Designing an Improved Method to Determine the Anhysteretic Curve in Soft Magnetic Materials via the Jiles-Atherton Model

ACTA PHYSICA POLONICA A No. 1 Vol. 146 (2024) Selec ed pape s p esen ed a he 14 h Symposium o Magne ic Measu emen s and Modelling SMMM'2023 Designing an Imp o ed Me hod o De e mine he Anhys e e ic Cu e in So Magne ic Ma e ials ia he JilesA he on Model Z. Roubal ∗ and V. Smejkal Depa men o Theo e ical and Expe imen al Elec ical Enginee ing, B no Uni e si y o Technology, Technická 12, 612 00 B no, Czech Republic Doi: 10.12693/APhysPolA.146.51 ∗ e-mail: oubalz@ u b .cz The a icle discusses a no el p ocedu e o measu ing he anhys e e ic cu e in so magne ic ma e ials. This cu e equen ly nds use in di e se applica ions, such as he JilesA he on hys e esis model. The ac ual me hod is cha ac e ized in de ail, including sample calcula ions o ma e ials exhibi ing a ious hys e esis loop shapes ( e i es, g ain-o ien ed s eel, and nanoc ys alline ma e ial). To illus a e he bene s o he p oposed app oach, he au ho s compa e he measu emen -based and he simula ed cu es, he la e being ob ained h ough an op imal in e lea ing o he model. opics: anhys e e ic cu e, JilesA he on hys e esis model, magne ic measu emen . 1. In oduc ion Hys e esis loop modeling embodies an impo an ask in he analysis o a ious elec ical ci cui s con- aining e omagne ic co es. The applica ions in- clude, o example, he ansien beha io o classic powe ans o me s and he esea ch o swi ching powe supplies. Se e al desc ip ions o hys e esis loops a e a ailable, including he P eisach ma h- ema ical model [1, 2] o he JilesA he on (J-A) o malism [3], o en implemen ed in SPICE-like sim- ula ion so wa e. The pa ame e de e mina ion p o- cedu e o he JilesA he on model p esen ed in [3] may di e ge in some cases. Thus, in a p e iously published expe imen al p ojec [4], some modica- ions o he classic JilesA he on model we e an- alyzed using he leas squa es me hod in o de o yield an imp o ed hys e esis loop o he nanoc ys- alline ma e ial VITROPERM 500F. He e, he an- hys e e ic magne iza ion cu e was ob ained as an a e age o he uppe and lowe pa s o he limi ing hys e esis loop and was he e o e no measu ed. The die ences be ween he measu ed loop and ha sim- ula ed ia he JilesA he on model we e compa ed. The p oblem wi h he JilesA he on model in ap- p oxima ing he VITROPERM 500F ma e ial es s in he apid ansi ion o sa u a ion, dissimila om he g adual ansi ion o he Lange in equa ion [4]. O he sou ces use die en ini ial app oxima ions o he JilesA he on model's pa ame e s [59]. An o e iew o he s a e o he a is p oposed in [10]. 2. Tools o he magne ic cu e measu emen s The se up shown in Fig. 1 allowed us o cha ac- e ize he p ima y magne iza ion cu e, he quasi- s a ic and dynamic hys e esis loop g oup, and he anhys e e ic cu e. The o iginal se o ins u- men s [11] was modied by using a mo e eec- i e analog- o-digi al (A/D) sampling de ice and a ol age- o-cu en ( V/I ) con e e o deli e high- quali y demagne iza ion o he sample and o measu e he anhys e e ic cu e eliably; in small ho ium samples, an au oma ically ze oed bue amplie is applicable [12]. A Siglen SDG2042X DDS gene a o wi h he ue o m echnology was u ilized o gene a e he equi ed wa e o m, p o id- ing a sucien esolu ion (14-bi ) o expose he demagne iza ion wa e o m. The gene a o hen ex- ci ed a V/I con e e , de eloped p e iously a he Depa men o Theo e ical and Expe imen al Elec- ical Enginee ing (DTEEE) o acili a e magne ic measu emen s. In he con e e , an OPA541 op- e