## 4 Results and Discussion: Finite Element Validation
### FEA Evaluation Original Designs (Skewless Machines)  
Figure 5 presents the electromagnetic torque waveform and spatial harmonic spectrum of both the four-pole and the six-pole SynRMs. Both 2D and 3D results are shown for comparison. From the results, the evaluation of the 2D and 3D models provide similar values, showing expected small differences, in accordance with the findings of [40]. The harmonic components of the electromagnetic torque for each machine are detailed in Figure 5b,d for the four-pole and six-pole machines, respectively. As expected, the largest harmonic component in both designs corresponds to the one generated by the stator slotting effect. Specifically, the highest magnitude harmonic of the four-pole machine corresponds to \(v=18\), and that of the six-pole machine corresponds to \(v=12\). Therefore, and according to the methodology proposed in Section 3, \(w=18\) for the four-pole SynRM and \(w=12\) for the six-pole machine. Table 2 shows the skew angle that should be considered to discrete skew the machine according to Equation 4, depending on the desired step number and aiming to mitigate the highest magnitude harmonic component. The following sections evaluate the impact of the proposed skew methodology on mitigating torque ripple for \(N=2\), \(N=3\) and \(N=4\). Figure 5: Electromagnetic torque waveform and spectrum of a skewness reference machine: (**a**) torque waveform of four-pole machine with two barriers per pole; (**b**) torque spatial harmonic content of four-pole machine with two barriers per pole; (**c**) torque waveform of six-pole machine with two barriers per pole; (**d**) torque spatial harmonic content of six-pole machine with two barriers per pole. Additionally, 2D and 3D simulation results are compared. This agrees with the estimations obtained from the reduction factor proposed in Equation (9), which can be simplified when evaluating \(N=2\) to:  
\[k_{\text{rs,$v$}}=\frac{\sqrt{2}}{2}\sqrt{1+\cos(\pi\frac{v}{w})}. \tag{10}\]  
According to Equation (10), the reduction of the main component of the electromagnetic torque (which has order \(w=18\)) should be maximum (\(k_{\text{rs,$v$}}=0\)). The difference lies in the fact that Equation (9) considers magnetically independent rotor slices, neglecting their interaction. Regardless of this, a significant reduction was observed in other harmonic components. In addition, for the four-pole machine it was found that the torque ripple harmonic component with \(v=12\) was also reduced to \(\sim\)45% of its original value, whilst other relevant harmonics of order \(v=6\) and \(v=36\) were not significantly affected. This agrees with the estimated reduction factor proposed in Equation (10), since \(k_{\text{rs,$6$}}=0.87\) \begin{table}
\begin{tabular}{c c c c c} \hline  & **Harmonic Order** & \begin{tabular}{c} **Mechanical Angle for** \\ **Two-Step Skew** \\ \end{tabular} & \
\begin{tabular}{c} **Mechanical Angle for** \\ **Three-Step Skew** \\ \end{tabular} &
\begin{tabular}{c} **Mechanical Angle for** \\ **Four-Step Skew** \\ \end{tabular} \\ \hline
2p2b & 18 th & \(5^{\circ}\) & 3.33\({}^{\circ}\) & 2.5\({}^{\circ}\) \\
3p2b & 12 nd & \(5^{\circ}\) & 3.33\({}^{\circ}\) & 2.5\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Skew angle to reduce a specific electromagnetic torque harmonic order by discrete skew. Figure 6: Comparison of electromagnetic torque waveform and harmonic content of a skewless machine vs. two-step skewed machine: (**a**) torque waveform of four-pole machine with two barriers per pole; (**b**) torque spatial harmonic content of four-pole machine with two barriers per pole; (**c**) torque waveform of six-pole machine with two barriers per pole; (**d**) torque spatial harmonic content of six-pole machine with two barriers per pole. The 2D and 3D simulation results are compared. \(k_{\text{rs,12}}=0.5\) and \(k_{\text{rs,36}}=1\).