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A general model of losses in soft magnetic materials
225
Journal of Magnetism and Magnetic Materials 26 (1982) 225-233 North-Holland Publishing Company
A GENERAL MODEL OF LOSSES IN SOFT MAGNETIC MATERIALS G. BERTOTTI, P. MAZZETTI * and G.P. SOARDO Istituto Elettrotecnico Nazionale Galileo Ferraris, IO125 Torino, Italy (Gruppo Nazionale Struttura della Materiade1 CNR)
A general magnetic loss model is presented which takes account with proper approximations of the spatial and temporal non-homogeneities and stochastic character of the elementary dissipative mechanisms. An expression is obtained in which dynamic losses in a lamination of thickness d are linked to the energy spectrum of the emf measured between two points on the sample surface at a distanced. A good fit is found between calculated and measured losses between about 1 and 100 Hz in G.O. SiFe, and it is shown that, from the obtained link, one can interpret the excess and non-linearity loss anomalies in terms of the stochastic characteristics of the magnetization processes and of their homogeneization with increasing magnetizing frequency.
1. Introduction The theoretical studies of magnetic losses based on the assumption that domain walls move in a continuous and uniform way, although permitting clariiication of some aspects of the excess and non-linearity loss anomalies, fail to provide a generally satisfactory account of these anomalies whenever the magnetization processes differ (as is all too often the case) from the model assumptions [l-3]. This is due to the fact that relevant contributions to energy dissipation also arise from stochastic magnetization processes, nonhomogeneities, correlation effects, which are completely disregarded by the said models. As known [2,3], if we introduce a function J(r, t) which represents the eddy current density at a time c in each volume element da located at t in the sample, the average loss per unit volume P is in principle given by the equation AT
IJ(r, t)12 dt,
(1)
where p is the material resistivity. In general J is a * Permanent address: Istituto di Fisica de1 Politecnico, Torino, Italy
0304-8853/82/0000-0000/$02.75
10129
0 1982 North-Holland
very complicated function oft and t, but its complete knowledge throughout space and time is not needed: in fact, for the actual loss calculation, only the integral (l/AT)JtT lJ12 dt must be known. Because of these considerations, and since we want to take into proper account the stochastic nature of the elementary magnetization processes, it is more convenient, both from a mathematical and a physical point of view, to handle the problem in the frequency domain rather than in the time domain by introducing the power spectra \ki(r, w), with i = 1,2,3, of the three components of J along the axes of an orthogonal reference system. The loss P can then be written in the form
P=$
Sdn 52
j=*,(,,,)dw. -0D
The eddy current spectra \ki’s are in general characterized by a line component (due to the magnetization processes which identically repeat themselves over many cycles) and by a continuous component (arising from the stochastic magnetization processes which randomly redistribute themselves over repeated cycles). The continuous component contributes both to the hysteresis loss and to the dynamic losses in excess of the ones evaluated according to models, in which wall
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G. Bertotti et al. / Losses in soft magnetic materials
motions are assumed perfectly regular. The line component instead corresponds to the losses resulting from these regular repetitive motions. Eq. (2) which is equivalent to (l), represents a simplification with respect to this former equation, since it does not require the knowledge of the phase spectra of J(r, t). The actual calculation of losses making use of eq. (2) is still a formidable task, since it requires knowing in detail the space and time correlation function of all magnetization processes. The really important aspect of this approach, in which losses are described within the frequency domain framework, is to permit the derivation of a somewhat approximated loss expression, which can be related to measurable quantities, representative of the stochastic features of the elementary magnetization processes. Actually, the aims of the present paper are the following ones: a) to discuss in general the relationship between magnetic losses and the statistical properties of domain walls dynamics; b) to introduce a model which takes into proper account the spatial and temporal non-homogeneities and correlations of the magnetization processes, permitting to derive from eq. (2) a simplified loss expression for a lamination sample: this expression will establish an important link between dynamic losses and the energy spectrum of the electromotive force induced between two point contacts placed across the lamination on its surface at a distance equal to the sample thickness. This quantity can be experimentally determined and is representative of the stochastic character of the elementary magnetization processes giving rise to losses; c) to show that from the rather good fit between theoretical and experimental loss values, and from the dependence on magnetizing frequency of the energy spectrum of this emf, it is possible to conclude that, while the excess loss anomaly mainly arises from correlation effects and non-homogeneities of the magnetization processes, the non-linearity anomaly of the power loss vs. frequency curve is essentially due to a progressive homogeneization of the magnetization processes throughout the lamination for increasing magnetization frequencies.
2. The statistical approach: some general comments Let’s consider a lamination sample with rectangular cross section of thickness d (along x) width I (along y), magnetized along the axis z (fig. 1). In our statistical approach the power spectra \ki(r, 0)‘s are assumed to be defined through ensemble averages over a set of identical laminations, characterized by the same average magnetization variations in time. With the made assumptions, the problem is a planar one, so that the space dependence of the Q&r, w)‘s is limited to the x and y coordinates. If the width of the lamination is much larger than its thickness, this dependence essentially reduces to x only. The behaviour of each spectrum 9&x, o) at any point of the specimen should be in principle derived from the decay of the eddy currents due to the motions of the domain walls. This motion can be represented by a sequence of elementary Barkhausen (B) jumps, strongly correlated with one another [4]. Let us first consider the hypothesis that no correlation exists among these elementary B jumps: this hypothesis, although physically unrealistic, permits us to discuss some basic concepts of the proposed model. 2.1. Independent Barkhausen jumps We first consider a volume element dS2in the midplane of the lamination at x = 0 (fig. 1). A single B jump occurring at a distance Af gives rise to a current density pulse in dS2, the energy spectrum of which has been shown to be well approximated by a Lorentzian, having an amplitude and a cut-off frequency depending on Ar [S]. Because of symmetry reasons, the B jumps occurring all over the sample will give rise to a sequence of pulses in dS2 of both signs, having a zero average value. Thus because of
Fig. 1. Cross section of lamination
and reference axes.
