Comprehensive Model of Magnetization Curve, Hysteresis Loops, and Losses in Any Direction in Grain-Oriented Fe Si

Transcription

1 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY Comprehensive Model of Magnetization Curve, Hysteresis Loops, and Losses in Any Direction in Grain-Oriented Fe Si F. Fiorillo, L. R. Dupré, Member, IEEE, C. Appino, and A. M. Rietto Abstract We report an investigation and theoretical assessment of the DC magnetic properties of high-permeability grain-oriented (GO) Fe Si laminations under variously directed applied fields. We verified that normal magnetization curves, hysteresis loops, and energy losses depend on the field direction according to the sample geometry. This is explainable in terms of specific 180 and 90 domain wall processes and magnetization rotations. We present a novel phenomenological theory of the magnetization curves and hysteresis losses in GO laminations, excited along a generic direction; the theory is based on the single crystal approximation and pre-emptive knowledge of the magnetic behavior of the material along the rolling (RD) and the transverse (TD) directions. This approach is consistent with the general structure of Néel s phase theory, with the additional consideration of hysteresis and losses. Epstein and cross-stacked sheet testing methods are the two base measuring configurations; all the other testing geometries (single sheet, disk, square) are expected to display intermediate behavior. The devised model provides, through a direct procedure, thorough and accurate prediction of magnetization curves and quasi-static losses in these two basic cases. Its application to the other geometries is equally possible, with only a limited amount of supplementary information. Index Terms Iron alloys, magnetic domains, magnetic hysteresis, magnetic losses, magnetic variables measurement, magnetization processes. I. INTRODUCTION THE PROPERTIES of grain-oriented (GO) Fe Si laminations magnetized along directions different from the rolling direction need sometimes to be considered in applications. Their knowledge is useful, for instance, when the calculation of fields and fluxes in joints and corners is required for the accurate design of transformer cores [1], [2] or when considering the use of GO laminations in large rotating machines [3]. But such practical circumstances have a connection with the classical problem of measuring and predicting the magnetization curve in a single crystal, the subject of celebrated experiments [4], [5], theoretically assessed by Néel through the so-called phase theory [6]. Néel s theory, focused on the magnetization curve of a zero coercivity material (anhysteretic curve), puts in the right perspective the role of the internal Manuscript received March 4, 2001; revised October 22, This work was partly carried out during a scientific stay of L. R. Dupré at IEN Galileo Ferraris, supported by FWO-Vlaanderen. F. Fiorillo, C. Appino, and A. M. Rietto are with the Istituto Elettrotecnico Nazionale Galileo Ferraris, I Torino, Italy ( fiorillo@ien.it). L. R. Dupré is with the Department of Electrical Engineering, University of Gent, B-9000 Gent, Belgium. Publisher Item Identifier S (02) field, the vectorial sum of applied field and demagnetizing field. The domain structure (i.e., the balance between the different phases) evolves with in a way depending on the shape of the crystal and its arrangement in the magnetic circuit. When talking of intrinsic magnetic behavior, one should therefore refer to few high-symmetry directions only. For the specific case of an Fe Si single crystal plate with (110) oriented surface, such directions are [ ] and [ ]. In a standard GO lamination they correspond, with a few degrees uncertainty, to the rolling (RD) and transverse (TD) directions, respectively. Attempts in the recent literature to theorize the magnetization curve in GO sheets along a generic direction, without disregarding the specific features of the low magnetization region (the undifferentiated mode I in Néel s phase theory), have indeed focused on the properties along RD and TD. Basically, these approaches call for some kind of interpolation between the RD and TD curves, in order to achieve the desired relationship at a generic angle to RD [7] [9]. Their empirical character, however, poses strong limitations on their general use and unphysical situations may occur when the RD and TD properties are considered to behave independently [2]. It has been suggested that the anhysteretic curve might provide, for any direction, sufficient information for numerical calculations in magnetic cores [10]. This curve, defined as the gradient of a suitable potential energy function (coenergy), is calculated using the RD and TD anhysteretic curves and assuming a suitable analytical expression for the equipotential curves, tailored to the specifically treated cases [10], [11]. Nontrivial numerical problems may arise, however, during differentiation of and the use of the anhysteretic curve at