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MATHEMATICAL MODELLING OF FERROMAGNETIC HYSTERESIS IN SOFT MAGNETIC MATERIALS K. Chwastek1, J. Szczygłowski1, W. Wilczyński2 1 Częstochowa University of Technology, Częstochowa, Poland 2 Electrotechnical Institute, Warsaw, Poland Abstract The paper discusses some physical concepts important for mathematical modelling of ferromagnetic hysteresis and their implementation in chosen macroscopic descriptions of the phenomenon. Introduction Recent advances in the design and emerging applications of electric machines stimulate the development of new engineering descriptions of hysteresis phenomenon and core losses [1-4]. Hysteresis is also the field of intensive research for solid state physicists. The paper focuses on several concepts advanced by physicists and implemented in practical descriptions of the phenomenon, used by electrical engineers in computer simulations of ferromagnetic cores. The effective field and the anhysteretic curve The cornerstone for contemporary understanding of ferromagnetism is the idea of effective field introduced by P. Weiss in 1907 [5]. According to the well known monograph [6, p. 130] “ … it is of invaluable importance in giving a simple and at the same time deep physical interpretation of the existence of spontaneous magnetization …”. Briefly speaking, the effective field is interpreted as the internal magnetic field experienced by magnetic moments within the material, different from the externally applied magnetic field. In the language of control theory, the effective field is the sum of the applied field and a certain positive feedback in the system, H eff H M , where the second term is referred to as the mean (or molecular) field [7,8]. It is remarkable that a similar concept of molecular interaction forces has been considered by J. D. van der Waals [9]. The effective field may be perceived as a manifestation of cooperative interactions between a large number of identical elements [10-14]. The concept has evolved during the years and many sophisticated descriptions useful for solid state physics have been derived from it, just to mention the Heinsenberg exchange model, the Ising model, the XY model etc. Some of the advances on the subject are summarized in a number of monographs, review papers and textbooks, cf. [15-17]. Their discussion is beyond the scope of the present paper. The seminal works by L. D. Landau and E. M. Lifschitz [18] have initiated the intensive research on the paramount role of the effective field in contemporary micromagnetics, what resulted in the development of magnetic memories [6, 19-24]. In the seventies and eighties of the last century one of the “hot” topics in physics was the study of universality, scaling and renormalization in the context of critical phenomena and phase transitions [25-27]. Connections of effective field concept with the theory of phase transitions are discussed in detail e.g. in Refs. [28, 29]. A generally accepted opinion is that the effective field theory might be an useful aproximation for the description of complex systems in those cases when fluctuations may be neglected. In macroscopic hysteresis models, which are of interest to engineers, the effective field concept is also useful, as the description may be expressed using a self-consistent relationship M (t ) H eff (t ) , where is the hysteresis operator and M is magnetization [30]. 1-62 In practical calculations it is convenient to invert the abovegiven relationship, thus to calculate the effective field from magnetization, subsequently external field strength is obtained. This approach is consistent with the conditions used for material characterization defined by international standards (ASTM, IEC 404), which assumes that ferromagnetic materials are examined for sine waveform of flux density rate (for soft magnetic materials this condition applies to magnetization rate as well, because M (t ) H (t ) and B(t ) 0 H (t ) M (t ) ). There exists a number of useful descriptions of hysteresis phenomenon, cf. [31]. The most important and widespread model is due to F. Preisach [32]. Its extensions including effective field have been introduced and discussed by E. Della Torre [33] and I. D. Mayergoyz [34]. This model is however purely phenomenological, it relies on a weighted superposition of outputs from abstract elementary relay components (“hysterons”), thus it is quite difficult to ascribe any physical meaning to hysterons. Some attempts to correlate or even interpret them to grains within the magnetic material may be found