## 5 Results and Discussion: Cogging Torque  
### DE: Evaluation of 24-Slot, 22-Pole PMSM and MPMSM  
In Figure 6, the cogging torque outcomes of the FE evaluation of a 24-slot, 22-pole PMSM and a 24-slot, 22-pole MPMSM are presented for DE. The results were extracted considering the rotor structure completes 360 mechanical degrees to correctly obtain the harmonic content of the cogging torque signal. However, only the main period (when eccentricity is present) is shown in the figure in order to provide a clearer visualization. In Figure 6a,c, the cogging torque waveforms are shown for a 24-slot, 22-pole MPMSM and PMSM respectively. In turn, Figure 6b,d present the harmonic spectrum of the cogging torque waveform for both machines, modular and conventional, when dynamic eccentricity (DE) is present.  
The main period of the cogging torque was found to be 1.36 mechanical degrees when no eccentricity is applied in both cases. This changes when DE is present: the main period changes to 15 degrees, which corresponds to the slot pitch. This can be explained since in conventional PMSMs, the cogging torque main period is defined by the periodicities of the stator and rotor structures when no eccentricity is present (native harmonic content) and is accounted by [34]:  
\[T_{\mathrm{T\;NHC}}=\frac{360}{\mathrm{LCM(Q_{s},2p)}}. \tag{2}\]  
This results in 1.36\({}^{\circ}\) for the 24-slot, 22-pole machine. However, when dimensional tolerances are affecting the machine, the stator/rotor periodicity breaks. For the case of DE, the minimum airgap is rotating with the rotor and the modules are stationary, and hence, the relative position between magnet symmetry axis and teeth symmetry axis is periodical and repeats each 360/\(Q_{\mathrm{s}}\) mechanical degrees. Therefore, the main cogging torque period should change to the additional harmonic components generated, given by:  
\[T_{T\;\mathrm{AHC\;DE}}=\frac{360}{Q_{\mathrm{s}}}, \tag{3}\]  
which happens to occur for the 24-slot, 22-pole machine and generates a significant increase in the peak-to-peak value of the cogging torque. The cogging torque after eccentricity  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(Q_{s}\)/\(2p\)ECC & \multicolumn{2}{c}{\(0\) mm} & \multicolumn{2}{c}{\(0.50\) mm} & \multicolumn{2}{c}{\(0.75\) mm} & \multicolumn{2}{c}{\(1.00\) mm} & \multicolumn{2}{c}{**Main HO**} \\ \hline  & Mod & Mon & Mod & Mon & Mod & Mon & Mod & Mon & Mod & Mon & Mod & Mon \\  & [N] & [N] & [N] & [N] & [N] & [N] & [N] & [N] & [N] & [N] & [-] & [-] \\
18S 12P & 0 & 0 & 4 & 2 & 12 & 4 & 23 & 6 & 36 & 8 & 12 & 12 \\
18S 20P & 0 & 0 & 81 & 83 & 160 & 165 & 236 & 244 & 307 & 318 & 20 & 20 \\
24S 20P & 0 & 0 & 2 & 1 & 5 & 3 & 10 & 6 & 20 & 12 & 20 & 20 \\
24S 22P & 0 & 0 & 11 & 12 & 22 & 24 & 37 & 40 & 55 & 60 & 22 & 22 \\
24S 28P & 0 & 0 & 1 & 0 & 2 & 1 & 3 & 2 & 7 & 5 & 28 & 28 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Radial force ripple for different slot/pole combinations of PMSM and MPMSM in the presence of SE.  
is therefore comprised of the native harmonic components (faultless) and the additional harmonic components (AHC) as per [34]:  
\[T_{\text{cogg}}(\alpha)=T_{\text{NHC}}(\alpha)+T_{\text{AHC}}(\alpha) \tag{4}\]  
From Figure 6, it can be noted that significant cogging torque components with \(\text{HO}=24\) and their multiples are generated with DE, which results in a peak-to-peak value increase of up to 400%. Nevertheless, this does not hold true for all evaluated slot/pole combinations.  
In summary, relevant features of the cogging torque results can be summarized by the following indicators, useful for evaluating and comparing the different slot/pole combinations of Table 2.  
* NHC of the cogging torque, measured in Nm;
* AHC of the cogging torque, measured in Nm;
* AHC of the cogging torque, measured in percentage of the NHC.