low fields may not be totally appropriate. A further problem with all these models is the tacit assumption that the prediction regards intrinsic magnetization curves, where the role of sample geometry and the associated demagnetizing effects are ignored. But the experiments in GO laminations made with different sample configurations and measuring devices (e.g., Epstein frame, single sheet/strip tester, rotational single sheet tester) provide, for a same material, different curves [12] [14], because depends on the sample geometry and the magnetic circuit. In the end, it is not clear which kind of experimental curve is to be predicted. This compounds with the inability of these models to predict hysteresis loops and losses. In the present paper, a novel comprehensive model of magnetization curve, hysteresis loops and energy losses in any direction in GO laminations is presented. This approach is consistent /02$ IEEE

2 1468 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002 Fig. 1. Fe (3 wt%)si (110)[001] single crystal, cut along the [1 10] direction (TD). (a) Antiparallel domain structure in the demagnetized state ([001] and [001] phases). (b) After application of a sufficiently high-td directed field, a balanced transition toward the [100] phase (white domains) and the [010] phase (dark-gray domains) is accomplished. The net magnetization is colinear with the field H. with the observed evolution of the domain structure and takes at face value the role of sample geometry. Two base measuring configurations and the related results are discussed in detail. In the first one, the sample is a narrow strip; in the second, it is an infinitely extended sheet. The first condition is realized by means of Epstein strips, tested alone or under parallel stacking in a conventional frame. As expected and experimentally verified, the high-transversal demagnetizing coefficient makes in this case applied field and macroscopic magnetization colinear, whatever the cutting angle of the strip [15]. Wide cross-stacked sheets conveniently emulate the second condition. It is shown that in both cases the theoretical prediction can be made exploiting no other data than those obtained in the RD and TD laminations. Other practical testing conditions (e.g., single sheet testing, circular sample) can be predicted as well, provided additional information is available (e.g., the sample demagnetizing coefficient). II. DOMAINS AND MAGNETIZATION PROCESS IN GO Fe Si SHEETS The conventional longitudinally cut GO Fe Si sheets are characterized by a regular pattern of 180 domain walls (DWs), basically directed along the RD, and subjected to back-and-forth motion under alternating fields. The domains evolve instead in a complex fashion when the field is applied along TD [16]. For a single crystal plate, such an evolution can be sketched as shown in Fig. 1. The demagnetized state is characterized, in absence of applied/residual stresses, by a regular slab-like domain pattern, with magnetization directed along [ ] and [ ] (i.e., RD). On increasing the field, a transition takes place, where the basic 180 domains transform, through 90 DW processes, into a pattern made of bulk domains, having symmetrically directed along [ ] and [ ] (i.e., making an angle of 45 with respect to the lamination plane), and of surface flux closing domains. When this novel domain structure, having no net magnetization normal to the sheet plane, occupies a fractional sample volume, the macroscopic magnetization value is, disregarding the flux-closing structure (1) Fig. 2. Major DC hysteresis loops measured on 0.30-mm-thick high-permeability grain-oriented Fe (3wt%)Si laminations (Epstein frame testing). RD strips cut along the rolling direction. TD strips cut along the transverse direction. The maximum magnetization value obtainable in the Fe (3wt%)Si plate at the end of this process is thus 1.42 T and further increase is obtained by moment rotations. The rotational contribution can be calculated, as a function of the applied field, according to standard methods [17]. A comparison between experimental RD and TD major loops is shown in Fig. 2. The sigmoid shape of the TD loop reflects the previously described domain transition, where the applied field must basically compensate for a magnetoelastic energy term. On comparing the anhysteretic magnetization curves associated with these loops, one can estimate such a magnetoelastic energy around 100 J/m. Looking at these curves from the schematic perspective of phase theory [6], it is realized that in the RD curve only mode I exists (two phases). For the TD curve, the sequence is mode I (four phases), mode III (two phases). For directions different from RD and TD, mode II (four phases) and mode IV (one phase) can take place, depending on the geometry of the sample. In any case, the RD and TD properties can be defined as intrinsic, in the sense that they are dependent on the material structure but independent of the sheet sample geometry, at least within the single