in the literature [35-37], but this conjecture is not fully justified. Another important model, but operating on a smaller length scale (closer to micromagnetics, so more interesting to physicists [12]) is the Stoner-Wohlfarth description [38]. The effective field concept has been examined for this description e.g. in Refs. [39-40]. The macroscopic model advanced by D. C. Jiles and D. L. Atherton [41] has gained a considerable interest in the last 25 years. The concept of effective field is inherent in the description, as H eff is the independent variable in the fundamental ODE of the model [42]. The basic idea behind the description is to offset magnetization from a theoretical sigmoidshaped curve, which describes the state of global thermodynamical equillibrium, cf. Fig. 1. Hysteresis is due to energy dissipation on the inclusions, voids, dislocations of crystalline structure in the material, commonly termed as pinning sites. Fig. 1. Major hysteresis loop and anhysteretic curve in the Jiles-Atherton model There is a number of problems with the seemingly simple description. Minor loops, i.e. those, which do not reach saturation (either symmetrical or asymmetrical ones) and more complex magnetization cycles are usually poorly represented, if the same parameter set as for the major loop is used [43]. The original parameter estimation procedure [44] is error-prone, so other methods have to be aplied [45]. Some problems have been caused by the introduction of supplementary relationships for the description of reversible magnetization phenomena [41, 45, 46]. The model developers did not notice the fact that the introduced relationships are merely approximations resulting from a Taylor expansion of more complicated formulas. An attempt to fix this problem has been proposed [47], where the concept of modulated contribution of reversible susceptibility, borrowed from the product Preisach model [48], has been incroporated in the description. The modified description resulted in an improved 1-63 description of minor loops. Anyhow there is yet another subtle problem with the reversibility issue, which cannot be so easily fixed. This is due to the very idea of D. C. Jiles and D. L. Atherton, that the magnetization (or its irreversible component) may be given with an ODE with respect to the effective field. Let us refer again to Figure 1. denotes simply the sign of variation of input variable ( sign(dH / dt ) for the simple model or sign(dM / dt ) for the inverse model, which corresponds to the conditions used for material characterization in normalized conditions). But M is introduced in order to remove the non-physical model behaviour after a sudden change of sign of the input signal. The original model formulation resulted in the appearance of negative susceptibility regions after the reversal point. The existence of this term may be justified with the fact, that right after the applied field changes its sign, only the reversible process contributes, because the irreversible process does not return any energy to the applied field [49]. M is aimed at switching off the irreversible component of susceptibility and it is given most consisely as M 0.51 signM an M irr dx / dt , where x denotes the input variable. Note, however, that in order to use the model, we have to track the M sign instantly. There are parts of the modelled hysteresis loop, which are described with the fundamental Jiles-Atherton model equation dM irr / dH eff M an M irr / k and there are other parts, which are not. In a certain sense the parameter M is introduced post-hoc into the formal derivation of the final relationship for dM / dH . The effective field in the Jiles-Atherton model has been extended in order to take into account some important physical phenomena: magnetostriction [42, 50-52] and eddy currents [53]. There were some attempts to consider and describe the effect of texture [54-56], anistropy of magnetic materials [37, 55-61] and steel microstructure [37, 62, 63]. A general conclusion may be drawn that the popularity of the description stems from the possibility to consider a number of physical phenomena and technological parameters like strain. A great merit of the formalism is the noticing of the role of the effective field and anhysteretic magnetization (or, more generally, reversible magnetization processes) in modelling. The simplicity of the description makes it a favourite “playground” for hysteresis modellers. However the Jiles-Atherton model is not necessarily useful for applications in electrical engineering calculations, what