# A Novel Design of Permanent Magnets for the Air Gap Magnetic Field of Hollow-Cup Motor
## 3 Finite Element Simulation Verification  
In this paper, in order to evaluate the width of flat section of the AMF, the parameter \(\tau\) is introduced, which is defined as the proportion of the top part to the half waveform width [25], as shown in Figure 4:  
\[\tau=\frac{\theta_{p}}{\alpha_{p}}, \tag{8}\]  
where \(\theta_{p}\) is the part of the AMF that lies above 99% of the maximum value \(B_{max}\) of the AMF in one cycle, and \(\alpha_{p}\) is the width of the half-cycle of the waveform. In order to clarify the optimum data of eccentric PMs, the parameters of a four-pole-pair HCM in the laboratory were analysed using the simulation method. The parameters of the HCM are shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline
**Item/Unit** & **Symbol** & **Parameter** \\ \hline Outer radius of ORC (mm) & \(R_{\alpha 1}\) & 69.2 \\ Inner radius of ORC (mm) & \(R_{\alpha}\) & 66 \\ PM thickness (mm) & \(h_{m}\) & 3.45 \\ Outer radius of IRC (mm) & \(R_{i1}\) & 57.3 \\ Inner radius of IRC (mm) & \(R_{i}\) & 53 \\ Air gap thickness (mm) & \(h_{air}\) & 4.55 \\ Pole pairs & \(p\) & 4 \\ PM remanence (T) & \(R_{r}\) & 1.2 \\ PM coercivity (A/m) & \(H_{cb}\) & 798,000 \\ Magnetising method & - & Parallel magnetisation \\ ORC and IRC material & - & DT4 \\ PMs material & - & NdFeB N35 \\ \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the motor. Figure 4: Schematic diagram of the percentage of the flat section of the AMF. A 2D simulation HCM model was constructed using finite element software; the mesh was dissected to appropriate grid elements according to its dimensions using the software mesh dissection tool. The magnetisation characteristic of the motor yoke is shown in Figure 5. The simulation result of the HCM with tile PMs is shown in Figure 6.  
The simulation results of the AMF are shown in Figure 7 when the samarium cobalt and NdFeB materials were used for the PMs. It can be seen that the samarium cobalt alloy material was slightly higher than the NdFeB material (by 1.8%), which indicates that the magnetic properties of the two materials were very close to each other. The AMF was analysed and compared with the waveform calculated using Equation (7), as shown in Figure 8. The analytical result was slightly larger than the simulation result (by 3.4%), but basically consistent, proving the correctness of the above analysis. The influence of the eccentricity value \(\lambda\) of the inner diameter of the PMs on the AMF waveform was analysed. The results of the AMF waveform under different eccentricity values (tile PMs, 2 mm, 3.5 mm, and 5 mm) are shown in Figure 9. It can be seen that, by adjusting the eccentricity of the inner diameter of the PM sheet, the middle part of the AMF waveform gradually became straight and the wave width gradually increased. Through calculation and simulation verification, when \(\lambda\) was 3.5 mm, the maximum wave width could be obtained. When \(\lambda\) was greater than 3.5 mm, the waveform of the PMs tended to sag, which led to a reduction in the value of \(\tau\). The comparison of AMF under different values \(\lambda\) = 0 and \(\lambda\) = 3.5 mm is shown in Figure 10. In summary, the AMF wave width \(\tau\) was 32.5% when \(\lambda\) = 0, and \(\tau\) was 57.8% when \(\lambda\) = 3.5 mm, representing an increase of 77.6%. In other words, adjusting the eccentricity of the inner diameter of the PMs could effectively increase the wave width. The flat section part of the AMF was made straighter, and the wave width ratio was higher, being closer to the trapezoidal shape. In addition, the problems caused by the mismatch between the AMF and the phase current waveform could be effectively reduced. Figure 5: Magnetisation characteristic of the motor yoke. Figure 8: Comparison of analytical result and FEA result. Figure 6: Magnetic field and magnetic line distribution of the HCM with tile PMs. Figure 4:0 FEA results of samarium cobalt alloy material and NdFeB material.