crystal approximation. Fig. 3 shows a Kerr effect image of domains and their evolution along a major half-loop in an Epstein strip cut at an angle 45 to RD (peak magnetization 1.25 T). Both 180 DW displacements, pertaining to the [ ] and [ ] phases, and 90 transitions to and from the [ ] and [ ] phases are observed. We conclude that, for a generic direction in the plane of the lamination, we have a mixture of the two base 180 and 90 processes. The fractional sample volume occupied by the 180 domains is when and decreases, because of the growth of the 90 phases, as. It is therefore natural, in trying to predict the magnetization curve under a generic applied field direction, to refer to the RD and TD magnetization curves. This has already been pro-

3 FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE 1469 Fig. 4. Parallel stacking (a) and X-stacking (b) of GO Fe Si laminations. In the parallel stacked strips, the internal field is H = H 0 H (H applied field, H demagnetizing field). With X-stacking, it is possible to emulate a two-dimensionally flux-closed magnetic circuit and H H. H and H are the components of H along [001] and [110], respectively. Fig. 3. Domain structure along a major half-loop in an Epstein strip cut at 45 to RD. The region shown is a 2 mm wide transverse band (see inset). Starting from the saturated state, filled with the 90 phases ([100] and [010] axes), transition toward the 180 phases ([001] axis) takes place while proceeding toward the demagnetized state. This process is reversed on going toward the oppositely saturated state. posed in the literature, where, however, the material is normally treated as a continuum, characterized by a tensorial permeability [7] [11]. This results in formal approaches, which, lacking a direct connection with the domain structure, have limited predicting capabilities. Since we can speak of intrinsic magnetization curves only for fields applied along either RD or TD, the measuring configuration must be specified. The generally used methods rely on the Epstein frame, the single strip, and the single sheet testers [18], [19]. Recently, the use of the rotational sheet tester, used under alternating regime, has become popular [12] [14], [19] [21]. Whatever the method, the sample shape and the induction control technique, the demagnetizing field, vectorially combined with the applied field, always affects magnetization curve and losses (with the exception of RD and TD testing). The only practical way to minimize the demagnetizing effects is to emulate an infinitely extended lamination, by cross-stacking sufficiently wide sheets ( -stacking). The role of sheet stacking on losses has been recognized since a long time, but the reason for the observed differences has never been clarified. It has been suggested, for example, that different flux transfer mechanisms at the corner joints in the Epstein frame are responsible for such differences [18], plainly in contrast with the fact that they persist in absence of joints (e.g., when using stacked strips on a yoke). The two base measuring conditions shown in Fig. 4 can then be envisaged for a generic cutting angle to RD: 1) narrow strip (i.e., Epstein) testing, where, orthogonal to the strip length (i.e., ) is so high to make the transversal magnetization component approximately zero; 2) -stacked sheets, where the two-dimensional (2-D) flux closure makes the demagnetizing field negligible, so that internal and applied field are approximately the same. All the other geometries are intermediate cases and can be subjected to modeling by a further assumption regarding the value of the demagnetizing coefficient. The results discussed in this work have been obtained on 0.30-mm-thick high-permeability GO Fe (3wt%)Si laminations, by means of a digitally controlled setup [22], under Epstein, -stack and single sheet testing (300 mm 120 mm sheet) configurations, the latter being taken as an example of intermediate sample geometry. III. THE MODEL AND ITS APPLICATION TO EXPERIMENTS It is shown in this section that general and accurate prediction of hysteresis loops, energy losses, and magnetization curves at any angle to RD can be made exploiting a limited amount of information. In particular, for the narrow-strip (Epstein) and -stack measuring conditions the RD and TD properties only are required. The general framework of the model is Néel s phase theory [6], applied to the specific case of Goss oriented laminations, with the important addition of hysteresis. It is assumed that the material is magnetized at a very low rate, so that the magnetization process can be considered as quasi-static. A. Epstein Strips It is experimentally verified that in the Epstein samples of conventional thickness (0.27 mm and higher), tested either in a conventional frame (parallel stacking) or as a single strip inserted in a flux-closing yoke, the condition and is satisfied (see Fig. 4). Let us analyze in this case the magnetization mode I, characterized by DW displacements only.

