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Basic principles of magnetization processes and origin of losses in soft magnetic materials
Journal of Magnetism and Magnetic Materials 112 (1992) 146-149 North-Holland
Invited paper
Basic principles of magnetization processes and origin of losses in soft magnetic materials G. Bertotti ~, F. Fiorillo ~ a n d P. Mazzetti b "lstituto Elettrotecnico Nazionale Galileo Ferraris and INFM-GNSM, 1-10125 7erino, Italy t, Dipartimento di Fisica del Politecnico, Torino, Italy
Domain wall dynamics can be investigated through properly designed experiments in well oriented single crystals, containing one or a few mobile 180° walls. Equations of motion are derived which can be specialized to describe extreme cases, such as quasi-static stochastic wall behavior and eddy-current-induced wall bowing. Concepts and results related to single-wall dynamics can then be exploited, through the use of statistical methods, to assess the phenomenology of eddy-current losses in ordinary materials, where complex domain structures exist. It turns out that the conventional concept of loss separation can be physically justified. A general theoretical framework is consequently worked out, which is solidly verified against power-loss experiments in crystalline and amorphous materials.
1. Introduction
The origin of power losses in ferromagnetic materials can be described by a simple statement: losses are due to Joule heating produced by the electric currents induced by magnetization changes. If the space and time dependence of the current density j ( r , t) is known, then the total power loss per unit volume is P = ( I j ( r , t) I 2)/tr, where tr is the electric conductivity and the angular brackets indicate space-time averaging. However, this very general definition does not provide a practical means of calculating losses because of the extremely complex behavior of j ( r , t) vs. r and t. A statistical approach to the magnztization process [1] permits one to simplify the whole probtcm, oas~cany providing a pI]yslcai rauonate for loss separation - a long-standing empirical tenet of the literature - in tt-xms of processes taking place over different spatial scales. The Correspondence to: Dr. G. Bertotti, IEN Galileo Ferraris, C.so M. d'Azeglio 42, 10125 Torino, Italy. Tel.: + 39-11-3488933; telefax: + 39-11-6507611.
hysteresis loss is associated with very localized Barkhausen jumps of individual wall segments, the classical loss is governed by the specimen geometry, while the excess loss strongly depends on the magnetic domain width and on the scale of microstructural inhomogeneities (e.g. grain dimension, fluctuations of internal stresses, crystallographic texture, etc.). Nevertheless, to describe and clarify in quantitative terms the conaection between excess losses, domain wall (DW) dynamics and microstructure remains a difficult task, in spite of its importance and interest from the fundamental and the applicative point of vl~.w. ",~ To tackle this problem, systems characterized oy very simple domain structures have often been considered in the literature. A satisfactory comprehension of the dynamic ..... " o f "lnUlVlUUZ41 ' " . . .L~WS .. oenav~or could actually represent a proper basis for treating the complex phenomenology exhibited by ordinary metallic soft magnetic materials. In this paper, we will briefly review some aspects of this problem, and we will discuss how the results obtained in simple systems can be generalized and applied to materials of applicative interest.