G. Bertotti et hl. /Losses in soft magnetic moteriols
the assumed independence of jumps, the power spectra Jli(x = 0, w) will be simply given by the sum of the energy spectra of the pulses occurring in a unit time, and will therefore be proportional to the magnetizing frequency f,. Taking account of eq. (2), we find that, from the hypothesis of statistically independent jumps, the power loss in a volume element da located in the midplane of the lamination, is proportional to f,,, and therefore simply coincident with the hysteresis loss. Before considering the case of volume elements da at x # 0, it is of interest to stress that practically the whole contribution to the integral of the power spectrum, and thus to the losses, is due to B jumps located at very short distances Ar from dSl, independently of the position of the considered volume. As shown by Allia and Vinai [5], at short distances the energy spectrum of the current density induced in da by a single B jump has an amplitude and a cut-off frequency which decreases as (I/&)’ , so that the energy loss contribution decreases as (l/&)4. At larger Ar values, the decay with distance is even stronger (at least as @/A#), since the B jump can be simply described by the inversion of a magnetic dipole. This conclusion is in agreement with the well known fact that the hysteresis loss is a volume effect, and does not depend on the shape of the specimen. From these considerations we can conclude that in an elementary volume dLI located at x # 0, the fluctuation noise of the current density is very much the same as for x = 0. However, the average value of the sequence of pulses is now different from zero, since at x # 0 different contributions arise from the upper and lower parts of the lamination. Actually, over the frequency range in which the skin effect is negligible, the ensemble average of the current density has the same time dependence of the derivative of the overall induction, with an amplitude proportional to x. As a consequence, for any x # 0 the power spectrum of the current density is characterized by: - a continuous component (closely identical to the x = 0 case) and again corresponding to hysteresis loss; - a line component, corresponding to dynamic loss, which, under the present hypothesis of no correlations among B jumps coincides with the so called classical loss, calculated from a uniform induction variation throughout the sample cross section.
221
2.2. Correlated Barkhausen jumps As pointed out, the absence of space and time correlation is completely unphysical, being ruled out by the very existence of Bloch walls and their mutual interactions. Correlation in general increases the total power associated with the current density spectrum and thus gives rise to excess magnetic losses. According to the previous discussion, an excess hysteresis loss should be mainly related to very short range correlation effects, while an excess dynamic loss must arise from macroscopic non-homogeneities of the magnetization processes. In principle, from the knowledge of the space-time correlation function for the elementary B jumps, eq. (2) should permit the exact calculation of losses in all cases. Some simplifications are, however, needed to mathematically handle the problem. The approximate model which will be developed in the following section will permit us to obtain the following results: - in the presence of domain walls characterized by smooth and continuous motions, theoretical loss expressions are obtained in very close agreement with the ones of other models [l]; - in the presence of more complex correlation processes, the theoretical loss expressions are found to have a form which permits relating them to appropriate experimental measurements of the local continuous and discontinuous behaviour of Bloch walls.
3. The statistical approach: an approximate model for the calculation of losses The present loss calculations will be limited to the case of antiparallel domain structures similar to the well known Pry and Bean model [ 11, but actually much more general, since no restrictions are posed as to the regularity of domain wall spacings and of their motions. In this case it is possible to develop an approximate calculation model which simplifies the previous general equations. This model can be further extended to more complex domain structures, but this is beyond the scope of the present paper [6]. The average loss per lamination unit length is calculated as the sum of losses generated in small circuits,
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G. Bertotti et al. /Losses in soft magnetic materials
d
1
I
Fig. 2. Geometry of ideal cylindrical circuits of squee,section of side 2x, in which the lamination is assumed to be divided for the approximate eddy current loss calculation.
instead of small volume elements as in the previous discussion. As shown in fig. 2, these circuits are chosen to have the shape of hollow cylinders, with square cross section of side 2x, thickness dx, and unitary length along the lamination axis, which is parallel to the field H. The sample cross section is ideally divided into a convenient number of these circuits, in each of which an eddy current density is assumed to be induced by the magnetic flux variation linked to the cylinder, as if each circuit were isolated from the rest of the material. The average loss in a cylinder of side 2x is then given by (3) where Qu (x) is the induction flux within the cylinder, u is the material conductivity, and the averaging is made both over time and the cylinder position over the sample cross section. The loss of the whole sample is obtained by multiplying (U(x)) by (M/4x2), ratio between the lamination’s and the cylinder’s cross sections, and integrating with respect to x from 0 to d/2. We therefore find for the total loss per lamination unit length
ones of other models (classical [7], Pry and Bean [l]), under identical magnetization conditions. In all cases one finds for A a value close to 1.2 [6]. Physically eq. (4) expresses the losses in terms of the behaviour of d@u (x)/dt at each point of the sample, taking into account the effect of spatial nonhomogeneities over various x ranges. It also takes proper account of the temporal stochastic character of the local magnetization processes, but a more convenient mathematical description can be given, also in this case, within the frequency domain framework. To this purpose, we will express the power losses making use of the power spectrum $(x, w) of the induction flux time derivative d@n(x)/dt, so that from eq. (4), letting A = 1.2 and dividing by (Id), the power loss per unit volume can be expressed as
P = 0.03750 s”’
(4)
In eq. (4) the factor A has been introduced to permit a better fit between the present calculations and the
jm
\k(x, o)dw
.
(5)
_-oo
As a final step, we introduce instead of the power spectrum \k(x, w) a function cp(x, w) defined by the relation 2n I S(x, w) I2 cp(x, 0) =
(
)’
T/2
where S(x, w) and 2n I S(x, w) I* , respectively, represent the Fourier transform and the energy spectrum of d@u (x)/dr in each half period T/2, and the averaging is performed over several half periods. In this way one has the advantage that while the integral over w of ~(x, w) is the same as for 9(x, w), no singularities are present in relation to the periodic components of the signal, and this will permit an easier interpretation of theoretical results in terms of experimentally measurable quantities. Therefore, eq. (5) using relation (6) becomes
P=O.O3750
7’ 0
=O.O312Aldo~'((~)>$. 0
$
0
$
j-
~(x, w)dw.
(7)
-cc
According to the previous discussion this equation should permit us to analyze and calculate in a somewhat approximate way the dynamic power losses due to all magnetization processes, taking account of their stochastic characteristics and of any source of nonhomogeneity. A significant test of the validity of the present theo-
G. Berrotri ethl. / Losses in soft magnetic materials
229
4. Excess dynamic losses and non-homogeneities of the magnetization
Fig. 3. Ratio of calculated to classical losses (excess loss anomaly factor) vs. domain spacing 2L over lamination thickness d, for maximum magnetization far (BM/Bs << 1) or close (BM/Bs = 1) to saturation. All calculations made assuming regular spacing 2L and uniform wall motions, for a triangular flux waveform (d@B/dt = const). Full curves: dynamic losses calculated from present approximate but generalized model. Dotted curves: from Pry and Bean equation under identical d@g/dt = const conditions (see also the Appendix).
retical results is obtained from the comparison of the dynamic losses as calculated from the proposed model (see also the appendix), and from the Pry and Bean equations under identical assumptions of regular spacing 2L and uniform motions of Bloch walls. In fig. 3 the curves of losses in excess of classical ones are reported as a function of 2Lfd as calculated both from the present and the Pry and Bean models: as seen, the agreement is better than +I 0% over a wide range of 2L/d values, in both limits of maximum induction BM equal or much smaller than
the saturation value BS. It can therefore be concluded that eq. (7) permits expression of dynamic losses with a very good approximation whatever the origin of the non-homogeneities of the magnetization processes giving rise to energy dissipation. In fact, in the Pry and Bean type models these non-homogeneous processes can only arise from the presence of equally spaced and uniformly moving domain walls, while eq. (7) also takes account of all other sources (present in actual physical cases) related to spatial and temporal irregular stochastic effects, for alI 2L/d average values.
