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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    4214

    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