## 2 Analytical Calculation Procedure
### Calculation of the Electromagnetic Torque  
The electromagnetic torque is created due to the electromagnetic coupling between the permanent magnet field and the armature fields [28].  
First of all, the distribution of the winding conductors is defined according to the star of slots method [29]. Employing the winding periodicity (\(t_{p}\)), which is the greatest common divisor (GCD) of the number of pole pairs and the number of slots, the number of pole pairs in a periodicity (\(p^{\prime}\)) and the number of slots in a periodicity (\(Q^{\prime}_{\rm S}\)) are calculated.  
\[p^{\prime}=\frac{p}{t_{p}} \tag{7}\]  
\[Q^{\prime}_{\rm S}=\frac{Q_{\rm S}}{t_{p}} \tag{8}\]  
Then, a star can be drawn, which has as many vectors as \(Q^{\prime}_{\rm S}\), and those vectors are enumerated according to \(p^{\prime}\). The conductors are placed following the sequence A+ /C\(-\)/B+ /A\(-\) /C+ /B\(-\). For example, Figure 2 shows the star of slots for a machine with 8 poles and 48 slots.  
Based on the star of slots drawn for a machine, the winding factor (\(K_{\rm w}\)) can be calculated, which represents the effectiveness of a coil in producing magnetomotive force. The harmonics of the winding factor can be calculated according to (9):  
\[K_{\rm wk}=\zeta_{k}\mathrm{e}^{-\mathrm{i}kt_{p}\gamma_{k}} \tag{9}\]  
where \(\zeta_{k}\) and \(\gamma_{k}\) are the amplitude and phase of order \(k\). For an ideal machine, the amplitude is unitary and the phase is kept constant between all the coils.  
Using the winding factor, the conductor distribution per phase (C) can be obtained, as shown in (10) for phase \(A\):  
\[C_{A}(\theta)=\sum_{k=-\infty}^{\infty}\frac{4 cN}{\pi ak}\sin\left(\frac{nt_{ p}\kappa}{2}\right)K_{\rm wk} \tag{10}\]  
where \(c\) is the number of coils per phase, \(N\) is the number of turns per coil, and \(\alpha\) is the pitch angle of the coils.  
Figure 2: Example of a star of slots, from [29].  
Then, the magnetomotive force (\(F\)) is calculated by means of (11):  
\[F_{A}(\theta)=\int\limits_{0}^{2\pi/p}C_{A}(\theta)\mathrm{d}\theta \tag{11}\]  
The flux linkage (\(\Psi\)) can be calculated in the function of the magnetic flux density (which is calculated as explained in Section 2.1) and the magnetomotive force, as shown in (12):  
\[\Psi_{\mathrm{A}}(t)=\frac{1}{2}pD_{\mathrm{si}}L_{\mathrm{s}}\int\limits_{0}^ {2\pi/p}B_{\mathrm{s}}(t,\theta)F_{\mathrm{A}}(\theta)\mathrm{d}\theta \tag{12}\]  
where \(D_{\mathrm{si}}\) and \(L_{\mathrm{s}}\) are the internal diameter and the length of the stator, respectively.  
Then, the electromotive force (\(E\)) can be calculated by means of (13):  
\[E_{A}(t)=\frac{\mathrm{d}\Psi_{\mathrm{A}}(t)}{\mathrm{d}t} \tag{13}\]  
Finally, the electromagnetic torque (\(T_{\mathrm{em}}\)) can be calculated considering the electro-motive force and the current (\(I\)) of the three phases, as expressed in (14):  
\[T_{\mathrm{em}}=\frac{E_{A}I_{A}+E_{B}I_{B}+E_{C}I_{C}}{\omega_{\mathrm{m}}} \tag{14}\]  
The total mechanical torque of the machine is composed of the electromagnetic torque and the cogging torque. However, this work is focused on the mean value of the total torque, which is not affected by the cogging torque. Therefore, the electromagnetic torque is the only component that is considered in the analytical development.