4 1470 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002 Since all possible phases are not energetically equivalent in the demagnetized state and coercivity cannot be disregarded, the assumption that in this mode the material is isotropic, as held in the phase theory, must be dropped. For given volume fractions and, the associated net magnetizations and are directed along the [ ] and [ ] axes (thus in the plane of the lamination), respectively (see Fig. 1), with the value of related to by (1). Their components along the transverse direction must be equal and opposite in sign. For a given cutting angle, the values of and are related to the values of and by the expressions from which (2) These magnetization values are referred to the whole sample volume. If we refer to the volume fractions and, we have the reduced magnetizations (3) (4) For an impressed sinusoidal time dependence of, having peak amplitude, the behaviors of are obtained through (1) (4), as shown in the example reported in Fig. 5 ( T). As previously stressed, the magnetization rate is assumed to be so low that dynamic effects in the domain wall process are negligible (no dynamic losses). This condition is respected in the present experiments, performed at frequencies equal to or lower than 1 Hz. Time appears in our equations in order to provide a parametric representation of the relationship between and. Notice, also, that always in the demagnetized state, according to direct observations (see Fig. 3). It is supposed that magnetization curves and DC energy losses under RD and TD excitation are known. The more detailed is such a knowledge, the more extended is the prediction. Let the DC losses be and for peak magnetization values and in the RD and TD cut strips, respectively. For any angle, we have from (3) and (4) The resulting energy loss is then obtained, according to our base assumptions, by summing up the two contributions deriving from the 180 and 90 DW processes (5) Fig. 5. GO Fe Si Epstein strip, cut at 60 to the rolling direction, with imposed sinusoidal magnetization J(t) and transverse magnetization J =0. The magnetization rate J(t) is sufficiently low to make all dynamic effects negligible. Calculated evolution along an hysteresis loop (J = 1.15 T) of: (a) sample volume fractions v (t) and v (t) occupied by the 180 and 90 phases, respectively, and (b) corresponding magnetizations J (t) and J (t). The quantity J (t) is normalized to the volume fraction v (t) (4). known, the hysteresis loop, for any angle and peak magnetization, can be predicted. The relationships (3) and (4), connecting the imposed with and, compound with the relationships concerning the applied field, the internal fields and, and the demagnetizing field (see Fig. 4). The latter is transversally directed and we can write (7) where the volume fraction is time averaged. If the corresponding RD and TD hysteresis loops, parametrically expressed as, are (6) from which (8)

5 FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE 1471 Fig. 6. Same sample and peak magnetization as in Fig. 5. (a) Time behaviors of the fields H (t) and H (t). They are determined from the behaviors of J (t) and J (t) shown in Fig. 5, according to the corresponding RD and TD loops. (b) Applied field H(t) and transverse demagnetizing field H (t). They are calculated through (7) and (8). H (t) is conventionally chosen to be positive when directed as shown in Fig. 4. Its amplitude and sign make J (t) and J (t) balance in such a way that always J 0. The time behaviors of and, derived from the known RD and TD loops of peak magnetization values and are shown, for the specific case of Fig. 5, in Fig. 6(a). Fig. 6(b) shows the corresponding time behaviors of and, calculated through (8). By associating with the imposed function, the desired hysteresis loop is eventually obtained, as shown, in comparison with the experiment, in Fig. 7(a). Another predictive result is provided in Fig. 7(b). A satisfactory correspondence between predicted and experimental loops is achieved, in general, in the investigated induction range T, for all the considered cutting angles The prediction of the hysteresis loop evidently incorporates that of the normal magnetization curve. Fig. 8 provides