a ional amplie (OA) wi h a p ecision-sensing esis o in a Howland ci cui is in eg a ed. The e- qui emen s comp ised a g ounded ou pu , s abili y o he ze o con e e (a p e equisi e o compensa- ion), and high ou pu esis ance. The s abili y a induc i e loads was ensu ed. Gene ally, he sample can be a o oid o an Eps ein ame. The cu en ex- ci es he magne izing winding N1 and is sensed a he shun RB . An elec onic uxme e is connec ed 51 Z. Roubal e al. Fig. 1. The measu ing se up. Fig. 2. The con en ional and he imp o ed algo i hms. o he seconda y measu ing winding N2 . The signals a e con e ed in o digi al o m using a 12-bi Pico- Scope 5242A oscilloscope. In quasi-hys e esis loop measu emen s, a 30 Hz low-pass l e is se o sup- p ess he 50 Hz mains in e e ence and o he spu i- ous eec s om, o example, he swi ching powe supply. The da a a e ans e ed ia a USB †1 in e - ace o a PC †2 , whe e he p ocessing is pe o med in MATLAB. The measu ing congu a ion is applicable up o a equency o 5 kHz. A highe equencies, a ol - age amplie appea s o be mo e ad an ageous han a V/I con e e , p o iding a ha monic exci a ion wa e o m o he magne ic ux densi y B . Then, measu emen s up o 100 kHz a e easible wi h a passi e in eg a o . 3. Imp o ing he algo i hm o measu e he anhys e e ic cu e The a ious algo i hms o measu e he anhys- e e ic cu e desc ibed in [13] a e e y demand- ing in e ms o he s abili y o he ze o o he elec onic uxme e . Mul iple measu emen a i- an s a e possible, assuming h ee o wo wind- ings. When a quasi-s a ic hys e esis loop equi ing a p ocessing ime o 40 s is measu ed, we can exe- cu e a p og am-based co ec ion o he measu ed  1 USB  Uni e sal Se ial Bus  2 PC  pe sonal compu e da a i he cu e is closed [11]. The s o ed ze o co ec ion is also usable in measu ing he ini ial magne iza ion cu e. This op ion canno be em- ployed in an anhys e e ic cu e o , o example, whe e he o al measu emen ime co esponds o 30 min and/o he samples a e e y small (an elec- onic uxme e ange o less han 3 mWb)  such a p ocedu e would ende he esul s gene ally inapplicable. Thus, in con as o he o iginal e sion o he poin -by-poin sequen ial measu emen , demagne- izing he measu ed sample and ze oing he elec- onic uxme e ook place be ween each measu e- men poin (Fig. 2). As a esul , he equi emen s o he s abili y o he elec onic uxme e ze o e- semble hose ha ela e o measu ing he ini ial magne iza ion cu e. The signal wa e o m o en- able a single poin measu emen o he anhys e esis cu e is desc ibed wi hin i1( ) = Imax e− A sin (2π ) + IDC 1−e− τ, (1) whe e Imax is he maximum ampli ude o he de- magne iza ion signal, A deno es he ampli ude de- cay cons an , e e s o he equency, IDC ep- esen s he se poin o he anhys e e ic cu e, and τ is he ime cons an o he se poin . Rega ding he expe imen ally p ese alues o ensu e an op i- mum demagne iza ion o he sample, he wa e o m is shown in Fig. 3. In he selec ed samples (including he o ien ed s eel Sonape m, T a oke S, NiZn e i e Ami- don 43, and nanoc ys alline ma e ial VITROP- ERM 500F), he measu ing signal pa ame e s ead Imax = 0.6 A, A= 0.06 , = 1 Hz, τ= 1 s. 52 Designing an Imp o ed Me hod o De e mine he Anhys e e ic... Fig. 3. The measu ed magne iza ion signal. Fu he , IDC equals ze o a demagne iza ion bu hen g adually inc eases un il he ma e ial is sa - u a ed. In addi ion o he uxme e ze o s abili y issues, we encoun e ed p oblems wi h he V/I con e e ose in ma e ials ha ing a nea ly o hogonal hys- e esis loop (T a oke S). The a o able design, how- e e , allowed he ose con e e V/I o exhibi an ose below 20 µ A. The se up also includes a he mocouple o mea- su e he empe a u e o he sample, simila o he scena io desc ibed in [14]. 