mor non-linar itrations. h calculatd magntizing currnt is compard with that obtaind by using tim-stpping mthod, and th good congruncy provs th validity of th proposd mthod. abl. Comparison btwn two diffrnt mthods. Mthod raditional Proposd Mmory/Mb im/s N h 4. Dcomposd Algorithm Whn th fixd-point rluctivity is computd according to (24), th nonlinar quation in (23) can b linarizd by ν which is spac-dpndnt and tim-indpndnt. hrfor harmonic solutions can b dcoupld and calculatd sparatly. Equation () can b dcomposd as follows, Figur 4. Gomtric structur of th 2-D modl. m i= ( S + ij k ) d N A, ij k kj, = K + P ( j=,2,..., m; k= 0,,2,..., N ) kj, kj, h (25)

5 Intrnational Journal of Enrgy and Powr Enginring 206; 5(-): d h k k ν ( i= ) = ν 0 ( k> ) 0 ν 0 ( k= ) = 0 k ω k 0 ( k ) (26) (27) whr k is th harmonic numbr. h dcoupld quation systm can b solvd squntially and concurrntly, updating harmonic solutions by Gauss-Sidl and Jacobi itrativ mthod rspctivly. 5. Conclusion h convrgnt spd of harmonic solutions highly dpnds on th dtrmination of optimal fixd point rluctivity in th fixd-point harmonic-balancd mthod. Diffrntial rluctivity can b usd to guarant th stabl and fast convrgnc of harmonic solutions. Du to th linarizd systm quation, harmonic solutions can b dcoupld and computd in paralll, which can improv th computational fficincy and rduc th mmory cost gratly. h proposd algorithm is mor fficint than th traditional harmonic-balancd mthod. Acknowldgmnt his work is supportd by th National Natural Scinc Foundation of China (Grant No ), Hbi Provinc Natural Scinc Foundation (Grant No. E ), Rsarch Fund for th Doctoral Program of Highr Education of China (Grant No ), and th Fundamntal Rsarch Funds for th Cntral Univrsitis (Grant No. 205MS82). Rfrncs [] E. Dlala, A. Blahcn, and A. Arkkio, Locally convrgnt fixd-point mthod for solving tim-stpping nonlinar fild problms, IEEE rans. Magn., vol.43, pp , [2] X. Zhao, L. Li, J. Lu, Z. Chng and. Lu, Charactristic analysis of th squar laminatd cor undr dc-biasd magntization by th fix-point harmonic-balancd mhtod, IEEE rans. Magn., vol. 48, no. 2, pp , 202. [3] O. Biro and K. Pris, An fficint tim domain mthod for nonlinar priodic ddy currnt problms, IEEE rans. Magn., vol. 42, no. 4, pp , [4] E. Dlala and A. Arkkio, Analysis of th convrgnc of th fixd-point mthod usd for solving nonlinar rotational magntic fild problms, IEEE rans. Magn., vol. 44, no. 4, pp , [5] S. Aussrhofr, O. Biro, and K. Pris, A stratgy to improv th convrgnc of th fixd-point mthod for nonlinar ddy currnt problms, IEEE rans. Magn., vol. 44, no. 6, pp , [6] G. Koczka, S. Aubrhofr, O. Biro and K. Pris, Optimal convrgnc of th fixd point mthod for nonlinar ddy currnt problms, IEEE rans. Magn., vol. 45, no. 3, pp , [7] X. Zhao, J. Lu, L. Li, Z. Chng and. Lu, Analysis of th saturatd lctromagntic dvics undr DC bias condition by th dcomposd harmonic balanc finit lmnt mthod, COMPEL., vol. 3, no. 2, pp , 202. [8] F. I. Hantila, G. Prda and M. Vasiliu, Polarization mthod for static fild IEEE rans. Magn., vol.36, no.4, pp , [9] X. Zhao, J. Lu, L. Li, Z. Chng and. Lu, Analysis of th DC Bias phnomnon by th harmonic balanc finit-lmnt mthod, IEEE rans. on Powr Dlivry., vol.26, no., pp , 20. [0] I. Ciric, and F. Hantila, An fficint harmonic mthod for solving nonlinar tim-priodic ddy-currnt problms, IEEE rans. Magn., vol. 43, no. 4, pp , [] P. Zhou, W. N. Fu, D. Lin, and Z. J. Cnds, Numrical modling of magntic dvics, IEEE rans. Magn., vol. 40, no. 4, pp , 2004.