a general overview of theoretical and experimental DC energy loss behaviors versus and. Discrepancies between theory and experiments are observed when the condition is not perfectly satisfied (e.g., very low and values) or rotations become important (e.g., high field values). It is Fig. 7. Comparison of predicted (dashed lines) and experimental hysteresis loops in Epstein strip samples with different cutting angles. The quantities involved in the derivation of the theoretical loop (a) are shown in Figs. 5 and 6. actually expected from the phase theory that, on increasing the field, mode I will end, for a given set of the direction cosines made by with the quaternary axes, at the so called ideal remanence value (Kaya s rule [23]), where it is superseded by the magnetization mode II. At the start of this mode, the two 180 phases have reduced to one (directed along [ ]) and three phases remain. The macroscopic magnetization is, asbefore, alignedwith the longitudinal field and the internal field is directed along the ternary axis [ ], to guarantee the survival of all three phases. The volumes and will balance accordingly and, following Néel [6], we shall write (9) (10) where is the modulus ofthe internal field. is the standard expression for the[ ] magnetization curve at high field strength

6 1472 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002 Fig. 8. Experimental (solid symbols and lines) and theoretical (dashed lines) quasi-static energy losses in GO Fe (3wt%)Si Epstein strips (parallel stacking). Eight different peak induction levels J and five different cutting angles have been considered. For induction values around and higher than 1.25 T the prediction may become less accurate, because hysteresis loss modeling does not account for the rotation of the magnetic moments. Fig. 9. Experimental (solid symbols and lines) and theoretical (dashed lines) quasi-static energy losses in GO Fe (3wt%)Si cross-stacked sheets versus cutting angle to RD. The experiments refer to 120-mm-wide, 300-mm-long sheets, tested in a flux-closing double-c laminated yoke. Notice the surge of losses for >45. in an Fe Si crystal [17] and, for the above mentioned symmetry reasons, the amplitudes of and are related by the equation (11) It is assumed that, for the present treatment of Epstein strips, mode II entirely covers the high field range of technical interest, excluding the approach to saturation, where it is known that the three-phase state displays anomalous features [16]. On the other hand, the transition between mode I and mode II is somewhat blurred in practice. The corresponding theoretical curves, separately shown in Fig. 12, can provide a convenient fit of the transition region when interpolated. B. -Stacked Laminations When the laminations are cross-stacked, the applied field and the internal field coincide, because, and the magnetization is always two-dimensional. But, as in the narrow strip case, the measured curve, involving the applied field and the longitudinal magnetization component, provides the basic physical information and defines the energy balance of the magnetic core. Let us, thus, analyze the magnetization process in mode I. As always, the demagnetized state is occupied by the two 180 phases. In the ideal case of perfect flux closure, single crystal sheet and absence of stresses, the 90 phases are inhibited for 54.7 (where 54.7 is the angle made by [ ] with RD) and. Mode I is thus associated with the 180 DW displacements and ends for. For a given applied field peak value, the RD hysteresis loop peaks at. The ensuing energy loss is (13) If 54.7 all four phases can coexist in mode I, but the 180 phases are still energetically favored in the demagnetized state. The components of the applied field along RD and TD are (14) and the associated DC hysteresis loops, having peak magnetizations and, respectively, are assumed to be known. Again, the 180 domain wall processes are considered to occur within the volume fraction. The values of the measured magnetization and the transverse magnetization are given by (2) (evidently now with ) with. Having thus determined from the RD and TD loops the correct amplitude and phase relationships between and, the macroscopic hysteresis loop at any