bergqvist - 1996 - a simple vector generalization of the jiles-atherton model of hysteresis.pdf
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8/11/2019 Bergqvist - 1996 - A simple vector generalization of the Jiles-Atherton model of hysteresis.pdf
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IE E E T R ANSAC T IONS ON M AGNE T IC S, V O L. 3 2 , NO. 5 SE PT E M B E R 1996
A
Simple Vector Generalization
of
t he
Jiles-Atherton Model of Hysteresis
h i d e r s
J .
Bergqvist
Electric Power Engi neering,Royal Iiistitute of Technology,
5-10044
Stockholm, Sweden
A
LstTact-
A
vector generalization
of
t he
Jiles-Atherton
model of ferromagnetic hysteresis is proposed. It gives a
differential equati on relating vector magnetization to vec-
tor magnetic f ield an d essentially retains t he s implici ty
of
the or ig inal scalar model . Th e model can handle both t ,he
isotropic an d anisotropic case a nd is equivalent. to t he scalar
model for unidirectional fields.
It
exhib its major features
of vector hysteresis such as rotational hysteresis and DC
magnetizat ion .
I. I NTR ODUC TI ON
HE
presence
of
hysteresis iii soft steels causes
losses
T ha t depend nont r iv ia l ly on the flux deiisity wa,ve-
form. Another problem associa .ted with hysteres is is the
r emanent magne t iza t ion a f te r the d i sconnec t ion of a de-
vice. Moreover,
in
for
ins tance r o ta t ing machines and th e
T-joints of trans form er cores, th e field is not restrain ed to
a fixed axis, giving rot ati on al hysteresis effects which com-
pl ica te the s i tua t ion f ur the r . To ca.lcula.te such effects, a
hysteres is model is needed that can
lie
used
t80
est$iiiiate
magii&iza.tion curv-es, prefe ral~l y for arb itr ary vect,orial
condit ions . Apa. r t f rom agreeing with exper imen ts , it , is
desirable for such a. iiiodel
to
be computationally efficient
and tha t the pa r am ete r s used t ,o char ac te rize
a
nmter ia l
are few a.nd s imple to de te r mine .
Ma ny different mod els for mag netic hysteres is have been
proposed over the years . Among the most widespread
approa,cl ies is that of Preisach- type inodels [ l] a n d t h e
St,oiier-Wohlfa,rtli model [a] n which the iiiediurn is
t r e a t e d a s an eiiseinble
of
independent par t ic les , ea.cli w ith
a s imple relay-l ike hysteres is mechan ism. By coii t ras t in
t,he Jiles-Atlierton (J-A) model [3] ,
a
differential equa.-
tioii between field a.iid t he microscopically averaged inag-
net~iza tioii s derived from tlie physical a,ssumptioii t h t ,
hysteresis ar i ses due to de f ec t s in the imte r ia l g iv ing a
friction-like force opposing inotion of doiimin walls.
The J-A model has sever a l a t t r ac t~ iv eeat,ures, nc.ludiag
t h e
use
of few a nd p hysical ly rela kd mat er ia l pa. rainet,ers ,
computational efficiency a.nd simplicity of
use.
However,
the mode l is only defiiied for th e one-diiiiensiona.1 case an d
does therefore no t t re at s i tuat ions involving va. r ia t ioi is in
the direct ion of
t,he
field. The aiin
of
this work is to pro-
pose
a
simp le forinalisin for generalizing t,he model to tw o
aiid three dimensions . Sect ion I1 contains
a
brief descrip-
t,ion
of
the scalar model . Sectmion
11
describes
tlie
proposed
vector generalizat,iori aiid
section
IV outl ines
soiiie
o i i ts
iiiost importa.n t, prop erties mid
shows
some coiiiparisoiis
w i th exper iments .
Maiiuscript received February
29,
1996.
This work
was
support,ed by the Swedish Research
Chuncil
for
Engineering Sciences.
4213
11.
T H I $
S C A L A R
M O D E L
In the J-A model , magne t iza t ion
M
i s represented as
t h e sum of tlie irreversible magnetization Mi d u e t o do-
m a i n wa.ll displacement and the revers ible magnetizat ion
mir due to domain w a l l bending .
