## 4 Electromagnetic Performance Analysis
### Torque Production Analysis  
The torque production of both proposed structures, consisting of the cogging torque and electromagnetic torque, is investigated. The cogging torque of the proposed HE-DSDPM structures excited by different \(I_{dc}\) is shown in Figure 9. The low cogging torque represents low tension between the magnet and rotor at the air gap while the electrical machine is started. The multi-tooth HE-DSDPM with all field currents is observed to have lower cogging torque than the conventional HE-DSDPM. Their deck-to-peck cogging torques are about 0.081, 0.064, and 0.238 Nm found at \(I_{dc}\) of \(-\)5, 0, and 5 A, respectively. This behaviour is exhibited that the multi-tooth machine indicates a lower starting tension because of the suitable magnetic flux path at the air gap from the structural multi-tooth modification with PM and stator arcs optimization.  
Figure 8: Open-circuit back-EMF of the proposed HE-DSDPM with various \(I_{dc}\): (**a**) waveform and (**b**) harmonic spectra.  
Figure 10a indicates the electromagnetic torque of the multi-tooth and conventional HE-DSDPM structures with \(I_{dc}\) variation. The \(I_{ac}\) of 8 A and current density of 2 A/mm\({}^{2}\) was fixed as electrical constraints in the load conditions. Based on no field current, the HE-DSDPM with a multi-tooth structure produces an electromagnetic torque of about 3.04 Nm and is especially 26.18% higher than the HE-DSDPM with a conventional structure because of the better back-EMF characteristic. The ripple torque of the multi-tooth HE-DSDPM has a minimum value of about 3.25% and is 4% less than that of the conventional one due to a lower THD percentage and cogging torque value.  
As shown in Figure 10b, the investigation of the average torque in the HE-DSDPM structures is proposed. It is seen that the variance average torque percentage of a multi-tooth HE-DSDPM with \(I_{dc}\) of +5 A indicates a 16.43% improvement from that of conventional HE-DSDPM. For \(I_{dc}\) of \(-5\) A excitation, the proposed structure shows a 20.73% lower variance average torque percentage when compared to the conventional structure. It is also represented that the multi-tooth HE-DSDPM has the smallest ripple torque with a minimum vibration of the machine structure. As described in the magnetic flux distribution section, it indicates that the strengthening flux distribution excited by \(I_{dc}\) of +5 A can produce the average torque improvement, while the wakening flux distribution excited by \(I_{dc}\) of \(-5\) A can produce minimum ripple torque for the HE-DSDPM with a multi-tooth structure. Therefore, the structural multi-tooth modification in HE-DSDPM can generate better electromagnetic torque, the lowest ripple torque and the best variance average torque percentage at all \(I_{dc}\).  
From the results, it can be concluded that the HE-DSDPM can generate better overall electromagnetic performance by using a structural multi-tooth modification since it enhances the magnetic flux path and the flux regulation. When compared to the conventional HE-DSDPM at \(I_{dc}\) of 0 A, it is found that the multi-tooth HE-DSDPM structure indicates a 24.54% higher back-EMF with about 3% lower THD percentage. Especially, the average torque provided by the proposed structure is 26.18% larger and 4% smaller ripple torque than the conventional one because of better back-EMF characteristics and the low cogging torque magnitude. Moreover, the multi-tooth HE-DSDPM with excited by \(I_{dc}\) of \(-5\) and +5 A exhibits the largest variance percentage of torque production than conventional HE-DSDPM because of the greater flux regulation quality. Hence, structural multi-tooth modification may be another important way to improve torque production in the other DSPM categories.  
Figure 9: Cogging torque of the proposed HE-DSDPM with various \(I_{dc}\).