## 2 Structural Multi-Tooth Modification of HE-DSDPM  
As shown in Figure 1a, the conventional HE-DSDPM with 12/10 (stator teeth/rotor pole) was initially proposed by [PERSON] [33]. Regarding the outstanding properties of this conventional structure as mentioned in the introduction, the structural modification of HE-DSDPM using a multi-tooth technique for torque production enhancement is presented, as inspired by [PERSON] [41].  
Based on the HE with dual-PM properties, even with the structural modification, its appearance resembles the HE-DSDPM. The dual-PM comprises the slot opening PM and the stator yoke PM as the source of the magnetic field in the machine. The slot opening PM acts as the main source, while the stator yoke PM regulates the magnetic flux production. The armature winding is utilized for excitation by a dc-biased sinusoidal current, divided into the ac-armature current for the production of the rotating armature field and the dc-excited current for regulation of the magnetic flux. The three-phase dc-biased sinusoidal current is expressed by  
\[\begin{array}{l}I_{A1}=\sqrt{2}I_{ac}sin(2\pi ft)+I_{dc}\\ I_{A2}=\sqrt{2}I_{ac}sin(2\pi ft)-I_{dc}\\ I_{B1}=\sqrt{2}I_{ac}sin(2\pi ft-2\pi/3)+I_{dc}\\ I_{B2}=\sqrt{2}I_{ac}sin(2\pi ft-2\pi/3)-I_{dc}\\ I_{C1}=\sqrt{2}I_{ac}sin(2\pi ft-4\pi/3)+I_{dc}\\ I_{C2}=\sqrt{2}I_{ac}sin(2\pi ft-4\pi/3)-I_{dc}\\ \end{array} \tag{1}\]  
where \(I_{ac}\) is the root-mean-square, rms, of the ac-armature current, \(I_{dc}\) is the dc-excited current, \(f\) is the electrical frequency.  
In this work, the stator/rotor combination of the HE-DSDPM is modified for the addition of a multi-tooth technique. The possibility of a stator/rotor combination can be expressed by  
\[N_{r}=(n\times N_{s})+1 \tag{2}\]  
where \(N_{r}\) is the number of rotor poles, \(N_{s}\) is the number of stator poles, and \(n\) is the number of stator teeth.  
Figure 1b shows the cross-section perspective of the HE-DSDPM with the multi-tooth structure. Its structure consists of the 6 big stator poles, \(N_{s}\) = 6, with 2 stator teeth, \(n\) = 2 and there are 13 salient rotor poles, \(N_{r}\) = 13. This stator/rotor combination has been claimed to produce strong electromagnetic performance because it shows a suitable magnetic flux path and high-frequency operation [41]. The PM number is reduced by half while the PM size is changed for the suitability of the structure. The three-phase concentrated armature winding is wound at the big stator pole. It can be seen that the slot winding area of the multi-tooth HE-DSDPM structure is enlarger than the conventional HE-DSDPM structure for better excitation by the dc-biased sinusoidal current. Consequently, the number of turns of the winding can be doubled compared to the conventional one.  
Therefore, the HE-DSDPM with the structural multi-tooth modification can illustrate a magnetic flux path improved by the proposed stator/rotor combination, magnetic resource improved by the adapted PM characteristics and flux regulation improved by the enlarged winding slot area. Then, the influence of PM and stator arcs, including \(\theta_{sp}\), \(\theta_{yp}\), \(\theta_{st}\) and \(\theta_{t}\), on torque production will be further studied since they relate to the multi-tooth modification in the HE-DSDPM.