## 4 Results and Discussion: Finite Element Validation
### Evaluation of Torque Ripple Reduction by Means of Four-Step Discrete Skew  
The comparison of the electromagnetic torque for the four-pole and six-pole machine is shown in Figure 8 when four-step skewing is applied. It can be observed that there is a significant reduction in the harmonic torque component to be mitigated, approximately 10% of its original magnitude, to a similar extent to the two-step and three-step skewing. This is in agreement with the estimations obtained from \begin{table}
\begin{tabular}{c c c c c c} \hline  & & **Skewless** & **Skewless** & **Three-Step Skew** & **Three-Step Skew** \\  & & **2D** & **3D** & **2D** & **3D** \\ \hline \multirow{2}{*}{Four-pole SynRM} & \(T_{\text{avg}}\) & 51.0 Nm & 50.2 Nm & 49.9 Nm & 48.7 Nm \\  & \(T_{\text{rp}}\) & 73.9\% & 72.1\% & 15.6\% & 18.3\% \\ \multirow{2}{*}{Six-pole SynRM} & \(T_{\text{avg}}\) & 54.7 Nm & 53.8 Nm & 51.5 Nm & 51.0 Nm \\  & \(T_{\text{rp}}\) & 49.2\% & 45.9\% & 11.8\% & 12.7\% \\ \hline \end{tabular}
\end{table} 
Table 6: Average torque and torque ripple comparison when applying three-step skewing, by means of 2D and 3D FEA simulations. Figure 8: Comparison of electromagnetic torque waveform and harmonic content of skewless machine vs. four-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. the reduction factor proposed in Equation (9), which, for the specific case of \(N=4\), can be simplified to:  
\[k_{\text{rs,$v$}}=\frac{\sqrt{2}}{2}\cos(\frac{\pi v}{2v})\sqrt{1+\cos(\frac{ \pi v}{2w})} \tag{12}\]  
Based on Equation (12), the reduction of the highest magnitude component of the electromagnetic torque, when \(v=w\), is maximum (\(k_{\text{rs,$w$}}=0\)). Additionally, for the four-pole machine it was found that the torque ripple harmonic components with \(v=12\) and \(v=36\) were reduced to \(\sim\)35% and -5% of their original value, respectively, whilst the other relevant harmonic of order \(v=6\) was not affected. This is in agreement with the reduction factor proposed in Equation (12) for \(N=3\), since \(k_{\text{rs,$6$}}=0.84\), \(k_{\text{rs,$12$}}=0.43\) and \(k_{\text{rs,$36$}}=0\). On the other hand, for the six-pole machine, it was observed that the harmonic components with \(v=6\), \(v=24\) and \(v=36\) were reduced to \(\sim\)74%, \(\sim\)12% and \(\sim\)6% of their original value, respectively. This is in agreement with the values provided by Equation (12), since \(k_{\text{rs,$6$}}=0.65\), \(k_{\text{rs,$24$}}=0\) and \(k_{\text{rs,$36$}}=0\). Although they are not relevant contributors, the trend of other harmonic components of both machines are in good agreement with the expression presented in Equation (12). A summary of these results is presented in Figure 8:0  
Four-step skewing resulted in a severe torque ripple reduction, as summarized in Table 8, which presents the results obtained by means of the 2D and 3D evaluations, respectively. The torque ripple reduction was greater than 75% in both analyzed designs and there was an expected slight reduction of the average torque. Similar to the two-step and three-step skewing cases, it may be appreciated that the 3D evaluation shows a worse outcome than the 2D assessment, since a lower ripple reduction and a higher mean torque reduction were achieved. Nevertheless, the 2D and 3D results were very similar; for this reason, assessing the proposed skew technique by means of 2D simulations as a preliminary stage design can be recommended. It is possible to observe that, when \(N=2\) and \(N=3\) are compared, there is a considerable improvement in the reduction of the torque ripple, and the behavior is different for \(N=3\) and \(N=4\), where the reduction is the same.