## 3 Finite Element Analysis of Loss Distribution
### Impact of Operating Speed on Loss Characteristics  
The loss-speed relationship curves presented in Figure 4 were obtained through extensive parametric simulations across the entire speed range (0-30,000 rpm) with 2000 rpm increments. For each operating point, steady-state electromagnetic simulations were performed with a time-step corresponding to 1\({}^{\circ}\) of mechanical rotation to ensure accurate capture of high-frequency harmonics. The raw simulation data were processed using synchronous demodulation to eliminate numerical noise while preserving the physical characteristics of loss variation. The power law relationships between losses and speed were established through least-squares regression analysis, with goodness-of-fit metrics exceeding R\({}^{2}\) > 0.98 for all components, confirming the validity of the derived scaling exponents. The minor discrepancies from theoretical scaling factors (e.g., \(\omega^{1.8}\) versus theoretical \(\omega^{2}\) for core losses) were confirmed through targeted experiments at selected operating points, validating that these deviations represent physical phenomena rather than numerical artifacts.  
The relationship between operating speed and loss characteristics in magnetic field-modulated motors exhibits complex nonlinear behavior that significantly impacts system efficiency and thermal management requirements. Analysis reveals distinctive scaling  
Figure 4: Impact of operating speed on loss characteristics.  
patterns across various loss components that dictate the overall performance of flywheel energy storage systems. Core losses demonstrate a speed dependency of approximately \(\omega^{1.8}\) rather than the theoretical \(\omega^{2}\) relationship due to material saturation effects at higher flux densities and frequencies. This deviation from classical theory is attributed to microstructural changes in soft magnetic materials under high-frequency excitation. Permanent magnet eddy current losses exhibit even more pronounced scaling behavior, increasing at approximately \(\omega^{2.3}\) due to the amplification of high-order spatial harmonics that penetrate deeper into conductive regions at elevated speeds. Notably, the relative contribution of these components shifts dramatically across the operating range, with core losses dominating at 57% of total losses during high-speed operation compared to just 28% at base speed. Copper losses follow a more complex pattern--initially decreasing with speed in the field-weakening region but subsequently increasing at very high speeds due to enhanced proximity effects. Mechanical losses, though reduced in vacuum environments, still scale approximately as \(\omega^{2.8}\) and account for up to 12% of total losses at maximum speed. As illustrated in Figure 5, the composite loss distribution creates a pronounced efficiency curve, with peak values occurring near rated speed and significant degradation at both extremes. The differential scaling behavior creates distinct thermal challenges, particularly at modulator interfaces where loss density increases by 45% between moderate and high-speed operation. These findings highlight the importance of component-specific loss analysis rather than simplified scaling models when designing thermal management systems for wide-speed-range flywheel energy storage applications.  
Figure 5: Loss density distribution along axial position at different speeds.