is due to the aforementioned low accuracy in representation of complex magnetization cycles and the necessity to introduce artificial “patches” to improve model behaviour. A much more promising approach is the one proposed by Bergqvist [64, 65] or its later modifications advanced by Henrotte et al. [66-68]. In the aforementioned descriptions the offset of loop branches from the anhysteretic curve is accomplished along the H-axis, not along the M-axis like in the Jiles-Atherton model. This eliminates the need to introduce M . Bergqvist has shown the compliance of the model with the first and the second laws of thermodynamics, expressed using the Clausius-Duhem inequality dQ / dt H M F 0 [64]. In the description he has proposed a smooth curve for the anhysteretic and a number of pseudoparticles to describe potential wells. Figure 2 depicts the basic concepts in the Bergqvist model. F(M) is the Helmholtz free energy. Its profile is convex, close to parabolic. The derivative of the free energy with respect to magnetization for the ideal parabolic case would yield the single-valued inverse anhysteretic curve. If the system were disrupted by a small potential well, the hysteresis would emerge. In the Figure a number of such potential wells is depicted. The author has investigated some formulas for calculation of distributions of pseudoparticles and has found that even using crude approximations it is possible to obtain realistic modelling results for the alternating and rotating excitation. 1-64 The hysteresis loop modelled using the Bergqvist model is a bit “jagged”, in order to obtain a smooth representation it is necessary to take a sufficient number of elementary pseudoparticles. An alternative is to avail of some other model, which is based on offseting the loop branches from the anhysteretic curve, a good candidate might be the Takács description, which shall be discussed in more detail in the subsequent section. It is interesting to take stock of the concept of anhysteretic curve in more detail. D. C. Jiles and D. L. Atherton have proposed to describe it with the modified Langevin function (“modified” means that its argument is the effective field). Langevin function m coth( x) 1 / x is in fact valid for gases. The Brillouin function given as m 2 J 1 /(2 J ) coth2 J 1x /(2 J ) 1 /(2 J ) coth x /(2 J ) should be more appropriate for solids [47]. Jiles has considered a number of different formulas for anhysteretic curve in dependence whether the magnetic material reveals isotropic properties ( m coth( x) 1 / x ), axial ( m tanh( x) ) or planar anisotropy ( m I 0' ( x ) / I 0 ( x) , I 0 is the modified Bessel function of the zeroth order [12, 59]. Using the identity I 0' ( x ) I1 ( x ) (formula 9.6.27 in [69]), the relationship may be further simplified. The three abovegiven functions are depicted in Figure 3. The Padé approximation to the inverse Langevin function 3 m2 proposed by A. Cohen [70], given as x m , which seems to be the simplest one for 1 m2 practical applications, is also shown. Fig. 2. Profile of free energy, its derivative with respect to magnetization and hysteresis loop described using the Bergqvist model Fig. 3. Functions appropriate for description of anhysteretic curve Yet another approach to describe the effect of anisotropy on the shape of the magnetization curves has been proposed in Ref. [48] and introduced into the Jiles-Atherton model in Ref. [60]. Gy. Kádár has considered the Brillouin function with a properly chosen quantum number J for the description of anhysteretic curve. Moreover a modulation factor R(m) dependent on the chosen form of anhysteretic has been introduced into the relationship 1-65 dM irr dm dM R(m) , where R(m) an expressed as the dH dH dx function of man itself. Analytical calculation of R(m) is possible only for man tanh( x) , i.e. J 0.5 , which corresponds to the extreme anisotropic, two-state case. In all other cases one has to resort to numerical methods to calculate R(m) for arbitrarily chosen values of J and x. for total susceptibility, Fig. 4. Dependence of R(m)/R(0) on m for several values of quantum number J [60] It is important to remark that the notion of anhysteretic curve is useful in practical applications e.g. removal of magnetic signatures of Navy vessels and submarines (the so-called “deperming” procedure). For a brief introduction to the subject, the Readers are referred to Refs. [71, 72]. Modelling We have chosen the phenomenological description advanced by J. Takács for modelling hysteresis loops in chosen core materials used in electrical