Summary

EMSolution provides MAGNET as a finite element model of magnets. If the magnetization distribution of a magnet is not uniform, the magnet can be subdivided and a magnetization vector can be given for each property, or a magnetization vector can be given for each element. This section introduces the functions to set the magnetization distribution by inputting a sine function, which represents the magnetization distribution as a combination of sine waves, and by inputting a mathematical expression, which represents the magnetization distribution as a mathematical formula.

Explanation

There are two types of magnetization orientations subject to sinusoidal function input: radial orientation (radial direction in cylindrical coordinate system) for a ring magnet shown in Fig. 1 (a) and parallel orientation (any direction in Cartesian coordinate system) shown in (b). Mathematical input can be applied to any magnet geometry.

The sine function input is described below. In the case of radial orientation in the cylindrical coordinate system shown in Fig. 1 (a), the radial ($r$-direction) component $M_r$ of magnetization is defined in the angular direction ($\theta$-direction). Suppose that the distribution is expressed as a Fourier series of sinusoidal functions as in equation (1). Generally, odd-order harmonics are used, where the magnetization distribution is symmetrical.

$${\displaystyle M_r(\theta )=M_{rs}\cdot \sum _{k=1,3,5,\cdot \cdot \cdot }^{k_{max}}{b}_k\sin ((\frac{m}{2})k(\theta –{\theta }_0)) (1)}$$

where $M_{rs}$ is the reference magnetization [$T$], $k$ is the harmonic order, $k_{max}$ is the maximum harmonic order, $b_k$ is the amplitude normalized by the reference magnetization of $k$-th order, and $m$ is the number of magnetic poles. Angle ${\theta }_0$ is the reference angle in the cylindrical coordinate system and corresponds to the position where the magnetization reverses (between poles). $\theta$ is the angle of the element center in the cylindrical coordinate system, and magnetization is assumed constant within the element. $m$ is the number of poles per 2$\pi$ and $\frac{m}{2}$ is the number of periods. The parallel orientation case shown in Fig. 1 (b) is expressed as in Eq. (2). The reference magnetization $M_s$ is a constant vector expressed in the local Cartesian coordinates of the respective physical property region. $\theta$ is the angle from the x-axis.

$${\displaystyle M(x,y,z)=M_s\cdot \sum _{k=1,3,5,\cdot \cdot \cdot }^{k_{max}}{b}_k\sin ((\frac{m}{2})k(\theta –{\theta }_0)) (2)}$$

As Example 1, we show an analysis of the model in Fig. 1 with a magnetization distribution set up. The calculation conditions are: reference magnetization $M_s$ = 1.0 T, maximum harmonic order $k_{max}$ = 1 (fundamental component only), its amplitude $b_1$ = 1, and number of poles $m$ = 4. The specific permeability of the magnet is 1.02, the specific permeability of the yoke is 1000, and the outside is air. In the arrangement shown in Fig. 1, if the x-axis direction of the local coordinate is the x-axis direction of the overall Cartesian coordinate, the reference angle is ${\theta }_0$ = $\frac{\pi }{4}$.

Fig. 2 shows the waveforms of the given magnetization $M_r$ and the analyzed magnetic flux density $B_r$ on the outer surface of the magnet. It can be seen that the magnetic flux density waveform on the outer surface of the magnet is nearly sinusoidal for radial orientation, while for parallel orientation, the magnetic flux density waveform is distorted between poles. The magnetization distribution (magnetization) and the magnetic flux density distribution (magnetic) are also shown in Fig. 3.

Next, as Example 2, assume the radially oriented ring magnet model of Fig. 1(a) with $M_{rs}$=1.0T, $k_{max}$=5, $b_1$=1.0, $b_3$=0.25, and $b_5$=0.05. Fig. 4 shows the magnetization waveform and the flux density waveform on the outer surface of the magnet obtained by the analysis. The magnetic field distribution changes according to the magnetization distribution including harmonic components.

How to use

To give the magnetization distribution as a function input, set the INPUT_TYPE option in the MAGNET input field to 2: Sine function input or 3: Formula input.

* MAGNET * SERIES_ID * TIME_ID * NO_MAT_IDS * INPUT_TYPE * MAGNET 1 1 4 2 or 3

The conditions shown in Example 2 are used below as input examples. Note that MAT_ID is as shown in Fig. 1.

