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. Ma con, M. áp, in:
P oc. 9 h
In . Con . 2012 ELEKTRO
, 2012, p. 460.
57
Z. Roubal e al.
[12] T. Hej mánek, Z. Roubal, in:
2021 13 h In .
Con . on Measu emen
, 2021, p. 228.
[13] M. Nowicki,
Ma e ials
11
, 2021 (2018).
[14] T. Cha ubin, M. U ba«ski, M. Nowicki, in:
Recen Ad ances in Sys ems, Con ol and
In o ma ion Technology: P oc. o he In .
Con . SCIT 2016
, 2016, p. 593.
[15] K. Chwas ek,
J. Physics D Appl. Phys.
43
,
015005 (2009).
[16] R. Szewczyk,
Ma e ials
7
, 5109 (2014).
58