## 5 Skewing Analysis
In Table 2, the analysis results are reported in several working points, increasing the torque. It can be noticed that, in the second step of the RFOA analysis, the rotor \(q\)-axis flux is significantly reduced. In Figure 7, stator and rotor \(dq\) flux linkage values are reported for each considered slice, imposing \(i_{\text{qq}}=-i_{\text{sq}}\). In Figure 7a, regarding the stator flux \(\lambda_{\text{sd}}\), moving the rotor anti-clockwise, the rotor current partially acts along the \(d\)-axis, increasing the magnetizing flux. \(q\)-axis stator flux behavior is reported in Figure 7b: moving the rotor electric load forward and backward, the \(q\)-axis magneto-motive force balance is less effective and \(\lambda_{\text{sq}}\) can increase. Figure 6: Example of 2D slice, used for the skewing analysis. The rotor and its electric load are rotated together, keeping the \(dq\) reference frame position fixed. As already mentioned, the rotor flux linkage \(\lambda_{\rm rf\!r}\) in several slices is considered a part of the overall flux linkage \(\lambda_{\rm rq}\). In other words, the transformation angle from rotor \(abc\) to \(dq\) is the same for each slice and it is defined by the angle between the rotor \(abc\) and \(dq\) in the central slice. The flux linkage \(\lambda_{\rm rf\!r}\), is linked with a theoretical winding with axis along \(\beta_{\rm r}\). When a central slice is considered, this winding is able to gather even the \(d\)-axis flux, as shown in Figure 8. \begin{table}
\begin{tabular}{l c c c c c} \hline \(\boldsymbol{T_{\rm dq}}\) & \(\boldsymbol{i_{\rm sd}}\) & \(\boldsymbol{i_{\rm sq}}\) & \(\boldsymbol{i_{\rm rq}}\) & \(\boldsymbol{\lambda_{\rm rq,1}}\) & \(\boldsymbol{\lambda_{\rm rq,2}}\) \\
**[N m]** & **[A]** & **[A ]** & **[A]** & **[mV s ]** & **[mV s ]** \\ \hline
8.48 & 4.64 & 3.21 & \(-\)3.20 & \(-\)7.6 & 0.07 \\
18.08 & 4.54 & 6.92 & \(-\)6.88 & \(-\)17.6 & 0.81 \\
28.56 & 4.47 & 11.08 & \(-\)11 & \(-\)30.6 & 3.85 \\
38.36 & 4.42 & 15.15 & \(-\)15.03 & \(-\)47.4 & 6.71 \\
46.26 & 4.41 & 18.64 & \(-\)18.47 & \(-\)65.6 & 6.41 \\
52.65 & 4.40 & 21.57 & \(-\)21.36 & \(-\)81.9 & 11.6 \\ \hline \end{tabular}
\end{table} 
Table 2: RFOA results with rotor skewing. Figure 8: Moving the rotor, the theoretical winding \(\beta_{\rm r}\) is able to link a portion of the \(d\)-axis flux, according to \(\sin\alpha\). Figure 7: Stator and rotor flux linkage behavior, with the motor moving forward and backward and its electric load. These results come from the first step of the RFOA procedure with \(i_{\rm rq}=-i_{\rm sq}\). (**a**) Stator \(d\)-axis flux linkage in several slices; (**b**) stator \(q\)-axis flux linkage in several slices; (**c**) rotor \(q\)-axis flux linkage in several slices. It is the flux linked with the winding with axis \(\beta_{\rm r}\) (\(\lambda_{\rm rq}=\lambda_{\rm rf\!r}\) for each slice). In Figure 8b, a forward rotation of the rotor and the \(\beta_{\rm r}\) winding is considered, which corresponds to \(\alpha_{\rm sl}<0\). The angle between the axes \(\beta_{\rm r}\) and \(d\) is called \(\alpha\), with  
\[\alpha=\frac{\pi}{2}-\alpha_{\rm sk}\]  
In the case of \(\alpha_{\rm sl}=0\), and \(i_{\rm rq}=-i_{\rm sq}\), the flux \(\lambda_{\tau\beta_{\rm r}}\) is:  
\[\lambda_{\tau\beta_{\rm r}}=L_{\rm r}i_{\rm rq}+L_{\rm m}i_{\rm sq}=i_{\rm rq} (L_{\rm r}-L_{\rm m})=L_{\sigma\rm r}i_{\rm rq} \tag{17}\]  
where \(i_{\tau\beta_{\rm r}}=i_{\rm q}\) and \(\lambda_{\tau\beta_{\rm r}}\) results negative, with \(L_{\sigma\rm r}\) as the rotor leakage inductance.