Classical dynamic losses result from a fully homogeneous magnetization process. Any magnetic domain structure breaks this homogeneity, and gives rise to an excess loss, the lowest limit of which, for a given pattern of rigid walls, is calculated by assuming smooth and equidistributed motions of all walls, as in the Pry and Bean model. In most practical cases these conditions are never fulfilled, so that losses should rigorously be calculated taking into account the irregularities of domain spacings and of wall motions. As already pointed out, eq. (7) provides the basis for this more rigorous calculation, if one knows the function cp(x, w). In the following we will show that if we limit the problem to the calculation of dynamic losses, cp(x, w) can be obtained from suitable measurements. Actually, for the dynamic loss calculation only the knowledge of cp(x, w) for x close to d/2 is needed. On the contrary, in order to calculate hysteresis losses, one should know this spectrum in the x + 0 limit, which starting from quantities measured outside the specimen, is impossible and would instead require a derivation from theoretical models. Let us now consider an experiment in which the electromotive force u(t) induced between two contacts A and B placed on the lamination surface at a distance d is measured (fig. 4). We want to establish a relation between u(f) (or actually its energy spectrum) and I&.X =d. o), and we therefore examine the following cases.
Fig. 4. Surface induced emf measured by placing contacts A and B at a distance d on lamination is related to dynamics of flux changes occurring in an ideal circuit of sided.
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G. Bertotti et al. /Losses in soft magnetic materials
4.1. The classicaccase Because of symmetry reasons u(r) is given by half the time derivative of the flux through the square circuit of side d: dL3 1 da&) - =-dt 2 dt
u(++
(8)
’
so that cp(d, w) is simply related to the spectrum cpv(d, a) of u(t) by M,
a) = +J”(d, 0).
(9)
Since in this case the induction write &!x, w) = $&(C&
w) = 6,s
B is uniform,
l&(d, w).
we can
(10)
Substituting this expression into eq. (7), the present model yields for the classical loss PC,
PC, = 0.30;
mation : the losses in excess to the classical ones now derive from long range non-homogeneities of the magnetization processes, and these effects, whatever their origins, are taken into account by the integral of cp,(d, w) over w which becomes larger than in the classical case.
J= cp,(d, w)dw. -cc
(11)
On the other hand, according to a rigorous derivation [7], the instantaneous classical loss for a lamination geometry is given by p = (ud2/12) ((M/dt)‘). Taking into account eq. (8), and since (v’(t)> = _f_‘Zcp,(d, w) dw, one finds for PC, an expression identical to eq. (1 l), but for the numerical factor which is equal to 0.33 instead of 0.30. Therefore, as anticipated, in the classical limit the present model provides an approximate loss evaluation differing by about 10% from the exact one (see also fig. 3). 4.2. In the presence of magnetic domains For more realistic cases in which the magnetization does not vary uniformly throughout the sample cross section, eq. (9) can still be retained as a good approximation. Some considerations are needed to accept also, in general, eq. (10): no problem exist in the low frequency range, where &2x, w) must always be proportional to the square of the flux variation (A@&x))~ as in the classical case. However, the cut-off frequency of the spectrum f, may in general change with x. We now have the following cases. 4.2.1. Small domain size In this case eq. (11) still represents a good approxi-
4.2.2. Large domain size If the average domain size 2L is much larger than the lamination thickness d (as is often the case in G.O. SiFe samples), it is reasonable to assume that f, a (l/x), f, being in fact inversely proportional to the time taken by a wall to sweep a distance x. Then the following expression should be used in eq. (7) +-
s
x4d +q(x, o) dw = 64&(d, w) dw s -0D d4 2x -0D = 32%
]- fi(d, w)dw . _m
(12)
One thus obtains for the dynamic loss in a sample unit volume the epxression
Pdv,, = 0.60 5
I- G(d, w)dw. _m
(13)
The present model then provides through eqs. (13) and (11) (which are very similar, differing only because of a numerical factor) general expressions for the calculation of losses, both for small and large domain sizes, taking account of all magnetization processes, and in particular of their stochastic character. These equations provide an important link between power losses and the energy spectrum (pv of the emf measurable between two points distant d on the lamination surface, which is representative of the spatial and temporal non-homogeneities and correlations of the magnetization processes giving rise to losses.
5. Comparison with experiments and discussion The previous theoretical results were compared with the ones of experiments performed on 3% SiFe single crystals, on which the following quantities were measured as a function of magnetizing frequency fnl:
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G. Bertotti et al. /Losses in soft magnetic materials
5. As is seen, a good agreement is obtained (again about 10%) between the absolute values of the
witbin
experimental
Fig. 5. Integral of energy spectrum of surface emf (see fig. 7) vs. magnetizing frequency fm.
a) power losses per cycle and per unit volume [8]; b) energy spectrum per semicycle of the surface emf u(t) PI; c)average spacing of Bloch walls in the demagnetized state. In fig. 5 the integral of the energy spectrum b) is reported as a function off,, while in fig. 6 the comparison is made between measured power losses and the ones calculated from eq. (13) and the data of fig.
0
I 0
20
I 40
fm(H,) 60
80
x)0
Fig. 6. Comparison of calculated and experimental power losses per cycle and unit mass on a (1 lO)[OOl] 3% SiFe single crystal [ 81 vs. magnetizing frequency fm: 1) experimental losses (from ref. [ 81); 2) losses calculated from present model, using eq. (13) and the data of fig. 5; 3) classical losses [6,7]; 4) experimental hysteresis losses.
