## 2 Theoretical Model of Loss Calculation and Thermal Analysis
### Comprehensive Loss Classification and Calculation Methods  
The accurate prediction and quantification of power losses in high-speed magnetic field-modulated motors for flywheel energy storage systems require a methodical classification framework and precise calculation methodologies. These losses can be systematically categorized into electromagnetic, mechanical, and additional losses, each with distinct physical mechanisms and mathematical representations, as shown in Table 2. The comprehensive assessment of these loss components is essential for thermal analysis, efficiency optimization, and reliability enhancement of the entire system [18]. Core losses represent a significant portion of the total electromagnetic losses and consist of three principal components: hysteresis, eddy current, and excess losses [19]. The modified Bertotti equation provides an effective mathematical framework for calculating these losses:  
\[P_{core}=P_{h}+P_{e}+P_{a}=k_{h}fB_{m}^{a}+k_{e}f^{2}B_{m}^{2}+k_{a}f^{1.5}B_{ m}^{1.5} \tag{1}\]  
where \(P_{h}\), \(P_{e}\), and \(P_{a}\) represent hysteresis, eddy current, and excess losses, respectively, with \(f\) denoting frequency, \(B_{m}\) signifying peak magnetic flux density, and \(k_{h}\), \(k_{e}\), \(k_{a}\), and \(\alpha\) serving as material-specific coefficients [20]. For high-frequency applications, this classical model requires further refinement to account for non-uniform flux distribution and magnetic saturation effects [21]. The coefficients kh, kc, and ke in Equation (1) were determined through standardized Epstein frame testing following IEEE Standard 393-1991 [22], with multiple excitation frequencies ranging from 50 Hz to 5 kHz to capture frequency-dependent effects accurately. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Loss Category** & **Loss Component** & **Physical Mechanism** & **Calculation Method** & **Reference** \\ \hline \multirow{3}{*}{Core Losses} & Hysteresis Loss & Magnetic domain wall movement & \(P_{h}=k_{h}fB_{m}^{a}\) & [14] \\  & Eddy Current Loss & Induced currents in core material & \(P_{e}=k_{e}f^{2}B_{m}^{2}\) & [13] \\  & Excess Loss & Micro-eddy currents & \(P_{a}=k_{a}f^{1.5}B_{m}^{1.5}\) & [13] \\ \hline \multirow{3}{*}{Copper Losses} & DC Resistive Loss & Joule heating in conductors & \(P_{cu,dc}=l_{mn}^{2}R_{dc}\) & [1] \\  & Skin Effect Loss & Current density redistribution & \(P_{skin}=k_{e}P_{cu,dc}\) & [15] \\ \hline \multirow{3}{*}{Mechanical Losses} & Proximity Effect Loss & Adjacent conductor interaction & \(P_{prox}=k_{p}P_{cu,dc}\) & [10] \\  & Bearing Friction Loss & Contact friction in bearings & \(P_{bearing}=0.5\mu F_{e}\omega d_{m}\) & [4] \\ \cline{1-1}  & Windage Loss & Air friction on rotating surfaces & \(P_{windage}=k_{w}\rho_{air}\omega^{3}r^{5}\) & [9] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comprehensive classification and calculation methods for losses in high-speed magnetic field-modulated motors. Copper losses in high-speed applications exhibit complex behavior due to skin and proximity effects, especially at elevated frequencies. The AC resistance factor \(k_{r}\) characterizes the ratio of AC to DC resistance:  
\[P_{cu}=k_{r}I_{rms}^{2}R_{dc}=k_{r}\rho l_{w}J^{2}V_{w} \tag{2}\]  
where \(\rho\) represents the conductor resistivity, \(l_{w}\) is the winding length, \(J\) is the current density, and \(V_{w}\) is the winding volume [2]. The AC resistance factor can be calculated using the Dowell equation or through finite element analysis for complex winding configurations [23]. Mechanical losses primarily consist of bearing friction and windage losses. In vacuum-sealed flywheel systems, bearing losses dominate and can be approximated by the following:  
\[P_{bearing}=0.5\mu F_{r}\omega d_{m} \tag{3}\]  
where \(\mu\) denotes the friction coefficient, \(F_{r}\) is the radial load, \(\omega\) is the angular velocity, and \(d_{m}\) is the bearing mean diameter [10].