## 4 Results and Discussion: Finite Element Validation
### Evaluation of Torque Ripple Reduction by Means of Three-Step Discrete Skew  
The comparison of the electromagnetic torque for the four-pole and six-pole machine is shown in Figure 7 when adopting a three-step discrete skew strategy. As in the case of the two-step skewing, there is a significant reduction of up to 93% of the highest magnitude harmonic component of the torque ripple. This is in accordance with the reduction factor derived in Equation (9), further simplified for cases with \(N=3\) as per:  
\[k_{\text{rs,}v}=\frac{1}{3}\bigg{[}1+2\cos\bigg{(}\frac{2\pi v}{3w}\bigg{)} \bigg{]} \tag{11}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Skewless** & **Skewless** & **Two-Step Skew** & **Two-Step Skew** \\  & & **2D** & **3D** & **2D** & **3D** \\ \hline \multirow{2}{*}{Four-pole SynRM} & \(T_{\text{avg}}\) & 51.0 Nm & 50.2 Nm & 50.0 Nm & 48.7 Nm \\  & \(T_{\text{rp}}\) & 73.9\% & 72.1\% & 20.2\% & 22.6\% \\ \multirow{2}{*}{Six-pole SynRM} & \(T_{\text{avg}}\) & 54.7 Nm & 53.8 Nm & 51.9 Nm & 50.5 Nm \\  & \(T_{\text{rp}}\) & 49.2\% & 45.9\% & 21.4\% & 19.2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average torque and torque ripple comparison when applying two-step skewing, by means of 2D and 3D FEA simulations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Harmonic Order** & \multicolumn{2}{c}{**Four-Pole SynRM**} & \multicolumn{2}{c}{**Six-Pole SynRM**} \\ \hline  & \(k_{\text{rs,p}}\) & \(k_{\text{rs,p}}\) & \(k_{\text{rs,p}}\) & \(k_{\text{rs,p}}\) \\  & **(Analytical)** & **(FEA)** & **(Analytical)** & **(FEA)** \\ \(v=6\) & 0.87 & 0.95 & 0.71 & 0.77 \\ \(v=12\) & 0.50 & 0.45 & 0.00 & 0.06 \\ \(v=18\) & 0.00 & 0.10 & - & - \\ \(v=24\) & - & - & 1.00 & 0.87 \\ \(v=30\) & - & - & - & - \\ \(v=36\) & 1.00 & 1.02 & 0.00 & 0.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Torque ripple harmonic component reduction as a result of two-step skewing. The 3D results were considered and relevant harmonic components analyzed. According to Equation (11), 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 \(\sim\)45% and \(\sim\)6% of their original value, respectively, whilst the other relevant harmonic of order \(v=6\) was not significantly affected. This agrees with the reduction factor proposed in Equation (11) for \(N=3\), since \(k_{\text{rs},6}=0.84\), \(k_{\text{rs},12}=0.45\) and \(k_{\text{rs},36}=0\). On the other hand, for the six-pole machine it was observed that the harmonic components with \(v=6\) and \(v=24\) were reduced to \(\sim\)70% and \(\sim\)12% of their original value, respectively, whilst the harmonic of order \(v=36\) was not visibly reduced. This agrees with the values provided by Equation (11), since \(k_{\text{rs},6}=0.67\), \(k_{\text{rs},24}=0\) and \(k_{\text{rs},36}=1\). Although they were not relevant contributors, the trend of other harmonic components of both machines were in good agreement with the expression presented in Equation (11). These findings are summarized in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline
**Harmonic Order** & \multicolumn{2}{c}{**Four-Pole SynRM**} & \multicolumn{2}{c}{**Six-Pole SynRM**} \\ \hline  & \(k_{\text{rs},p}\) & \(k_{\text{rs},p}\) & \(k_{\text{rs},p}\) & \(k_{\text{rs},p}\) \\  & **(Analytical)** & **(FEA)** & **(Analytical)** & **(FEA)** \\ \(v=6\) & 0.84 & 0.95 & 0.67 & 0.70 \\ \(v=12\) & 0.45 & 0.45 & 0.00 & 0.06 \\ \(v=18\) & 0.00 & 0.09 & - & - \\ \(v=24\) & - & - & 0.00 & 0.12 \\ \(v=30\) & - & - & - & - \\ \(v=36\) & 0.00 & 0.06 & 1.00 & 0.90 \\ \hline \end{tabular}
\end{table} 
Table 5: Torque ripple harmonic component reduction as a result of three-step skewing. The 3D results are considered and the relevant harmonic components analyzed.