and calculated
losses over a wide fm
(l-100 Hz), without introducing any arbitrary parameter. This permits us to conclude that the model presented in this paper, besides permitting us to describe with good approximation the case of uniform wall motions specifically treated by other models, takes into proper account the stochastic character of the magnetization processes. The obtained link between dynamic magnetic losses and the energy spectrum of the emf induced on the lamination surface between appropriately chosen points is important, since this spectrum provides complete and general information to be used in predicting and disnussing the behaviour of power losses in magnetic laminations. In fact, the area of this spectrum is directly proportional to the dynamic losses, while, as later further discussed, its shape is strictly related to the stochastic characteristics of the magnetization processes giving rise to the power dissipation. This relationship between losses and the energy spectrum of the surface emf will probably prove very useful in more detailed investigations on the separation of various loss sources. In fact, by appropriate experiments one may study the instantaneous behaviour of losses along the dynamic loop, that is investigate the contribution to losses from different magnetization processes. The suggestion arising from the present results is to evaluate the energy dissipated at various points along the magnetization cycle by performing spectral measurements of the corresponding surface emfs. The third type of measurements performed to test the present model, concerns the variation of Bloch wall average spacing with magnetizing frequency. Domain refinement is found to partly account for the non-linear behaviour of losses vs. f, only above 1O20 Hz [8,9]. The pronounced non-linearity observed below these frequencies must instead be attributed to other mechanisms, which essentially involve a spatial homogeneization of the magnetization processes with increasing frequency, mainly due to increasing effects of eddy-currents. In fact, at low magnetizing frequencies and for fields close to the coercive value, small randomly distributed obstacles, such as impurities, stresses and dislocations, may block some walls forcing other ones to move faster so that the average flux may range
G. Bertotti et al. /Losses in soft magnetic materials
232
ICP
10’
d2dHz) I
I
I
loo
I
I
,
lo’
I
I
I
102
I
I
I
D3
I
I
I@
Fig. 7. Energy spectrum of emf detected with experimental set up of fig. 4 vs. analysis frequency fa = (w/2a) for various magnetizing frequencies fm (0.02 to 100 Hz). Measurements (reported from ref. [8]) refer to a (llO)[OOl] single crystal of 3% SiFe.
obey the externally imposed waveform. At higher fm’s, as the eddy currents damping forces become dominant, the influence of these localized hindrances on domain wall motion is clearly strongly reduced and an homogeneization of magnetization processes occurs [4,10]. Other experiments have already been performed [ 1 l] which confirm this smoothing-out of the magnetization processes at increasing frequencies. Domain multiplication is not the only mechanism responsible for this smoothing-out effect, especially in the lower frequency range, as can be seen from the variations .of the surface emf energy spectrum with magnetizing frequency. In fig. 7 we report from ref. [8] some energy spectra of surface emf for different fm’s.Wall motions are seen to be much more irregular at low than at high magnetizing frequency, as shown by the difference of slopes of the tails of the spectra taken at increasing fm’s. The slower decay of the spectra observed at low f,,, corresponds to a very large spread of the times needed by the wall to sweep beneath the point contacts A and B of fig. 4. On the contrary, the increasing steepness of these tails at higher fm’s, makes the spectra more and more similaito the ones relative to ideal rectangular emf pulses, as one would expect for large domains obeying exactly the Pry and Bean assumptions of regular uniform motion. Domain multiplication alone would imply rather similar rec-
tangular type spectra at all fm’s or possibly in the limit of drastic domain size reductions, an attenuation of the steepness of the spectrum tail at higher fm’s. Just the opposite behaviour is observed pointing to the presence of other homogeneization mechanisms besides the often invoked one related to domain walls refinement . Another interesting point concerns the fact that all spectra of fig. 7 tend to the same value in the limit of zero analyzing frequency f,, independently off,. Theoretically this value corresponds, but for the numerical factor (1/8~), to the square of the magnetic flux variation (A@u(d))2 over half a period in a square section of side d. This is easily verfied by noting that the time integral of the induced emf u(r) over T/2 must always be equal to the one detected in a classical case, that is to (A@B(d)/2) (see also eq. (8)) and reminding us that the low frequency value of the energy spectrum of u(t) is just equal to (1/2n) I@* u(t) dt 12. The experiments confirm this simple prediction, as shown by the behaviour at low analyzing frequencies of the spectra of fig. 7, the common value of which for f,+ 0 is indeed found to be equal to the expected (A@~(d))~/8n.
6. Conclusions
A model has been proposed which permits derivation of a general expression of magnetic losses, which takes into account, in principle all contributions to energy dissipation deriving from continuous and stochastic magnetization processes. In this expression an important link is established between dynamic power losses and the energy spectrum of the emf induced on the lamination surface between points having a distance across its section equal to sample thickness. This spectrum actually provides all the relevant statistical information on the magnetization processes giving rise to losses. The area of this spectrum is in fact proportional to the dynamic losses, while its shape is strictly related to the stochastic characteristics of the dissipative mechanisms. Experiments confirm that the variations of the characteristics of this spectrum with magnetizing frequency are strictly related to the behaviour of losses, and point to the role played by the progressive homogeneization of magnetization processes with increasing f, in determining the non-
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G. Bertotti et al. /Losses in soft magnetic materials
linearity of the power loss curves. One can conclude that further studies of these variations will actually permit the performance of detailed investigations of the elementary processes and of their correlation effects which are responsible for the excess and nonlinearity anomalies of power losses in soft magnetic laminations.
In this equation N is an integer defined by the condition: N<$ti
<(N+l)
and the constants (Yand p assume the following values: I)forBM/B,
<
II) for Bu/Bs = 1: Acknowledgement The present work has been partially supported by the European Coal and Steel Community.
(A-2)
o=2and/3=
1.
The values of losses obtained from eq. (A.l) are compared in fig. 3 with the ones obtained from the Pry and Bean model [ 1] under the same d@/dt = const conditions (triangular flux waveform). These equations will also be given and derived in detail in ref. [6].
Appendix By applying the model proposed in this paper to the case of antiparallel domains with constant spacing 2L and moving uniformly and continuously, according to the Pry and Bean assumptions [ 11, a general expression of dynamic losses can be found which is a function of 2L/d (d being the lamination thickness) and of the maximum induction BM reached during cycling. A detailed derivation of this expression will be given elsewhere [6]. In this appendix we will only give the form of this equation on which is based the comparison between the Pry and Bean and present model calculations reported graphically in fig. 3. Let P$ be the rigorously calculated classical loss and B, the saturation induction. Then from the present model, and for a triangular flux waveform (d@/dt = const), we find for the dynamic power losses:
+lv(Iv+
1) (A.1)
References [l] R.H. Pry and C.P. Bean, J. Appl. Phys. 29 (1958) 532. [ 21 J.W. Shilling and G.L. Houze, Jr., IEEE Trans. Magn. MAC-10 (1974) 195. [3] A. Ferro and G.P. Soardo, J. Magn. Magn. Mat. 19 (1980) 6. [4] P. Mazzetti, IEEE Trans. Magn. MAC-14 (1978) 758. [S] P. Allis and F. Vmai, J. Appl. Phys. 48 (1977) 4649. [6] G. Bertotti, F. Fiorillo, P. Mazzetti and G.P. Soardo, to be published. [ 71 R.M. Bozorth, Ferromagnetism (Van Nostrand, New York, 1951) p. 778. [ 81 G. Bertotti, F. Fiorlllo and M.P. Sassi, IEEE Trans. Magn. MAC-17 (1981) in press. [ 91 K.J. Overshott, IEEE Trans. Magn. MAG-12 (1976) 840. [lo] W.J. Carr, Jr., J. Appl. Phys. 30s (1959) 90s. [ 111 A. Ferro and G. Montalenti, eds., IENGF-ECSC Report (March 1981) unpublished.
