## 5 Study Case: Optimized SynRM for Minimum Ripple Torque  
A 36-slot, six-pole, three-barrier SynRM was selected as the case study for this section. The aim was to show the applicability of the proposed method to an optimized machine, with the particularity that its torque ripple harmonic content does not have a predominant component. The machine design was optimized using ANSYS commercial software, including Electronic Desktop for electromagnetic analysis, DesignModeler for geometry parameterization, and Workbench for optimization. The optimization process was conducted using a combination of multi-objective genetic algorithms (MOGA) and FEA simulations. During the optimization process, the machine was supplied with a current of 20 A/mm\({}^{2}\), which is approximately three times the rated current listed in Table 9. This answers to the known fact that selecting a current between two and three times the rated current can improve the insensitivity of torque ripple to load variations [41]. The objective functions were set to minimize torque ripple, maximize power factor, and maintain average torque above a specified value. The primary constraint during optimization was to keep the machine's insulation ratio within the defined limits of 0.35 to 0.45, as per the literature [42]. Geometric constraints were established to ensure feasible rotor geometries and to prevent errors or unrealistic solutions during optimization. The upper and lower limits for the rotor's geometrical parameters (objective variables) were properly defined. The optimization results are presented in Figure 9. The optimal design was selected based on a trade-off between average torque and torque ripple, as it is not possible to achieve the lowest torque ripple and highest average torque simultaneously. In this case, the machine with the lowest torque ripple was selected. To choose a suitable skew angle for reducing the torque ripple to acceptable values, a FFT was performed on the electromagnetic torque for one period of the torque ripple, as described in Section 3. The harmonic components of both designs under different load conditions are shown in Figure 10. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Stator outer diameter & \(D_{\text{so}}\) & 246 & mm \\ Stator inner diameter & \(D_{\text{si}}\) & 161.4 & mm \\ Rotor outer diameter & \(D_{\text{ro}}\) & 160.4 & mm \\ Rotor inner diameter & \(D_{\text{ri}}\) & 70 & mm \\ Tooth height & \(h_{\text{t}}\) & 22.8 & mm \\ Toot width & \(b_{\text{t}}\) & 9 & mm \\ Air-gap length & \(g\) & 0.5 & mm \\ Stack length & \(l_{\text{st}}\) & 120 & mm \\ Number of slots & \(Q_{\text{s}}\) & 36 & - \\ Number of turns & \(N_{\text{s}}\) & 20 & - \\ Number of pole pairs & \(p\) & 3 & \\ Synchronous speed & \(n\) & 5000 & rpm \\ Rated current density & \(J_{n}\) & 7.5 & A/mm\({}^{2}\) \\ Stacking factor & \(k_{s}\) & 0.95 & mm \\ Lamination thickness & \(e\) & 0.5 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Main data of the study-case machine subject to optimization. From Figure 10, it can be noted that, when the current density is below 10 A/mm\({}^{2}\), the second harmonic is predominant, which is mainly related to stator slotting. When the current density is above 10 A/mm\({}^{2}\), the third harmonic instead is predominant, which could render traditional methods useless based on mitigating the contribution of slotting effect on the torque ripple. In consequence, a customized skew angle is calculated from the proposed methodology, summarized in Table 10. This analysis considered the first, second, and third harmonics for mitigation using two-step, three-step, four-step, and five-step skewing.