## 6 Results and Discussion: Back-Emf  
### DE: Evaluation on a 24-Slot, 22-Pole PMSM and MPMSM  
In Figure 8, the back-emf 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 one full turn to correctly obtain the harmonic content of the back-emf signal, which is depicted in the figures. In Figure 8a,c, the three-phase back-emf waveforms are presented for a 24-slot, 22-pole MPMSM and PMSM, respectively. In turn, Figure 8b,d show the harmonic spectrum of the back-emf for both machines, modular and conventional, when DE is present. From Figure 8, it can be noted that the back-emf waveform is composed of a high magnitude fundamental component and a third-order harmonic component of low magnitude. In addition, it is evident that eccentricity does not have a significant impact on the back-emf magnitude, and no significant unbalance between phases is observed either, contrary to what was suggested in [30]. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(Q_{s}\)/\(2p\)ECC & \(HO_{NHC}\) & \(\Delta(Q_{s},2p)\) & \multicolumn{2}{c}{**0.25 mm**} & \multicolumn{2}{c}{**0.50 mm**} & \multicolumn{2}{c}{**0.75 mm**} & \multicolumn{2}{c}{**1.00 mm**} \\ \hline  & & & Mod [\%] & Mon [\%] & Mod [\%] & Mon [\%] & Mod [\%] & Mon [\%] & Mod [\%] & Mon [\%] \\
18S 12P & 72 & 6 & 0.19 & 0.22 & 0.54 & 0.66 & 1.17 & 1.31 & 2.05 & 2.19 \\
18S 20P & 180 & 2 & 28.92 & 24.87 & 117.38 & 107.98 & 260.77 & 244.69 & 462.91 & 439.94 \\
24S 20P & 120 & 4 & 0.41 & 0.83 & 1.52 & 2.89 & 4.63 & 7.72 & 11.22 & 17.86 \\
24S 22P & 264 & 2 & 12.31 & 16.10 & 67.51 & 87.25 & 178.98 & 210.45 & 373.15 & 417.45 \\
24S 28P & 168 & 4 & 0.55 & 0.59 & 1.46 & 2.72 & 4.37 & 9.48 & 10.14 & 25.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Relative cogging torque increase of different PMSM and MPMSM slot/pole combinations in the presence of SE. A significant back-emf unbalance was observed in [30] when evaluating eccentricity on a 9-slot, 8-pole PMSM with conventional stator core. This can be explained by analyzing the spatial distribution of phases of that 9-slot, 8-pole machine when compared to an 8-slot, 12-pole PMSM, which are represented in Figure 9a,b, respectively. Figure 8: FE evaluation of the back-emf in the presence of DE for a 24-slot, 22-pole PMSM and MPMSM: (**a**) back-emf waveform for the modular machine, (**b**) back-emf spectrum for the modular machine, (**c**) back-emf waveform for the monolithic machine, (**d**) back-emf spectrum for the monolithic machine. Figure 9: Schematics of (**a**) a 9-slot, 8-pole PMSM with conventional stator core subject to eccentricity. In this case, the phase windings form a single group of conductors (light blue oval) distributed in a specific part of the stator circumference and (**b**) an 18-slot, 12-pole PMSM with conventional stator core subject to eccentricity. In this case, the phase windings form six group of conductors (light blue oval) evenly distributed in the stator circumference, damping the potential back-emf unbalance. For the case of the 9-slot, 8-pole PMSM, when eccentricity is applied, the minimum airgap position can increase or decrease the back-emf magnitude of a phase depending on how close it is to the phase windings, since it can strengthen or weaken the linked flux. As can be seen from the schematics of Figure 9a, phase A is strengthened as the airgap in front of phase A is smaller, and the back-emf of phase B and C should be lower since the equivalent airgap is larger than the original. In that case, the phase windings are not evenly distributed in the stator circumference, which translates into an unbalanced back-emf waveform when eccentricity is present. By the contrary, in the case of the 18-slot, 12-pole machine shown in Figure 9b, the phase windings are forming six groups of conductors evenly distributed in the stator circumference.