Journal of Magnetism and Magnetic Materials 26 (1982) 225-233 North-Holland Publishing Company
A GENERAL MODEL OF LOSSES IN SOFT MAGNETIC MATERIALS G. BERTOTTI, P. MAZZETTI * and G.P. SOARDO Istituto Elettrotecnico Nazionale Galileo Ferraris, IO125 Torino, Italy (Gruppo Nazionale Struttura della Materiade1 CNR)
A general magnetic loss model is presented which takes account with proper approximations of the spatial and temporal non-homogeneities and stochastic character of the elementary dissipative mechanisms. An expression is obtained in which dynamic losses in a lamination of thickness d are linked to the energy spectrum of the emf measured between two points on the sample surface at a distanced. A good fit is found between calculated and measured losses between about 1 and 100 Hz in G.O. SiFe, and it is shown that, from the obtained link, one can interpret the excess and non-linearity loss anomalies in terms of the stochastic characteristics of the magnetization processes and of their homogeneization with increasing magnetizing frequency.
1. Introduction The theoretical studies of magnetic losses based on the assumption that domain walls move in a continuous and uniform way, although permitting clariiication of some aspects of the excess and non-linearity loss anomalies, fail to provide a generally satisfactory account of these anomalies whenever the magnetization processes differ (as is all too often the case) from the model assumptions [l-3]. This is due to the fact that relevant contributions to energy dissipation also arise from stochastic magnetization processes, nonhomogeneities, correlation effects, which are completely disregarded by the said models. As known [2,3], if we introduce a function J(r, t) which represents the eddy current density at a time c in each volume element da located at t in the sample, the average loss per unit volume P is in principle given by the equation AT
IJ(r, t)12 dt,
(1)
where p is the material resistivity. In general J is a * Permanent address: Istituto di Fisica de1 Politecnico, Torino, Italy
0304-8853/82/0000-0000/$02.75
10129
0 1982 North-Holland
very complicated function oft and t, but its complete knowledge throughout space and time is not needed: in fact, for the actual loss calculation, only the integral (l/AT)JtT lJ12 dt must be known. Because of these considerations, and since we want to take into proper account the stochastic nature of the elementary magnetization processes, it is more convenient, both from a mathematical and a physical point of view, to handle the problem in the frequency domain rather than in the time domain by introducing the power spectra \ki(r, w), with i = 1,2,3, of the three components of J along the axes of an orthogonal reference system. The loss P can then be written in the form
P=$
Sdn 52
j=*,(,,,)dw. -0D
The eddy current spectra \ki’s are in general characterized by a line component (due to the magnetization processes which identically repeat themselves over many cycles) and by a continuous component (arising from the stochastic magnetization processes which randomly redistribute themselves over repeated cycles). The continuous component contributes both to the hysteresis loss and to the dynamic losses in excess of the ones evaluated according to models, in which wall
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G. Bertotti et al. / Losses in soft magnetic materials
motions are assumed perfectly regular. The line component instead corresponds to the losses resulting from these regular repetitive motions. Eq. (2) which is equivalent to (l), represents a simplification with respect to this former equation, since it does not require the knowledge of the phase spectra of J(r, t). The actual calculation of losses making use of eq. (2) is still a formidable task, since it requires knowing in detail the space and time correlation function of all magnetization processes. The really important aspect of this approach, in which losses are described within the frequency domain framework, is to permit the derivation of a somewhat approximated loss expression, which can be related to measurable quantities, representative of the stochastic features of the elementary magnetization processes. Actually, the aims of the present paper are the following ones: a) to discuss in general the relationship between magnetic losses and the statistical properties of domain walls dynamics; b) to introduce a model which takes into proper account the spatial and temporal non-homogeneities and correlations of the magnetization processes, permitting to derive from eq. (2) a simplified loss expression for a lamination sample: this expression will establish an important link between dynamic losses and the energy spectrum of the electromotive force induced between two point contacts placed across the lamination on its surface at a distance equal to the sample thickness. This quantity can be experimentally determined and is representative of the stochastic character of the elementary magnetization processes giving rise to losses; c) to show that from the rather good fit between theoretical and experimental loss values, and from the dependence on magnetizing frequency of the energy spectrum of this emf, it is possible to conclude that, while the excess loss anomaly mainly arises from correlation effects and non-homogeneities of the magnetization processes, the non-linearity anomaly of the power loss vs. frequency curve is essentially due to a progressive homogeneization of the magnetization processes throughout the lamination for increasing magnetization frequencies.
2. The statistical approach: some general comments Let’s consider a lamination sample with rectangular cross section of thickness d (along x) width I (along y), magnetized along the axis z (fig. 1). In our statistical approach the power spectra \ki(r, 0)‘s are assumed to be defined through ensemble averages over a set of identical laminations, characterized by the same average magnetization variations in time. With the made assumptions, the problem is a planar one, so that the space dependence of the Q&r, w)‘s is limited to the x and y coordinates. If the width of the lamination is much larger than its thickness, this dependence essentially reduces to x only. The behaviour of each spectrum 9&x, o) at any point of the specimen should be in principle derived from the decay of the eddy currents due to the motions of the domain walls. This motion can be represented by a sequence of elementary Barkhausen (B) jumps, strongly correlated with one another [4]. Let us first consider the hypothesis that no correlation exists among these elementary B jumps: this hypothesis, although physically unrealistic, permits us to discuss some basic concepts of the proposed model. 2.1. Independent Barkhausen jumps We first consider a volume element dS2in the midplane of the lamination at x = 0 (fig. 1). A single B jump occurring at a distance Af gives rise to a current density pulse in dS2, the energy spectrum of which has been shown to be well approximated by a Lorentzian, having an amplitude and a cut-off frequency depending on Ar [S]. Because of symmetry reasons, the B jumps occurring all over the sample will give rise to a sequence of pulses in dS2 of both signs, having a zero average value. Thus because of
Fig. 1. Cross section of lamination
and reference axes.
