Conductivity and AC magnetic loss of Nd–Fe–B sintered magnet

1. Introduction

An Nd–Fe–B sintered magnet is one of the essential constituent elements of motors. Decrease of remanence due to the temperature rise (thermal demagnetization) is a significant issue in designing high-efficiency motors with high power-weight ratio. To reduce eddy-current loss and suppress thermal demagnetization, a magnet is commonly divided to small segments. Therefore, it is indispensable to accurately grasp conductivity and AC loss property of the permanent magnet for the appropriate magnet segmentation [1–3]. In this study, three kinds of anisotropic permanent magnets with different remanence B _r and coercivity H _c are investigated. Each permanent magnet is not coated. For each magnet, two specimens with different alignment directions are prepared and their conductivities and AC losses are measured. Furthermore, the influence of magnetization ratio on conductivity and AC loss is examined.

2. Measurement of conductivity

Table  1 shows the specifications of specimens used. “No.” means the grade of the prepared permanent magnet. L and C indicate that the alignment direction is parallel and vertical to the longitudinal direction of the specimen, respectively. The conductivity of each specimen is measured by using the four-terminal method. Furthermore, to investigate the influence of magnetizing on conductivity, unmagnetized specimens are also measured. To confirm the reproducibility, three measurements are carried out for the same specimen. The specimen with a low aspect ratio of 5 mm × 5 mm × 50 mm is used so that the current density is uniformly distributed in the specimen. Potential differences V s at the positions of ±10, ±15, ±20, ±25 and ±30 mm from the center of the specimen are measured by imposing a direct current I (2.5 A, 2.75 A, 3.0 A) to the specimen. The average of V∕I for the three current values is taken as the value of V∕I at each measurement distance. The measurement results are linearly approximated by the least squares method. Then, the conductivity is calculated from the slope.

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Fig. 1.

Measurement results of the conductivity.

Figure 1 shows the measurement results of the conductivity σ for Nos. 1L and 1C. It seems that the magnetizing does not affect the conductivity. The average conductivities of magnetized Nos. 1L and 1C are 622.32 kS/m and 766.12 kS/m, respectively. The molecules in an Nd–Fe–B permanent magnet have a layered structure and the magnetization (alignment) direction (L direction) is parallel to the laminated direction. Because of the layered structure, the conductivity in the L direction is lower than that in the in-plane direction (C direction). This measurement has sufficient accuracy because the variation in three measurements is within ±0.6%. Figure 2 compares the conductivities of three kinds of magnetized permanent magnets. The deviation between them is very small.

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Fig. 2.

Comparison of conductivity.

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Fig. 3.

AC loss property.

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Fig. 4.

Frequency dependence of W∕f.

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Fig. 5.

Loss separation.

Size of specimens used for measuring the AC loss is 20 mm × 20 mm × 50 mm. Exciting direction is the longitudinal direction of specimen. For exciting the specimens, a solenoid is used in which 189 turns of 𝜙 1 mm insulated wire are wound in the range of 200 mm on a square-shaped winding frame of which the inner and outer sizes are 30 mm and 36 mm, respectively. The average flux density b in the cross-sectional are of the specimen is detected by a B-coil. 22 turns of 𝜙 0.2 mm insulated wire is wound in the range of 5 mm at the center of specimen as the B-coil. The magnetic field strength h is detected by a H-coil. 𝜙 0.06 mm insulated wire is wound at 0.08 mm pitch in the range of 5 mm around a frame whose thickness and width are 1 mm and 5 mm, respectively. Its area-turns is 3.88368 × 10⁻⁴ m² turns. The distance between the center of the H-coil and the surface of the specimen is about 1 mm. Maximum flux density applied to the specimen is controlled at 10 mT. The exciting frequency is changed from 50 Hz to 10 kHz. The number of sampling points per period is 1024.

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Fig. 6.

Comparison of AC loss.

Figures 3 and 4 show the frequency dependence of AC loss W and the AC loss per one magnetization cycle W∕f for Nos. 1L and 1C, respectively. The alignment direction has the influence on the AC loss. In the high frequency region, W∕f does not change in proportion to f because of the large skin effect. Furthermore, the relative permeability of unmagnetized No. 1L is larger than the others. Therefore, the influence of the skin effect appears conspicuously. Figure 5 shows the results of loss separation for No. 1. The hysteresis loss W _h is calculated using the hysteresis loss coefficient k _h obtained by linear approximation of AC losses measured at 50, 100 and 200 Hz. The eddy-current loss W _e is calculated by subtracting W _h from the measured total AC loss. It is confirmed that W _h is extremely smaller than W _e for magnetized Nos. 1L and 1C. However, W _h for unmagnetized No. 1L is not negligibly small.