given peak value of the applied field is obtained, together with the loss. Fig. 9 compares the so-calculated hysteresis losses with experiments in the range T T. Two examples of loop fitting are shown in Fig. 10. On increasing the field, mode I ends, for 54.7, when the 180 DW displacements are completed and the [ ] saturated phase remains. Mode IV follows, where the [ ] directed magnetization rotates in the (110) plane toward the direction of [Fig. 11(a)]. The portion of the normal (i.e., anhysteretic) magnetization curve associated with this mode is calculated by finding, for each value of, the equilibrium direction of the vector. To this end, the sum of anisotropy energy (12) with is and the associated hysteresis loop (15)

7 FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE 1473 TABLE I ANGLE ' MADE BY THE [100] AND [0 10]AXES WITH THE DIRECTION OF THE APPLIED FIELD WHEN THIS FORMS AN ANGLE WITH RD [I.E., [001], SEE Fig. 11(b)]. ' AND ARE RELATED BY (18) and Zeeman energy (16) where are the direction cosines of and is the angle made by with the [ ] axis, is minimized. From the value of at equilibrium, the component of along the direction of (17) is determined. When 54.7, the magnetization process succeeding mode I is the symmetric rotation of the magnetization vectors belonging to the axes [ ] and [ ] toward the direction of (mode III) [Fig. 11(b)]. The angle to be covered by the rotation of to reach the saturated state is (18) and is given, as a function of, in Table I. With reference to Fig. 11, the direction cosines of are Fig. 10. Examples of experimental hysteresis loops in X-stacked sheets (solid lines) and their theoretical prediction (dashed lines). Notice the ladder-like loop with = 75. In this case the 90 transitions start when the 180 DW processes are largely accomplished. Fig. 11. Rotation of the magnetization vector J in absence of demagnetizing fields (X-stacking) under (a) mode IV and (b) mode III. The field H is applied along a direction making an angle to RD and lying in the (110) plane. Process (a) is accomplished by the J vector belonging to the [001] easy axis and occurs for < Process (b) involves the J vectors belonging to [0 10] and [100] and occurs for > The magnetization component along H is J = J 1 cos(' 0 ), where ' is given by (18) (Table I). and the anisotropy energy interaction energy is The usual minimization procedure of the equilibrium angle and the magnetization (19) is provided by (15). The field (20) provides (21) Fig. 12 provides a general overview of the experimental normal magnetization curves under the Epstein and -stack measuring configurations here discussed. A comfortable agreement between theory and experiment can be observed. C. Single Sheets/Circular Samples The variety of measuring configurations suggested for the measurement of the properties of GO Fe Si laminations in any direction lead to a variety of results. They are expected to lie

8 1474 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002 Fig. 13. Experimental (symbols) and theoretical (lines) normal magnetization curves for = 30 under three measuring configurations. The SST experiments have been carried out on a 120-mm-wide and 300-mm-long sheet. The applied field is directed along the sheet length and the flux is closed at the ends by means of a double laminated C-yoke. The 180 and 90 phases evolve according to the behavior of the internal fields Fig. 12. The experimental normal magnetization curves in parallel stacked Epstein strips and X-stacked sheets (symbols). They are compared with theoretical predictions (lines) obtained by considering both domain wall displacements and rotation of the magnetization. Solid symbols and lines: X-stacked laminations. Open symbols and dashed lines: parallel stacked Epstein strips. The theoretical curves have been obtained at low and intermediate induction values (mode I) exploiting the experimental RD and TD curves. At higher inductions the magnetization rotations have been considered (modes II, III, IV). Separate fitting lines are provided for these two induction regions in the Epstein samples. where the magnetization given by (23), transverse to the applied field, is (24) between the ones obtained in the narrow strips and those