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Magnetic Field Analysis of Permanent Magnet Motor with Magnetoanisotropic Materials Nd–Fe–B M

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Magnetic Field Analysis of Permanent Magnet Motor with Magnetoanisotropic Materials Nd–Fe–B M IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003 1373 Magnetic Field Analysis of Permanent Magnet Motor With Magnetoanisotropic Materials Nd–Fe–B M. Enokizono, Member, IEEE, S. Takahashi, and T. Kiyohara Abstract—In this paper, we propose the method to analyze the magnetization distribution in magnetoanisotropic materials by using the finite-element method considering the improved variable magnetization and Stoner–Wohlfarth model. By using this method, furthermore, the effect of the eddy currents induced in permanent magnets was analyzed. From the analyzed result, it is clarified how the magnetization distribution affects the performance of the surface permanent magnet-type motors. Index Terms—Anisotropic material, demagnetization factor tensor, magnetic field analysis, magnetization, Nd–Fe–B, perma- nent magnet motor, Stoner–Wohlfarth model. Fig. 1. Definition of vector relation and notations. I. INTRODUCTION II. FORMULATION N GENERAL, magnetic properties of the permanent mag- I nets must be explained with the vector relation between mag- A. Calculation of Initial Magnetization Process netic field strength vector and the magnetization vector in When an external magnetic field strength vector induces materials. The magnetic field analysis on products made from the vector, the volumetric free energy can be expressed by the hard magnetic material with uniaxial anisotropy usually re- the following: quires a large number of data from two-dimensional magnetiza- tion curves, because vector and vector are not always par- (1) allel to each other in the material. As such analysis is tedious, a smaller number of material data is more desirable. Furthermore, though it is necessary to obtain the inside distribution of the In the calculation of the free energy, the vector char- vector in permanent magnets when we carry out the analysis of acteristic curve is required. This curve involves the effect of the electrical machinery, it cannot be given in fact. It is impossible material shape. Therefore, the curve has to be obtained at every to measure the inside distribution of the vector. The distri- element and consider the demagnetization factor. It can be ex- bution must be treated as an unknown value [1]. However, up to pressed by the following: now, it has been treated as a given value. In the case of the strong hard magnetic materials such as Nd–Fe–B alloy having a strong anisotropy, it is impossible to directly measure the inside distri- bution of vector in the arbitrary direction. In order to analyze such problems, the demagnetization factor tensor method was (2) developed [3]. It is significant that the analytical method for the magnetizing process of the anisotropic hard magnetic material where, is the demagnetizing field vector and is the is established. By using this method, we can treat the magneti- effective field vector; subscripts “ ” and “ ” are the magnetic zation vector quantity, in other words, the absolute value of field components of the magnetizing easy direction and hard magnetization and the directional angle. direction, respectively. This paper gives the new magnetic field analysis method, To calculate (2), the curve is required and is ex- which requires only two kinds of data from the magnetization pressed by two kinds of characteristic curves, and curves measured along the easy direction and hard direction of curve. The principal axes of the demagnetization the hard magnetic material. factor are named the -axis and the -axis. A general spheroidal magnetic body in the uniform effective field is shown in Fig. 1. The self-energy is written as follows: Manuscript received June 18, 2002. The authors are with the Department of Electrical and Electronic Engi- neering, the Faculty of Engineering, Oita University, Oita 870-1192, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TMAG.2003.810422 (3) 0018-9464/03$17.00 © 2003 IEEE 1374 IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003 Fig. 3. Residual magnetization process. where and are given by [1], is the length to -axis, Fig. 2. Rectangular element. the length of the element “ ” for a polygon element, and is the number of apex on rectangular elements, as shown in Fig. 2. The where and are the magnetic susceptibility of the easy and relationship between and can be expressed by means of hard direction, respectively. The and are the the coefficient of demagnetization factor tensor as follows: demagnetizing factor in the -axis and -axis. The , , and are expressed as shown in Fig. 1. The -axis makes an angle from the easy axis. The total energy is minimized by the following conditions: , when . Then the following can be obtained: when (10) This tensor was used instead of the arbitrary vector curve, which could not be measured. (4) C. Calculation of Residual