A general statistical approach to the problem of eddy current losses

Journal of Magnetism and Magnetic Materials 41 (1984) 253-260 North-Holland, Amsterdam

253

INVITED PAPER A GENERAL STATISTICAL APPROACH TO THE PROBLEM OF EDDY CURRENT LOSSES G. BERTOTTI lstituto Elettrotecnico Nazionale Galileo Ferraris, Gruppo Nazionale Struttura della Materia del C.N.R., I-10125 Torino, Italy

The lines along which a general statistical theory of losses can be developed are discussed. The magnetization rate l ( r , t) in the sample cross-section is described as a random sequence of elementary magnetization jumps, and the loss behavior is expressed in terms of the space-time correlation properties of the jump sequence. In the important case where these correlation effects are of the Markov type, the loss becomes a function of the transition density M(Ar, At), representing the conditional probability density that two subsequent correlated jumps are separated by a distance Ar and a time delay At. M(Ar, At) provides a direct link between the loss behavior and the microscopic dynamics of the magnetization process. Some applications of the theory to the prediction of measured excess loss anomalies in soft magnetic materials are discussed.

1. Introduction

The mechanism by which energy is dissipated in a conducting ferromagnet during the magnetization process is essentially the decay through the Joule effect of the eddy currents induced in the material by the variations of magnetization. If P is the average loss per unit volume and j(r, t) the eddy current density, the simple relation holds P=

lim f~T/2 ar-.oo -aT/2

o

2

'

(1)

where V is the sample volume and o represents the electrical conductivity of the material. Though the previous statement and eq. (1) give, in principle, a clear and general solution to the problem of the physical origin of magnetic losses, the prediction of the actual loss behavior corresponding to given experimental conditions is, in practice, an extremely difficult task. Actually, in order to predict, through Maxwell equations, the behavior of j(r, t), we need a detailed knowledge of the magnetization rate ;r(r, t) in the whole material, an information which is by no means simple to obtain. First important progress began to be achieved when it was recognized that the loss behavior is mainly governed by the dynamic properties of well defined macroscopic objects, namely the Bloch walls which separate different magnetic domains

[1-4]. Starting from the results obtained by Williams, Shockley and Kittd [5], several authors have investigated the relationship between losses and wall dynamics, by treating the Bloch walls simply as rigid objects [5-7], or taking also into account the wall flexibility [8-11]. In spite of their importance, however, the models based on deterministic descriptions of the wall motion fail to provide a general characterization of eddy current losses. Great difficulties arise, for instance, in the interpretation of the strongly nonlinear loss anomalies observed in thin iron laminations (see fig. 1), where the presence of a very fine domain structure would seem to imply a purely 130 "~

Iron poly~ystat

A : 55s O =9'K~mJ/kg

d :Ol rnm Imax =13 T 90

~oI...01.--E~-~ 9-"

I00 fm (Hz) Fig. 1. Behavior of the loss per cycle P/f. vs. the magnetizing frequency fro, measured on an iron lamination of thickness d = 0.1 ram. The experimental set-up and procedure used in the loss measurement are described in ref. [12]. The dashed line represents the best fit curve obtained from eq. (13). The corresponding values of the parameters A and B are indicated.

0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

0

20

t,0

60

G. Bertotti / Statistical approach to eddy current losses

254 50 3*/oSiFe 010} [001] singte cryslal ~0

A:Ss B : 0.9 mJ/kg

d : 028mm lmax : t72 T

~ ~-I ~ ~.o- ~

~30 uJ

JIY

.o"

10 I

I

I

20

40

60

I

80 fm (Hz)

100

Fig. 2. Behavior of the loss per cyle P/fm vs. the magnetizing frequency fro, measured on a 3% SiFe single crystal of thickness d = 0.28 mm (from ref. [12]). The dashed line represens the best fit curve obtained from eq. (13). The corresponding values of the parameters A and B are indicated.

