IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 2, FEBRUARY 2010 187

Comparison of Magnetic Field Analysis Methods ConsideringMagnetic Anisotropy

Teruyuki Tamaki�, Keisuke Fujisaki�, Kiyoshi Wajima�, and Koji Fujiwara�

Technical Development Bureau, Nippon Steel Corporation, Chiba 293-8511, JapanDepartment of Electrical Engineering, Doshisha University, Kyoto 610-0321, Japan

The magnetic field analysis results on grain-oriented silicon steel sheet and nongrain-oriented silicon steel sheet by the isotropic method,the two-axis isotropic method, the two-axis anisotropic method, the anisotropic method, and the �� anisotropic method, which are2-D magnetic analysis methods, are compared. In analyzing the magnetic field in the isotropic material, the isotropic method is moreappropriate than the two-axis isotropic method. For the anisotropic material in which the magnetic flux density is much less than sat-uration magnetic flux density, the two-axis anisotropic method is more preferable than the anisotropic method, because the result ofthe two-axis anisotropic method is similar to that of the �� anisotropic method. However, for the analysis in high magnetic flux densitynear the saturation magnetic flux density, the anisotropic method is more suitable than the two-axis anisotropic method.

Index Terms—Anisotropy, magnetic field analysis, silicon steel, 2-D magnetic property.

I. INTRODUCTION

T HE silicon steel sheets which are used for the cores ofelectrical equipment such as generators, transformers, and

motors are developing for saving energy [1]. However, it is noteasy to take full advantage of ferromagnetic property of steel[2], because the steel sheets have magnetic anisotropy and mag-netic saturation [3], [4]. The magnetic field analysis consideringthe anisotropy and the saturation is useful for getting great per-formance out of the silicon steel sheets. Many analysis methodsfor the electrical equipment made of the silicon steel sheets areproposed [5], [6]. Most of the models are 2-D ones, becausemagnetic flux in the sheet is almost parallel to the sheet planes.

Analysis based on more data including the magnetic proper-ties under the conditions of rotational magnetic flux density isthought to be more accurate. However, there is problem that ittakes much time to acquire such data and to calculate it. More-over, there is another problem that the calculation of magneticfield analysis on anisotropic material sometimes does not con-verge [7]. One of the reasons for the difficulty of the conver-gence is that the data include the measurement error.

In this paper, in order to avoid the problems, we compare themagnetic field distributions calculated by various conventionalproposed analysis methods on grain-oriented silicon steel sheet(GO) and nongrain-oriented silicon steel sheet (NO).

II. ANALYSIS METHODS AND MODEL

The magnetic field distributions calculated by the five anal-ysis methods, which are (i) isotropic method, (ii) two-axisisotropic method, (iii) two-axis anisotropic method, (iv)anisotropic method [8], and (v) anisotropic method [9],are investigated by using finite element method [10]. In theisotropic method, only one averaged - property is usedand it is assumed that magnetic flux density, , is parallel tomagnetic field, . In the two-axis isotropic method, the isseparated to components on axes of easy magnetization and

Manuscript received June 20, 2009; revised September 04, 2009; acceptedSeptember 15, 2009. Current version published January 20, 2010. Corre-sponding author: T. Tamaki (e-mail: tamaki.teruyuki@nsc.co.jp).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2009.2033558

Fig. 1. Relation of magnetic flux density, ���, and magnetic field, ��� . ���is amplitude of ���, and � is direction of ��� against rolling direction (��,X-axis). ��� is amplitude of ��� , and � is direction of ��� against rollingdirection (x-axis). � is angle between��� and���, that is � -� . TD (Yaxis)is the perpendicular to ��.

Fig. 2. Analysis model.

hard one, and each component of the is calculated from eachcomponent of the using the one averaged - property. Inthe two-axis anisotropic method, it is assumed that - prop-erties on axes of easy magnetization and hard one are different.The and the are separated to two components in the sameway as two-axis isotropic method. In the anisotropic method,

- properties are different in respective directions and it isassumed that the is parallel to the . In the anisotropicmethod, the angle, , between and is considered. The

anisotropic method is thought to get the best solution infive methods. Fig. 1 shows the relation of the and the inthe anisotropic method. In the anisotropic material suchas silicon steel sheet, the direction of magnetic field, , isgenerally different from that of magnetic flux density, .