0304-8853/92/$05.00 O 1992 - Elsevier Science Publishers B.V. All rights reserved
G. Bertotti et al. / Losses in soft magnetic materials
2. Basic aspects of domain wall dynamics The investigation of the DW behavior in (100) picture-frame single crystals containing one 180° DW extending all along the four frame limbs [2-4] can provide basic reference to studies of complex systems. Measurements under feedbackcontrolled constant DW velocity v have shown that the dependence of v on the applied magnetic field H is in agreement with the linear law
kv=H-Hc,
(1)
where H c is the coercive field experienced by the moving DW and the constant k is calculated by solving Maxwell's equations for a plane wall moving in a rectangular bar [2]. Far-reaching developments of eq. (1) can be envisaged, to the point that many phenomena associated with the magnetization process in ordinary materials can eventually be assessed through convenient application of it. For example, it is found that eq. (1) can be extrapolated to describe the DW behavior at arbitrarily small velocities, where the intrinsically stochastic nature of H c, related to the random character of the DW interaction with the pinning centers, plays a major role. This extrapolation leads to a theory of the Barkhausen effect which turns dot to be in striking agceement with experiments and provides a physically convincing interpretation of the Barkhausen effect in terms of DW dynamics [5]. An equally important generalization of eq. (1) to the case of many-wall systems can be made, with impo:tant consequences as to the prediction of power losses. Before considering in detail this problem, which, is left to the next section, we will briefly discuss the case where the assumption of a rigid wall, implied by eq. (1), has to be dropped. 'This occurs when, under the action of the applied field, the wall velocity tends to become so high that ioternal degrees of freedom are excited and novel effects have to be considered. The best known aspect of these complications is the so-called DW bowing, a distortion of the DW shape resulting from the competition between the DW surface tension and the pressure of the eddy-current fields generated by the DW motion itself. The problem can be investi-
147
gated numerically, by discretizing the DW as a set of segments in mutual interaction [6]. This approach has led to predictions in good agreement with loss measurements performed in thick (d = 0.55 ram) (100) [001] single-crystal plates [7], but could not reproduce loss results obtained in somewhat thinner (110) [001] crystals [8,9]. A different approach can be attempted by analytically describing the DW shape through a suitable differential equation [10,11], which can be solved by numerical methods. The investigation of the influence of DW internal degrees of freedom on DW dynamics can be pursued to a high degree of detail and accuracy by making use of magnetooptic methods based on the Kerr effect. By employing one or two polarized laser beams in order to measure the time dependence of the surface displacement of a given DW segment, of different parts of the same DW, or of different DWs, it is possible to obtain accurate information on important features like the value of local dynamic losses in different specimen volumes, the onset of DW bowing and its dependence on surface pinning effects, the propagation of dynamic perturbations along a DW, wall-wall interaction fields, differences between surface and bulk Barkhausen effect [11-13].
3. Loss behavior in soft magnetic materials The above-mentioned complex domain wall processes have a fundamental interest but can often be disregarded for the materials and the experimental conditions of practical interest. In fact, many features of tke loss behavior in ordinary materials can be discussed and interpreted through models based on eq.(1). The fundamental problem one has to deal with in this case can be expressed by the following question: if eq.(1) provides a correct evaluation of the dynamic less for an individual, isolated DW, can the total loss ' be expressed simply as {total 1O s,~= {loss of a single DW} × {number of simultaneously active DWs}? The general answer to this question is negative, because the loss is a quadratic function of the eddy current density, and any superposition of the eddy-current patterns generated by
148
G. Bertotti et al. / Losses in soft magnetic materials
different active DWs will result in some loss change. Evaluating these superposition effects is not simple. Pry and Bean did solve this problem for an array of regularly spaced and uniformly moving 180° DWs in a plane lamination, showing that the dynamic losses are in this case controlled by the ratio between domain width and lamination thickness [14]. This is, however, too an ideal model, for the domain structure in real materials can be very complex and the DW motion quite irregular. In order to deal properly with such features, one has to adopt a different attitude, and make use of statistical concepts and methods. It is then possible to develop a sort of statistical version of the Pry and Bean model [1], whose basic assumptions can be briefly summarized as follows: (1) Magnetization changes are due to the contributions of a certain number n of simultaneously active correlation regions (magnetic objects (MOs)), distributed randomly in the specimen cross-section. Each region may correspond to a single DW or to a group of tightly correlated DWs. Under these conditions, the excess loss Pc, can be expressed in terms of the corresponding excess field H e as H e = 4trGSBf/n,
H e = PJ4Bf,
(2)
where G = 0.136, S is the specimen cross-sectional area, B is the peak induction and f is the magnetizing frequency. From the point of view of the above-mentioned eddy-current superposition effects, this result permits one to state that {total loss}-- {loss of a single MO} × {number of simultaneously active MOs} +{classical loss}. Thus, random space distribution of the MOs implies that the superposition effects give rise to the classical loss term. This result can be lefined by a more detailed description of MO dynamics [15]. tzj i ne number of active MOs is an increasing function n ( H e) of the excess field H e. It is expected that this functional dependence, rather than a direct dependence of n on f a n d / o r B, has a major physical meaning. It reflects the fact that the existence of a random distribution of local pinning ahd nucleation fields, which are progressively exceeded by the applied field, is the
basic mechanism controlling the magnetization dynamics with varying B and f. In this frame of interpretation, attempts to clarify the physical connection between dynamic losses and microstructure should basically address the problem of predicting the shape of the function n ( H e) from some microstructural model. An appealing property of this theoretical approach lies in the fact that the loss behavior appears much simpler when reinterpreted in terms of the n ( H e) behavior. In particular, it is now well assessed that for most iron-based crystalline and amorphous alloys n ( H e) is a simple linear function of H e [ 16-18] n(.h~) = n o + H J V o.
