## 3 Optimal Design of the FCPM Model
### Optimal Design of Basic FCPM Model Using GA (Step III)  
GA was used to determine the optimal design point that satisfies the objective function and constraints [28; 29]. Figure 6 shows the convergence process for the design variables. It can be seen that the magnet angle x1 and the polar angle x2 converge to 58.3 degrees and 39.4 degrees, respectively. Figure 6: Convergence results of the design variables: (**a**) magnet angle, (**b**) polar angle. Figure 7 shows the convergence process for the objective functions, and each objective function converges to the optimal point. The number of convergences was set to a total of 200 in consideration of the time to converge to the optimal point. Figure 8 shows the change in the shape of the core through optimization and the optimal points of the design variables calculated through the optimization process are shown in Table 6. As a result, the magnetic angle of the individual magnets is reduced, and the pole angles are widened (Figure 8a). Therefore, as shown in the Figure 8b, the basic model partially has a very high magnetic flux density distribution, but it can be confirmed that the optimal model has a wide distribution of magnetic flux density (on the left is a basic FCPM model and on the right is an optimal FCPM model). Through this, it can be confirmed that the asymmetric back-EMF waveform has been changed to a symmetric structure. Figure 7: Convergence results of the objective function: (**a**) y1 (magnitude of back-EMF), (**b**) y2 (total harmonic distortion (THD)), (**c**) y3 (peak to peak value of cogging torque). Figure 9 shows a comparison of the analysis under no-load conditions between the basic and optimal FCPM models. As shown in Figure 9, through optimization, the back-EMF waveform of the optimized model was improved, and through the analysis of the harmonic order, it can be confirmed that the THD also improved. The cogging torque due to the combination of the pole slots must generate a total of six ripples when rotating one round electrically. However, the basic model did not show periodicity owing to the influence of the asymmetric back-EMF. In the case of the optimal model, the periodicity of the cogging torque appears, and it can be seen that the magnitude of the ripple is also reduced. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **x1 (Magnet Angle)** & **x2 (Polar Angle)** & **Magnet Thickness** \\  & **[Deg.]** & **[Deg.]** & **[mm]** \\ \hline Basic model & 70.0 & 26.5 & 3.0 \\ Optimal model & 58.3 & 39.4 & 3.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimal points of the design variable. Figure 8: Comparison of rotor assembly shape and flux density under no-load condition: (**a**) the rotor assembly shape of basic FCPM model and optimal FCPM model (1/4 model), (**b**) flux density under no-load condition. Table 7 shows a comparison of the optimization results under the no-load condition. Through optimization, it can be seen that the back-EMF is increased by 14.5%, and the THD of back-EMF and cogging torque are reduced by 38.3% and 25.4%, respectively. Figure 10 shows the torque waveform and flux density between the basic and optimal FCPM models at the rated load. As shown in Figure 10a, it can be seen that the torque ripple of the optimal FCPM model is greatly improved through optimization. Additionally, as shown in Figure 10b,c, it can be seen that the magnetic flux density in the rotor core is lowered. Through this, it can be estimated that the iron loss of the optimal FCPM model is reduced. The analysis results of the optimal model using a 2D FEM are shown in Table 8 to compare with the characteristics of the basic FCPM model. Through optimization, it can be seen that the torque ripple is reduced by 73.8%, and the efficiency is increased by 2.4%.