The 2-D static magnetic field analysis in various methods ismade on the analysis model shown in Fig. 2. The sample of80 mm 80 mm is set in the coil of 200 mm length in X-axis

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188 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 2, FEBRUARY 2010

Fig. 3. Measured and calculated 2-D magnetization property of (A) 35G165 and (B) 35A300 in JIS for respective analysis methods under alternating flux densitycondition. � is amplitude of magnetic flux density, ���, and � is angle of ��� against rolling direction. � is amplitude of magnetic field, ��� , and � is angle of��� against rolling direction. The solid thick lines and the planes of deep color are measured data. The broken thick lines of �-��� � � are calculated data whichis the average of the solid thick line of �-��� � �. The broken thick lines of � -��� � � are also calculated data which is the same as the solid thick lines of� -��� � �. The planes of faint color are calculated data using the solid or broken thick lines.

direction and 80 mm width in Y-axis direction. The direction ofexternal magnetic field by coil is X-axis direction. The analysisarea is 1600 mm 1600 mm. The mesh number is about 20 000.

The examined samples are two virtual materials made of ei-ther the GO or the NO. The axis of easy magnetization in outerarea of the sample is X-axis, and the axis of easy magnetizationin center area of 30 mm 30 mm is inclined at an angle of 30degrees of X-axis.

The total number of coils is 800 turns, and the electric currentof coil for the GO is 100 A and one for the NO is 84.85 A. In thiscurrent condition, the average magnetic flux densities in bothGO sample and NO sample are 0.64 T in the isotropic method.

The rough values can be read from the - relations ofmethod (i) in Fig. 3.

Fig. 3 shows the measured and calculated magnetizationproperties of (A) 35G165 in JIS and (B) 35A300 in JIS forrespective analysis methods under alternating flux densitycondition. The solid thick lines and the planes of deep colorare measured data. The broken thick lines of - arecalculated data which is the average of the solid thick line of

- . The broken thick lines of - are alsocalculated data which is the same as the solid thick lines of

- . The planes of faint color are calculated data usingthe solid or broken thick lines. The measurement at any direc-

TAMAKI et al.: COMPARISON OF MAGNETIC FIELD ANALYSIS METHODS CONSIDERING MAGNETIC ANISOTROPY 189

Fig. 4. Analysis results of (a) amplitude of magnetic flux density,�, and (b) angle between��� and���� � , by (i) isotropic method, (ii) two-axis isotropic method,(iii) two-axis anisotropic method, (iv) � anisotropic method, and (v) � anisotropic method for (A) 35G165 and (B) 35A300 in JIS. The arrows indicate the axesof easy magnetization.

tion is done by 2-D magnetic property measurement apparatus[11], [12].

The - properties for (iii) two-axis anisotropic method onthe axes of easy magnetization, degree, and hard mag-netization, degree, are measured. The - propertiesfor (i) isotropic method at all angles and (ii) two-axis isotropicmethod on the axes of easy magnetization and hard one are av-erages of the measured two - properties for (iii) two-axisanisotropic method on the axes of easy magnetization and hardone. All of - properties and - properties for(v) anisotropic method are measured. All of -properties for (iv) anisotropic method are the same as thosefor (v) anisotropic method. By using only the above-men-tioned properties, analysis in each method can be done.

III. RESULTS AND DISCUSSION

Fig. 4 shows the analysis results on (a) distribution of ampli-tude of magnetic flux density, , and (b) angle, , between

and for GO and NO.For the isotropic method, the symmetry is taken as a mea-

sure of quality of calculation. The result of (i) isotropic methodis considered to be quite expected, because every result of themethod (i) is symmetric against the horizontal axis although thecenter part in the sample is inclined. However, the result of (ii)two-axis isotropic method is anisotropic in spite of isotropicsample. The reason of anisotropy in the method (ii) is to sep-arate and to components of axes of easy magnetization

and hard one. The method (ii) is not appropriate for isotropicmaterial.