(3)
The field V0 plays a basic role in the theory. As a parameter related to the distribution of the internal coercive fields, it permits one to address quite effectively the relationship between dynamic losses and microstructure. For example, by investigating the behavior of V0 in non-oriented silicon steel, it was possible to quantitatively express the excess loss as a function of hysteresis loss and average grain size, with no adjustable parameters [19,20]. In any case the theory predicts a dependence of excess losses on B and f which is in agreement with the results obtained in grain-oriented and non-oriented SiFe [16]. In addition, it can be generalized to predict losses under distorted flux conditions, with no additional parameters. This is confirmed by experiments in grain-oriented, non-oriented and amorphous laminations [21]. The principal objective to be pursued by future studies should be the development of microstructural models capable of predicting the behavior of the function n ( H e) in different materials and under various microstructural conditions.
References [1] G. Bertotti, J. Appl. Phys. 57 (1985) 2110. [2] H.J. Williams, W. Shockley and C. Kittel, Phys. Rcv. 80 (1950) 1090. [3] G. Hellmiss and L. Storm, IEEE Trans. Magn. MAG-10 (1974) 36. [4] W. Grosse-Nobis and W. S,'hoerfelder, Physica B 80 (1975) 407.
G. Bertotti et al. / Losses in soft magnetic materials
[5] B. Alessandro, C. Beatrice, G. Bertotti and A. Montorsi, J. Appl. Phys. 68 (1990) 2901. [6] J.E.L. Bishop, IEEE Trans. Magn. MAG-16 (1980) 129. [7] K. Narita and M. Imamura, IEEE Trans. Magn. MAG-15 (1979) 981. [8] F. Fiorilio, results reported in G. Bertotti, J. Appl. Phys. 55 (1984) 4348. [9] W.M. Swift, J.W. Shilling, S.K. Bhate and F.J. Young, IEEE Trans. Magn. MAG-10 (1974) 810. [10] W.J. Carr, Jr., J. Appl. Phys. 47 (1976) 4176. [11] C. Beatrice, P. Mazzetti and F. Vinai, IEEE Trans. Magn. MAG-26 (1990) 1412. [12] M. Celasco, A. Masoero, P. Mazzetti and A. Stepanescu, J. Magn. Magn. Mater. 54-57 (1986) 859. [13] M. Celasco, A. Masoero, P. Mazzetti, G. Montalenti and
149
A. Stepanescu, in: Proc. 8th Intern. Conf. on Noise in Physical Systems (North-Holland, Amsterdam, 1986) p. 259. [14] R.H. Pry and C.P. Bean, J. Appl. Phys. 29 (1958) 532. [15] J.E.L. Bishop, IEEE Trans. Magn. MAG-26 (1990) 2217. [16] G. Bertotti, IEEE Trans. Magn. MAG-24 (1988) 621. [17] G.E. Fish, G.F. Chang and R. Bye, J. Appl. Phys. 64 (1988) 5370. [18] M. Pott-Langemeyer, W. Riehemann and W. Heye, J. Magn. Magn. Mater. 112 (1992) 284. [19] G. Ban and G. Bertotti, J. Appl. Phys. 64 (1988) 5361. [20] G. Bertotti, J. de Phys. 46 (1985) C6-389. [21] F. FiorJllo and A. Novikov, IEEE Trans. Magn. MAG-26 (1990) 2904.
