Frequency dependence magnetic materials

of hysteresis

curves in conducting

D. C. Jiles Ames Laboratory, Iowa State Universiq, Ames, Iowa 50011 (Received 12 January 1994; accepted for publication 4 August 1994) An extension of the hysteresis model has been developed that takes into account the effects on the hysteresis curves of eddy currents in electrically conducting media. In the derivation presented it is assumed that the frequency of the applied field is low enough (or the thickness of the material medium small enough) that the skin effect can be ignored so that the magnetic field penetrates uniformly throughout the material. In this case, the dc hysteresis equation is extended by the addition of a classical eddy-current-loss term depending on (i) the rate of change of magnetization with time, (ii) the resistivity of the material, and (iii) the shape of the specimen; and on an anomalous (or excess) eddy-current-loss term which depends on (~@ldf)~‘~. In the limit, as the frequency of the magnetic field tends to zero, the frequency-dependent hysteresis curve approaches the dc curve.

I. INTRODUCTION The hysteresis model equation used in this work has been described previous1y.r” The hysteresis equation can be used to describe both major (symmetric) loops and minor (asymmetric) 100~s.~ Previously, the model equation has been extended to take into account the change in shape of hysteresis loops under a constant applied stress.5 The earlier work on the development of the hysteresis model was concerned with quasistatic or dc hysteresis loops. The model is based on the concept of the anhysteretic, which provides a global optimum macroscopic state for the magnetization of a material subjected to a magnetic field. The equation for the anhysteretic magnetization was derived, and this resulted in the following expression: M,,(H) where qx)

=M,B

H+ aM,(H) a f

i

2

is the Langevin function,

2&?$) = coth x- l/x.

(2)

In IQ. (l), M,(H) is the anhysteretic magnetization under an applied field H, M, is the saturation magnetization, a is a coupling coefficient which arises from the exchange interaction, and a = kBT/puo(m), where kB is Boltzmann’s constant (1.38X1O-23 J K-r), T is the temperature in K, ,uo is the permeability of free space (4rX 10V7 H ml), and (m) is the effective domain size. This (m) is the size of a typical domain in a classical ferromagnetic Ising array with a meanfield interaction which obeys the Langevin equation. The parameter a is therefore proportional to the effective domain density and the absolute temperature. The equation for hysteresis can be derived from an energy-balance equation in which the magnetic energy supplied, for example, to an initially demagnetized material, can appear either as a change in total magnetization M (magnetostatic energy), or be dissipated due to irreversible changes in magnetization Mi, (hysteresis loss). If there is no dissipation (hysteresis) then, by definition, the magnetization must J. Appl. Phys. 76 (IO), 15 November 1994

follow the anhysteretic {hysteresis-free) curve. In the presence of hysteresis, the energy-balance equation is

PO I M dH, I Ma,dff,=~uo +p,k&l-c)

(3)

where the first term of the right-hand side is the contribution to the magnetostatic energy, and the second term on the right-hand side is the dissipation loss due to pinning. In this equation H,= H + CYM, the coefficient k is the pinning parameter which determines the amount of energy dissipated, and 6 is a directional parameter which ensures that energy is always lost through dissipation. S= + 1 when dHldt> 0 and S= - 1 when dHldtMi= and dH/dt>O) and when the magnetization is above the anhysteretic and the field is decreasing (M,l g

in (A7)

Now it has also been shown elsewhere2 that M=Min+M,,,

w>

=(l-c)M,+cM,,

(A9)

and therefore for small values of c it follows that dMldM,*l, and therefore Eq. (A7) becomes equal to the equation for the irreversible differential susceptibility that has been quoted beforeP namely

dMirr(W dH

energy supplied=change in magnetostatic energy

~oMan(H)dHe=

dMi#) dH

M,iH)=MiH)+k~

dMir#O

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences, under Contract No. W-7405ENG-82. The author wishes to thank Dr. G. Bertotti for useful discussions which have led to a more general exposition of the eddy current losses within the hysteresis model presented here, and Magnetics Technology, Inc. for use of proprietary hysteresis modeling software.

I

total magnetization M, whereas the hysteresis loss depends on Mi, . As we shall see, compared with earlier derivations, this merely results in a scaling correction to k, which amounts to multiplying k by a constant. Differentiating the above equation,

M&O -MirriW = [kLV(l -c)]

- a[M,(H)

iAl01

-Mb(H)]

and when summed with the reversible differential susceptibility, dMrw dH=c

dManV-0 dMiAf8 dH dH

(All)

’

so that the total differential susceptibility becomes d:f’

-(1-c>

ks M.&O-MirrW

~(1-c) +c dM,iH) dH

4ManW) - Mirr(H)1 ’

6412)

Another possibility that has not been fully explored is what happens when c is not small, so that dMldMh#l, and the approximations leading to Eq. (A12) cannot be made. The only known case in this situation is c= 1, when the solution is simply M(H)=M,(H). D. C. Jiles

‘D. C. Jiles and D. L. Atherton, J. Appl. Phys. 552115 (1984). ‘D. C. JiIes and D. L. Atherton, J. Magn. Magn. Mater. 61, 48 (1986). 3D. C. JiIes and J. B. Thoelke, IEEE Trans. Magn. MAG-253928 (1989). 4D. C JiIes, IEEE ‘Darts. Magn. MAG28, 2603 (1992). 5D. A. Kaminski, D. C. Jiles, and M. J. Sablik, J. Magn. Magn. Mater. 104, 382 (1992). “D. C. JiIes, J. B. Thoelke, and M. K. Devine, IEEE Trans. Magn. MAG28, 27 (1992). 7D. C. Jiles, in 37th Annual Conference on Magnetism and Magnetic Materials, Houston, 1992; J. Appl. Phys. 73, 5854 (1993).

J. Appl. Phys., Vol. 76, No. 10, 15 November 1994

8D. C. Jibes, IEEE Trans. Magn. MAG-29, 3490 (1993). 9S. Cbikazumi, Physics of Magnetism (Wiley, New York, 1964), p. 321. “D. C. JiIes, Introduction to Magnetism and Magnetic Materials (Chapman and HalI, London, 1991), p. 271. “G. Bertotti, J. Appl. Phys. 55, 4339 (1984). “G. Bertotti, J. Appl. Phys. 57, 2210 (1985). t3G. Bertotti, IEEE Trans. Magn. MAG-24, 621 (1988). 14F. Fiorillo and A. Novikov, IEEE Trans. Magn. MAG-26, 2904 (1990). “Design Manual for Tape Wound Cores (Magnet& Inc., 1987), p. 37.

D. C. Jiles

5855