classical dynamic loss, as well as in highly oriented 3% SiFe single crystals [12] (see fig. 2), where the presence of a well defined bar-like domain structure would suggest the validity of a simple Pry and Bean.model [6]. These inconsistencies have a precise meaning. Actually, due to the great complexity of the involved interactions, the real character of the magnetization process is rather stochastic than deterministic, so that models which disregard this essential aspect of the problem are often destined to provide inaccurate predictions. We expect that, in a satisfactory theory of losses, statistical concepts should play a quite fundamental role. The basic idea on which we can found a general statistical theory of losses is fairly simple. Barkhausen noise experiments [13,14] clearly put in evidence that the motion of the domain walls during the magnetization process is highly discontinuous and naturally lead to describe the domain structure dynamics as a space-time stochastic process, made up of a random sequence of elementary magnetization changes, each corresponding to a sudden, localized jump of a domain wall segment in the material [14-16]. If we consider the eddy current pattern generated by a single jump around itself, it is not difficult to evaluate the related energy loss [17,18]. However, when many jumps take place, the total loss is not simply given by the sum of the elementary losses associated with the individual jumps. In fact, this sum merely corresponds to the hysteresis loss contribution. Being,

as shown by eq. (1), quadratic in the eddy current density, the total loss is also strongly affected by the degree of superposition of the eddy current patterns generated by different jumps. These superposition effects, which are evidently governed by the space-time correlation properties of the jump sequence, are at the origin of the dynamic loss contributions. Understanding the physical nature and the general properties of this relationship between the loss and the correlation properties of the magnetization process is the main task of a statistical theory of losses. A first attempt in this direction was made by Mazzetti [19,20], who developed an approximate model of loss calculation, applicable to the case of a sample of cylindrical shape and subsequently extended by Bertotti et al. [12,21,22] to the more realistic case of a sheet geometry. The results obtained by these authors have suggested a new interesting approach to the experimental investigation of 10ss mechanisms [12,23-25]. From the theoretical point of view, however, their model is heavily founded on intuitive and approximate arguments which allow only an indirect insight into the real nature of the connection between losses and space-time correlation effects. A general statistical theory of losses, based on more rigorous physical arguments, has been proposed recently by Bertotti [18,26]. It is the aim of the present paper to review and discuss the basic principles and applications of this theory. 2.

Theory

2.1. The model

The first step in the development of a general statistical theory of losses is to work out the general connection existing between the average loss P and the magnetization rate jr(r, t). The mathematical procedure by which, starting from Maxwell equations, this can be done is thoroughly discussed in refs. [18,26] and can be summarized, very briefly, by the following points: It is assumed that each elementary magnetization change is always directed along the longitudinal axis of the sample; - The transient response of the material to a local -

G. Bertotti / Statistical approach to eddy current losses

magnetization change is assumed to be controlled by the reversible permeabilit.y #; - The random character of I(r, t) with respect to both r and t is conveniently dealt with in the space (k, o~) of space-time Fourier transforms. As a final result, eq. (1) turns out to be equivalent to the equation

p=o4 ~

f_o da~ ,kl 2 St(k, ~), 2~r ikl4 + (60o#)2

(2)

where S represents the area of the sample crosssection and the summation with respect to k runs over a discrete set of values determined by the boundary conditions of the problem. Si(k, ~) represents the statistical power spectrum [27] of jr(r, t), with r varying in the sample cross-section S [28]. Eq. (2) is the basic equation connecting the loss behavior with the space-time correlation properties of the magnetization process. As anticipated in the introduction, the magnetization rate jr(r, t) can suitably be described as a random sequence of elementary magnetization changes jr(~)(r, t; r~, ti), taking place at different positions r/and times t~: jr(r, t ) = ~"~I(S)(r, t; ~, t,). i

(3)

Through eqs. (2) and (3), the loss calculation is reduced to the typical statistical problem of evaluating the power spectrum Si(k, ~) of the stochastic process jr(r, t) as a function of the shape and correlation properties of the elementary magnetization events from which jr is built up.