In
tlie ideal, or lossless
ca.se, the relation between M and magne tic f ie ld
H
would
follow
a
single-va.lued curve called the anh yste retic curve
Man. Th e origin of 1iy:steresis is assum ed to be defects
in the mater ial which pin t l ie dom ain walls and obstruct
changes in Mi.
A
change in
M I ,
hereby pr oduces an en-
ergy
loss pr opor t iona l to the magni tude
of
t l ie change,
Here
R
is denoted th e pinnin g coeffic ient an d is re lated to
t h e density of pinning defects in the medium. A conse-
quence of (1) is tha t i f
M i
changes cyclically with an a.m-
pl i tude ki h e
net
loss
per
cycle is Q =
RldMil = 4 k l k i .
I t is well know n that f or r e al m a k r i a l s , t h e loss is well ap-
pr oximated by
a.
dependence of t h e t y p e
Q
- A l p where
/? is in
tlie
range 1.5
-
2.5. Ey. (1) ma y therefore overes-
tii-na.te th e loss a r e a of small hystteresis loop s.
Dur ing a , i i iagnet , izat ion process , the ma,gnetizat ion en-
ergy is tlle diiference bet,weeii
tlie
energy which would be
obtained in th e lossless case minus t,he energy
loss,
giving
tl ie energy balance equatio n
Mi dH
-
I
Man
d H , 1
IdMij
/
(2)
Here
H e
G H
+
aM
i , s
the effective field experienced
by iiidividual inagnet. ic inoiiients, with a t l ie mater ial-
dependent , mean field ~o ns t~ a i i t . ubs t i tu t ing IdMi
=
6 dA J i / dN , )
dH here
5
gn
d H , / d t ) ,
Sett ing the i i i tegrands on bo th s ides equal gives
which can be rewrits tena:j
4)
(5)
There is a.n inconsis tency here ~ v h i c h as been noted in
t he
past,
[4] which is
t h t
if for instlance MI
< Jan,
a n d
clH, < 0, hen
dM,/clH, < 0
which contradicts
ea,rlier
as-
su mp tio ns a.nd is also uiilihysical be1ia.viour. M etho ds for
mod ifying th e equa tion tcl a.void a negative s lope have been
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proposed [4] , [5].
There
;einains
t h e
quest ion of why this
effect, which looks like a math emat ica, l necess i t,y, emerges
i n t h e
first
place . I t
is
l iere suggested that the
reason
is
tlie
fact t,ha.t (4)
does
not necessarily follow from (3) s ince
t ,he integra t ion l imit ,s
on
t h e two s ides of t he eq uali ty s ign
in
3 )
ar e no t necessa r i ly equa l . The condi t ion
used
for
set8t8iiig
y
(3)
was no t tha. t the f ield H e wa.s t,he
same
for
the two processes but ra . ther that
t h e
supplied energy wa.s
t l ie s am e . A n inclica.tioii tha t there is
a
difference is t,lia.t if
me c.oiisider the case of a. cyclica.lly va.rying ina.giietization
wit11 amplit,ucIe
A ,
and (numer ical ly ) solve (5) th e result-
ing hysteresis loss will usually differ from
the one
predicted
b y
(1). On th e oth er ha.nc1, (5) does give agreement with
a loss expression
d Q
= k6 dMi.
Here we ado pt th e modi f ica t ion used in [SI which is
t o
assume
t,ha.t whenever
Man
M i )dH
0,
x)+
=
0
if
z
5
0.
ence hetmeeii
Pi
n d Adan,giving
T h e reversible compo ncii t is propor t ,ioi ia l to tshe differ-
dM,-=
c c11vIan
ClMi),
7 )
where c i s an ad jus tab le d imensionles s pa r am ete r .