Sine function input: INPUT_TYPE=2

For sine function input, use equation (1) or (2). Equations (1) and (2) show only odd-order components, but even-order components are also supported as input.
First, input the sine function. Input NO_ORDERS for the number of harmonic orders, NO_POLES for the number of poles m, and ANGLE(deg) for the reference angle θ0 in unit deg. Next, set the ORDER of the harmonic order k and the AMPLITUDE of its amplitude bk to the number of NO_ORDERS.

* MAGNET * SERIES_ID * TIME_ID * NO_MAT_IDS * INPUT_TYPE * MAGNET 1 1 4 2 * NO_ORDERS * NO_POLES * ANGLE(deg) * 3 4 45 * ORDER * AMPLITUDE * 1 1.0 3 0.25 5 0.05

Then, enter the MAT_ID to assign the entered sine function, the COORD_ID for the coordinates defining the magnetization direction, and the reference magnetization $M_s$ for NO_MAT_IDS.

Radial orientation

Since a local cylindrical coordinate system is defined and used, COORD_ID=1 is specified for each MAT_ID and the reference magnetization of the local cylindrical coordinate system is entered as (MR, MT, MZ)=(1.0, 0.0, 0.0). Here, the same coordinate system is used for each MAT_ID, but a different coordinate system may be used for each. If MT≠0.0 or MZ≠0.0, you can set the reference magnetization vector in any direction in the cylindrical coordinate system.

* MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 101 1 1.0 0.0 0.0 * MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 102 1 1.0 0.0 0.0 * MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 103 1 1.0 0.0 0.0 * MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 104 1 1.0 0.0 0.0

Here, COORD_ID=1 for the local cylindrical coordinate system is defined in the COORDINATE setting as follows. TYPE=2 represents a cylindrical coordinate system, where EX_X=1 is defined to align the radial direction of angle zero in the local cylindrical coordinate system with the x-axis direction of the global Cartesian coordinate system.

* COORD_ID * TYPE * X0 * Y0 * Z0 * 1 2 0 0 0 * * EX_X * EX_Y * EX_Z * 1 0 0 * * EZ_X * EZ_Y * EZ_Z * 0 0 1

Parallel orientation

Since the coordinate system to be defined is a Cartesian coordinate system, COORD_ID=0, which is the ID of the entire Cartesian coordinate system, is specified for each MAT_ID. For MAT_ID=101, where the magnetic pole center is the x-axis in the global Cartesian coordinate system, set x-axis positive magnetization vector M(x,y,z)=(1,0,0). For other MAT_ID=102, 103, and 104, set the magnetization direction vector to y-axis positive = (0,1,0), x-axis negative = (-1,0,0), and y-axis negative = (0,-1,0) for one round, respectively.

* MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 101 0 1.0 0.0 0.0 * MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 102 0 0.0 1.0 0.0 * MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 103 0 -1.0 0.0 0.0 * MAT_ID * COORD_ID * MX(MR) * MY (MT) * MZ * 104 0 0.0 -1.0 0.0

Mathematical formula input: INPUT_TYPE=3

In the formula input, each component MX, MY, and MZ of the magnetization M(x,y,z) is given by a formula. Please refer to “Appendix 1: Mathematical Input” in the EMSolution Handbook for details on functions that can be used in mathematical input. In the case of formula input, the coordinate system defining magnetization and that of the magnet shape coincide, so it is possible to give magnetization distribution to a magnet of arbitrary shape. First, enter the magnetization distribution function. The formula to be entered here is the function including the reference magnetization $M_s$ in equation (2). Next, enter MAT_ID for the property ID that assigns the magnetization distribution function you entered and COORD_ID for the coordinate ID that defines the magnetization direction NO_MAT_IDS times.

Radial orientation

Enter the magnetization of the local cylindrical coordinate system in the formulas. Note that MX and MY correspond to MR and MT, and y corresponds to $\theta$, which is set to MR here. All MAT_IDs use the same magnetization distribution function, so repeatedly enter the next MAT_ID to be assigned and the coordinate ID that defines the magnetization NO_MAT_IDS times. Note that since a local cylindrical coordinate system is defined and used, COORD_ID=1 is specified here for each MAT_ID as its ID. Here, PI=$\pi$ is a reserved variable.