255

jumps per unit time and cross-section area, IJr(S)(k, a~)l2 is the average energy spectrum of a single jump and s(cr°ss)(k, 60) includes all the contributions coming from the ensemble average of the cross products I(S)*(k, o~; r/, ti)'I(S)(k, ~; ry, tj), with i 4:j. Since p is proportional to the magnetizing frequency fm, when the first term of eq. (4) is inserted in eq. (2), a loss contribution per cycle independent of fm is obtained. This is the hysteresis lOSS, pthyst), associated through IJr(S)(k, ~)12 with the kinetic properties of the individual elementary jumps. The information on the correlation properties of different jumps is carried by S(Cr°s~)(k, w). If we assume that no correlation effects take place, i.e. the magnetization process proceeds through statistically independent jumps, then it is found that the loss contribution associated with S(Cr°~)(k, ~) is simply coincident with the classical loss p(dass). On the other hand, when we introduce space-time correlation effects in order to build up the real domain structure dynamics of the system, we obtain that further excess loss contributions are added to the classical loss. Of great physical interest is the case where these correlation effects are described in terms of a Markov process [27]. Actually, in the case of a chain of Markov correlated events, S(C~°s~)(k, w) is fully determined by one function only, namely the transition amplitude M(Ar, At) giving the condition probability density that two subsequent correlated jumps take place at a relative distance Ar and with a time delay At. Eq. (2) becomes then

P = P(hyst)+p(class)+F(M(Ar, At)},

(5)

2.2. Loss separation A first interesting result of the present statistical approach concerns the subject of the separation of losses. It turns out that the usual classification of eddy current losses in terms of the hysteresis, classical, excess or anomalous contributions is a natural consequence of the general mathematical structure of St(k, ~) [18,26]. Actually, eq. (3) and the general definition of power spectrum imply that

st(k,

=

)12+

(4)

where ~, is the average number of magnetization

where the exact form of the functional F can be found in ref. [18]. The result expressed by eq. (5) has important physical implications. Actually, the functional F provides a direct bridge between the loss behavior and the microscopic dynamics of the magnetization process, characterized by the transition density M(Ar, At). The main interest will lie now in the development of physical models of the magnetization process, by which the principal properties of M(Ar, At) corresponding to different conditions of domain structure, magnetizing frequency, etc. can be predicted.

256

G. Bertotti / Statistical approach to eddy current losses

3. Applications The general relationship established between the loss and the transition density M(Ar, At) permits quite different approaches to the loss problem. On the one hand, we can assume a kinematic viewpoint, in which we postulate, by a priori considerations, different behaviors of M(Ar, At), in order to calculate, through eq. (5), the related loss and see which description is in better agreement with the experimental results. On the other hand, we can try a more difficult but valuable dynamic approach, in which we deduce the properties of M(Ar, At) directly from an investigation of the magnetic interactions controlling the evolution of the domain structure.

3.1. Kinematicapproach: the Pry and Bean model As a typical example of kinematic description, we shall consider the Pry and Bean model [6], for instance in the case of a magnetization cycle far below saturation. As shown in ref. [18], eq. (5) predicts the exact value of the Pry and Bean loss when M(Ar, At) is assumed to be of the form

M(Ar, A t ) = ~ - ~qS(Ax-2qL) -M

×

exp(-At/-~) __ 0(At), At

(6)

where 2L is the wall spacing, 2M + 1 ( >> 1) represents the number of walls in the sample cross section, ~-t is the average time interval between subsequent elementary jumps and 0(At) is the Heaviside step function. Eq. (6), which simply states that, given a jump, the next one will take place, with equal probability, on any of the existing walls, clearly puts in evidence the highly idealized character of the Pry and Bean model. In particular, if we want to associate the correlation effects described by M(Ar, At) with real physical interactions taking place between subsequent jumps, then the fact that all the delta functions in eq. (6) have the same amplitude implies that we are actually deahng with a nonrealistic correlation of infinite range between the domain walls. Some generalization and modification of eq. (6) seems

therefore necessary, in order to achieve a more suitable description of the domain structure dynamics [29].