4. The JilesA he on (J-A) hys e esis loop model The ini ial equa ion o cha ac e ize his model is one ha exposes he beha io in he magne ic ma e ial a he domain le el. Mo e conc e ely, he equa ion embodies a die en ial desc ip ion ha changes he ou pu acco ding o he a ying di ec- ion o he inpu a iable, namely he magne ic eld s eng h. The o al magne iza ion M is hen gi en by M=Mi +M e , (2) whe e Mi is he i e e sible and M e he e- e sible magne iza ion. When he magne iza ion changes, i e e sible shi s occu ; hese a e dened by dMi dH=Man −Mi kδ −α(Man−Mi ). (3) In (3), Man and Mi deno e he lossless (an- hys e e ic) and he i e e sible magne iza ion, e- spec i ely; k is he pa ame e de e mining he cu e b oadening (i.e., he hys e esis losses); δ ep esen s he sign pa ame e ; and α ep esen s he molecula eld pa ame e [3]. The sign unc- ion δ ollows he change in he di ec ion o he magne ic eld s eng h and is hus specied ia δ=(+1, o dH d >0, −1, o dH d <0. (4) Lossless magne iza ion is an ideal p ocess whe e no dis u bances in he c ys al la ice (causing he losses) occu du ing he magne iza ion; i s ac ual p og ess is hus de e mined by he displacemen o he domain walls and he o a ion o he spon a- neous magne iza ion o he domains in he di ec ion o he ex e nal eld. This dependence is mos o en gi en by he Lange in unc ion Man =Msa co h He a−a He = Msa co h H+αM a−a H+αM , (5) whe e Ms is he sa u a ion magne iza ion (a cha ac- e is ic o each ma e ial, empe a u e-dependen ), a [A/m] deno es he empe a u e-dependen shape pa ame e , and He s ands o he o al magne ic eld s eng h; his s eng h is ob ained om he sum o he ex e nal eld H and he in e nal (Weiss) eld, which is −α imes he magne iza ion M . The pa ame e α akes on alues o he o de o app oxi- ma ely 10−3 o 10−7 . As p oposed in [5], (5) was de- i ed o pa amagne ic ma e ials and hus does no always app oxima e he wa e o m exac ly. Then, in some cases, o he dependencies a e used, such as he B illouin unc ion gi en by Man =Msa ·2J+1 2Jcosh 2J+1 2J He a −1 2Jcosh 1 2J He a, (6) whe e J [-] is he quan um numbe , a quan i y ha akes disc e e alues om 0.5 o ∞ [15], and α [A/m] has a meaning die en om ha in (5). In gene al, such a unc ion can be any mono onic inc easing unc ion passing h ough ze o and limi - ing o ∓Ms o He going o ∓∞ . I he wa e o m is measu ed, he ob ained alues can be applied. The las pa o (2) is e e sible magne iza ion, exp essed in he model as he die ence be ween he lossless and he i e e sible magne iza ion, which is a enua ed [2], i.e., M e =c(Man −Mi ), (7) whe e he pa ame e c belongs o he in e al 0< c < 1 . The esul ing o mula, which shows he magne iza ion change wi h he magne ic eld s eng h a ia ion, is o med by a de i a i e o (1), an addi ion o (2), and a de i a i e o (3); he e o e, dM dH=dMi dH+dM e dH= dMi dH+cdMan dH−dMi dH= (1−c)Man−Mi kδ −α(Man−Mi )+cdMan dH. (8) The magne iza ion wa e o m M co esponding o he inpu wa e o m o he magne ic eld s eng h H is hen compu ed simila ly o he p ocedu e 53 Z. Roubal e al. Fig. 4. The measu ing hys e esis loop g oup, ini- ial magne iza ion cu e (a), and anhys e e ic cu e o he Sonape m ma e ial (b). Fig. 5. The simula ion hys e esis loop, ini ial mag- ne iza ion cu e (a), and anhys e e ic cu e o he Sonape m ma e ial (b). Fig. 6. Compa ison o measu ed and simula ed limi ing hys e esis loop o he Sonape m ma e ial. The e o is calcula