## 5 Results and Discussion: Cogging Torque
### DE: Comparison of Slot/Pole Combinations for PMSMs and MPMSMs  
In Table 9, the natural harmonic component of cogging torque for different DE magnitudes is presented. Both the modular and monolithic machines are evaluated considering different slot/pole combinations. From Table 9, it may be noted that the cogging torque magnitude is very different depending on the slot/pole combination and it does not have a clear sedation with respect to the slot count or pole count separately. In this regard, it  
Figure 6: FE evaluation of cogging torque in the presence of DE for the 24-slot, 22-pole PMSM and MPMSM: (**a**) cogging torque waveform for the modular machine, (**b**) cogging torque spectrum for the modular machine, (**c**) cogging torque waveform for the monolithic machine, (**d**) cogging torque spectrum for the monolithic machine. is commonly accepted in faultless machines to consider the cogging torque HO (HO\({}_{\text{NHC}}\)) as a comparative indicator for cogging torque magnitude ([35], see (2)). The higher the HO, the lower the main period of the cogging torque and the lower its magnitude should be. As can be seen from Table 9, the NHC of the cogging torque is barely affected by DE, and HO\({}_{\text{NHC}}\) is able to compare slot/pole combinations regardless of the eccentricity magnitude. It may draw the attention that a significant difference in the NHC magnitude can be observed for the 24-slot, 20-pole and the 24-slot, 28-pole machines when comparing the modular and the monolithic machine. This is related to a stator/rotor periodicity break generated by the modular structure of U-shape MPMSMs. In those cases, HO\({}_{\text{NHC}}\) should be corrected to include the periodicity provided by the modular stator core. As an example, in the case of the 24-slot, 28-pole machine, the main period of the cogging torque is 84 for the modular machine, which explains the difference regarding the monolithic machine. In Table 10, the peak-to-peak value of the additional harmonic components of cogging torque for different DE magnitudes is summarized. Both the modular and monolithic machines are evaluated considering different slot/pole combinations. From Table 10, it is clear that the generated AHC of cogging torque is very different from a one slot/pole combination to another. Nevertheless, and contrary to what was observed for NHC, the AHC does not scale with HO\({}_{\text{NHC}}\). This can be explained by analyzing airgap flux density. As developed in [33], a simple analytical model of the additional airgap flux density radial and circumferential components accounting for the eccentricity can be expressed by:  
\[B_{\text{r}_{\text{r}_{\text{r}_{\text{r}_{\text{r}_{\text{r}_{\text{r}_{ \text{r}_{\text{r}_{\text{r}_{\text{r}_{\text{r}_{\text{r}_{\text{r}_{ \text{r}}}}}}}}}}}}}}}}=B_{\text{sr}}\lambda_{\text{e}cc} \tag{5}\]  
\[B_{\text{sr}}=B_{\text{sr}}\lambda_{\text{e}cc} \tag{6}\]  
where \(B_{\text{sr}}\) and \(B_{\text{sr}\alpha}\) are the radial and circumferential air flux densities (without the presence of eccentricity) with harmonic order \(n\), and \(\lambda_{\text{e}cc}\) is the real component of the equivalent complex permeance that represent the influence of eccentricity given by:  
\[\lambda_{\text{e}cc}=1+\sum_{v}\lambda_{v}\cos(v\alpha-\chi v\omega t) \tag{7}\]  
where \(\lambda_{v}\) is the magnitude of the \(v\)-th order component of equivalent permeance representing the eccentricity and \(\chi\) allows to select between SE (\(\chi=0\)) and DE (\(\chi=1\)). Since the stator slotting effect generates field harmonics of order \(n=mp\pm\mu\mathcal{Q}_{s}\), then the additional field harmonics due to eccentricity are \(n\pm v\). By virtue of this, there are two scenarios in which eccentricity have an impact on the cogging torque:  
* If both the additional flux density due to eccentricity and the flux density without eccentricity share harmonic spatial orders, eccentricity has an influence on the cogging torque;