For the anisotropic method, the analyzed result of themethod (v) is taken as a measure of quality of calculation by themethods (iii) and (iv). The magnetic flux density in the outerarea is easy to flow in the direction of X-axis, because the easyaxis is the X-axis. However, the magnetic flux density in thecenter area tends to flow in the direction inclined at an angle ofthirty degrees from the X-axis. The distribution of angelby (iii) two-axis anisotropic method is similar to that by (v)

anisotropic method, although the angel the is notdirectly taken account of in the method (iii). All the results ofthe methods (iii) and (v) show much similar patterns.

It is obvious that the (iv) anisotropic method has no dis-tribution of the angle, , from method definition. However,patchy pattern is appeared, which is much different from themethod (iii) and (v) for the GO. It can be speculated that thepatchy pattern is caused by the large anisotropy of the GO, be-cause the patchy pattern does not appear in the NO. In the NO,the distribution of the by method (iv) is different from that bythe methods (iii) and (v).

In Fig. 3, 2-D magnetization properties, which are trans-formed as functions of amplitude of magnetic flux density, ,and angle of flux density against rolling direction, , is shown.

The - properties at all amplitudes of magnetic flux den-sity for (i) isotropic method and (iv) anisotropic method ismade so that is equal to , because magnetic flux density,

, is parallel to magnetic field, .

190 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 2, FEBRUARY 2010

The properties which are - properties and- properties for (ii) two-axis isotropic method

and (iii) two-axis anisotropic method except the propertieson axes of easy magnetization and hard one are calculated byfollowing equations:

(1)

(2)

(3)

where and denote functions and represent - propertieson the axes of easy magnetization and hard one respectively.

In (i) isotropic method, the is not function of the . Theis equal to the , and not function of the . However, in

(ii) two-axis isotropic method, the and the are functionsof the and the . It means nonisotropy.

We compared the magnetic properties of the methods (iii),(iv), and (v) in Fig. 3.

The properties of - are much similar in the methods(iv) and (v). At the high magnetic flux density near saturationmagnetic flux density, - of the methods (iii) is muchdifferent from that of the methods (iv) and (v). In this analysismodel, because the magnetic flux density is much less than sat-uration magnetic flux density, it is considered that the effect ofthis difference is not appeared.

The properties of - are a little similar in methods(iii) and (v). It makes the similar distribution of angel bymethod (iii) and (v) in Fig. 4. However, the change of theagainst in the range of 0 to about 20 degree and of 0 to about1.5 T in the method (iii) is slower than that in the method (v).Therefore, the distribution of the by the methods (iii) and(v) in Fig. 4 is not same pattern qualitatively.

IV. CONCLUSION

The magnetic field analysis results on the grain-oriented sil-icon steel sheet and the nongrain-oriented silicon steel sheet bythe isotropic method, the two-axis isotropic method, the two-axis anisotropic method, the anisotropic method, and theanisotropic method are compared.

In analyzing the magnetic field in the isotropic material, theisotropic method is more appropriate than the two-axis isotropicmethod.

For the anisotropic material in which the magnetic fluxdensity is much less than saturation magnetic flux density,the two-axis anisotropic method is more preferable than the

anisotropic method, because the result of the two-axisanisotropic method is similar to that of the anisotropicmethod. However, for the analysis in high magnetic flux densitynear the saturation magnetic flux density, the anisotropicmethod would be more preferable than the two-axis anisotropicmethod, because the - curves between the axes of easymagnetization and hard one, which are calculated from themare much different from the measured - curves in respectiveangles.

ACKNOWLEDGMENT

The authors would like to thank Dr. S. Yasuhiro (KyoeiElectronics Corporation) and Mr. S. Yoshida (Addwin) for theircomputational technical support of this work.

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