ed o he uppe pa o he hys e esis loop. desc ibed in [4]: Fi s , he lossless magne iza ion alue is de e mined, acco ding o op ions such as ha om (5). Nex , he change o he i e- e sible magne iza ion is es ablished (3). Now, he e e sible magne iza ion (7) is calcula ed, allowing us o exp ess he esul ing magne iza ion change acco ding o (8). Howe e , he p e ious alues ha e al eady been en e ed, which means ha he calcula ion can be ca ied ou di ec ly. The dis- ad an age is ha an i e a i e me hod needs o be employed in he calcula ion, because he calcu- la ed alue de e mined om he de i a i e o he magne iza ion and i s p eceding alues appea s in he esul . Du ing he p ocedu e, we ha e o check whe he he e e sible magne iza ion (2) is smalle han he lossless magne iza ion in he  s quad an and, simila ly, in he hi d quad an when he mag- ne iza ion eld s eng h dec eases om he op o he cu e. I he condi ion is no applied, he magne iza- ion inc eases whe e he magne ic eld in ensi y is educed om he op o he loop, a p ocess ha does no co espond o he ac ual beha io o he magne ic ma e ial. In iew o he abo e de ails, i can be concluded ha ob aining he model pa am- e e s is no a simple ask. I is possible o use he 54 Designing an Imp o ed Me hod o De e mine he Anhys e e ic... Fig. 7. The measu ing hys e esis loop g oup, ini- ial magne iza ion cu e (a), and anhys e e ic cu e o he T a oke S ma e ial (b). Fig. 8. The simula ion hys e esis loop, ini ial mag- ne iza ion cu e (a), and anhys e e ic cu e o he T a oke S ma e ial (b). Fig. 9. Compa ison o measu ed and simula ed limi ing hys e esis loop o he T a oke S ma e ial. The e o is calcula ed o he uppe pa o he hys e esis loop. es ima ion o ini ial model alues om [3] and [6] and hen pe o m hei pa ame ic uning using he leas squa es me hod o he bes cu e  ing. 5. The measu ed anhys e e ic cu e as compa ed wi h he model All o he quasi-s a ic hys e esis loops we e mea- su ed o a pe iod o 40 s, when he inuence o eddy cu en s can be igno ed. The exci a ion signal H was ha monic. A sample o an olde o i- en ed Sonape m ail was used o e i y he ag ee- men o he anhys e e ic cu e measu ed ia a mod- ied algo i hm and op imized o he bes  o he hys e esis loop g oup and he p ima y magne iza- ion cu e (Fig. 4). Such a scena io was employed due o he ansi ion o sa u a ion being mo e g ad- ual han ha o he mode n o ien ed shee s. The o oidal sample had an ou e diame e o 110 mm, an inne diame e o 70 mm, and a heigh o 20 mm. The magne izing and he measu ing winding, N1 and N2 , had 100 and 50 u ns, espec i ely. 55 Z. Roubal e al. Fig. 10. The hys e esis loop g oup, ini ial magne- iza ion cu e (a), and anhys e e ic cu e o he MnZn e i e ma e ial Amidon 43 (b). Fig. 11. The simula ion hys e esis loop, ini ial magne iza ion cu e (a), and anhys e e ic cu e o he e i e ma e ial Amidon 43 (b). Fig. 12. Compa ison o measu ed and simula ed limi ing hys e esis loop o he e i e ma e ial Amidon 43. The op imal pa ame e s o he JilesA he on model we e de e mined in MATLAB by means o he leas squa es me hod (Fig. 5). The p ocedu e in- dica ed good ag eemen wi h he Lange in unc ion and sa is ac o y p og ession o he op imized anhys- e e ic cu e; he la ges die ences we e ound in low alues o Hmax (Fig. 6). Subsequen ly, a calcula ion was pe o med o he mo e mode n HI-B o ien ed silicon s eel T a oke S, showing ha he J-A model wi h he Lange in unc ion is p ac ically unable o exp ess he shape o he ec angula hys e esis loop (Figs. 79). Using a measu ed anhys e e ic loop did no yield a be e app oxima ion. The o oidal sample had an ou e diame e o 150 mm, an inne diame e o 110 mm, and a heigh o 30 mm. The magne izing winding N1 and he measu ing winding N2 had 150 and 100 u ns, espec i ely. The NiZn e i e Amidon 43 possesses a hys e esis loop wi h a specic shape (Fig. 10). The ini ial pe - meabili y egion is well cha ac e ized, bu he sha p sides o he hys e esis loop a e no isualized a all in he esul (Figs. 11 and 12). Again, he measu ed anhys e e ic loop appea s o be s eepe a he lowe alues o H . 