3.2. Dynamic approach Of course, the most valuable result, from a physical point of view, would be to obtain the properties of M(Ar, At) corresponding to different conditions of domain structure, magnetizing frequency, etc. directly from an analysis of the microscopic interactions controlling the magnetization dynamics. Here, we shall discuss a simple model concerning materials with a very fine domain structure which, although somewhat schematic, can nevertheless provide valuable information on the loss behavior. As anticipated in the introduction, the loss models of the type of the Pry and Bean one cannot give a satisfactory account of the loss anomalies observed in materials having a very fine domain structure, like polycrystalline iron (see fig. 1). Actually, the basic parameter involved in all these models is the ratio between the wall spacing and the sample thickness and, as a general result, it is expected that, when this ratio becomes much smaller than unity, any loss anomaly vanishes and the material approaches a purely classical loss behavior. Such disagreement between theory and experiments comes from the fact that Pry and Bean-like models take only account of the geometrical effects related to the spatial configuration of the walls but completely disregard any dynamic effect related to the local interactions which control the propagation of a domain structure rearrangement. Such effects can instead be properly taken into account when the behavior of the transition density M(Ar, At) is investigated. As a starting point, it is important to recognize the strict connection existing between M(Ar, At) and the Barkhausen (B.) noise properties. Actually, starting from the theory of the B. noise spectrum developed by Mazzetti and Montalenti [15], the statistical properties of the B. signal have usually been interpreted in terms of time correlation effects between subsequent elementary domain wall jumps, and characterized in terms of the conditional probability density P(At) that two subsequent jumps are separated by a time interval

G. Bertotti "/Statistical approach to eddy current losses

At. According to this definition, P ( A t ) is no more than the space integral of M(Ar, At) over the sample cross-section: P(At) =

fsd2ArM(Ar,

At),

(7)

so that any information on P ( A t ) obtained from B. noise experiments actually concerns also M( Ar, At). A striking feature put in evidence by B. noise investigations is the tendency of the elementary magnetization jumps to cluster into large avalanches. As shown in ref. [15], this clustering effect is well explained by a function P ( A t ) of the form [ C exp( - At/T o)

P(At)= [ 0

~0

+ C1 exp( - At~e1) ] 0(At), T1

(8)

l

where ¢0 represents the average time interval between subsequent events in a cluster, while ¢1 >> T0 is the average time separation between the end of a cluster and the beginning of the following one. According to eq. (7), M(Ar, At) will therefore be given by a generalization of eq. (8) of the form

m ( Ar, At) = [Corn(At )

I

e x p ( - At/e0) T0

C1 e x p ( - A / / ¢ l ) ] 0 ( A t ) . + S ¢1

]

(9)

m ( A r ) describes the local space correlation properties of the clustering process and is expected, in general, to be different from 0 only in some limited range IArl ,< r~. On the other hand, no space dependence has been introduced in the second term of eq. (9). This means that the magnetization clusters are assumed to originate at random positions in the sample cross-section, as expected in the case of a material with a very fine domain structure. The main physical point concerning eq. (9) is thus that the elementary magnetization events are clustered both in space and time, as a consequence of the strong local interactions which couple the motion of different wall segments. It is evident that the space and time concentration of many

257

magnetization jumps greatly favours the local superposition of eddy currents and is therefore expected to give rise to some excess loss. Eqs. (5) and (9) permit a quantitative evaluation of this effect. Detailed calculations [29], which we have not the space to report here, lead to the final result p

= p(hyst) + p(class) +

o'[a ( rc, ¢o) A~/¢0,

(10)

where I represents the average magnetization rate, A~ is the average flux variation associated with an elementary jump and the dependence of the function G(r c, %) on both rc and ~0 is so weak and it can be fairly approximated by a constant value = 0.5.