Eqs (6) a.ncl 7 ) ) a n
b e
combined in to a. s ing leequa t , ion
a.s follows: O n o n e h a n d , d M
= d M c l M r =
I-C)Mi +
c clnCIa,,
giv ing
i
1 c
iik
dild
= -[ Man Mi)
dN,] ccZM,,,
O n t l i e o the r hand , (1 - c)(Ad,, M , ) = Mal, [Mi +
c(Adal, M1)] = Ma, M , so
(8)
1
6 k
dM = -[(Man M ) N e ] ++ c dM
T h i s e q u a t i o n ca.n be used to inc r ementa l ly de te r mine Ad
for a.ny ar b i t r a r y va r ia t ion of H or vice versa. .
111. THE
V E C T O R
G E N E R A L I Z A T I O N
The
sca1a.r m od el is a, hyste resis mo de l of t,he so-called
Diiliein t8ype [6] which means tlie relat ion
het,weeii
f ip lc l
N aiid magne t iza t ion
Ad can
lie expressed on t,he forni
dM = f H , Ad, c / H / l c Z H I
ctN.
The vect,or geiieralization
proposed here consis ts of modifying the dif ferential
equa-
t ion t80
a. vector D u h e i n model, i .e . an expression of t h e
t,ype
dl2
=
YjH, J ? dl?/jclI?l)
dg The modif ica. t ion
t8reat8s
h e
hysteresis iiiechanism for the isotropic, and
aii-
isot,ropic cases. No a.t, tem pt is made here however,
t o
inoclel t he anliys teret ic , a nisotropic cha, racter is t ic .
In
tlie
sc,ala.r
m o d e l , t h e m a g n e t i z a ti o n s t r i ve s
t owards
tlie anliysteretic,
value
but, is hindered
from
rea.ching it clue
to pinning . Th us , t ,he dif ference Mall iVl ar, as s trated
in [ : ] h e in te r pr eted as t8he or ce co inpel li iig c loma .i ii ~ ~ a l l s
to mm:e a n d
k
a.s
a.
resistance to cliaiiges. In the vector
case , one may s imi la r ly as sume tha t the force is
M , aiid t8he esis tance to changes through pi i lni i ig can
be
represented by k w h i c h i n t h e m i s o t r o p i c ca.se m a y
b e
a
symmetr ic t ensor
of
tlie second
rank
reflecting possible
diffeyent pinn ing stren gt,h s in different dire ctio ns while in
the isotropic case i t is a sca,la.r . I t is here convenient to
introduce th e a .uxil ia .ry quan ti ty
-
-
CJIianges in A?, ar e
assuinec]
par a l le l l to gf,
so
d M i / l d M ~ ]= gf / l gf ] .lso, i n t he s d a r c ase, Adi
changes oiily when th e f ie ld is incremeii ted in th e direc-
t ion
of
the for ce. I n t he vector case, i t seems n a t u r a l t o
expec t th a t changes
occur
only w hen gf clz
>
0
a n d are
then
pr opor t io i ia l to th i s q uant i ty . These concepts l ead
to
t h e e q u a t i o n
wliich gives t he i r revers ible component .
version of
(7)
is obviously to
set
Coiiceriiiiig the reversible coinpoiieiit ,
a
s imple vector
dA?,
=
Z
(dGall n?i)
(11)
where
7 may , l ike
k , be a.
sy mm etr ic tensor of t i ie second
ran k, c lue to dif ferent cap acity for dom ain w al l bending in
different directions.
Similar t o t he scalar case, (10) a.nd
(11)
can he rear -
ranged t o r e ad
t
CCG
= gflffl-qff
dl-i,]+ + 7 n?,,
(12)
where *-I
j
=
k ( fan n?)
(13)
F r o m
(12)
a n d
(13))
n?
can
be
evaluated for
any
var iat ion
of
N .
h e c:qua.tioii co nta ins no referenc e
to
t h e number of
dii i iensions and t ,~ ie valuat ion of c ln? for any giveii dXf is
s imple and f a .s t, . I n t e r m of computa t iona l s impl ic i ty aiid
speed , th e m odel coinpa. res fa .vorably to for ins tan ce vector
Preisach models
[1]
and the S toner - Wohl f a r th mode l [a]
Because dee
=
dI? + d A? , t,he equa t ion i s impl ic i t
wlien & f 0. T h i s does however not seein
to
be
a
serious
prohlein.
