## 3 Results and Discussions
### Design and Analysis of AFPMSM  
Two coreless AFPMSM were designed (described in Tables 1 and 2) by three methods, namely an analytical method (Equations (6)-(12)), and a combination of an analytical method and mathematical models in explicit form and FEM (Ansys Maxwell 3D). Results of the design by analytical method are shown in Table 3, where different stator, rotor and PM thicknesses were considered. Table 4 shows the results of combination of the analytical method and mathematical models in explicit form and Table 5 shows the results gained by FEM as well as the comparison of the results of all three methods. Table 5 shows that all three used methods produce similar results with minor deviations at certain points. For the selected parameters of the AFPMSM, the matching of the results of the torque size of the analytical method and FEM is 102.98% and between the mathematical model in explicit form and FEM is 98.54%. Matching of the induced voltage between the analytical method and FEM is 93.62% and between the mathematical model in explicit form and FEM is 97.91%. The procedure of designing was repeated for the 20-pole AFPMSM, described in Table 2 with the same rotor disk thicknesses used for the AFPMSM in Table 1. A new mathematical model with new input data for different pole number was produced for this purpose by the same procedure as presented in the previous section. Table 6 shows the comparison of the torque size, calculated by using three different methods for designing AFPMSMs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \ \begin{tabular}{c} **Analytical** \\ **Method** \\ \end{tabular}  & \multicolumn{2}{c}{**FEM**} & \multicolumn{2}{c}{ \begin{tabular}{c} **Analytical Method and** \\ **Mathematical Models** \\ \end{tabular}  } & \multicolumn{2}{c}{**Matching (\%)**} \\ \hline \(\boldsymbol{h_{\text{TM}}}\) & \(\boldsymbol{d_{\text{s}}}\) (mm) & \(  
For the selected parameters, the matching of the results between the analytical method and FEM is 132.14% and between the mathematical model in explicit form and FEM is 95.43%. It can be seen that the mathematical model in explicit form produces more accurate results for PM thicknesses \(h_{\text{PM}}=5\) mm and \(h_{\text{PM}}=7\) mm in comparison with the analytical method. The reason for that can be found in the fact that the analytical method does not consider the PM span, but it works under the assumption that the PM surface and air gap surface are the same. For \(h_{\text{PM}}=10\) mm, neither the analytical method nor mathematical model in explicit form produce accurate results compared to the gained FEM results. The reason for this is in the magnetic flux leakage, since the PM thickness is larger than the distance between adjacent PMs on the same rotor disk (\(h_{\text{PM}}>h_{\text{PM}}\)). These dimensions are also shown in Figure 9.  
Figure 9 shows the 2D model of the coreless double-sided AFPMSM with dimensions and flux paths, where Path 1 represents the mean path of the magnetic flux and Paths 2 and 3 represent the magnetic flux leakage. PM dimensions and the stator and rotor disk thicknesses influence the size of the magnetic flux and its leakage. If the PM thickness is too small or stator thickness is too high, the magnetic flux leakage can increase in Path 2 and in the case of higher PM spans in Path 3. In the case of \(h_{\text{PM}}=10\) mm, the magnetic flux leakage between adjacent PMs on the same rotor disk increases (Path 3). This leakage is also increased by increasing the stator thickness, due to the magnetic flux path enclosing in Path 3 and the decreasing share of magnetic flux in Path 1. In Table 6, it can also be seen that the developed mathematical model in explicit form produces accurate results for the stator thickness \(d_{\text{s}}=20\) mm, which represents twice the PM thickness.