engineering [73, 74]. The Takács approach is based on the assumption that there is a hyperbolic tangent mapping between the variables involved in the description of hysteresis loop. The model has a number of advantages: it is purely phenomenological in nature, therefore its users may easily adjust it to their needs, it is able to describe a number of important phenomena observed experimentally by solid state physicists: minor loops, transients, accomodation, existence of a unique anhysteretic curve [75], etc. The variable presented at the abscissa axis has been identified previously as equivalent to the effective field [76, 77], this concept has been extended recently in Refs. [78, 79]. It was assumed that the effective field might include an additional term related to magnetization rate (which, generally, might be fractional [80, 81]). Therefore the extended description is able to describe dynamic hysteresis loops, which are affected by the effect of eddy currents generated in the conductive core material [82]. The modified description has been identified [79] as a specific case of the famous Chua model1. The relationship describing the major hysteresis loop (and other symmetric loops) using the Takács model may be written as follows: H (t ) H c 0 bTIP M (t ) M s tanh eff A 1 (1) L. Chua is the pioneer of research on dynamic hysteresis modelling [83, 84]. 1-66 where M (t ) denotes the instant value of magnetization, M s is saturation magnetization, A is a model parameter, H c 0 is coercive field strength in quasi-static conditions, whereas bTIP is introduced in order to match the ascending and the descending loop branches at loop tip H TIP H c 0 H TIP H c 0 tanh eff , bTIP 0.5M s tanh eff A A (2) where H effTIP denotes the effective field at loop tip, when the dynamic effects vanish. Generally, we assume the dynamic effects may be incorporated in the definition of the effective field H eff (t ) H (t ) M (t ) dM / dt , (3) where is the Weiss coupling coefficient, whereas is a new model parameter. The exponent may be in most cases considered fractional (an estimate may be obtained from the fitting of H c H c ( f ) dependence, cf. Fig. 5. This piece of information may be considered useful during estimation of some model parameters. Actually the values of fractional exponent vary in dependence on flux density level, thus the word ,,multifractional” might be more appropriate for the description of energy dissipation due to eddy currents in soft magnetic materials). A number of formulas similar to (1) for other more complicated magnetization cycles have been derived and discussed in detail by the model developer [73, 74]. Fig. 5. Fitting of the dependence H c H c ( f ) for the amorphous Metglas core. Dots – measurement In Figures 6 and 7 some exemplary results depicting the behaviour of the aforedescribed Takács model with effective field, which includes additional fractional term, are shown. Modelling was carried out for two chosen core materials used in electrical engineering: an amorphous glass (Metglas 2605 TCA) core and a sheet of grain-oriented electrical steel, grade ET 120-27. Model parameters were determined in quasi-static excitation conditions. Their values were as follows: 8.2 105 [-], A 71.3 [A/m], H c 0 2.14 [A/m], M s 9.55 105 [A/m] for Metglas 2.4 105 [-], A 35.9 [A/m], H c 0 9.29 [A/m], M s 1.44 106 [A/m] for the GO steel ( 1.17 [A/m/Hz ], 0.68 @ Bm 1.8 [T] ). These values were next used for prediction of loop shapes under dynamic excitation conditions. Despite some slight discrepancies in the shapes of the measured and the modelled 1-67 loop shapes, in particular for the grain-oriented steel sheet, it may be stated that the model predicts the change of loop shape upon increase in excitation frequency with an accuracy acceptable for engineering purposes. Fig. 6. Modelling results for the amorphous Metglas core Fig. 6. Modelling results for the grain-oriented steel sheet 1-68 Conclusions Some concepts important in hysteresis modelling (effective field, anhysteretic magnetization) have been discussed. Some of deficiencies of the Jiles-Atherton description in this context have been pointed out. As an alternative to the Jiles-Atherton model our proposal is to focus on the Takács approach. Exemplary modelling of hysteresis loops for chosen core materials used in electrical engineering has been carried out. Acknowledgements The work of Krzysztof Chwastek has been carried out within the framework of research grant N N510 702540 from National Research Centre. 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