Oiie
possible technique is for ins tance to first
cliecIi if 2f .
(cGi
+ ii: .
Z
dn/7,,) > 0. If not,, tlieii
dnJ7 =
c .
cl Ga1,;
otherwise the in (12)
is
ignor ed and the
resultiiig 1inea.r equa.tion wit h respec t to d f is solved.
-
I V .
S O M E
MODEL
PROPERTIES
I f the f ie ld is res trained to
a,ny
fixed
axis , the vector
model (12) behaves l ike th e scalar model (8). Th is is c lear
if one considers t he fa.c, t, hat,
fo r
example in b o t h (6) a n d
the 1-D case of ( l o ) ,
d h f i >
0 if a,iid oiily if both
dN
>
0 a n d Ma,
>
M a i d t h e n h a s t h e m i n e v a lu e i n b o t h
ca.ses. T h i s ineaiis t h a t if t h e sca.1a.r
model
is c i i is idered to
lx sat isfactory or unidirectional f ields, tlie vect,or model
sliould autoiiia,tica.llyh e as well
if
t,liefield
does
not devia te
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2 0
2
. 0.0
r: -1.0
i
-2.0
Field [ N m l Field [ A h ]
Fig. 1. Typical scalar behaviour. Material parame ters
M S
=
2 . 1 T
a = 50 Afm,
k
=
S
A/m,
c
= 0.1,
y =
k /Ms .
far f rom being unidirect ional . Fig. 1i l lus trates typical
one- d imens ional behaviour w i th an anhys te r e tic f unc tion
M a n H )=
M s ( c o t h ( H / a ) - / H ) . Th e lef t curve shows
a
case when H is cycled with s lowly decreasing ampli tude,
while the r ight curve shows a case when the field is the
sum
of tw o components , one w i th smal le r ampl ihde and
higher f r equency than the o the r . I n the l a t t e r case , the
minor loop excurs ions dev iate , notably f rom being closed
unl ike in exper iments . This accomm oda t ion may be seen
as
a
f law in the mode l which is thus not ent irely well sui ted
to such processes . A modif ied model giving closed minor
loops was presented by J i les in [7], but i t does no t s eem
possible to apply th e technique to this vector mo del .
We next consider the al ternat ing and rotat iona l hystere-
sis loss. In gen eral , for
a
cyclic process, the net supplied
work dn?, is t h e
loss
per cycle . T he al terna t ing loss
is the loss emerging when M goes back a ,nd for th along
a
fixed axis and is often expressed as a function of magne-
t iza . t ion ampli tude,
Q a l t ( M ) .
Incidental ly, dur ing a sa tu-
r a t ion cyc le, i t a lw ays holds th a t
(Adan M ) lH, 20
a n d
t,he loss, in a.ccorda.nce wit h
l ) ,
ur ns o u t to be exac t ly
Qait(Ms) =
4Ms
I C (14)
c2 result which can be used as
a
m e t h o d t o d e t e r m i ne k
f r om exper iments .
It
i s r obus t and s imple and does no t
involve any o the r m a te r ia l pa r am ete r s except M, . For
an an iso t r opic mate r ia l ,
k
can be s imilar ly determined
f r om the los s in t he m ain d i r ec t ions . Th e r o ta t iona l hys-
teresis loss
is
the loss that appears i f the f ie ld is rotat ing
w i th cons tan t ampl i tu de . I n th e i so t r op ic ca .se, the mag-
netization too will rotate wit81ia c o n s ta n t a m p l i t u d e b u t
lag behind by some angle
6
giving
a
net loss per cycle
Qrot
=
Z T H M s in 4. Fig. 2a shows compar isons between
ca lcula ted a l te r m t ing and r o ta t iona l loss and exper iments
performed on
a
sam pl e of unoriented silicon steel using
a
rotat ional s ingle sheet tes ter [8]. Th e sam ple w as s l igh t ly
anisotropic and al te rnat ing measurements were therefore
taken in th e tw o ma in d i r ec tions .
