## 5 Results and Discussion: Cogging Torque
### DE: Comparison of Slot/Pole Combinations for PMSMs and MPMSMs
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(Q_{s}\)/\(2p\)ECC & HO\({}_{\text{NHC}}\) & 0 mm (Faultless) & 0.25 mm & 0.50 mm & 0.75 mm & 1.00 mm \\ \hline  & & Mod & Mon & Mod & Mon & Mod & Mon & Mod & Mon & Mod & Mon \\  & & [Nm] & [Nm] & [Nm] & [Nm] & [Nm] & [Nm] & [Nm] & [Nm] & [Nm] \\
18S 12P & 72 & 19.11 & 22.42 & 19.13 & 22.44 & 19.17 & 22.46 & 19.24 & 22.49 & 19.34 & 22.54 \\
18S 20P & 180 & 0.63 & 0.66 & 0.62 & 0.65 & 0.59 & 0.62 & 0.54 & 0.56 & 0.47 & 0.48 \\
24S 20P & 120 & 3.92 & 2.70 & 3.91 & 2.71 & 3.88 & 2.77 & 3.84 & 2.86 & 3.83 & 2.99 \\
24S 22P & 264 & 0.27 & 0.25 & 0.27 & 0.25 & 0.26 & 0.25 & 0.25 & 0.24 & 0.24 \\
24S 28P & 168 & 2.28 & 0.72 & 2.28 & 0.73 & 2.30 & 0.74 & 2.34 & 0.77 & 2.40 & 0.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Cogging torque NHC of different PMSM and MPMSM-slot/pole combinations in the presence of DE. * If the additional flux density due to eccentricity has a component of spatial order \(l\) resulting from different values of \(n\) and \(v\), then eccentricity can also contribute to the cogging torque (for instance, if \(l=n_{1}\pm v_{1}=n_{2}\pm v_{2}\));
* In the case of machines having \(2p=Q_{s}\pm 2\) (18-slot, 20-pole and 22-slot, 24-pole), the interaction between spatial harmonics from \(n\) and \(n\pm 2\) generate the AHC; in the case of machines that have \(2p=Q_{s}\pm 4\) (24-slot, 20-pole and 28-slot, 24-pole), the interaction between spatial harmonics from \(n\) and \(n\pm 4\) generate the AHC; in the case of machines that have \(2p=Q_{s}\pm 6\) (18-slot, 12-pole), the interaction between spatial harmonics from \(n\) and \(n\pm 6\) generate the AHC. It can be noted that several field harmonic components interact in machines with \(2p=Q_{s}\pm 2\), and interacting harmonic components decrease as the difference between the slot number and the pole number is higher. The result of this analysis is that slot/pole combinations with the slot number being close to the pole number (\(2p=Q_{s}\pm 2\)) should have a more significant effect of eccentricity on cogging torque that machines with it slot number very different from its pole number (\(2p=Q_{s}\pm 4\) and \(2p=Q_{s}\pm 6\)). This is verified from the results presented in Table 10.  
The results of the 18-slot, 12-pole machines versus that of the 24-slot, 28-pole machines may draw attention since they do not exactly follow this tendency: the 18-slot, 12-pole machine has higher absolute AHC than the 24-slot, 28-pole machine, but it has a higher difference between the slot count and the pole count. However, the analysis must consider the proportion between the AHC generated by eccentricity and the NHC. From Table 9, it can be recalled that the 18-slot, 12-pole machine has a significantly higher cogging torque NHC than the 24-slot, 28-pole machine, which translates into AHC being proportionally lower than that of the 24-slot, 28-pole machine. Table 11 is created to account for this proportion and presents the relative cogging torque increase of the different slot/pole combinations in the presence of DE. It may be appreciated that the relative cogging torque increase effectively depends on the difference between the slot count and the pole count, as indicated in the spatial field harmonic analysis. Cogging torque of modular and monolithic machines have a similar response to eccentricity, although some slot/pole combinations have mild differences due to the break of stator/rotor periodicity, as indicated in [18].