## 5 Optimization of SP-PMVM
### Optimize \(k_{rt}\) and \(h_{m}\)  
To obtain the region of maximum average torque with a variety of optimal PM size, the parameters \(k_{rt}\) and \(h_{m}\) are investigated by the FEM calculation. Based on optimized dimension of the SP-PMVM in the last two steps, its average torque, cogging torque, and torque ripple are investigated when \(k_{rt}\) and \(h_{m}\) are in the range of 0.1-0.5 and 3-15 mm, respectively. Other parameters are kept the same as the original ones which are given in Table 1.  
The average torque at different \(k_{rt}\) and \(h_{m}\) is shown in Figure 8. It demonstrates that the area of maximum \(T_{seg}\) can be obtained is in the region \(\gamma\), which is defined between the contour Line a and b. Line a corresponds to the parameters of \(k_{rt}\) and \(h_{m}\) in the range of 0.2-0.4 and from 10-15 mm, respectively. \begin{table}
\begin{tabular}{c c c} \hline
**Items** & **Initial Value** & **Optimized Value** \\ \hline \(k_{rt}\) & 0.46 & 0.46 \\ \(k_{rt}\) & 0.3 & 0.3 \\ \(\beta_{mce}\) (deg) & 1 & 0 \\ \(T_{seg}\) (N.m) & \(750\) & 800 \\ \(T_{seg}\) (N.m) & 10 & 10.5 \\ \(T_{seg}\) (\%) & 1.33 & 1.31 \\ \(h_{m}\) (mm) & 10 & 10 \\ \hline \end{tabular}
\end{table} 
Table 2: Optimized Dimension of SP-PMVM. Figure 8: Average torque at different \(k_{rt}\) and \(h_{m}\). Figure 7: Average torque, cogging torque, and torque ripple at different \(\beta_{mce}\). The outer air-gap flux density of the conventional PMVM and the SP-PMVM are calculated by a time-stepping finite-element method (TS-FEM) at no-load, as shown in Figure 12. It can be noted that the flux density of the conventional PMVM is much smaller than that of the SP-PMVM with the same number of PMs. Quantitatively, the maximum outer air-gap flux density of the conventional PMVM and SP-PMVM are 0.944 T and 1.8 T, respectively. The flux linkage of the conventional PMVM at no-load is much smaller that of the SP-PMVM, as shown in Figure 13. The flux linkage of the SP-PMVM is almost twice that of the conventional PMVM. It should be noted that the power factor of the proposed SP-PMVM is 0.87, as listed in Table 3, which is much larger compared with that of the conventional PMVM (about 0.4). One of the reasons of the improved flux density, the flux linkage and the power factor is that the magnetic flux leakage of the SP-PMVM is much less than the conventional PMVM owing to the shift angle between the inner stator and outer stator of the SP-PMVM. Other reasons may include that the PMVM model based on the original design by other researchers, whereas the SP-PMVM is optimized. It should also be noticed that the data calculated by 2D TS-FEM might be higher than the actual value because the end effect is not considered. Figure 11: Magnetic field distribution: (**a**) conventional PMVM; and (**b**) SP-PMVM. Figure 12: Air gap flux density. Figure 14 shows the no-load cogging torque waveform of the two machines, while the speed of the motor is set as 1 rpm. The peak to peak values of cogging torque _T_cog_ and the torque ripple percentage _T_ripple_ of both machines are listed in Table 3. It can be seen that the _T_ripple_ of the proposed SP-PMVM is 1.31%, which is much lower than that of the conventional PMVM. Therefore, the cogging torque and torque ripple can be reduced by using the proposed design structure. Additionally, the steady-state average torque \(T\_avg\) is shown in Figure 15. It can be found that the averaged steady-state torque of proposed SP-PMVM is 800 N\(\cdot\)m, while for the conventional PMVM is 235 N\(\cdot\)m, namely, the torque-handling ability of the proposed machine is 319% higher than the conventional PMVM. According to above analysis, the power factor, torque, and efficiency are improved to a great extent. The reasons for lower torque ripple, improved output torque, and efficiency may include,  
* The SP-PMVM has two air gaps instead of one; therefore, the active area of the machine is almost doubled.