## 5 Thermal Analysis  
Improving the cooling method is an effective way to improve machine performance. In this paper, an efficient cooling method was adopted to improve the performance of HSPMSMs, as shown in Figure 11. A spiral waterway was used in the cooling water loop installed in the stator shell. The airway was set in the stator slots with axial ventilation. Table 2 shows the heat conductivity coefficients of the adopted materials in these machines. Ambient temperature was set to be 20 \({}^{\circ}\)C. As radiation heat translation slightly influences rotor temperature in nonvacuum environments, its effect was ignored in this paper. According to heat transfer theory, the convective heat transfer coefficient between rotor surface and air gap of the machines could be calculated with  
\[c_{a}\ =\ \frac{N_{u}\lambda}{\delta}, \tag{10}\]  
where \(c_{a}\) is the heat transfer coefficient between rotor surface and air gap; \(\lambda\) is the air thermal conductivity; and \(N_{u}\) is the Nusselt number, which is a dimensionless number to represent the strength of convective heat transfer. Considering the HS rotation of the rotors, air flow in the air gap is accelerated in the tangential direction. The Nusselt number can be calculated with the following experimental correlation:  
\[N_{u}\ =\ \frac{0.3387R_{ea}{}^{1/2}{P_{r}}^{1/3}}{\left[1+\left(0.0468/P_{r} \right)^{2/3}\right]^{1/4}}, \tag{11}\]  
where \(P_{r}\) is the air Prandtl number at room temperature, which was set to be 0.692; and \(R_{ea}\) is the Reynolds number. According to Equations (10) and (11), the convection heat transfer  
Figure 11: Cooling method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Loss Distribution** & **M1** & **M2** & **M3** & **M4** \\ \hline Core loss (W) & 212.34 & 115.87 & 237.54 & 242.12 \\ Copper loss (\(W\)) & 202.58 & 208.05 & 208.05 & 208.05 \\ Windage loss (W) & 37.25 & 58.24 & 97.6 & 97.6 \\ Rotor loss (W) & 45.12 & 15.11 & 5.9 & 18.95 \\ Total loss (W) & 495.73 & 397.27 & 594.09 & 566.72 \\ Efficiency & 97.39\% & 97.91\% & 97.11\% & 97.01\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Loss distribution of HSPMSMs with different rotors. coefficient for the waterway in shell and the axial ventilation was 4058 and 651 W/m\({}^{2}\)/K, respectively. To reduce the risk of irreversible PM demagnetization, the thermal field of these HSPMSMs was analyzed. The permissible maximal working temperature of the adopted PM needed to be below 180 \({}^{\circ}\)C. Therefore, for safety operation, the maximal operation temperature was limited to no more than 150 \({}^{\circ}\)C. The temperature distribution of these rotors at the rated load state were obtained on the basis of FEM, as shown in Figure 12. All these HSPMSMs were within the allowable temperature increase, and temperature distribution was generally consistent with loss distribution. M1 had the highest temperature increase due to its highest loss, while M3 had the lowest temperature increase. Maximal temperature that occurred at the end of PM in M4 was 123 \({}^{\circ}\)C.  
The electromagnetic properties of PMs are highly sensitive to temperature, and temperature increase can cause irreversible demagnetization. Therefore, it is important to compare the temperature-dependent properties of different HSPMSMs. The \(B\)-\(H\) curve was derived from [24]. The permeability of the PM recoil line at any temperature could be calculated with a series of discrete data at reference temperature \(T_{0}\):  
\[\mu(T)\ =\ \frac{\partial B_{i}(T,H)}{\partial H}\bigg{|}_{H=0}+\mu_{0}\ =\ \frac{P(T)}{Q(T)}\mu_{i}(T_{0})+\mu_{0}, \tag{12}\]  
Figure 12: Temperature distribution on rotor of HSPMSMs at rate load. (**a**) M1; (**b**) M2; (**c**) M3; (**d**) M4.