## 5 Skewing Analysis  
The skewing effect analysis is performed connecting the multi-slice theory [11; 12] with the procedure described above. The continuous skewing is approximated as a stepped skewing, considering that in each slice the rotor is unskewed but properly rotated with respect to central reference position. For the induction motor also, the electric loading has to shift together with the rotor. The main meaning is that, with the skewing, the same bar current is distributed in a larger portion of space, depending to the skewing angle, interacting in different ways with \begin{table}
\begin{tabular}{l c c c c c c} \hline \(T_{\rm dq}\) & \(i_{\rm sd}\) & \(i_{\rm sq}\) & \(i_{\rm rq,2}\) & \(\lambda_{\rm rd}\) & \(\lambda_{\rmrq,1}\) & \(\lambda_{\rmrq,2}\) \\ \([\mathbf{N}\mathbf{m}]\) & \([\mathbf{A}]\) & \([\mathbf{A}]\) & \([\mathbf{A}]\) & \([\mathbf{V}\mathbf{s}]\) & \([\mathbf{mV}\mathbf{s}]\) & \([\mathbf{mV}\mathbf{s}]\) \\ \hline
8.3 & 3.11 & 2.3 & 2.29 & 1.75 & 9.2 & 0.098 \\
16.6 & 3.08 & 4.61 & 4.57 & 1.74 & 19.1 & 0.88 \\
24.8 & 3.04 & 6.91 & 6.86 & 1.73 & 29.9 & 3.0 \\
32.9 & 2.99 & 9.19 & 9.12 & 1.71 & 42.9 & 2.9 \\
40.8 & 2.95 & 11.55 & 11.45 & 1.69 & 59.4 & 5.6 \\
48.6 & 2.92 & 13.88 & 13.72 & 1.67 & 79.2 & 26 \\ \hline \end{tabular}
\end{table} 
Table 1: Results of the two-steps procedure, applied in different working points. Figure 5: Second simulation field solution and related equivalent circuit, linked to the flux linkage–current relationships. (**a**) Second MS FEA field solution, imposed current density and flux lines are shown; (**b**) equivalent circuit related to the second MS FEA. the stator field in the different slices. The novelty here is that the rotor-induced current is predicted using the MS FEAs procedure, avoiding the time-stepping analysis. In the following example, five 2D slices have been considered [14]. In this analysis, the IM exhibits a skewing angle of o slot. The angle of each slice, with respect to the \(dq\) reference frame, is defined as:  
\[\alpha_{\text{sl}}=\alpha_{\text{sk}}\frac{2M-M_{\text{tot}}-1}{2M_{\text{tot}}} \tag{16}\]  
where \(\alpha_{\text{sk}}\) is the rotor skewing angle, \(M\) is the considered slice, \(M_{\text{tot}}\) is the total number of slices. The first considered slice is already reported in Figure 4a. In the other slices, the \(dq\) reference frame remains the same, together with the stator current vector. The rotor and its electric load are rotated backward or forward according to the slice angle \(\alpha_{\text{sl}}\). An example of a 2D slice is reported in Figure 6. The rotor skewing is represented by the reference frame \(\alpha_{r}\beta_{r}\) rotation, forward and backward. The field solutions are post-processed in the same way as for the slice aligned to the \(dq\) reference frame (\(\alpha_{\text{sl}}=0\)): in particular, the angle between \(\alpha_{r}\beta_{r}\) and \(dq\), used in the Park transformation, to achieve stator and rotor \(\lambda_{\text{d}}\) and \(\lambda_{\text{q}}\), is the same for each 2D FE problem. In other words, the rotor flux \(\lambda_{r\beta_{r}}\) is considered valid even in the \(dq\) reference frame (\(\lambda_{r\beta_{r}}=\lambda_{\text{rq}}\)) for each slice, and \(i_{\text{rq}}=i_{r\beta_{r}}\). Basically, with skewing, the RFO Analysis method works in the same way as described in Section 4. In the first step of the procedure, stator and rotor current space vectors are imposed as in (11). A series of simulations is carried out, rotating backward and forward the rotor and its electric load. Stator and rotor \(dq\) flux linkages are mediated to achieve values, taking into account the skewing. A first set of inductances can be derived from the field solution, using (13) and the average value of the fluxes. Then, for the second step, the rotor current is imposed according to (14). A further set of slice simulations provides the new value of the rotor \(q\)-axis flux linkage; with skewing, the average value, considering each slice, has to be close to zero.