AbstractâThe hysteresis loop of technical barium ferrite was measured at room temperature applying the external field parallel and perpendicular to the ...
IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 5, SEPTEMBER 2000
3357
Hysteresis Modeling of Anisotropic Barium Ferrite R. Grössinger, H. Hauser, L. Stoleriu, and P. Andrei
Abstract—The hysteresis loop of technical barium ferrite was measured at room temperature applying the external field parallel and perpendicular to the preferential axis. The hysteresis loop was analyzed using the Energetic as well as the Preisach model. The parameters of the Energetic model allow a physical interpretation of the results. Index Terms—Barium anisotropy.
ferrite,
hysteresis
model,
of the SD and their behavior is determined by a probability function. The total energy consists of reversible and irreversible contributions and the magnetization curve is calculated by the parameter equations:
shape
(1)
I. INTRODUCTION
B
ARIUM FERRITE is a standard permanent magnet material, where all magnetic parameters (magnetization, anisotropy) are well known [1], [2]. This type of material consists of single domains and is produced by a sintering technique. The magnetization process is nucleation determined is dominant—the [2]. The first order anisotropy constant material exhibits an easy -axis over the whole temperature range. Because this material is well known it is a good candidate for modeling the hysteresis loop. The following phenomenlogical models have been investigated: i) Energetic model [3]. In the present work the parameters used in the energetic model will be defined and interpreted especially for hard magnetic materials. ii) Preisach model [4].
(2) is the domain (particle) volume with a positive component to is the domain volume at field reversal, is the field reversal parameter (initial magnetization: 1, saturation: 2) [7]. The model parameters are and for a reversible energy contribution due to loss–free magnetization [e.g., rotation of domain (particle) magnetization]. The constant is obtained by solving the following equation:
(3) II. EXPERIMENTAL AND MODELS In the present work the hysteresis loop of a technical barium ferrite magnet was measured at room temperature in a pulsed parallel field hysteresigraph [5], applying the external field and perpendicular to the preferential axis. The anisotropy field was measured using the SPD method [6]. A. Energetic Model The energetic model of ferromagnetic hysteresis calculates the magnetic state by minimizing the total energy function. The magnetic material is supposed to consist of a huge number of statistical domains (SD) or statistical particles (SP), which are assumed to be all of the same, constant volume [7]. Statistical means that the domain (particle) magnetizations are distributed over several easy directions , given by anisotropic contribuare the related voltions (crystalline, shape, strain). umes of domains oriented in the direction . The distribution Manuscript received February 15, 2000; revised May 15, 2000. This work was supported by the Austrian Science Foundation under Project PHY-13152. R. Grössinger is with the Inst. f. Experimentalphysik, Techn. Univ. Vienna; Wiedner Hauptstr. 8-10, A-1040 Austria. H. Hauser, L. Stoleriu, and P. Andrei are with the Inst. f. Werkstoffe d. Elektrotechnik, Techn. Univ. Vienna, Austria. Publisher Item Identifier S 0018-9464(00)08260-1.
is equal to 1 for an ideally with aligned material and smaller than 1 for all other cases (for a ) and uniaxial isotropic material (4) There are two constants for the irreversible energy loss is related to the density of nucleation centers in the domain case and to the amount of reversible rotations with respect to switching in the particle case (5) represents the energy loss of an irreversible domain wall and jump (or switching particle) (6) is proportional to the energy loss per unit volume which is for a permanent magnet in principle the energy product
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 5, SEPTEMBER 2000
Fig. 1. Magnetization curve of the barium ferrite applying the field parallel to the preferential axis–comparison with the fit.
The average (total) demagnetizing factor (the inner and the geis ( susceptibility at the coercivity): ometrical) (8)
Fig. 2. Magnetization curve of the barium ferrite applying the field perpendicular to the preferential axis—comparison with the fit. TABLE I (a) EXPERIMENTALLY DETERMINED HYSTERESIS PARAMERERS (Fig. 1). (b) HYSTERISES PARAMETERS DUE TO THE ENERGETIC MODEL. THE FIRST LINE WAS OBTAINED FOR THE FIELD PARALLEL, THE SECOND LINE FOR THE FIELD PERPENDICULAR TO THE PREFERENTIAL AXIS
In contrast to mathematical models, these constants have their physical meaning. They can be derived from coercivity [8]: (9) and from the initial susceptibility
[9]: (10) (11)
is the sum of magnetocrystalline and shape anisotropy enare the demagnetizing factors parallel ergy density. (perpendicular) to the preferential axis. The saturation field (where ) can be written as (12) corresponding to the law of approach to saturation and the recan be used to identify the constant manence magnetization in (3). For a hard magnetic material the magnetoeleastic term can be neglected. The only remaining fit parameters and , are without dimension proportional to geometric ratios (e.g., pinning site/domain wall and reversible magnetization rotation, respectively). The equations above explain the meaning of the model parameters. In Figs. 1 and 2 the results of the modeling procedure are shown, using (1) and (2). The model parameters are given in Table I(a) and (b). They have been identified for applying the external field parallel to the preferential axis. The parameters for the perpendicular direction [(8) and (11)]. are calculated only by varying can be compared with the energy The constant stored in such a magnet—the value of about product 32 kJ/m is a little bit high for a barium ferrite magnet. The saturation field has to be compared with the anisotropy —it should be between 2–3. . The anisotropy field field
as measured at room temperature is 1.32 MA/m. This which seems also delivers a coefficient meaningful. If one calculates from the initial susceptibility the anisotropy energy one obtains about 1.4 J/m —this value is one order of magnitude smaller then the energy density J/m . This can be explained that the initial due to susceptibility is not only due to rotational process in the grains of the sample. Comparing the parameters with the nucleation model [10] delivers the following interpretation. There the coercivity can be simply described as: (13) describes a parameter which is the product of grain alignis the nucleation ment and the quality of the grain surface. field which is equal to the anisotropy field when higher order anisotropy constants can be neglected. The second term describes the stray field due to neighboring grains. Comparing : with the Energetic model one can calculate (14) value of 0.166 and for This delivers for the parallel case a value of 0.27. The difference can the perpendicular case a be understood by the fact that the ferrite forms plates as particles value gives an value of 0.21 respecafter grinding. This tively 0.26. All these values are physically very reasonable.