obtained in the -stacked sheets, but their prediction requires the supplementary knowledge of the demagnetizing coefficient. It is understood that in most experimental arrangements the demagnetizing field is not uniform and so will be the magnetization. We shall consider here the representative cases of a rectangular/square single sheet, cut at an angle to RD, flux-closed by a yoke along the applied field direction [12] and of a circular sample, inserted in a 2-D measuring setup [23]. Let be the average (magnetometric) demagnetizing coefficient, transverse to the applied field in the rectangular/square sheet and isotropic in the circular specimen. We wish to determine the hysteresis loop when a field of peak amplitude, making an angle with RD, is applied. The parametric equations hold (22) In open samples (e.g., circular specimens) is the field measured with the use of a sensor (e.g., flat -winding, Hall probe) placed on the sheet surface. By introducing (24) in (23), we obtain that the corresponding RD and TD loops have peak magnetization values satisfying the condition (25) and is accordingly found through (22). On leaving the magnetization mode I at high fields, the magnetization curve may become difficult to predict, lacking and the special constraints on their direction observed in narrow strips and -stacked sheets. To simplify the matter, we might adopt in the high field range a suitable interpolation between the Epstein and -stack curves, which, as induced by the observation of Fig. 12, is the less imprecise the larger is the cutting angle. An example of experimental and predicted normal magnetization curves in a 120-mm-wide, 300-mm-long single sheet, tested in a double-c flux-closing yoke, is provided in Fig. 13. The corresponding average demagnetizing factor is estimated to be.

9 FIORILLO et al.: COMPREHENSIVE MODEL OF MAGNETIZATION CURVE 1475 IV. CONCLUSION The DC magnetization curves, hysteresis loops, and energy losses in Fe Si grain-oriented laminations can be predicted for a generic field direction through a method based on the exploitation of the material properties along the rolling and the transverse directions. Such directions are the only ones for which properties can be defined independent of the specific sample shape. For a generic direction, the measuring conditions (i.e., sample geometry) must be specified. Two base conditions have been identified: Epstein strip samples and cross-stacked sheets; all the other cases (e.g., single sheets, squares, disks) are expected to display intermediate behaviors. While the proposed interpretative framework relies on the concepts of Néel s phase theory, hysteresis loops and losses are theoretically treated here for the first time and the shortcomings of previous literature approaches are overcome. Energy losses and hysteresis loops can be meaningfully predicted in the induction range where the magnetization process is mostly accomplished by means of domain wall displacements (typically up to around at intermediate field orientations). The normal magnetization curve is calculated, starting at the end of the Rayleigh region, up to applied fields of the order of 10 A/m. The model is conceptually simple and its application requires elementary calculations. Based on the single crystal approximation, it is expected to display a predicting accuracy dependent on the actual textural quality of the laminations. The present experiments show that high-permeability laminations are endowed with the required textural perfection, but it is expected that application of the model to the conventional grain-oriented alloys, with their 7 misorientation angle of [ ] with respect to the rolling direction, should equally provide good results. ACKNOWLEDGMENT [10] T. Pera, F. Ossart, and T. 