Residual Magnetic Flux Density Distribution Calculation Considering Effect of Aligning Field for Anisotropic Bonded NdFeB Magnets

Abstract
This paper presents a calculation method of residual magnetic flux density distribution for a four-pole anisotropic bonded NdFeB permanent magnet (PM) considering the effect of aligning magnetic field during the forming process. To manufacture the anisotropic bonded NdFeB magnet, the magnet powder needs to be aligned with a proper aligning field before magnetizing. Therefore, it is necessary to analyze the magnetizing process based on the aligning field analysis to determine accurate distribution of residual magnetic flux density ( Br ) for the anisotropic bonded NdFeB PM. In order to estimate the B r distribution of the anisotropic bonded NdFeB magnet, an analysis method by combining the external electric circuit equation coupled with the transient finite element method and the scalar Jiles-Atherton hysteresis model is proposed.
Keywords
Aligning magnetic field ; Anisotropic bonded NdFeB magnet ; Finite element method ; Forming process ; Magnetizing analysis

1. Introduction
In order to design high performance permanent magnet (PM) machine, the PM type selection is essential. Nowadays, the PM machine utilizing NdFeB magnet is popular due to its high efficiency, high power density, and simple structure [1 , 2] . On one hand, according to the manufacturing process, the NdFeB magnet can be classified into sintered and bonded types. On the other hand, according to the characteristics of magnet powders, it can be classified into isotropic and anisotropic ones. Among different types of PMs, the anisotropic bonded NdFeB magnets are more attractive to the low power applications. The main reason is that they have lower cost around 150 $/kg and higher structure flexibility than the sintered ones (200 $/kg). Furthermore, they have relatively higher B r (~0.98 T) and maximum magnetic energy product ( BH ) max (~175 kJ/m 3 ) than the conventional isotropic bonded NdFeB magnets with B r of 0.71 T and ( BH ) max of 80 kJ/m 3.
To align the anisotropic magnet powders, the manufacture of the anisotropic bonded NdFeB magnet requires a proper and enough strong magnetic fields which is named aligning magnetic field. The aligning magnetic field, which is generated by a four-pole electromagnet, can be either a radial pattern or a polar pattern . Following the molding process, the magnet powder, which is uniformly distributed and fixed with the resin in the PM, is aligned with the aligning magnetic field. However, it is not fully magnetized and has to be magnetized by impulse magnetization at a considerably high field level.

Aligning Field for Anisotropic Bonded NdFeB Magnets

Aligning Field for Anisotropic Bonded NdFeB Magnets

Although the magnetic property of the anisotropic bonded NdFeB magnet such as B r , mainly depends on the orientation ratio of particles decided by the aligning magnetic field, there does not exist any general guidance for magnetizing the anisotropic bonded NdFeB magnet.
In the forgoing researches, there are some contributions that investigate the effect of aligning magnetic field for the anisotropic bonded NdFeB magnet. However, authors only explained the effect of aligning magnetic field on the magnet itself, the effect on the magnetizing process was not mentioned. authors described the effect of aligning magnetic field on the magnetizing process for the anisotropic bonded NdFeB magnet. However, the detailed description was not given in their papers. Furthermore, in all of these contributions, the hysteresis property of magnet was not considered.
In this paper, a numerical method which combines the finite element method (FEM) is proposed to predict B r distribution of the anisotropic bonded NdFeB magnet. Before magnetizing analysis, the aligning field analysis is carried out and the local magnetic property of the PM determined by the aligning magnetic field is calculated.
2. Molding Process Analysis of Anisotropic Bonded NdFeB PM with Aligning Magnetic Field
The anisotropic bonded NdFeB magnets are developed with anisotropic magnet powders, which are manufactured through the dynamic hydrogenation decomposition, desorption recombination (d-HDDR) treatment, and the resin compound. The mixture, and then, is filled with 2500-4000kg/m 3 filling density in the molding tool. As the temperature of the molding tool increases around 100-150℃ under the pressure of 0.4 GPa, the compound melts and the magnet powders suspend in the semi-liquid. Meanwhile, surrounding the molding tool, a four-pole DC electromagnet is assembled to generate a strong magnetic field so that the magnet particles can be enforced to rotate and attempt to align the direction of the magnetic field vector. This DC magnetic field is called aligning magnetic field. The aligning mechanism of the magnet powder is sketched in Fig. 1 . After that, the semi-liquid component is cooled until room temperature and the prepared magnet is manufactured.

