## 2 Theoretical Model of Loss Calculation and Thermal Analysis
### Comprehensive Loss Classification and Calculation Methods
As indicated in Table 2, windage losses become significant at atmospheric pressure but are substantially reduced in vacuum environments [24]. These loss calculation formulas incorporate several assumptions that define their applicable ranges and potential limitations. The modified Bertotti Equation (1) assumes that the material's hysteresis behavior follows a power law relationship with flux density, which is valid for flux densities below 1.7 T, where saturation effects remain moderate. Beyond this threshold, the equation may underestimate hysteresis losses by up to 15% as verified through our material testing. The AC resistance calculation in Equation (2) assumes uniform current distribution within conductor bundles and neglects the strand-level circulating currents that may occur in complex winding configurations. This simplification introduces a maximum error of 8% at the highest operating frequency (5 kHz), based on comparison with detailed finite element conductor models. For mechanical loss estimation, Equation (3) assumes laminar flow conditions in the bearing lubricant and a linear relationship between the friction coefficient and load, which holds valid for the operating temperature range of 20-120 \({}^{\circ}\)C. All data processing steps applied to the simulation results used standard techniques, including 5-point moving average smoothing for noise reduction in time domain waveforms and fast Fourier transform with Hanning window for harmonic analysis. These processing methods preserve the fundamental characteristics of the data while improving visualization clarity. Additional losses encompass stray load losses, magnetic wedge losses, and harmonic-induced losses that manifest during high-speed operation [25]. These losses, while individually smaller in magnitude, collectively constitute a non-negligible portion of the total loss profile and require specialized analytical or numerical methods for accurate quantification [26]. The comprehensive calculation of all loss components, as summarized in Table 2, provides the foundation for thermal analysis and efficiency optimization in high-speed magnetic field-modulated motors for flywheel energy storage applications. Loss calculations were implemented using specialized formulations for each loss component. Core losses were calculated using the modified Bertotti equation with frequency-dependent coefficients as shown in Equation (1), with finite element field solutions providing spatially resolved flux density distributions at each time step. Copper losses incorporated both DC resistance effects and AC effects using the Dowell equation (Equation (2)) with strand-level eddy current modeling for capturing proximity effects in the end-winding regions. Permanent magnet and modulator eddy current losses were calculated directly from induced current densities in conducting regions using \(\mathrm{J}^{2}/\sigma\) integration. Mechanical losses, including bearing friction and windage losses, were calculated using analytical expressions (Equation (3)) calibrated against experimental measurements at different vacuum levels and rotation speeds. This comprehensive loss calculation approach was validated through comparison with direct calorimetric measurements, demonstrating agreement within 4.2% across the entire operating range.