## 2 Skewing Effects
### Finite Element Analysis  
In the analyzed model, there are totally 626,641 node elements, 6175 line elements, 112,807 face elements, and 464,299 volume elements. To improve the accuracy of the 3D FEM analysis, all elements are second order mesh. Mesh distribution and magnetization of the radial and axial PMs of the investigated model with a skewing angle of 75\({}^{\circ}\) in the different planes are presented in Figure 2.  
When the proposed model is skewed by 75\({}^{\circ}\) and operated under the rated condition (vrms = 0.25 m/s), it seems that there is a saturation phenomenon on the iron teeth and spacers, with a flux density of around 2.0 T (Figure 3). In the original model without skewing effect, the maximum electromagnetic force, and maximum and average power were 325.8 N, 51.4 and 18.3 W, respectively. In an extreme case, when the PMs are skewed by 120\({}^{\circ}\), the maximum force is reduced by 43.9% to 182.8 N. However, the maximum and average power are also significantly decreased to 31.6 and 13.0 W. In terms of the detent force, although it is relatively smaller than the electromagnetic force, the ratio of reduction for different skewing angles is similar.  
Comparison of the machine's performance for different skewing angles is shown in Figures 4 and 5. From Table 2, the performance of the full 8-slot 8-pole model can be easily estimated by multiplying with 8. There are two cases which can both limit electromagnetic force to be less than 2000 N and achieve the design targets for output power, including skewing angles of 75\({}^{\circ}\), and 120\({}^{\circ}\). For example, when the skewing angle is 120\({}^{\circ}\), the electromagnetic force, maximum and average power of the full model are about 1462.64 N, 252.80, and 104.32 W, respectively.  
Figure 3: Flux density (skewing angle 75\({}^{\circ}\)).  
Figure 2: Mesh and magnetization of the 1/8 model with a skewing angle of 75\({}^{\circ}\).  
Figure 4: Output power for different skewing angles.  
Figure 5: Detent force and electromagnetic force for different skewing angles.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Skewing Angle**} & \multicolumn{2}{c}{**Output Power (W)**} & \multicolumn{2}{c}{**Force (N)**} \\ \cline{2-5}  & **Maximum** & **Average** & **Detent** & **Electromagnetic** \\ \hline \(0^{\circ}\) & 51.43 & 18.25 & 171.19 & 325.82 \\ \(30^{\circ}\) & 50.15 & 17.84 & 160.08 & 305.75 \\ \(30^{\circ}\) & (97.5\%) & (97.8\%) & (93.5\%) & (93.8\%) \\ \(45^{\circ}\) & 47.89 & 17.37 & 149.91 & 284.26 \\ \((85.1\%)\) & (95.2\%) & (87.6\%) & (87.2\%) \\ \(60^{\circ}\) & 45.33 & 16.69 & 135.57 & 260.65 \\ \((87.5\%)\) & (91.5\%) & (79.2\%) & (80.0\%) \\ \(75^{\circ}\) & 42.28 & 15.87 & 121.37 & 236.35 \\ \((88.1\%)\) & (87.0\%) & (70.9\%) & (72.5\%) \\ \(120^{\circ}\) & 31.60 & 13.04 & 82.56 & 182.83 \\ \((61.4\%)\) & (71.5\%) & (48.2\%) & (56.1\%) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance of the 1/8 model for different skewing angles.  
It can be concluded that although the classical approach using the skewing effect can dramatically reduce the detent force and electromagnetic force, the output power is decreased accordingly. In reality, this approach is also difficult for manufacturers to fabricate.