56 Designing an Imp o ed Me hod o De e mine he Anhys e e ic... TABLE I The JilesA he on model's pa ame e s ela ed o he indi idual ma e ials. Ma e ial Ms [A/m] a [A/m] α ( ×10−6)k [A/m] c [ − ] Sonampe m 1034500 10.3 37.276 23.75 0.23 T a oke S 1909800 3.5 8.8587 14.0 0.10 Amidon 45 238732 16.5 100 24.5 0.01 Vi ope m 500F   1 0.75 0.01 Fig. 13. The measu ing hys e esis loop g oup o Vi ope m 500F (a) and he simula ion ela ing o he measu ing anhys e e ic cu e (b). The o oidal sample had an ou e diame e o 73.7 mm, an inne diame e o 38.9 mm, and a heigh o 12.7 mm. The magne izing winding N1 and he measu ing winding N2 had 135 and 44 u ns, espec i ely. The Vi ope m 500F nanoc ys alline ma e ial ex- hibi s a specic shape o he hys e esis loop, is e y na ow, and sa u a es quickly (Fig. 13). He e, he measu ed anhys e e ic loop should be ideally em- ployed in he model o deli e a e y small de ia- ion be ween he measu ed and he simula ed hys- e esis loops. The o oidal sample showed an ou e diame e o 30 mm, an inne diame e o 20 mm, and a heigh o 10 mm. The magne izing winding N1 had 8 u ns, and he measu ing winding N2 had 200 u ns. 6. Conclusions The a icle p esen s an algo i hm o measu e an anhys e e ic loop, ou lining he ypical hys e esis loop shapes (a smoo h ansi ion o sa u a ion in he Sonape m ma e ial, sha p ansi ion and igh - angled hys e esis loop in T a oke S, sphe ical shape in Amidon 43, and a na ow hys e esis loop and sha p ansi ion o sa u a ion in Vi ope m 500F). The cha ac e is ics o he ma e ials in ol ed in he p ojec a e compa ed in e ms o hei cha ac e is- ics and capabili ies (Table I). By ex ension, he au- ho s discuss he possible limi s o he J-A hys e esis loop model. Gene ally, in s ongly aniso opic ma- e ials such as Vi ope m 500F, he Lange in unc- ion is unsui able, as i was de i ed o iso opic ma e ials. Fu u e esea ch is planned o e i y ou desc ip ion, modi ying he cha ac e iza ion o he anhys e e ic cu e in acco dance wi h he guidelines p oposed in [16]. The p ocedu es exposed he ein in- ol ed an anhys e e ic cu e in he o m o a able, ensu ing he bes app oxima ion o he Vi ope m 500F ma e ial. The app oxima ion e o o he lim- i ing hys e esis loop was below 0.8%. Re e ences [1] J. Eichle , M. No ák, M. Ko²ek, Ac a Phys. Pol. A 136 , 713 (2019). [2] M. No ák, Ac a Phys. Pol. A 136 , 731 (2019). [3] D. Jiles, J. Thoelke, M. De ine, IEEE T ans. Magn. 28 , 27 (1992). [4] Z. Roubal, V. Smejkal, in: 2013 9 h In . Con . on Measu emen , 2013, p. 127. [5] M. No ák, Ph.D. hesis, Technical Uni e - si y o Libe ec, Libe ec (Czech Republic) 2003. [6] D. Lede e , H. Iga ashi, A. Kos , T. Honma, IEEE T ans. Magn. 35 , 1211 (1999). [7] K. He gli, H. Ma ouani, M. Zidi, Phys. B Cond. Ma e 549 , 74 (2018). [8] K. Chwas ek, J. Szczyglowski, Ma h. Com- pu . Simula . 71 , 206 (2006). [9] G. Xue, H. Bai, T. Li, Z. Ren, X. Liu, C. Lu, Ma hema ics 10 , 4431 (2022). [10] R. Szewczyk, AIP Con . P oc. 2131 , 020045 (2019). [11] Z. Roubal, P. 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