Frequency Dependence of Hysteresis Curves in " Non-Conducting " Magnetic Materials

The problem of modelling the frequency dependence of hysteresis in magnetic materials is approached in a new way. The de magnetization curve, or hysteresis loop, is assumed to be the equilibrium position for the bulk magnetization. All microscopic processes which occur under the action of a time-varying field can be averaged to give a time dependent displacement from the equilibrium, AM. In this paper, we examine the case where eddy current effects do not play a signillcant role, that is to say the model applies to "nonconducting" media. It is shown that AM obeys 8 d8aped simple harmonic motion eqnation. This means that the time dependence of the displacement magnetization AM is the Laplace transform of the field waveform. This enables the time dependence of the magnetization to be modelled, once the dc hysteresis curve is known, with only two additional materials parameters of relaxation time and natural freqmency. The model emerges as a natural extension of the theory of hysteresis.


Frequency Dependence of Hysteresis Curves in "Non-Conducting" Magnetic Materials
means that the time dependence of the displacement magnetization AM is the Laplace transform of the field waveform.This enables the time dependence of the magnetization to be modelled, once the dc hysteresis curve is known, with only two additional materials parameters of relaxation time and natural freqmency.
The model emerges as a natural extension of the theory of hysteresis.

INTRODU"
The magnetization curves, or hysteresis loops of femmagnetic materials change as a function of the frequency and waveform of the applied magnetic field.Most measurements of hygteresis are performed under dc, or quasi dc, conditions.In most applications the material is subjected to an ac field.ILhedBeames between the dc andac hysteaesiscurves deped on a number of factors including the electrical conductivity and permeability of the material, the rate at which the magnetic moments can rotate into the field, the fiquency of the applied field and its waveform, whether sinusoidal, triangular OrsQplaeWave.
'his paper introduces a new approach to modelling the changes in magnetic hysteresis curves as a function of fiequency.The key development has been the identification of diffcrrntial equations representing the time dependence of hysteresis curves in two principal cases.lhese are the effects of eddy currents in electrically conducting magnetic media and the effects of magnetic relaxation in nonconducting media.
In this paper, we will discuss only the latter.
In previous work, a phenomenological model of hysteresis date, the theory has only dealt with time independent hysteresis.The equations can, however, be extended to account for time &pendent effects.

Non-Conducting Media
In "nonconducting" media, we assume that the effects of eddy currents can be ignored.If we consider these averaged domain wall movements, it is clearly apparent that they cease once the magnetization M(t) at a given time t has reached the dc magnetization curve, which we will denote w.

lim M(t) = K(H) (1)
Equally clearly K(H) is a function of the magnetic field H, which must be described by a time independent hysteresis function.The value of &(H) is uniquely dehed by the magnetic field history of the specimen and is obtained by calculating the value of the bulk magnetization that would be achieved when all transients in the magnetization process have been completed.This means that M,,,,(H) is represented by the value of bulk magnetization on the quasi& hysteresis loop for a given sequence of field reversals and the prevailing magnetic field strength.In other words, &(H) is path dependent but is time independent.It can be modelled using the equations given in earlier papers for describing dc hysteresis loops [l], or by other descriptions of time independent hysteresis loops, such as Preisach models.Given this result, and the damped harmonic nature of the change in magnetization, the displacement magnetization obeys a differential equation of the form, t-b- -". .