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Nakata, Effects of shape on samples of silicon-iron on the directions of magnetic field and flux density, in Non-Linear Electromagnetic Systems, A. J. Moses and A. Basak, Eds, Amsterdam, The Netherlands: IOS Press, 1996, pp [16] A. Hubert and R. Schäfer, Magnetic Domains, Berlin, Germany: Springer, 1998, p [17] F. Brailsford, Magnetic Materials, London, U.K.: Methuen, 1960, p. 54. [18] A. J. Moses and P. S. Phillips, Effect of stacking methods on Epsteinsquare power loss measurements, Proc. Inst. Elect. Eng., vol. 124, pp , [19] T. Nakata, N. Takahashi, Y. Kawase, and M. Nakano, Influence of lamination orientation and stacking on magnetic characteristics of grain-oriented silicon steel laminations, IEEE Trans. Magn., vol. 20, pp , [20] N. Nencib, S. Spornic, A. Kedous-Lebouc, and B. Cornut, Macroscopic anisotropy characterization of Si-Fe using a rotational single sheet tester, IEEE Trans. Magn., vol. 31, pp , [21] N. Baumgartner, H. Pfützner, and G. Krismanic, Practical relevance of the hard direction of h.g.o. Si-Fe, J. Magn. Magn. Mater., vol , pp , [22] L. R. Dupré, F. Fiorillo, J. Melkebeek, A. M. Rietto, and C. Appino, Loss versus cutting angle in grain-oriented Fe-Si laminations, J. Magn. Magn. Mater., vol , pp , [23] S. Kaya, On the remanence of iron single crystals, Z. Physik, vol. 84, pp , [24] L. R. Dupré, F. Fiorillo, C. Appino, A. M. Rietto, and J. Melkebeek, Rotational loss separation in grain-oriented Fe-Si, J. Appl. Phys., vol. 87, pp , The authors wish to thank Mr. S. Rocco and Mr. E. Genova, who provided substantial help in the experimental activity. REFERENCES [1] A. J. Moses and B. Thomas, The spatial variation of localized power loss in two practical transformer T-joints, IEEE Trans. Magn., vol. 9, pp , [2] T. Nakata, N. Takahashi, K. Fujiwara, and M. Nakano, Numerical analysis of flux distributions in cores made of anisotropic materials, J. Magn. Magn. Mater., vol. 133, pp , [3] G. Neidhoefer and A. 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Magn., vol. 19, pp , Fausto Fiorillo was born in Bricherasio, Italy, in He received the degree in physics from the University of Torino, Italy, in Since then, he has been with the Materials Department at Istituto Elettrotecnico Nazionale Galileo Ferraris (IEN), Torino, where he holds the position of Research Director. From 1997 to 2000, he was responsible in chief of the whole scientific activity of IEN. His research activity has covered many aspects of the properties of magnetic materials, their physical interpretation, and the related measuring techniques, including problems in standardization. He has devoted special attention to the study of the magnetization process, energy losses, and structural properties of soft magnetic laminations. He is author/co-author of some 130 scientific papers. Dr. Fiorillo is a Member of the Italian Physical Society and past-chairman of the International Committee of the Soft Magnetic Materials Conference. Luc R. Dupre (M 01) was born in He graduated in electrical and mechanical engineering in 1989 and received the Ph.D. degree in applied sciences in 1995, both from the University of Gent, Belgium. He joined the Department of Electrical Power Engineering, Laboratory of Electrical Machines and Power Electronics, University of Gent, in 1989 as a Research Assistant. Since 1996, he has been a postdoctoral Researcher for the Fund of Scientific Research-Flanders. His research interests are mainly focused on numerical methods for electromagnetics, especially in electrical machines, modeling, and characterization of magnetic materials.

10 1476 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 3, MAY 2002 Carlo Appino was born in Biella, Italy, in He received the degree in physics from the University of Torino, Italy, in 1985, and the Ph.D. degree in physics from the Politecnico of Torino in He has carried out his scientific activity at the Materials Department of Istituto Elettrotecnico Nazionale Galileo Ferraris (IEN), Torino, where he has held the position of researcher since His main research activity has been devoted to magnetization processes in amorphous and crystalline materials. Anna Maria Rietto was born in Moncalieri, Italy, in She received the degree in physics from the University of Torino, Italy, in She has been with the Istituto Elettrotecnico Nazionale Galileo Ferraris (IEN), Torino, since For about 20 years, her research activity has dealt primarily with dielectric materials and their physical properties. Since the 1980s, she has been involved with research in soft magnetic laminations, especially magnetic losses and hysteresis under alternating and rotational magnetic fields.