At this point, in order to compare eq. (10) with experiments, we need to determine how %, the average time interval between subsequent events in a cluster, is expected to depend on the magnetizing frequency. A simple, though somewhat schematic, model is sufficient to provide the basic features of this dependence. Let us consider two generic subsequent jumps W0 and W in a cluster. The correlation existing between them corresponds to the fact that W0 perturbs the stability of the sorrounding domain structure, eventually giving rise, after a time interval of the order of %, to the new jump W. If we call H ~ ) the local coercive field at the point W, we can imagine that the magnetic field H w ( t ) acting on W starts, at the time t = 0 where W0 takes place, from a value Hw(0 ) = *rn0) ' W < rnc) "*W , and then increases in time until when, after a time interval To, it reaches the value Hw(%) = H~'

(11)

and W takes place. Two main causes contribute to the time increase of Hw(t). On the one hand, the external field increases at a rate proportional to the magnetizing frequency, giving a contribution O~fmt. On the other hand, the perturbation brought about by W0 tends to increase H w from the value Htw°) up to some final value Htw°)+ A H w > H ~ ). In order to understand how this increase will take place, let us note that the propagation of the perturbation caused by W0 will involve the development of new flux paths in the material, a process which is controlled by the decay of the eddy currents around W0. We know [17,18] that, for a time interval of

258

G. Bertotti / Statistical approach to eddy current losses

the order of the duration z of W0, large eddy currents concentrated in a small region around W0 completely shield the flux variation carried by W0, so that the surrounding domain structure actually cannot "know" that Wo has taken place. The static flux variation brought about by W0 begins to emerge only for t > T, since then the eddy currents begin to decay with a law that, for large times, becomes of the 1/t type. This suggests that also the related increase of H w will follow, for t > T, a 1/t dependence, and leads to the final equation

H w ( t ) = H ~ ) + a f m t + ( 1 - T / t ) A H w.

(12)

From eqs. (11) and (12), the dependence of TO on fm can easily be determined. When the result is inserted in eq. (10), the loss expression is obtained P = e(hyst) + P(dasS' + Bfm(vfl + 2Afm - 1 ) ,

(13)

where A and B are proper combinations of the parameters appearing in eqs. (10)-(12). As shown by the dashed line of fig. 1, eq. (13) gives a quite good account of the nonlinear loss anomaly observed in thin iron samples. From the best fit values of A and B it is obtained that the ratio z/T0 attains values ranging from -~ 10 -3 to = 10 -~ when fm increases from 0 to 102 Hz. Since, according to this interpretation, the same mechanism, i.e. the clustering of wall jumps, is at the origin of both the loss anomaly and the B. noise, we expect that some relation should exist between the parameters A and B of the loss model and the B. noise properties. It can be shown that this is actually the case. In particular, the quantity B turns out to be simply proportional to the B. noise power measured at low magnetizing frequencies. The comparison of loss and noise results can therefore provide a valuable test of the consistency of the proposed loss model [29]. In fig. 2, eq. (13) has been used to fit the loss curve measured on a 3% SiFe(ll0) [0011 single crystal [12], obtaining a remarkably good fitting. Of course, the model previously discussed, being heavily based on the assumption of a very fine domain structure, is not directly applicable to the case of oriented materials with a large wall spacing. However, the fact that eq. (13) works nevertheless very well is not an accident. Actually, a loss

expression identical to eq. (13) is obtained also in the case of oriented materials when it is assumed that the applied field determines not only the velocity of the single moving walls, but also the total surface of active walls [30] participating in the magnetization process at different magnetizing frequencies. From this point of view, eq. (13) turns out to be a suitable generalization of the Pry and Bean model, capable of taking into account, in some approximate but unified way, various mechanisms, such as irregularities in the wall velocities [2,12,31], domain multiplication [31-34] and wall bowing [8-11,35], which strongly affect the loss behavior of oriented materials. Actually, all these mechanisms can be interpreted as different ways by which the system tends, with increasing magnetizing frequency, to increase the total active wall surface in the material. A thorough discussion about this important point lies, however, outside the scope of the present paper, and will be given elsewhere [29]. Here, we only wish to stress that, apart from the details of specific models, the striking similarity of the dynamic loss curves represented in figs. 1 and 2, and the fact that the same equation can describe them very well, is the result of a common underlying mechanism, namely the competition between the external field, applied uniformly in the sample cross-section, and the eddy current counterfields, locally acting only on the regions where magnetization variations take place.