IC
was fi t to th e measured
a l te r na ting loss , l eav ing no ad jus tab le pa r ame te r
for
t h e
rotat ional loss . I t is seen th at th e calculated rotat ional loss
is very close to a l inear dependence 27ICA4 which appears
t o b e
a
good appr oximat ion in the r ange
0.5-1.5
T, b u t
an
ov er e s th a t i on f or low and ve ry h igh fields .
In
par t icular ,
the exper ime ntal ly observed dekrease of the Ql.ok(M) near
sat ,urat ioii is no t given by t ,he model . If k were a, func-
tion of n?, his phenomenon could be r eproduced bu t th i s
I
0 0 0 5 1 0
1 5
2 0
Magnetiration [TI
Field [Ah]
Fig. 2 .
a)
Measured and calculated rotational and alternating hys-
teresis loss vs magnetization amphtude. Alternating measure-
ments in rolling (RD) and transversal (TD ) direction b) Illus-
tration of DC demagnetization.
might r equi r e addi t iona l ma te r ia l pa r amete r s .
Another well documented aspect of vector hysteres is is
tha t o f DC magn etizat ion. Th is refers t80 he pr oper ty
that i f
a
f ie ld is applied in one direct ion a nd t hen remo ved,
the resulting remanence is erased if the field is increased
along the perpendicular direct ion. T he model exhibi ts this
pr oper ty
as
is i l lus trated. in Figure 2b.
V.
CONCLUSIONS
A vector general izat ion of the J i les-Ather ton mode l of
hysteresis has been proposed. The re
is
no ev idence tha t
i t represents the best poss ible such general izat io n, but ha.s
the a t t r ac t ive f ea tur es of mainta in ing the conceptua l s im-
plici ty of the scalar model an d inheren tly exhibi t ing such
aspects of vector hysteresis as
D C
demagne t iza t ion and r o-
tat io nal hysteres is. Because t ,he mode l exhibi ts s ignif icant
accomm odation of small hysteresis loops a nd
a
monoton-
ically increasing rotat ional loss, i t ap pea rs unlikely t ha t
i t is
as
accur a te
as
some vector Preisach models , e .g. [9].
However, due t o i ts compu tat ional ef fic iency, small num-
ber of hysteret ic mater ial parameters , and s implici ty of
use, it could still be useful, especially for systems where
hysteresis ef fects are not the predom inant concern.
RE~FERENCES
[l] I. D. Mayergoye, Matheroatieal models of hy s t e re s i s , Springer,
New York, 1991.
[ Z ] E. C. Stoner and E.
P.
Viohlfarth, A mechanism of magnetic
hysteresis in heterogenous alloys, Trans. R.
Soc .
London, vol.
240, pp, 599, 194s.
[3]
D . C. Jiles and D.L. Atherton, Theory of ferromag netic hys-
t,eresis,J .
M a p
Magn. Mate r . , vol. 61, pp. 48 986.
[4]Kenneth H. Carpenter, A differential equation approach to mi-.
nor loops in th e Jiles-Atherton hysteresis model, IEEE Trans.
M a g . , vol. 27 pp.
4404,
1991.
[5]
D . C . Jiles, J. B. Thoelke. and M. E< Devine, Numerical deter-
mina tion of hysteresis para mete rs for the modelin g of magnetic
properties using th e theory of ferromagnetic hysteresis,
IEEE
Trans. Mag., vol. 28, pp. 27
1992.
[6] Augusto Visintin, Diflerent ia l m o d e l s
of
hysteresis, Springer,
Berlin, 1994.
[7] D.
C. Jiles, A self consistent generalized model for the calcula-
tion of minor loop excursions in the theory of hysteresis, IEEE
Trans. Mag., vol. 28 pp. 2602,
1992.
[8] S. A Lundgren, A J. Bergqvist, and S. G. Engdahl, A system
for dynamic measurements of magnetomeclianical properties pf
1995 Conf.
model of hysteresis, J .
A p p l .
Phus., vol. 73, pp. 5824, 1993.
ax-L;tr,trily exc; t ed
silicon-iron
sheet s
in
Proc. of the
ISEM
[9] A. A . Adly and I. D. Mayergoyz, A new vector Preisach-type