Fast Computation of CMH Model

2 Pages·2001·0.13 MB·English

by  Patel, Umesh D.

Fast Computation of the CMH Model
Umesh D. Patel* and Edward Della Torre**
*Goddard Space Flight Center, Code 564, Greenbelt. MD 20771 USA, e-mail:[email protected]
**George Washington University, DC 20052 USA. e-mail:[email protected]
Abstract-- A fast differential equation approach for the DOK model the squareness of the material. In the CMH model. XR is given
has been extented to the CMH model. Also, a cobweb technique for by
calculating the CMH model is also presented. The two techniques are
contrasted from the point of view of fle.vdbility and computation lime. 3m R da_ am Rda _)mR dH c3mR
- 4 - _ -+._ (3)
ZR Oa÷ dh Oa dh _H dh Oh
INTRODUCTION In the following analysis, we will use
mR=a+f(H)+a_f(H), (4)
where
The CMH model is a moving Preisach model in which the
f(n)= 1-e -_ . (5)
reversible component of the magnetization is state-dependent.
Itcan be shown that
Unlike the DOK model, it accurately predicts the variation in
zero-field susceptibility with magnetization. It has not been a± = Jjexp(_ 7hi)p(h_,hi)dh, dh i. (6)
used much in the past because the DOK model, which is
Q=:t:I
much easier to calculate, is accurate enough in most cases to
From [1] the irreversible susceptibility for increasing
describe the magnetization process and a fast simple operative field is given by
algorithm was not available.
_ , 0+,t)h+,,L
This paper presents two techniques, the differential
equation method [1] and the cobweb method [2], which have
gt O. 2c_ e erf r.q/_ (7)
been previously presented for the DOK model, for calculating
the CMH model and contrasts them. In addition an
identification strategy for the CMH model will be presented.
The speed increase of the two techniques results from
computing the magnetization by simply adding the change in and for decreasing operative field is given by
magnetization due to any change in applied field to the
previously computed magnetization, instead of integrating 1 2_'_ "LerfL("_ +2h-t<-h,
Xt = _e
over the Preisach plane at each step of the computation. z..v[2
(8)
The differential equation method calculates the
susceptibility in closed form for Gaussian Preisach functions.
This limits the technique to simple magnetization
distributions that are integrable. However, it is a variable
step-size technique that is not limited by the discretization Using (1) and (6), we can derive da./dh for increasing
operative field as
error in the magnetization and distributes pseudo-hysterons
over the two hysterons switching at the same time, leading to
da. e(r'_i) !*-_'*_"._:F ((l+2)h+xh'+2r2Y"
a smooth magnetization curve. Changing the error criterion
during acalculation is difficult in the midst of acalculation. (9)
t. ,,,
THE DIFFERENTIAL EQUATION METHOD [,_+2h+xh, + ' -
-err _r y
We assume that the Preisach function of the CMH model and for decreasing operative field as
is the standard moving model of the form I(
i
da. e(r:":l ,....... :1: h_+ _.-x'h, +-1 r:y
elh,,h_:_ (h_-L) 2 (h +_1): _l)
2a_ 2a:
[ { r,q/2 (lO)
We compute the susceptibility using (
z:M,.[sz, (l+x_-x_, +!
-er] _ r ;"
r_
l
where Z1 is the susceptibhity of the irreversible component,
Similarly, we can derive da/dh for increasing operative
XR is the susceptibility of the reversible component and S is
field as
Similarly, the operative co-ordinates h, and hk for a+ are
2 " given by
dh .a__z :':- ""rfI I11_
L
(16)
hi =-7.'o',: +a, cosR_/- 2log0 - p)
-err 2 and for a. the operative co-ordinates are given by
- h_- Ah- xh, _-I r:7
h_=h, +a, sinai/- 21og(l-'p)
and for decreasing field as (I7)
hi = _o',:"or, cosO_/-2log(I p)
da_ e9:_':1 "-h I- +_r y The contribution of any single hysteron to the total a÷ or a. is
-- = r42 (t2)
dh 2--a-_x_ :' Ier't" equal to exp( _ cr 2/2 )/mn.
MODEL PARAMETER IDENTIFICATION
--e? .¢._
The parameters X, _, and _ are the same as in [I]. The
An identification method will be presented for the CMH
reversible susceptibility is given by
model which is a modification of the DOK identification
a'.f(H )-a'_f(H )+ (a+f'(H )-a_f(- H))
algorithm [3]. All parameters are computed in essentially the
-_:_SMsZ, (a+f'(H )-a_ f(- H)) same way as in the DOK model except for 7. In the DOK
model 7 is the reversible zero-field susceptibility that is
2' = _-....[l+.a(1-S_'t,_4s(a÷-.f.'(H)-a-f(-H}} I o3)
independent of the magnetic state. Since the reversible
The sum of (7) and (13) gives the total susceptibility and then susceptibility for zero field is now a function of
the next magnetization can be computed by magnetization. Thus, we recommend identification of _/ by
M(H + ,5I"t )= m(H )+ 2",5tl. (14) fitting the descending major loop near positive saturation.
There are several problems with this technique. The first
problem is that the stability of this method requires small step
in H under certain conditions. The second problem is that DISCUSSION AND CONCLUSION
when the reversible magnetization is close to saturation the
irreversible magnetization can get to be very large due to the
exponential increase of fill). This can be overcome by setting In previous models, the squareness S and were the same
the magnetization to saturation when it gets close to for the entire distribution. This made it difficult to satisfy the
saturation. However, this puts a glitch in the solution which crossover condition for particles with large hi. In the cobweb
may be undesirable. model it is possible to decrease S or for particles with large hi.
We have presented two fast techniques for computing the
magnetization expected for the CMH model, both of which
produce the same results and are fast. We will present results
THE COBWEB METHOD for both methods of computation.
The cobweb grid in Preisach plane was developed to ACKNOWLEDGEMENTS
achieve higher speed and accuracy of numerical computation
for irreversible magnetization [2]. Here, the CMH model is
implemented based on the cobweb grid systems for We would like to thank Ann Reimers and Shailendra Garg for
useful discussions.
irreversible magnetization and a. and a variables for the
reversible magnetization.
REFERENCES
The cobweb method distributes hysterons uniformly in 8
in the interval (0,2x) and in p in (0,1) on m x n grid. The
[1]. A. Reimers and E. Della Ton-e, "Fast Preisach based
relationship between these co-ordinates and the operative co-
magnetization model and fast inverse hysteresis model,"
ordinates h, and hkfor irreversible magnetization are given by
IEEE Trans. Magn., 34, Nov. 1998, pp.3857-3866.
hk = hk + O'_ sin 04- 2 log o - p) [21. O. Alejos and E. Della TorTe, "'Improving numerical
(151 simulations of Preisach models for accuracy and speed,"
hi = cri cosS-J- 2 log O-/9) presented at INTERMAG, Toronto, Canada, April %14,
The contribution of any single hysteron to the total moment is 2000. IEEE Trans. Ma_n., (in press).
equal to that of all the others and is equal to Ms/ran. Thus, the [31. E. Della TorTe, "'Parameter identification of complete-
maximum error in M is half that and by choosing mn large moving hysteresis mode[ using major loop data," IEEE Trans.
enough the error can be made arbitrarily small :daen.. 30, Nov. 1994, pp.4987-5000.