Invited paper
Basic principles of magnetization processes and origin of losses in soft magnetic materials G. Bertotti ~, F. Fiorillo ~ a n d P. Mazzetti b "lstituto Elettrotecnico Nazionale Galileo Ferraris and INFM-GNSM, 1-10125 7erino, Italy t, Dipartimento di Fisica del Politecnico, Torino, Italy
Domain wall dynamics can be investigated through properly designed experiments in well oriented single crystals, containing one or a few mobile 180° walls. Equations of motion are derived which can be specialized to describe extreme cases, such as quasi-static stochastic wall behavior and eddy-current-induced wall bowing. Concepts and results related to single-wall dynamics can then be exploited, through the use of statistical methods, to assess the phenomenology of eddy-current losses in ordinary materials, where complex domain structures exist. It turns out that the conventional concept of loss separation can be physically justified. A general theoretical framework is consequently worked out, which is solidly verified against power-loss experiments in crystalline and amorphous materials.
1. Introduction
The origin of power losses in ferromagnetic materials can be described by a simple statement: losses are due to Joule heating produced by the electric currents induced by magnetization changes. If the space and time dependence of the current density j ( r , t) is known, then the total power loss per unit volume is P = ( I j ( r , t) I 2)/tr, where tr is the electric conductivity and the angular brackets indicate space-time averaging. However, this very general definition does not provide a practical means of calculating losses because of the extremely complex behavior of j ( r , t) vs. r and t. A statistical approach to the magnztization process [1] permits one to simplify the whole probtcm, oas~cany providing a pI]yslcai rauonate for loss separation - a long-standing empirical tenet of the literature - in tt-xms of processes taking place over different spatial scales. The Correspondence to: Dr. G. Bertotti, IEN Galileo Ferraris, C.so M. d'Azeglio 42, 10125 Torino, Italy. Tel.: + 39-11-3488933; telefax: + 39-11-6507611.
hysteresis loss is associated with very localized Barkhausen jumps of individual wall segments, the classical loss is governed by the specimen geometry, while the excess loss strongly depends on the magnetic domain width and on the scale of microstructural inhomogeneities (e.g. grain dimension, fluctuations of internal stresses, crystallographic texture, etc.). Nevertheless, to describe and clarify in quantitative terms the conaection between excess losses, domain wall (DW) dynamics and microstructure remains a difficult task, in spite of its importance and interest from the fundamental and the applicative point of vl~.w. ",~ To tackle this problem, systems characterized oy very simple domain structures have often been considered in the literature. A satisfactory comprehension of the dynamic ..... " o f "lnUlVlUUZ41 ' " . . .L~WS .. oenav~or could actually represent a proper basis for treating the complex phenomenology exhibited by ordinary metallic soft magnetic materials. In this paper, we will briefly review some aspects of this problem, and we will discuss how the results obtained in simple systems can be generalized and applied to materials of applicative interest.