Values ofparameters
A study has been made Of manganese zinc ferrite Philips series 3C81 ferrite material).The dc hysteresis parameters have been given previously [l], together with a C O m m S O n of the modelled and measured hysteresis curves.These 30AOm-1, a = 10-5, = o.55.The V e a b f i Q at 1 kHz was almost identical to the quasi static initial permeability.From these results it can be seen that there is an iacrease in coercivity, hysteresis loss and remanence with increasing frecluency.An increase in initial permeability is predicted by the model, and this is followed by a decmse at where is the gyromagnetic ratio (0.22~10~ rads.m*s-l-A-l).pi is the relative initial permeability, which is dimensionless frequencies, both of which are observed in practice, (&-I = %i the initial susceptibility), Ms iS the SiituriltiOn changes in initial permability with fresuency for two magnetization, 6 is the wall thickness and d denotes the different sets of model parameters are shown in tables 1 and 2.
The actual values of o, can be OM experimentally from Table 1 .The main contribution of the present work consists of the incorporation of hysteretic effects into the equations of motion given first by Landau and Lifschitz [7] for the description of time dependent magnetization in magnetic matexiah.
The ikqmcy dependent hysteresis curves h f m consist of two independent contributions to the magnetization.These are the dc hysteresis curve, which represents the locus of quilibrium magnetization as a function of field, and the displacement magnetization, which obeys the damped harmonic motion quation.

723 the material to be determined from the initial susceptibility 4Ooo 300 55 1 and the damping coefficient X in the differential equation of
3 measurements of the initial susceptibility of materials as a function of frequency.The frequencies at which the maximum of the initial susceptibility occurred in the two materials m g a n e ẽ zinc fed& x 8 o studied here were v, = 0.768~106~'~ (* = 4.83~106 rads1) 3.60~106 rads1) for the 3C81 Mn-Zn femte [8].v(kilz) bcr is the critical value of X, which is the value of A 1732 separating conditions under which wall resonance, rather than #x)o 1500 1040 wall relaxation, occur.This enables the natural frequency of 3Ooo 600 solving equation (4) give increasingly values obtained with the following d l m e t e r s : rounded hysteresis loops as the frecluency of excitation is in-Br43 Tesla, *27AOm-l, k=30AOm-l, a=5x10-5, creased ~. 8 3 x l O b d .~-~ -srith . .for the 3C80 Mn-Zn ferrite, and vr = 0.573~106 (or = Freauencv Msasm4 related by the equation 10 2100 (&lo%) 202%The resonance frequency %and the natural !i-equency o, are lo00 2500 cd.41, m.22x1~-8s-l (.F=9x10-9s),

Fig. 1 .Fig. 2 .
Fig. 1.Model hystaesis loop for 3C81 ferrite at 1 kHz.This curve is identical to the dc hysteresis curve on this scale.

is that it
has been developed [ 1.21 based on the conside" of energy loss due to, among other factors, domain wall pinning.The advantage of this model over others

3]. In this work, the time
This approximation works well for high frequency femtes [ The change in magnetization in the low field region, where hysteresis occurs, is determined primarily by domain wall motion.The domain wall motion can itself be described by a second order linear differential equation, as discussed by D(lring 141 and later by ChiLaZumi [SI.This applies on the micromagnetic scale of a single domain wall, however, the concept can be scaled to describe the macroscopic magnetization changes, since the change in bulk dM magnetization with time,is simply the average over the d t entire material of the individual domain wall movements.

be used to prove this: once the magne
magnetic moments inside the material can oscillate in the the Larmor precession frequency of the electron spins under tization reaches &(€I), it is in equilibrium and SO them is no net face on the domain walls.this phenomenon is that it relam the -yt frequency a, of domain wall motion to the initial permeability of the hysteresis curve via the equation Ihe damping meter A determines the Of Of the magnetization to an external field, and from the equation this can be expressed in tenns of an equivalent relaxation time 't = lh.For this material 't * lo4 Sec.
-AM(t,H)+2X d2 zAM(t.H)+on2AM(t,H) d = 0 dt2 and since the he derivatives of &(H) are the Of based on the paramem given above.Figs.1-3 show the hysteresis c w e s of the modelled ferrite material under the action of a sinusoidal magnetic field of amplitude 100 A/m (1.25 Oe), at of 1, 50, and kHz.

Table 2 .
-v u Of hysteresis curves in non-conducting materials has been presented.The model is based on the second order linear differential equation of motion of domain walls, which is averaged to describe the behavior of the whole material.The result is a differential equation describing the displacement magnetization AM = M(t) -hL(H) where k ( H ) is the locus of points on the dc hysteresis curve.
CONCLUSIONSA model for descn'bing the 6requen~y dependence