Magnetization As the applied magnetizing field decreases, both and decrease. Finally, when the becomes zero, the and (5) change the and , respectively, as shown in Fig. 3. The Utilizing two kinds of magnetization curves, the residual angle is the angle due to the residual magnetization and curves, it can solve these two simultaneous equa- from the easy direction. It occurs when the direction of tions. These curves can be obtained by the measurement of is not parallel to the easy direction. The relationship between two directions, “ ” and “ ”. Therefore, the initial magnetizing and can be expressed experimentally as follows [4]: process can be analyzed in the above procedure. Equation (5) is (11) the Stoner–Wohlfarth equation (2) with two kinds of anisotropic fields, too: the shape magnetic anisotropic field and the intrinsic where, is the square ratio in the easy direc- anisotropic field. tion. The residual magnetization is given from the Stoner–Wohl- farth equation (2) by the following: B. Calculation of Demagnetization Factor Tensor As shown in Fig. 2, when the rectangular element is magne- (12) tized uniformly, the magnetic charge induced in an element “ ” is expressed by the following: (6) The rectangular element is more useful than the triangular ele- (13) ment [1]. The demagnetizing field component and which the magnetic charge makes for the point where is the anisotropic magnetic field strength to the are shown as follows: magnetization . The value of can be obtained by using the Stoner–Wohlfarth equation. In this way, the magnetizing process can be analyzed in all the elements. As a result, the final (7) inside distribution of the residual magnetization in the perma- nent magnets can be analyzed, and then the magnetic field in- duced by the permanent magnets can be calculated. By introducing Kirchhof’s law, the exciting circuit equation (8) for calculating the distribution of initial magnetization is given as follows [5], [6]: (9) (14) ENOKIZONO et al.: MAGNETIC FIELD ANALYSIS OF PERMANENT MAGNET MOTOR 1375 (a) (b) (c) Fig. 6. Top part of teeth. (a) Model 1. (b) Model 2. (c) Model 3. Fig. 4. Analytical model of magnetizer. Fig. 7. Initial magnetization curve of Nd–Fe–B material. Fig. 5. Analytical model of surface-type permanent magnet motor. where is the interlinkage flux to the winding and is the initial electric charge of the capacitor. The current can be expressed using the charge as . D. Calculation of Demagnetization Process For the with the from the easy axis, when the external Fig. 8. Equivalent circuit of magnetizer. field applies to direction of angle , the is shown as follows: (15) (16) In the demagnetization curve of the easy axis, is a gradient of the demagnetization characteristic curve in the second quadrant. Fig. 9. Flux distribution of magnetizer. E. Fundamental Equation of Permanent Magnet Motor initial magnetizing curve of two directions, as shown in Fig. 7, The fundamental equation of the surface permanent typed are used in this analysis. Fig. 8 shows the equivalent circuit motor is given by of the magnetizer and the capacitance is 2400 , the total resistance is 0.031 . The charging voltage is 3187 V. III. ANALYTICAL RESULTS (17) Fig. 9 shows the flux distribution in the magnetizer at mag- where is the magnetic vector potential. and are the netizing state (Model 1). It looks like a successful magnetizing reluctivity and the conductivity, respectively. Fig. 4 shows a state from the flux distribution. However, as shown in Fig. 10, full model of a magnetizer. The magnetizer is used to magne- the value of the eddy current density induced in permanent tize the four poles hard magnetic materials, Nd–Fe–B alloy. magnet was different from each other at each time steps. The The pole pieces are made of steel, and its conductivity is time change of the eddy current was drawn at positions A, B, S/m. Fig. 5 shows the model of the permanent and C as shown in Fig. 11. The eddy current increases in the magnet motor. Fig. 6(a), (b), and (c) shows magnetizer models, boundary neighborhood of the magnetic pole. Fig. 12 shows which have the different shape of the top part of teeth. The the inside distributions of the magnetization vector after the 1376 IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003 Fig. 10. Value of eddy current density with time. Fig. 14. Characteristic curve of the cogging torque. motor was analyzed. Fig. 14 shows that the characteristic curve of the cogging torque was obtained for each model. The least pulsation of cogging torque was obtained when the magnetizer Model 2 was used. IV. CONCLUSION Fig. 11. Calculated points. In this paper, finite-element method introduced the improved VSWM method considering eddy current for anisotropic perma- nent magnet was presented. We have carried out the magnetic field analysis of the permanent magnet motor considering the distribution of the residual magnetization vector. As a result, it has been shown that it is important to consider the magnetiza- tion process of permanent magnets.
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