G. Bertotti et hl. /Losses in soft magnetic moteriols
the assumed independence of jumps, the power spectra Jli(x = 0, w) will be simply given by the sum of the energy spectra of the pulses occurring in a unit time, and will therefore be proportional to the magnetizing frequency f,. Taking account of eq. (2), we find that, from the hypothesis of statistically independent jumps, the power loss in a volume element da located in the midplane of the lamination, is proportional to f,,, and therefore simply coincident with the hysteresis loss. Before considering the case of volume elements da at x # 0, it is of interest to stress that practically the whole contribution to the integral of the power spectrum, and thus to the losses, is due to B jumps located at very short distances Ar from dSl, independently of the position of the considered volume. As shown by Allia and Vinai [5], at short distances the energy spectrum of the current density induced in da by a single B jump has an amplitude and a cut-off frequency which decreases as (I/&)’ , so that the energy loss contribution decreases as (l/&)4. At larger Ar values, the decay with distance is even stronger (at least as @/A#), since the B jump can be simply described by the inversion of a magnetic dipole. This conclusion is in agreement with the well known fact that the hysteresis loss is a volume effect, and does not depend on the shape of the specimen. From these considerations we can conclude that in an elementary volume dLI located at x # 0, the fluctuation noise of the current density is very much the same as for x = 0. However, the average value of the sequence of pulses is now different from zero, since at x # 0 different contributions arise from the upper and lower parts of the lamination. Actually, over the frequency range in which the skin effect is negligible, the ensemble average of the current density has the same time dependence of the derivative of the overall induction, with an amplitude proportional to x. As a consequence, for any x # 0 the power spectrum of the current density is characterized by: - a continuous component (closely identical to the x = 0 case) and again corresponding to hysteresis loss; - a line component, corresponding to dynamic loss, which, under the present hypothesis of no correlations among B jumps coincides with the so called classical loss, calculated from a uniform induction variation throughout the sample cross section.
221
2.2. Correlated Barkhausen jumps As pointed out, the absence of space and time correlation is completely unphysical, being ruled out by the very existence of Bloch walls and their mutual interactions. Correlation in general increases the total power associated with the current density spectrum and thus gives rise to excess magnetic losses. According to the previous discussion, an excess hysteresis loss should be mainly related to very short range correlation effects, while an excess dynamic loss must arise from macroscopic non-homogeneities of the magnetization processes. In principle, from the knowledge of the space-time correlation function for the elementary B jumps, eq. (2) should permit the exact calculation of losses in all cases. Some simplifications are, however, needed to mathematically handle the problem. The approximate model which will be developed in the following section will permit us to obtain the following results: - in the presence of domain walls characterized by smooth and continuous motions, theoretical loss expressions are obtained in very close agreement with the ones of other models [l]; - in the presence of more complex correlation processes, the theoretical loss expressions are found to have a form which permits relating them to appropriate experimental measurements of the local continuous and discontinuous behaviour of Bloch walls.
3. The statistical approach: an approximate model for the calculation of losses The present loss calculations will be limited to the case of antiparallel domain structures similar to the well known Pry and Bean model [ 11, but actually much more general, since no restrictions are posed as to the regularity of domain wall spacings and of their motions. In this case it is possible to develop an approximate calculation model which simplifies the previous general equations. This model can be further extended to more complex domain structures, but this is beyond the scope of the present paper [6]. The average loss per lamination unit length is calculated as the sum of losses generated in small circuits,
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G. Bertotti et al. /Losses in soft magnetic materials
d
1
I
Fig. 2. Geometry of ideal cylindrical circuits of squee,section of side 2x, in which the lamination is assumed to be divided for the approximate eddy current loss calculation.
instead of small volume elements as in the previous discussion. As shown in fig. 2, these circuits are chosen to have the shape of hollow cylinders, with square cross section of side 2x, thickness dx, and unitary length along the lamination axis, which is parallel to the field H. The sample cross section is ideally divided into a convenient number of these circuits, in each of which an eddy current density is assumed to be induced by the magnetic flux variation linked to the cylinder, as if each circuit were isolated from the rest of the material. The average loss in a cylinder of side 2x is then given by (3) where Qu (x) is the induction flux within the cylinder, u is the material conductivity, and the averaging is made both over time and the cylinder position over the sample cross section. The loss of the whole sample is obtained by multiplying (U(x)) by (M/4x2), ratio between the lamination’s and the cylinder’s cross sections, and integrating with respect to x from 0 to d/2. We therefore find for the total loss per lamination unit length
ones of other models (classical [7], Pry and Bean [l]), under identical magnetization conditions. In all cases one finds for A a value close to 1.2 [6]. Physically eq. (4) expresses the losses in terms of the behaviour of d@u (x)/dt at each point of the sample, taking into account the effect of spatial nonhomogeneities over various x ranges. It also takes proper account of the temporal stochastic character of the local magnetization processes, but a more convenient mathematical description can be given, also in this case, within the frequency domain framework. To this purpose, we will express the power losses making use of the power spectrum $(x, w) of the induction flux time derivative d@n(x)/dt, so that from eq. (4), letting A = 1.2 and dividing by (Id), the power loss per unit volume can be expressed as
P = 0.03750 s”’
(4)
In eq. (4) the factor A has been introduced to permit a better fit between the present calculations and the
jm
\k(x, o)dw
.
(5)
_-oo
As a final step, we introduce instead of the power spectrum \k(x, w) a function cp(x, w) defined by the relation 2n I S(x, w) I2 cp(x, 0) =
(
)’
T/2
where S(x, w) and 2n I S(x, w) I* , respectively, represent the Fourier transform and the energy spectrum of d@u (x)/dr in each half period T/2, and the averaging is performed over several half periods. In this way one has the advantage that while the integral over w of ~(x, w) is the same as for 9(x, w), no singularities are present in relation to the periodic components of the signal, and this will permit an easier interpretation of theoretical results in terms of experimentally measurable quantities. Therefore, eq. (5) using relation (6) becomes
P=O.O3750
7’ 0
=O.O312Aldo~'((~)>$. 0
$
0
$
j-
~(x, w)dw.
(7)
-cc
According to the previous discussion this equation should permit us to analyze and calculate in a somewhat approximate way the dynamic power losses due to all magnetization processes, taking account of their stochastic characteristics and of any source of nonhomogeneity. A significant test of the validity of the present theo-
G. Berrotri ethl. / Losses in soft magnetic materials
229
4. Excess dynamic losses and non-homogeneities of the magnetization
Fig. 3. Ratio of calculated to classical losses (excess loss anomaly factor) vs. domain spacing 2L over lamination thickness d, for maximum magnetization far (BM/Bs << 1) or close (BM/Bs = 1) to saturation. All calculations made assuming regular spacing 2L and uniform wall motions, for a triangular flux waveform (d@B/dt = const). Full curves: dynamic losses calculated from present approximate but generalized model. Dotted curves: from Pry and Bean equation under identical d@g/dt = const conditions (see also the Appendix).
retical results is obtained from the comparison of the dynamic losses as calculated from the proposed model (see also the appendix), and from the Pry and Bean equations under identical assumptions of regular spacing 2L and uniform motions of Bloch walls. In fig. 3 the curves of losses in excess of classical ones are reported as a function of 2Lfd as calculated both from the present and the Pry and Bean models: as seen, the agreement is better than +I 0% over a wide range of 2L/d values, in both limits of maximum induction BM equal or much smaller than
the saturation value BS. It can therefore be concluded that eq. (7) permits expression of dynamic losses with a very good approximation whatever the origin of the non-homogeneities of the magnetization processes giving rise to energy dissipation. In fact, in the Pry and Bean type models these non-homogeneous processes can only arise from the presence of equally spaced and uniformly moving domain walls, while eq. (7) also takes account of all other sources (present in actual physical cases) related to spatial and temporal irregular stochastic effects, for alI 2L/d average values.