GRÖSSINGER et al.: HYSTERESIS MODELING OF ANISOTROPIC BARIUM FERRITE
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B. The Generalized Preisach Moving Model (GPM) In the Generalized Preisach Model (GP) [11], [12], one obtains the magnetization curves by computing the irreversible and the reversible components of the magnetic moment as functions of the applied field.
(15) are elementary operators over the ensemble of elwhere ementary hysteresis hysterons represented by rectangular hysand as up and down switching values, teresis loops with saturation values defined by respectively, and with if if and if are a set of step operators with the following if and if . properties: is the Preisach distribution function, which is assumed to be the product of a Gaussian distribution in the interaction field and a log-normal distribution in the coercive field
Fig. 3. 3D plot of the internal field given in kA/m.
H
versus
H
dependence. The axes are
and represents there the local effective anisotropy and interaction field, respectively. Fig. 3 shows the results of this type of analysis as obtained for the barium ferrite magnet. The which indicates no distribution shows a maximum at A/m. This interaction field (as mean value) and at value is much smaller then the magnetocrystalline anisotropy field of this material. The symmetry of the distribution indicates a homogenous magnetization process as can be expected for a single phased material. A deeper interpretation of the Preisach parameters is not yet possible. For this purpose more such comparisons are in preparation.
(16) III. CONCLUSION are the distribution standard deviations and where is the center of the distribution; is the true reversible component of the model and is given by (17) where is a fit parameter. The GPM model is taking into account that the magnetization itself contributes to the further magnetization of the magnetic sample by the mean of a supplementary magnetic field which is proportional to the magnetization. This way one may, in principle, separate the random interaction effects, described by the Preisach distribution, from the mean-field interactions described by the moving term. The effective value of the field in the sample is calculated in the iterative process: (18) where is the iteration index, is the applied field and is the moving parameter. The value of the real field applied to the particle is and . satisfying the condition This leads finally to surfaces that the saturation magnetization of each segment is the same.
The major hysteresis loops of particular media have been measured and can be modeled parallel and perpendicular to the preferential axis by changing only the effective demagnetizing , and of the Enfactor. The six parameters ergetic model allow a reasonable interpretation of the results. The Preisach distribution indicates a homogenous magnetization process without interaction field. REFERENCES [1] L. Michalowsky, Ferritwerkstoffe. Berlin: Akademie Verlag, 1985. [2] H. Kojima, Ferromagn. Materials, E. P. Wohlfarth, Ed., 1982, vol. 3, ch. 5. [3] H. Hauser, “Energetic model of ferromagnetic hysteresis,” J. Appl. Phys., vol. 75, no. 5, pp. 2584–2597, 1994. [4] G. Bertotti, “Energetic and thermodynamic aspects of hysteresis,” Phys. Rev. Let., vol. 76, pp. 1739–1742, 1996. [5] R. Grössinger, X. C. Kou, and M. Katter, “Hard magnetic materials in pulsed fields,” Physica B, vol. 177, pp. 219–222, 1992. [6] G. Asti and S. Rinaldi, “Singular points in the magnetization curve of a polycrystalline ferromagnet,” J. Appl. Phys., vol. 45, pp. 3600–3610, 1974. [7] H. Hauser and P. L. Fulmek, “Hysteresis calculations by statistical behavior of rarticles of high density,” JMMM, vol. 155, pp. 34–36, 1996. [8] M. Kersten, Probleme der Technischen Magnetisierungskurve. Berlin: Springer Verlag, 1938, pp. 42–72. [9] E. Kondorsky, “Hc and irreversible changes in magnetization,” Z. Phys. Sov., vol. 11, pp. 597–620, 1937. [10] H. Kronmüller, Supermagnets; Hard Magnetic Materials, G. J. Long and F. Grandjean, Eds. Dordrecht: Kluwer Academic, 1991, ch. 19. [11] F. Preisach, “Über die magnetische Nachwirkung,” Z. Phys, vol. 94, pp. 277–303, 1935. [12] I. D. Mayergoyz, Mathematical Models of Hysteresis. Berlin, Germany: Springer-Verlag, 1991.
