## 3 Electromagnetic Performances Comparison and Analysis
### Voltage Regulation Ratio  
The back EMF vector trajectory of four cases using FEM is shown in Figure 12, and the curves of back EMF amplitude with the change in the initial phase angle are shown in Figure 13.  
In Figure 13, it can be observed that the 12/7 design has relatively the best flux controlling ability, as its types A and B component's amplitude are the closest among all  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **Type** & **Order** & \begin{tabular}{c} **Flux Density** \\ \(B_{m}(T)\) \\ \end{tabular} & \begin{tabular}{c} **Winding Factor** \\ \(k_{w}\) \\ \end{tabular} & \
\begin{tabular}{c} **Angular Speed** \\ \(\omega\) \\ \end{tabular} &
\begin{tabular}{c} **Induced Voltage** \\ \(\mathsf{E}\mathsf{\propto}B_{m}k_{w}\omega\) \\ \end{tabular} \\ \hline \multirow{3}{*}{12/7} & A & 7 & 0.7178 & 0.933 & \(\Omega\) & 0.6697\(\Omega\) \\  & B1 & 19 & 0.0221 & 0.933 & 0.3864\(\Omega\) & 0.0076\(\Omega\) \\  & B2 & 5 & 0.4704 & 0.933 & 1.4\(\Omega\) & 0.6144\(\Omega\) \\ \hline \multirow{3}{*}{12/8} & A & 8 & 0.6679 & 0.866 & \(\Omega\) & 0.5784\(\Omega\) \\  & B1 & 20 & 0.0560 & 0.866 & 0.4\(\Omega\) & 0.0194\(\Omega\) \\  & B2 & 4 & 0.4642 & 0.866 & 2\(\Omega\) & 0.8040\(\Omega\) \\ \hline \multirow{3}{*}{12/10} & A & 10 & 0.2203 & 0.866 & \(\Omega\) & 0.1907\(\Omega\) \\  & B1 & 22 & 0.0528 & 0.866 & 0.454\(\Omega\) & 0.0208\(\Omega\) \\  & B2 & 2 & 0.2929 & 0.866 & 5\(\Omega\) & 1.2683\(\Omega\) \\ \hline \multirow{3}{*}{12/11} & A & 11 & 0.1753 & 0.933 & \(\Omega\) & 0.1636\(\Omega\) \\  & B1 & 23 & 0.0220 & 0.933 & 0.4783\(\Omega\) & 0.0098\(\Omega\) \\ \cline{1-1}  & B2 & 1 & 0.1342 & 0.933 & 11\(\Omega\) & 1.3770\(\Omega\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Contribution of the harmonic components to back EMF by the analytical method.  
Figure 12: Back EMF vector trajectory of four cases by FEM.  
Figure 13: The curves of back EMF amplitude vs. the initial phase angle.  
four cases, as shown in Table 4. The 12/8 design is similar to the 12/7 configuration with notable drawbacks: Its type B component's amplitude is higher than the type A component, which makes the vector trajectory have a specific portion on the left side of y-axis, as shown in Figure 12. For the 12/10 and 12/11 designs, in Table 4, it can be observed that the amplitude of the type B component is significantly higher than type A. As a result, although the type B component vector can rotate, the variation of the total back EMF amplitude is significantly smaller, which causes these two designs to have relatively weak flux controllability, as shown in Figure 13.  
To value the ability of flux control, the voltage regulation ratio (VRR) is introduced in this paper. The voltage regulation ratio of the analytical method and FEM is summarized and compared in Table 5, where \(E_{A}\) and \(E_{B}\) denote the back EMF amplitude of types A and B components, and \(E_{max}\) and \(E_{min}\) denote the maximum and the minimum of the back EMF amplitude. The voltage regulation ratio is defined as follows:  
\[VRR=\frac{E_{max}-E_{min}}{E_{max}}\cdot 100\% \tag{39}\]  
The larger the VRR, the better the flux controllability that the design has. In Table 5, it can be found that the results of the analytical method agree with the results of the FEM, thus proving the validity of the operating principles. The VRR of the 12/7 design is significantly higher than its counterparts.  
From the above analysis, a design guideline can be given: The best flux controllability can be obtained when the rotor pole pair number is slightly higher than half of the stator teeth number.