0304-8853/92/$05.00 O 1992 - Elsevier Science Publishers B.V. All rights reserved
G. Bertotti et al. / Losses in soft magnetic materials
2. Basic aspects of domain wall dynamics The investigation of the DW behavior in (100) picture-frame single crystals containing one 180° DW extending all along the four frame limbs [2-4] can provide basic reference to studies of complex systems. Measurements under feedbackcontrolled constant DW velocity v have shown that the dependence of v on the applied magnetic field H is in agreement with the linear law
kv=H-Hc,
(1)
where H c is the coercive field experienced by the moving DW and the constant k is calculated by solving Maxwell's equations for a plane wall moving in a rectangular bar [2]. Far-reaching developments of eq. (1) can be envisaged, to the point that many phenomena associated with the magnetization process in ordinary materials can eventually be assessed through convenient application of it. For example, it is found that eq. (1) can be extrapolated to describe the DW behavior at arbitrarily small velocities, where the intrinsically stochastic nature of H c, related to the random character of the DW interaction with the pinning centers, plays a major role. This extrapolation leads to a theory of the Barkhausen effect which turns dot to be in striking agceement with experiments and provides a physically convincing interpretation of the Barkhausen effect in terms of DW dynamics [5]. An equally important generalization of eq. (1) to the case of many-wall systems can be made, with impo:tant consequences as to the prediction of power losses. Before considering in detail this problem, which, is left to the next section, we will briefly discuss the case where the assumption of a rigid wall, implied by eq. (1), has to be dropped. 'This occurs when, under the action of the applied field, the wall velocity tends to become so high that ioternal degrees of freedom are excited and novel effects have to be considered. The best known aspect of these complications is the so-called DW bowing, a distortion of the DW shape resulting from the competition between the DW surface tension and the pressure of the eddy-current fields generated by the DW motion itself. The problem can be investi-
147
gated numerically, by discretizing the DW as a set of segments in mutual interaction [6]. This approach has led to predictions in good agreement with loss measurements performed in thick (d = 0.55 ram) (100) [001] single-crystal plates [7], but could not reproduce loss results obtained in somewhat thinner (110) [001] crystals [8,9]. A different approach can be attempted by analytically describing the DW shape through a suitable differential equation [10,11], which can be solved by numerical methods. The investigation of the influence of DW internal degrees of freedom on DW dynamics can be pursued to a high degree of detail and accuracy by making use of magnetooptic methods based on the Kerr effect. By employing one or two polarized laser beams in order to measure the time dependence of the surface displacement of a given DW segment, of different parts of the same DW, or of different DWs, it is possible to obtain accurate information on important features like the value of local dynamic losses in different specimen volumes, the onset of DW bowing and its dependence on surface pinning effects, the propagation of dynamic perturbations along a DW, wall-wall interaction fields, differences between surface and bulk Barkhausen effect [11-13].
3. Loss behavior in soft magnetic materials The above-mentioned complex domain wall processes have a fundamental interest but can often be disregarded for the materials and the experimental conditions of practical interest. In fact, many features of tke loss behavior in ordinary materials can be discussed and interpreted through models based on eq.(1). The fundamental problem one has to deal with in this case can be expressed by the following question: if eq.(1) provides a correct evaluation of the dynamic less for an individual, isolated DW, can the total loss ' be expressed simply as {total 1O s,~= {loss of a single DW} × {number of simultaneously active DWs}? The general answer to this question is negative, because the loss is a quadratic function of the eddy current density, and any superposition of the eddy-current patterns generated by
148
G. Bertotti et al. / Losses in soft magnetic materials
different active DWs will result in some loss change. Evaluating these superposition effects is not simple. Pry and Bean did solve this problem for an array of regularly spaced and uniformly moving 180° DWs in a plane lamination, showing that the dynamic losses are in this case controlled by the ratio between domain width and lamination thickness [14]. This is, however, too an ideal model, for the domain structure in real materials can be very complex and the DW motion quite irregular. In order to deal properly with such features, one has to adopt a different attitude, and make use of statistical concepts and methods. It is then possible to develop a sort of statistical version of the Pry and Bean model [1], whose basic assumptions can be briefly summarized as follows: (1) Magnetization changes are due to the contributions of a certain number n of simultaneously active correlation regions (magnetic objects (MOs)), distributed randomly in the specimen cross-section. Each region may correspond to a single DW or to a group of tightly correlated DWs. Under these conditions, the excess loss Pc, can be expressed in terms of the corresponding excess field H e as H e = 4trGSBf/n,
H e = PJ4Bf,
(2)
where G = 0.136, S is the specimen cross-sectional area, B is the peak induction and f is the magnetizing frequency. From the point of view of the above-mentioned eddy-current superposition effects, this result permits one to state that {total loss}-- {loss of a single MO} × {number of simultaneously active MOs} +{classical loss}. Thus, random space distribution of the MOs implies that the superposition effects give rise to the classical loss term. This result can be lefined by a more detailed description of MO dynamics [15]. tzj i ne number of active MOs is an increasing function n ( H e) of the excess field H e. It is expected that this functional dependence, rather than a direct dependence of n on f a n d / o r B, has a major physical meaning. It reflects the fact that the existence of a random distribution of local pinning ahd nucleation fields, which are progressively exceeded by the applied field, is the
basic mechanism controlling the magnetization dynamics with varying B and f. In this frame of interpretation, attempts to clarify the physical connection between dynamic losses and microstructure should basically address the problem of predicting the shape of the function n ( H e) from some microstructural model. An appealing property of this theoretical approach lies in the fact that the loss behavior appears much simpler when reinterpreted in terms of the n ( H e) behavior. In particular, it is now well assessed that for most iron-based crystalline and amorphous alloys n ( H e) is a simple linear function of H e [ 16-18] n(.h~) = n o + H J V o.
