## 4 Losses Analysis
### Mechanical Loss  
As the rotor is supported by airfoil bearing, and there is no mechanical contact between bearing and shaft, windage loss is the main component of mechanical loss. Windage loss can be calculated by the method proposed in [23]:  
\[P_{wl}\ =\ kC_{f}\tau p\omega^{3}r^{4}l, \tag{4}\]  
where \(k\) is the roughness coefficient, and its value is 1.0 for smooth surfaces; \(\rho\) is the air density; \(\omega\) is rotor angular velocity; \(l\) is rotor axial length; and \(C_{f}\) is the friction coefficient, which depends on the cylinder radius of \(r\).  
The effect of tangential air flow as high-speed rotor rotation inside air gap can be expressed by Reynolds number  
\[R_{et}\ =\ \rho\omega r\delta/\mu, \tag{5}\]  
Figure 9: Eddy-current loss distribution in rotor at rated state (160 krpm). (**a**) M1; (**b**) M2; (**c**) M3; (**d**) M4.  
where \(\mu\) is air dynamic viscosity, \(\delta\) is the radial air gap length, and \(R_{et}\) is the Reynolds number.  
For axial flow with additional cooling gas inside the air gap, the Reynolds number is expressed as  
\[R_{en}\ =\ 2\rho Q\delta/\mu S_{ag}, \tag{6}\]  
where \(Q\) is the flow rate of the cooling air, and \(S_{ag}\) is the cross-sectional area of the air gap.  
The friction coefficient within different flow speed levels of cooling air is calculated according to measurements in [16]:  
\[C_{f}\ =\ 0.515\frac{(\delta/r)^{0.3}}{R_{et}{}^{0.5}},500<R_{et}<10^{4}, \tag{7}\]  
\[C_{f}\ =\ 0.0325\frac{(\delta/r)^{0.3}}{R_{et}{}^{0.2}},10^{4}<R_{et}, \tag{8}\]  
Windage loss induced by an additional axial flow of cooling air can be expressed as:  
\[P_{ax}\ =\ 0.67\pi\rho\Big{(}(r+\delta)^{3}-r^{3}\Big{)}v_{a}k_{a}r\omega^{2}, \tag{9}\]  
where \(v_{a}\) is air flow velocity, and \(k_{a}\) is the coefficient related to rotor roughness and stator surface, which was set to be 0.48 for the smooth surface. At a temperature of 100 \({}^{\circ}\)C and standard pressure of 0.1013 MPa, dynamic viscosity was \(1.88\times 10^{-5}\) Ns/m\({}^{2}\), air density was 1.29 kg/m\({}^{3}\), and cooling air flow velocity was 15 m/s.  
Windage loss is significantly influenced by cooling air velocity. Figure 10 shows rotor radius versus windage loss at different cooling air velocity levels. Windage loss with cooling air was higher than that without cooling air. As air velocity increases, windage loss also greatly increases. Therefore, for better performance at high-speed operation, appropriate axial air flow is required.  
On the basis of the above analysis, the loss distribution and efficiency of HSPMSMs with different rotor structures at the rated state were obtained, as shown in Table 3. For these HSPMSMs, core and copper losses were the main components, but windage loss cannot be ignored. The total loss and efficiency of M2 were superior to those of other HSPMSMs, which means that M2 achieved better performance from this aspect. Although the efficiency of M1 was high, its rotor loss was also the highest. The main reason is that its rotor loss density was the highest due to its titanium alloy sleeve, which is a major  
Figure 10: Windage loss under different cooling conditions.  
challenge for HS applications. The efficiency of M3 and M4 was not as good due to the higher core loss.