Classical dynamic losses result from a fully homogeneous magnetization process. Any magnetic domain structure breaks this homogeneity, and gives rise to an excess loss, the lowest limit of which, for a given pattern of rigid walls, is calculated by assuming smooth and equidistributed motions of all walls, as in the Pry and Bean model. In most practical cases these conditions are never fulfilled, so that losses should rigorously be calculated taking into account the irregularities of domain spacings and of wall motions. As already pointed out, eq. (7) provides the basis for this more rigorous calculation, if one knows the function cp(x, w). In the following we will show that if we limit the problem to the calculation of dynamic losses, cp(x, w) can be obtained from suitable measurements. Actually, for the dynamic loss calculation only the knowledge of cp(x, w) for x close to d/2 is needed. On the contrary, in order to calculate hysteresis losses, one should know this spectrum in the x + 0 limit, which starting from quantities measured outside the specimen, is impossible and would instead require a derivation from theoretical models. Let us now consider an experiment in which the electromotive force u(t) induced between two contacts A and B placed on the lamination surface at a distance d is measured (fig. 4). We want to establish a relation between u(f) (or actually its energy spectrum) and I&.X =d. o), and we therefore examine the following cases.
Fig. 4. Surface induced emf measured by placing contacts A and B at a distance d on lamination is related to dynamics of flux changes occurring in an ideal circuit of sided.
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G. Bertotti et al. /Losses in soft magnetic materials
4.1. The classicaccase Because of symmetry reasons u(r) is given by half the time derivative of the flux through the square circuit of side d: dL3 1 da&) - =-dt 2 dt
u(++
(8)
’
so that cp(d, w) is simply related to the spectrum cpv(d, a) of u(t) by M,
a) = +J”(d, 0).
(9)
Since in this case the induction write &!x, w) = $&(C&
w) = 6,s
B is uniform,
l&(d, w).
we can
(10)
Substituting this expression into eq. (7), the present model yields for the classical loss PC,
PC, = 0.30;
mation : the losses in excess to the classical ones now derive from long range non-homogeneities of the magnetization processes, and these effects, whatever their origins, are taken into account by the integral of cp,(d, w) over w which becomes larger than in the classical case.
J= cp,(d, w)dw. -cc
(11)
On the other hand, according to a rigorous derivation [7], the instantaneous classical loss for a lamination geometry is given by p = (ud2/12) ((M/dt)‘). Taking into account eq. (8), and since (v’(t)> = _f_‘Zcp,(d, w) dw, one finds for PC, an expression identical to eq. (1 l), but for the numerical factor which is equal to 0.33 instead of 0.30. Therefore, as anticipated, in the classical limit the present model provides an approximate loss evaluation differing by about 10% from the exact one (see also fig. 3). 4.2. In the presence of magnetic domains For more realistic cases in which the magnetization does not vary uniformly throughout the sample cross section, eq. (9) can still be retained as a good approximation. Some considerations are needed to accept also, in general, eq. (10): no problem exist in the low frequency range, where &2x, w) must always be proportional to the square of the flux variation (A@&x))~ as in the classical case. However, the cut-off frequency of the spectrum f, may in general change with x. We now have the following cases. 4.2.1. Small domain size In this case eq. (11) still represents a good approxi-
4.2.2. Large domain size If the average domain size 2L is much larger than the lamination thickness d (as is often the case in G.O. SiFe samples), it is reasonable to assume that f, a (l/x), f, being in fact inversely proportional to the time taken by a wall to sweep a distance x. Then the following expression should be used in eq. (7) +-
s
x4d +q(x, o) dw = 64&(d, w) dw s -0D d4 2x -0D = 32%
]- fi(d, w)dw . _m
(12)
One thus obtains for the dynamic loss in a sample unit volume the epxression
Pdv,, = 0.60 5
I- G(d, w)dw. _m
(13)
The present model then provides through eqs. (13) and (11) (which are very similar, differing only because of a numerical factor) general expressions for the calculation of losses, both for small and large domain sizes, taking account of all magnetization processes, and in particular of their stochastic character. These equations provide an important link between power losses and the energy spectrum (pv of the emf measurable between two points distant d on the lamination surface, which is representative of the spatial and temporal non-homogeneities and correlations of the magnetization processes giving rise to losses.
5. Comparison with experiments and discussion The previous theoretical results were compared with the ones of experiments performed on 3% SiFe single crystals, on which the following quantities were measured as a function of magnetizing frequency fnl:
231
G. Bertotti et al. /Losses in soft magnetic materials
5. As is seen, a good agreement is obtained (again about 10%) between the absolute values of the
witbin
experimental
Fig. 5. Integral of energy spectrum of surface emf (see fig. 7) vs. magnetizing frequency fm.
a) power losses per cycle and per unit volume [8]; b) energy spectrum per semicycle of the surface emf u(t) PI; c)average spacing of Bloch walls in the demagnetized state. In fig. 5 the integral of the energy spectrum b) is reported as a function off,, while in fig. 6 the comparison is made between measured power losses and the ones calculated from eq. (13) and the data of fig.
0
I 0
20
I 40
fm(H,) 60
80
x)0
Fig. 6. Comparison of calculated and experimental power losses per cycle and unit mass on a (1 lO)[OOl] 3% SiFe single crystal [ 81 vs. magnetizing frequency fm: 1) experimental losses (from ref. [ 81); 2) losses calculated from present model, using eq. (13) and the data of fig. 5; 3) classical losses [6,7]; 4) experimental hysteresis losses.