(3)
The field V0 plays a basic role in the theory. As a parameter related to the distribution of the internal coercive fields, it permits one to address quite effectively the relationship between dynamic losses and microstructure. For example, by investigating the behavior of V0 in non-oriented silicon steel, it was possible to quantitatively express the excess loss as a function of hysteresis loss and average grain size, with no adjustable parameters [19,20]. In any case the theory predicts a dependence of excess losses on B and f which is in agreement with the results obtained in grain-oriented and non-oriented SiFe [16]. In addition, it can be generalized to predict losses under distorted flux conditions, with no additional parameters. This is confirmed by experiments in grain-oriented, non-oriented and amorphous laminations [21]. The principal objective to be pursued by future studies should be the development of microstructural models capable of predicting the behavior of the function n ( H e) in different materials and under various microstructural conditions.
References [1] G. Bertotti, J. Appl. Phys. 57 (1985) 2110. [2] H.J. Williams, W. Shockley and C. Kittel, Phys. Rcv. 80 (1950) 1090. [3] G. Hellmiss and L. Storm, IEEE Trans. Magn. MAG-10 (1974) 36. [4] W. Grosse-Nobis and W. S,'hoerfelder, Physica B 80 (1975) 407.
G. Bertotti et al. / Losses in soft magnetic materials
[5] B. Alessandro, C. Beatrice, G. Bertotti and A. Montorsi, J. Appl. Phys. 68 (1990) 2901. [6] J.E.L. Bishop, IEEE Trans. Magn. MAG-16 (1980) 129. [7] K. Narita and M. Imamura, IEEE Trans. Magn. MAG-15 (1979) 981. [8] F. Fiorilio, results reported in G. Bertotti, J. Appl. Phys. 55 (1984) 4348. [9] W.M. Swift, J.W. Shilling, S.K. Bhate and F.J. Young, IEEE Trans. Magn. MAG-10 (1974) 810. [10] W.J. Carr, Jr., J. Appl. Phys. 47 (1976) 4176. [11] C. Beatrice, P. Mazzetti and F. Vinai, IEEE Trans. Magn. MAG-26 (1990) 1412. [12] M. Celasco, A. Masoero, P. Mazzetti and A. Stepanescu, J. Magn. Magn. Mater. 54-57 (1986) 859. [13] M. Celasco, A. Masoero, P. Mazzetti, G. Montalenti and
149
A. Stepanescu, in: Proc. 8th Intern. Conf. on Noise in Physical Systems (North-Holland, Amsterdam, 1986) p. 259. [14] R.H. Pry and C.P. Bean, J. Appl. Phys. 29 (1958) 532. [15] J.E.L. Bishop, IEEE Trans. Magn. MAG-26 (1990) 2217. [16] G. Bertotti, IEEE Trans. Magn. MAG-24 (1988) 621. [17] G.E. Fish, G.F. Chang and R. Bye, J. Appl. Phys. 64 (1988) 5370. [18] M. Pott-Langemeyer, W. Riehemann and W. Heye, J. Magn. Magn. Mater. 112 (1992) 284. [19] G. Ban and G. Bertotti, J. Appl. Phys. 64 (1988) 5361. [20] G. Bertotti, J. de Phys. 46 (1985) C6-389. [21] F. FiorJllo and A. Novikov, IEEE Trans. Magn. MAG-26 (1990) 2904.