3.3. Experimental investigations of loss mechanisms As discussed in section 2, the general connection existing between the loss behavior and the magnetization dynamics, expressed by eq. (2), involves the power spectrum St(k, to) of the magnetization rate )(r, t). Suitable statistical models of the magnetization process, as the ones discussed in section 3.2, are certainly useful in order to predict the expected properties of S), but it is undoubtedly of great importane also to obtain some direct experimental information on S), through proper measurements of the local magnetization behavior in different regions of the sample cross section. The correct way to perform this experimental investigation is suggested by the loss models of Mazzetti [19,20] and Bertotti et al. [12,21,22], al-

G. Bertotti / Statistical approach to eddy current losses

ready mentioned in the introduction. Actually, on the one h a n d it can be shown [36] that these models are no more than an approximate form of the more rigorous theory discussed in section 2. O n the other hand, they lead to a direct relationship between the behavior of the dynamic loss and experimentally measurable quantities, namely the e.m.f.s detected on the sample surface over distances of the order of the sample thickness d [12,37,38], or the fluctuations of intensity induced by the Kerr effect in a laser beam reflected by a region of the sample surface of linear dimensions of the order of d [23,24,39]. These types of measurements have been, so far, mainly applied to the investigation of oriented materials, providing interesting information on the connection between the total loss and various local features of the magnetization process, like the presence of irregularities in the velocities of different walls or the local values of the magnetization rate and wall spacing in different grains [12,23,24]. 4. C o n c l u s i o n s In the previous section, we have discussed how a statistical approach to the problem of eddy current losses can help to improve our understanding of the physical origin of loss mechanisms. The basic idea of the statistical theory briefly reviewed in section 2 is to describe the magnetization process as a r a n d o m sequence of elementary magnetization j u m p s and express the loss behavior in terms of the s p a c e - t i m e correlation properties of the j u m p sequence. In the important case where such correlation properties are described in terms of a M a r k o v process, a direct link between the loss behavior and the microscopic dynamics of the magnetization process is obtained. This link is expressed by eq. (5) and basically involves the transition amplitude M ( A r , At), representing the conditional probability density that two subsequent correlated j u m p s take place at a relative distance A r and with a time delay At. Evidently, a great physical interest is attached to the development of suitable statistical models of the magnetization process, by which the main properties of M ( A r , At) corresponding to different conditions of domain structure, magnetizing frequency, etc.

259

can be predicted, F r o m this point of view, the simple examples discussed in section 3 can serve as useful hints to the future development of more satisfactory rigorous and general models. References

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260

[29] [30] [31] [32] [33] [34]

G. Bertotti / Statistical approach to eddy current losses

side S must be extended to the whole x - y plane. The correct procedure to be followed and its interesting relations with the classical loss are discussed in ref. [26]. G. Bertotti, J. Appl. Phys. to be published. H. Winner, W. Grosse-Nobis and M. Reinhardt, J. Magn. Magn. Mat. 41 (1984) 285. J.W. Shilling, IEEE Trans. Magn. MAG-9 (1973) 351. T.R. Hailer and J.J. Kramer, J. Appl. Phys. 41 (1970) 1034. T.R. Hailer and J.J. Kramer, J. Appl. Phys. 41 (1970) 1036. J.V.S. Morgan and K.J. Overshott, J. Appl. Phys. 53 (1982) 8293.

[35] K. Narita and M. Imamura, IEEE Trans. Magn. MAG-15 (1979) 981. [36] G. Bertotti, F. Fiorillo, P. Mazzetti and G.P. Soardo, Approximate and rigorous models of losses in magnetic laminations, presented at SMM6 conf., but not published in these Proceedings. [37] G. Bertotti, F. Fiorillo and M.P. Sassi, J. Magn. Magn. Mat. 26 (1982) 234. [38] G. Bertotti and F. Fiorillo, J. Magn. Magn. Mat. 41 (1984) 303. [39] M. Celasco, A. Masoero, A. Stepanescu and P. Mazzetti, J. Magn. Magn. Mat. 41 (1984) 295.