## 3 Analytical Method Derivation for Discrete Skewing  
The main period of torque ripple of three-phase winding machines is 60 electrical degrees [39], which therefore dictates the period of the torque ripple harmonic of order \(v\), derived in electrical degrees by:  
\[\mathrm{T}_{v,\mathrm{elec}}=\frac{360^{\circ}}{v} \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Stator outer diameter & \(D_{\mathrm{so}}\) & 245 & mm \\ Stator inner diameter & \(D_{\mathrm{si}}\) & 161.4 & mm \\ Rotor outer diameter & \(D_{\mathrm{ro}}\) & 160.4 & mm \\ Rotor inner diameter & \(D_{\mathrm{ri}}\) & 70 & mm \\ Tooth height & \(h_{\mathrm{t}}\) & 22.8 & mm \\ Tooth width & \(b_{\mathrm{t}}\) & 9 & mm \\ Air-gap length & \(g\) & 0.5 & mm \\ Stack length & \(l_{\mathrm{st}}\) & 120 & mm \\ Turns per slot & \(N_{\mathrm{s}}\) & 20 & - \\ Number of slots & \(Q_{\mathrm{s}}\) & 36 & - \\ Speed & \(n\) & 3000 & rpm \\ Current Density & \(J\) & 10 & A/mm\({}^{2}\) \\ Current angle & \(a_{\mathrm{i}}^{\mathrm{e}}\) & 60 & electric degrees \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main data of the selected machines. Figure 3: Three-dimensional schematics of (**a**) four-pole synchronous reluctance motor with two barriers per pole; and (**b**) six-pole synchronous reluctance motor with two barriers per pole. Three-phase stator windings are highlighted in red, green, and blue, corresponding to each phase. Rotor in both machines is shown as an exploded view. The aim of the discrete skewing is to consider \(N\) machine slices, rotated with respect to each other by a specific angle, so that each slice contributes to different torque harmonics that will ultimately modulate the torque waveform. This superposition has to be adjusted with the aim of mitigating undesired harmonic components of the resulting torque waveform. The \(v\)-th order harmonic of the torque ripple can be expressed as a term of the Fourier series expansion of the electromagnetic torque as:  
\[\mathrm{T_{ripple,v}}(\theta_{r,e})=A_{v}\cdot\cos\left(\frac{\pi}{180}\theta_ {r,e}v+\phi_{v}\right), \tag{2}\]  
where \(A_{v}\) is the amplitude of the \(v\)-th harmonic of the torque ripple, \(\theta_{r,e}\) is the rotor position in electrical degrees, \(\phi_{v}\) is the phase shift of the torque waveform, and \(v=1,2,3\),... In this sense, and following the concept of balanced multiphase systems, if the \(w\)-th harmonic of the torque ripple wants to be mitigated, then the electrical angle between each one of the N slices of the machine is proposed as:  
\[\theta_{w,\mathrm{elec}}=\frac{360^{\circ}}{Nw} \tag{3}\]  
For example, if a two-step skew is adopted, then the machine rotor will be comprised of two halves, each one contributing with torque waveforms that are shifted one with respect to the other. If the electrical shift is \(\theta_{w,\mathrm{elec}}\), then the positive semi-cycles of the \(w\)-th harmonic of the torque ripple of one half of the rotor will compensate the negative semi-cycles of the other half, hence mitigating the undesired component. The electrical angle between the \(N\) slices of the machine is translated into the skew angle (\(\theta_{\mathrm{s}}\)), which corresponds to the mechanical angle in which two consecutive rotor slices are rotated one with respect to the other so as to mitigate the \(w\)-th order harmonic of the torque ripple (Equation (4)). \[\theta_{\mathrm{s}}=\frac{360^{\circ}}{pNw} \tag{4}\]  
In Figure 4, the proposed methodology is schematized, comprising the decomposition of the electromagnetic torque waveform in harmonic components, the selection of a high-magnitude undesired component, and the calculation of \(\theta_{\mathrm{s}}\) to mitigate that undesired harmonic depending on the adopted number of slides \(N\).