## 6 Optimization Results and Discussion
During the optimization, the slot shape became more rounded, which reduced the slot scattering field and the saturation of the machine, and increased its power factor. Furthermore, the stator yoke length is decreased, which decreased the stator core losses and compensated an increase in winding losses. Figure 7 shows the losses in various parts in the FRMs at rated power. Figure 8 reports the torque ripple and its harmonic spectrum of the single-phase FRM at the rated mode. It is obvious that the first harmonic is reduced, while the second harmonic is increased more significantly after the optimization. As a result, torque ripple is slightly increased. Triplen harmonics (third and sixth) are significantly suppressed. As for the next triplen harmonic (ninth), its value is very small in both cases. Triplen harmonic suppression results in decreasing the torque ripple of the three-phase machine. Figure 9 shows the torque ripple curves of the three single-phase machines. The blue line in Figure 9 corresponds to the portion of the graph (Figure 8a) in the range from 0 to 30\({}^{\circ}\) that belongs to the first single-phase FRM. The green line corresponds to a portion from 30\({}^{\circ}\) to 60\({}^{\circ}\) that belongs to the second single-phase FRM. The red line corresponds to a portion from 60\({}^{\circ}\) to 90\({}^{\circ}\) that belongs to the third single-phase FRMs. The combined image of the blue, green, and red lines following each other forms the graph for a single-phase FRM, as shown in Figure 8a. The torque ripple of the three-phase FRM composed of the three single-phase FRMs has a period of 30\({}^{\circ}\). The total torque of the three-phase FRM (defined as the sum of the three graphs (Figure 9) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Before** & **After** \\ \hline The stator core mass, kg & 1.1 & 1.2 \\ The rotor core mass, kg & 0.22 & 0.3 \\ The core winding mass, kg & 0.5 & 0.293 \\ The magnets mass, kg & 0.096 & 0.103 \\ Total active mass, kg & 1.91 & 1.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Active material masses of the three-phase FRM before and after optimization. Figure 6: Magnetic flux density field when the rotor teeth are over the stator slot: (**a**) Before optimization; (**b**) after optimization. Figures 11-13 illustrate the waveform of FRM line supply voltage, current, and instantaneous power, before and after optimization. Figure 11: Comparison of line voltage waveforms of the three-phase FRMs at rated torque. Figure 12: Comparison of phase current waveforms of the three-phase FRMs at rated torque. Figure 10: (**a**) The three-phase FRMs’ torque ripple; (**b**) its Fourier spectrum. As a result of optimization, the voltage waveform is less sinusoidal, which could be due to THD not being considered as part of the target function. However, it should be noted that a more complete use of voltage is in evidence. Before the optimization, the maximum voltage is 469 V, and the minimum voltage is \(-554\) V. A converter for the FRM before the optimization should be selected based on maximum voltage magnitude, and therefore, the positive half-wave is underused. After the FRM optimization, the maximum and minimum voltages are 552 V and \(-552\) V, i.e., the values are equal in magnitude. Furthermore, sinusoidal current amplitude was decreased from 16.1 A to 13.7 A. This results in a 15% decrease in the required converter power.