Figure 6 compares the AC losses of three kinds of permanent magnets. In the magnetized specimens, the difference in their AC losses is very small, and the difference in grade has little effect on AC loss properties. This is because the conductivities of the three kinds of specimens are nearly the same as shown in Fig. 2. In the unmagnetized specimens, there is a difference in AC loss properties between L specimens. It is considered that hysteresis loss in each specimen with different grade affects loss properties in unmagnetized specimens.

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Fig. 7.

Comparison of AC loss properties by magnetization ratio.

The AC losses of No. 1L having various magnetization ratios of 93%, 58%, 26% and 12% are measured to investigate the influence of the magnetization ratio on the loss property. We adjust the magnetization ratio by the excitation current value when magnetizing the permanent magnets. Figure 7 shows the measurement results of the loss property for each magnetization ratio. From Fig. 7(a), it can be seen that the loss decreases as the magnetization ratio decreases in the high frequency region. In Fig. 7(b), the region in which W∕f change in proportion to f differs depending on the magnetization ratio. Therefore, the influence of the skin effect becomes larger as the magnetization ratio becomes smaller. As a result, the eddy current loss becomes smaller. From Fig. 7(c), it can be seen that the hysteresis loss increases as the magnetization ratio decreases. Therefore, when the magnetization ratio of the permanent magnet is not 100%, there is a possibility to take hysteresis loss into consideration as the AC loss of the permanent magnet.

In order to verify the measurement accuracy, eddy current loss of a copper specimen with a size of 20 mm × 20 mm × 50 mm is measured at B _m = 5 mT and compared with the analysis result calculated by using the finite element method (FEM). The exciting frequency is 50, 100, 200, 400 and 600 Hz, the conductivity of the copper specimen is 5.68182 × 10⁴ kS∕m, and the density is 8920 kg/m³. Figure 8 shows the comparison between the analysis and the measurement results. The analysis result is roughly in agreement with the measurement result. It is considered that sufficient measurement accuracy is obtained.

Eddy-current loss analysis of the permanent magnets is carried out by using the conductivity obtained in the chapter 2. Figure 9 shows the calculated eddy-current losses of the permanent magnets with a size of 20 mm × 20 mm × 50 mm. Analysis results have a similar tendency to the measurement results shown in Fig. 3.

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Fig. 8.

Eddy-current loss of a copper specimen.

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Fig. 9.

Analysis results of eddy current loss in permanent magnets.

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Fig. 10.

Analysis model.

A magnetic field analysis of a permanent magnet synchronous motor by considering anisotropy of conductivity is carried out. Figure 10 and Table  2 show an analysis model and analysis conditions. The conductivity of the alignment direction is 622.32 kS/m and the conductivity of the vertical direction to the alignment direction is 766.12 kS/m. The iron losses are compared between the case of taking anisotropy into account (anisotropy) and the isotropic cases in which the conductivities are 766.12 kS/m (isotropy C) and 622.32 kS/m (isotropy L), respectively. Figures 11 and 12 show the calculated iron losses and the direction of eddy current in the permanent magnet. Comparing the anisotropy and isotropy C model, almost the same results are obtained regardless of considering the anisotropy of the permanent magnet because the magnetic flux generally passes parallel to the alignment direction. In addition, isotropy L model underestimates the eddy current loss in the permanent magnet about 18.1% compared with isotropy C model. The difference in conductivity in respective alignment directions directly affects the difference of eddy current losses.

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Fig. 11.

Analysis result of iron loss.

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Fig. 12.

Direction of eddy current.

This paper measured the conductivity and AC loss of three grades of permanent magnets with the different alignment direction. To investigate the influence of magnetizing on conductivity and loss property, unmagnetized specimens were also measured. As a result, it is confirmed in the range investigated in this paper that the conductivity does not change depending on the difference in grade and the magnetizing. Therefore, the conductivity of the permanent magnet can be measured by using unmagnetized specimens. In addition, the difference of conductivity caused by the difference of alignment direction affects the AC loss properties. Furthermore, the smaller the magnetization ratio becomes, the greater the influence of hysteresis loss and skin effect become. Therefore, it is considered that they should be taken into account when the permanent magnets are not fully magnetized.