## 3 Influence of PM and stator Arcs on Torque Production
### Stator Arcs  
Figure 5 shows the influence of \(\theta_{st}\) and \(\theta_{t}\) on torque production for the multi-tooth HE-DSDPM. It is seen that the average torque increases with the \(\theta_{st}\) reduction between 34 and 57 degrees as a consequence of the behaviour of \(\theta_{sp}\) variation explained above, shown in Figure 5a. However, the \(\theta_{t}\) variation indicates a minimal impact on the average torque since the big stator pole of the proposed machine structure has suitable magnetic permeance to sufficiently contain the magnetic flux produced by both PMs. Figure 5b presents that the low ripple torque is acquired by the small \(\theta_{st}\) between 28 and 40 degrees, whereas the various \(\theta_{t}\) results in the inconsistent magnitude of the ripple torque due to unstable magnetic flux density at the air gap. Thus, the optimal stator arc candidate is \(\theta_{st}\) of 36 degrees and \(\theta_{t}\) of 23 degrees because it exists a suitable magnetic permeance and provides high average torque with low ripple torque.  
Figure 4: Influence of PM arcs on torque production for a multi-tooth HE-DSDPM: (**a**) average torque and (**b**) ripple torque.  
Figure 3: Parametric model of a multi-tooth HE-DSDPM.  
From the optimization results, it is concluded that the suitable PM and stator arcs are \(\theta_{sp}\) of 23 degrees, \(\theta_{sp}\) of 1.75 degrees, \(\theta_{st}\) of 36 degrees and \(\theta_{t}\) of 23 degrees. The multi-tooth HE-DSDPM using these suitable arcs can produce large average torque and low ripple torque because of its suitable symmetrical magnetic flux path with the best MMF and magnetic permeability. Thus, this proposed HE-DSDPM with the suitable PM and stator arcs is chosen as the optimal structure in this work. All design specifications of the optimal multi-tooth HE-DSDPM and the conventional HE-DSDPM are given in Table 2. Then, the electromagnetic performance of the optimal multi-tooth HE-DSDPM will be deeply analysed to compare it with conventional HE-DSDPM.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Main Parameter**} & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**Conventional**} & **Optimal** \\  & & **HE-DSDPM** & **Multi-Tooth** \\  & & [29] & **HE-DSDPM** \\ \hline Number of stator poles & pole & 12 & 6 \\ Number of stator tooth & teeth & - & 2 \\ Number of rotor poles & pole & 10 & 13 \\ Rate speed & rpm & & 1000 \\ Outer stator radius, \(R_{so}\) & mm & & 50 \\ Inner stator radius, \(R_{si}\) & mm & & 30 \\ Stator teeth radius, \(R_{st}\) & mm & & 34 \\ Outer rotor radius, \(R_{r0}\) & mm & & 29.5 \\ Inner rotor radius, \(R_{ri}\) & mm & & 10 \\ Air gap length & mm & & 0.5 \\ Stack length & mm & & 50 \\ Stator yoke PM height & mm & & 5 \\ Slot opening PM height & mm & & 4 \\ Rotor pole arc, \(\theta_{r}\) & degrees & 12 \\ Slot opening PM arc, \(\theta_{sp}\) & degrees & 11.5 & 23 \\ Stator yoke PM arc, \(\theta_{sp}\) & degrees & 3 & 1.75 \\ Stator tip arc, \(\theta_{st}\) & degrees & - & 36 \\ Stator teeth arc, \(\theta_{t}\) & degrees & 18.5 & 23 \\ Number of turns/coil & degrees & 28 & 56 \\ Slot area/coil & mm\({}^{2}\) & 121 & 264 \\ Steel type & - & Silicon steel 1010 \\ PM type & - & NdFeB—N35 \\ Magnetic flux remanent & T & 1.23 \\ Armature current & A & 8 \\ Current density & A/mm\({}^{2}\) & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Design specifications of optimal multi-tooth HE-DSDPM and conventional HE-DSDPM.  
Figure 5: Influence of stator arcs on torque production for a multi-tooth HE-DSDPM: (**a**) average torque and (**b**) ripple torque.