To investigate the effect of the aligning magnetic field on magnetic properties of the anisotropic bonded NdFeB magnet, the anisotropic bonded NdFeB magnet samples with different aligning fields are prepared. The magnetic properties are measured by using the PM hysteresis loop measurement instrument. From Fig. 3 , it can be seen that with increasing the aligning magnetic field, the residual magnetic flux density B r and maximum magnetic energy product ( BH ) max are enhanced, especially at the region of a low aligning field. Even though the intrinsic coercive force ( i H c ) of the magnet is slightly reduced, it can be said that the increasing aligning field can improve the magnetic properties of the magnet. Moreover, the orientation of the anisotropic bonded NdFeB magnet is determined by the aligning magnetic field. It should be noted that the prepared anisotropic bonded NdFeB magnet does not have magnetism after forming process. Therefore, before using, the magnet has to be magnetized by applying an enough strong impulse magnetic field. This impulse magnetic field is called magnetizing field. Before the magnetizing process analysis, the aligning magnetic field based on the magnetostatic FEM is analyzed firstly. In this analysis, the PM region is taken as air and the permeability of this region equals to the permeability of vacuum μ 0 .

where the B comp and the B mag are magnitudes of effective components of B and magnetizing field, respectively, and dalign is the direction vector of the aligning magnetic field.
Not only the direction but also the magnitude of the aligning magnetic field has effect on the magnetic properties of the magnet. Therefore, it is necessary to distinguish magnetic properties based on the magnitude of the aligning magnetic field. In this article, four different cases are considered, as shown in Fig. 5 . For example, for case IV, there is the largest aligning magnetic field around 2.5 T, so the corresponding measured initial magnetization curve and demagnetization curves are selected and applied. For case I, in the transition region of PM, the aligning magnetic field is almost zero, so the initial magnetization curve and demagnetization curves which are measured according to this case, is selected. Apart from case I and IV, the magnetic properties named case II with aligning magnetic field of 0.5 T and case III with the aligning magnetic field of 1.5 T are also applied. In order to combine the analysis of aligning field with magnetizing analysis, the post process of the aligning magnetic field analysis is applied and the flow chart of this process is shown in Fig. 6 .

3. Calculation of Br Distribution Considering Effect of Forming Process
After forming process, the prepared magnet is required to be fully magnetized by a considerably strong magnetizing field which is generated by a capacitor discharge impulse magnetizer. The fundamental equation for magnetizing analysis of PM is treated as an unknown voltage source initial value problem coupled with an external electric circuit equation. Due to the relatively low conductivity of the prepared bonded magnet material around 2.9×10 3 S/m and the laminated back yoke for the magnetizer, the effects of eddy current in the analysis are ignored without loss of accuracy. The field governing equation in the FEM is written as follows:

where v is the medium reluctivity, A is the magnetic vector potential, J 0 is the applied current density determined by the external electric circuit equation, and M is the magnetization of the magnet.
When the transient FEM coupling with external electric circuit is applied, the current is unknown. The voltage equation of the equivalent electric circuit shown in Fig. 7 derived from Kirchhoff’s voltage law is shown as follows:

In order to simplify the analysis process, after achieving the maximum B for each element, the magnetization in the PM is assumed to increase or decrease monotonically. The overall algorithm can be summarized as follows:

Step 1.Analyze magnetization during the ascent stage of magnetic field according to initial magnetization curves;
Step 2.After achieving maximum Bin Step 1, the magnetization is calculated from the hysteresis loops modeled by the Jiles-Atherton (J-A) hysteresis model.

The magnetizing process is analyzed step by step until the discharge current decreasing to zero, then the magnetization of PM is recorded and treated as residual magnetization of PM ( M r ). The M r saved in each element of PM can be used to the following numerical analysis.
In order to analyze the magnetizing process of the magnet, the magnetization needs to be estimated at each time step.
At the beginning, the magnetization is determined by initial magnetization curves based on the magnitude of aligning field. Fig. 8 shows the measured initial magnetization curves under different aligning fields. According to the level of aligning field in the specific element, the magnetization is calculated by the linear interpolation or extrapolation of measured curves.

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