* MAGNET * SERIES_ID * TIME_ID * NO_MAT_IDS * INPUT_TYPE * MAGNET 1 1 4 3 * FUNCTION MX(x,y,z) * x(m) or x(m) * MX(x,y,z)=sin(2*(y-(PI/4)))+0.25*sin(6*(y-(PI/4)))+0.05*sin(10*(y-(PI/4)));; * FUNCTION MY(x,y,z) * y(m) or y(rad) * MY(x,y,z)=0;; * FUNCTION MZ(x,y,z) * z(m) or z(m) * MZ(x,y,z)=0;; * MAT_ID * COORD_ID * 101 1 * MAT_ID * COORD_ID * 102 1 * MAT_ID * COORD_ID * 103 1 * MAT_ID * COORD_ID * 104 1

Parallel alignment

In order to have a distribution in the parallel direction, it is necessary to define the angle $\theta$ in the formula input and give it to the formula. Furthermore, since the magnetization distribution function has different magnetization components and signs for each MAT_ID, input the magnetization components Mx, My, and Mz for each MAT_ID repeatedly NO_MAT_IDS times. The coordinate system to be defined is a Cartesian coordinate system, so COORD_ID=0, which is the ID of the entire Cartesian coordinate system, is specified for all MAT_IDs. $\theta$ is defined as theta(x,y) variable.

* MAGNET * SERIES_ID * TIME_ID * NO_MAT_IDS * INPUT_TYPE * MAGNET 1 1 4 3 * FUNCTION MX(x,y,z) * x(m) or x(m) * theta(x,y) = atan2(y,x); MX(x,y,z)=sin(2*(theta(x,y)-(PI/4)))+0.25*sin(6*(theta(x,y)-(PI/4)))+0.05*sin(10*(theta(x,y)-(PI/4)));; *FUNCTION MY(x,y,z) * y(m) or y(rad) * MY(x,y,z)=0;; *FUNCTION MZ(x,y,z) * y(m) or y(rad) * MZ(x,y,z)=0;; * MAT_ID * COORD_ID * 101 0 *FUNCTION MX(x,y,z) * y(m) or y(rad) * MX(x,y,z)=0;; *FUNCTION MY(x,y,z) * y(m) or y(rad) * theta(x,y) = atan2(y,x); MY(x,y,z)=sin(2*(theta(x,y)-(PI/4)))+0.25*sin(6*(theta(x,y)-(PI/4)))+0.05*sin(10*(theta(x,y)-(PI/4)));; *FUNCTION MZ(x,y,z) * y(m) or y(rad) * MZ(x,y,z)=0;; * MAT_ID * COORD_ID * 102 0 *FUNCTION MX(x,y,z) * y(m) or y(rad) * theta(x,y) = atan2(y,x); MX(x,y,z)=sin(2*(theta(x,y)-(PI/4)))+0.25*sin(6*(theta(x,y)-(PI/4)))+0.05*sin(10*(theta(x,y)-(PI/4)));; *FUNCTION MY(x,y,z) * y(m) or y(rad) * MY(x,y,z)=0;; *FUNCTION MZ(x,y,z) * y(m) or y(rad) * MZ(x,y,z)=0;; * MAT_ID * COORD_ID * 103 0 *FUNCTION MX(x,y,z) * y(m) or y(rad) * MX(x,y,z)=0;; *FUNCTION MY(x,y,z) * y(m) or y(rad) * theta(x,y) = atan2(y,x); MY(x,y,z)=sin(2*(theta(x,y)-(PI/4)))+0.25*sin(6*(theta(x,y)-(PI/4)))+0.05*sin(10*(theta(x,y)-(PI/4)));; *FUNCTION MZ(x,y,z) * y(m) or y(rad) * MZ(x,y,z)=0;; * MAT_ID * COORD_ID * 104 0

Here, the overall Cartesian coordinates are used as the Cartesian coordinate system, resulting in different magnetization distributions for each MAT_ID. However, the same distribution function can be used for all MAT_IDs by defining the appropriate local Cartesian coordinates for each MAT_ID. If you want to give a magnetization distribution to a bar-shaped magnet, you can do so by using the Cartesian coordinate system and entering the magnetization distribution function as a function of x, y, z.

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　　Sample data DL

・ input_radial_sin：
     Input file with radial orientation, sine function input, maximum order 1.
     (The condition of maximum order 5 is commented out.)
・ input_radial_eq：
     Input file with radial orientation, mathematical input, maximum order 1.
・ input_parallel_sin：
    Input file with parallel orientation, sine function input, maximum order 1.
・ input_parallel_eq：
   Input file with parallel orientation, mathematical input, file with maximum order 1.
・ pre_geom.neu：2D mesh file