## 6 Optimization Results and Discussion  
The optimization process, using the optimization procedure presented in the previous section, was applied to the 2D FEM. The DC bus voltage and the fundamental frequency at the rated value are 560 V and 1867 Hz, respectively. The optimization was stopped after 128 objective function calls. At that, the ratio of the maximum value of the objective function in the simplex to the minimum one is less than 0.5%. Figure 5: Indication of some geometrical parameters. Some motor characteristics before and after the optimization are provided in Table 3.  
In Table 3, \(TR3_{i}\) is torque ripple in the three-phase FRM, \(i\) stands for the different modes, and \(TR_{i}\) is torque ripple in the single-phase FRM. The optimization criteria values, as well as the logarithm of the ratios before and after optimization, are shown in Table 4. The logarithm is used to account for the \"compound interest effect.\"  
As a result of optimization, the \(K_{1}\) optimization criterion increased by about 4%, which resulted in a slight decrease in efficiency. However, a significant reduction in torque ripple in the three-phase FRM was achieved. Other optimization parameters also yielded some improvements. The drop in efficiency as a result of optimization is more pronounced under a reduced load compared to the nominal mode. This effect is explained by the fact that the contribution of the average (over a lengthy time period) mechanical and electric power values at underload is lower compared to the rated mode. When another \(K_{1}\) criterion is selected, the optimization result can be different. Torque ripple in the three-phase machine is significantly reduced (by more than four times for all three modes), which justifies a certain decrease in efficiency. Torque ripple in the single-phase machine is increased somewhat in modes 2 and 3; however, taking into account the reduction in the same parameter at the most under-loaded mode, the \(K_{4}\) criterion is also reduced. Figure 6 shows the FRM geometry and magnetic flux density at the rated mode before and after optimization. The position of the rotor when the rotor tooth is located in front of the stator slot opening is obvious in Figure 6. Both PM poles shown in the figure are magnetized equally in the same direction. However, the current flowing through the winding amplifies one magnetic pole and weakens the other magnetic pole. The rotor rotates in the direction of the strongest magnetic pole. A similar situation \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Before** & **After** & **Log (After/Before)** \\ \hline \(K_{1}\) & 0.09839 & 0.10238 & 0.04 \\ \(K_{2}\), W & 7780 & 6610 & \(-\)0.163 \\ \(K_{3}\) & 2.03 & 0.863 & \(-\)0.855 \\ \(K_{4}\) & 15.2 & 14.8 & \(-\)0.027 \\ Objective function, W & 1431.7 & 823.09 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimization criteria. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Before** & **After** \\ \hline The stator inner diameter, mm & 27.6 & 29.5 \\ Input torque at rated speed, N-m & 1.53 & 1.53 \\ Number of turns in the slot & 47 & 50 \\ Current at rated mode, RMS, A & 11.4 & 9.7 \\ Efficiency at mode No. 1, \% & 88.4 & 87.4 \\ Efficiency at mode No. 2, \% & 90.4 & 89.7 \\ Efficiency at mode No. 3, \% & 90.3 & 90.1 \\ \(\eta\), \% & 90.2 & 89.8 \\ \(TR3_{1}\), \% & 231 & 38.6 \\ \(TR3_{2}\), \% & 117 & 21.5 \\ \(TR3_{3}\), \% & 63.4 & 14.5 \\ \(TR_{1}\), \% & 827 & 768 \\ \(TR_{2}\), \% & 411 & 424 \\ \(TR_{3}\), \% & 282 & 291 \\ Required convertor power (\(occurs when rotor teeth are located in front of the middle part of the stator teeth, i.e., over the junction of magnetic poles of different polarities. The FRM before and after optimization has roughly the same mass as indicated in Table 5. The increased mass of the rotor and the stator magnetic cores is compensated by the decrease in winding mass.