and calculated
losses over a wide fm
(l-100 Hz), without introducing any arbitrary parameter. This permits us to conclude that the model presented in this paper, besides permitting us to describe with good approximation the case of uniform wall motions specifically treated by other models, takes into proper account the stochastic character of the magnetization processes. The obtained link between dynamic magnetic losses and the energy spectrum of the emf induced on the lamination surface between appropriately chosen points is important, since this spectrum provides complete and general information to be used in predicting and disnussing the behaviour of power losses in magnetic laminations. In fact, the area of this spectrum is directly proportional to the dynamic losses, while, as later further discussed, its shape is strictly related to the stochastic characteristics of the magnetization processes giving rise to the power dissipation. This relationship between losses and the energy spectrum of the surface emf will probably prove very useful in more detailed investigations on the separation of various loss sources. In fact, by appropriate experiments one may study the instantaneous behaviour of losses along the dynamic loop, that is investigate the contribution to losses from different magnetization processes. The suggestion arising from the present results is to evaluate the energy dissipated at various points along the magnetization cycle by performing spectral measurements of the corresponding surface emfs. The third type of measurements performed to test the present model, concerns the variation of Bloch wall average spacing with magnetizing frequency. Domain refinement is found to partly account for the non-linear behaviour of losses vs. f, only above 1O20 Hz [8,9]. The pronounced non-linearity observed below these frequencies must instead be attributed to other mechanisms, which essentially involve a spatial homogeneization of the magnetization processes with increasing frequency, mainly due to increasing effects of eddy-currents. In fact, at low magnetizing frequencies and for fields close to the coercive value, small randomly distributed obstacles, such as impurities, stresses and dislocations, may block some walls forcing other ones to move faster so that the average flux may range
G. Bertotti et al. /Losses in soft magnetic materials
232
ICP
10’
d2dHz) I
I
I
loo
I
I
,
lo’
I
I
I
102
I
I
I
D3
I
I
I@
Fig. 7. Energy spectrum of emf detected with experimental set up of fig. 4 vs. analysis frequency fa = (w/2a) for various magnetizing frequencies fm (0.02 to 100 Hz). Measurements (reported from ref. [8]) refer to a (llO)[OOl] single crystal of 3% SiFe.
obey the externally imposed waveform. At higher fm’s, as the eddy currents damping forces become dominant, the influence of these localized hindrances on domain wall motion is clearly strongly reduced and an homogeneization of magnetization processes occurs [4,10]. Other experiments have already been performed [ 1 l] which confirm this smoothing-out of the magnetization processes at increasing frequencies. Domain multiplication is not the only mechanism responsible for this smoothing-out effect, especially in the lower frequency range, as can be seen from the variations .of the surface emf energy spectrum with magnetizing frequency. In fig. 7 we report from ref. [8] some energy spectra of surface emf for different fm’s.Wall motions are seen to be much more irregular at low than at high magnetizing frequency, as shown by the difference of slopes of the tails of the spectra taken at increasing fm’s. The slower decay of the spectra observed at low f,,, corresponds to a very large spread of the times needed by the wall to sweep beneath the point contacts A and B of fig. 4. On the contrary, the increasing steepness of these tails at higher fm’s, makes the spectra more and more similaito the ones relative to ideal rectangular emf pulses, as one would expect for large domains obeying exactly the Pry and Bean assumptions of regular uniform motion. Domain multiplication alone would imply rather similar rec-
tangular type spectra at all fm’s or possibly in the limit of drastic domain size reductions, an attenuation of the steepness of the spectrum tail at higher fm’s. Just the opposite behaviour is observed pointing to the presence of other homogeneization mechanisms besides the often invoked one related to domain walls refinement . Another interesting point concerns the fact that all spectra of fig. 7 tend to the same value in the limit of zero analyzing frequency f,, independently off,. Theoretically this value corresponds, but for the numerical factor (1/8~), to the square of the magnetic flux variation (A@u(d))2 over half a period in a square section of side d. This is easily verfied by noting that the time integral of the induced emf u(r) over T/2 must always be equal to the one detected in a classical case, that is to (A@B(d)/2) (see also eq. (8)) and reminding us that the low frequency value of the energy spectrum of u(t) is just equal to (1/2n) I@* u(t) dt 12. The experiments confirm this simple prediction, as shown by the behaviour at low analyzing frequencies of the spectra of fig. 7, the common value of which for f,+ 0 is indeed found to be equal to the expected (A@~(d))~/8n.
6. Conclusions
A model has been proposed which permits derivation of a general expression of magnetic losses, which takes into account, in principle all contributions to energy dissipation deriving from continuous and stochastic magnetization processes. In this expression an important link is established between dynamic power losses and the energy spectrum of the emf induced on the lamination surface between points having a distance across its section equal to sample thickness. This spectrum actually provides all the relevant statistical information on the magnetization processes giving rise to losses. The area of this spectrum is in fact proportional to the dynamic losses, while its shape is strictly related to the stochastic characteristics of the dissipative mechanisms. Experiments confirm that the variations of the characteristics of this spectrum with magnetizing frequency are strictly related to the behaviour of losses, and point to the role played by the progressive homogeneization of magnetization processes with increasing f, in determining the non-
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G. Bertotti et al. /Losses in soft magnetic materials
linearity of the power loss curves. One can conclude that further studies of these variations will actually permit the performance of detailed investigations of the elementary processes and of their correlation effects which are responsible for the excess and nonlinearity anomalies of power losses in soft magnetic laminations.
In this equation N is an integer defined by the condition: N<$ti
<(N+l)
and the constants (Yand p assume the following values: I)forBM/B,
<
II) for Bu/Bs = 1: Acknowledgement The present work has been partially supported by the European Coal and Steel Community.
(A-2)
o=2and/3=
1.
The values of losses obtained from eq. (A.l) are compared in fig. 3 with the ones obtained from the Pry and Bean model [ 1] under the same d@/dt = const conditions (triangular flux waveform). These equations will also be given and derived in detail in ref. [6].
Appendix By applying the model proposed in this paper to the case of antiparallel domains with constant spacing 2L and moving uniformly and continuously, according to the Pry and Bean assumptions [ 11, a general expression of dynamic losses can be found which is a function of 2L/d (d being the lamination thickness) and of the maximum induction BM reached during cycling. A detailed derivation of this expression will be given elsewhere [6]. In this appendix we will only give the form of this equation on which is based the comparison between the Pry and Bean and present model calculations reported graphically in fig. 3. Let P$ be the rigorously calculated classical loss and B, the saturation induction. Then from the present model, and for a triangular flux waveform (d@/dt = const), we find for the dynamic power losses:
+lv(Iv+
1) (A.1)
References [l] R.H. Pry and C.P. Bean, J. Appl. Phys. 29 (1958) 532. [ 21 J.W. Shilling and G.L. Houze, Jr., IEEE Trans. Magn. MAC-10 (1974) 195. [3] A. Ferro and G.P. Soardo, J. Magn. Magn. Mat. 19 (1980) 6. [4] P. Mazzetti, IEEE Trans. Magn. MAC-14 (1978) 758. [S] P. Allis and F. Vmai, J. Appl. Phys. 48 (1977) 4649. [6] G. Bertotti, F. Fiorillo, P. Mazzetti and G.P. Soardo, to be published. [ 71 R.M. Bozorth, Ferromagnetism (Van Nostrand, New York, 1951) p. 778. [ 81 G. Bertotti, F. Fiorlllo and M.P. Sassi, IEEE Trans. Magn. MAC-17 (1981) in press. [ 91 K.J. Overshott, IEEE Trans. Magn. MAG-12 (1976) 840. [lo] W.J. Carr, Jr., J. Appl. Phys. 30s (1959) 90s. [ 111 A. Ferro and G. Montalenti, eds., IENGF-ECSC Report (March 1981) unpublished.