## 3 Finite Element Analysis of Loss Distribution
### Comparative Analysis of Loss Distribution in Different Components  
The finite element analysis reveals distinct patterns of electromagnetic losses across the magnetic field-modulated motor components, with significant implications for thermal management and efficiency optimization. Figure 3 presents a comprehensive visualization of the loss distribution patterns across various components of the high-speed magnetic field-modulated motor used in flywheel energy storage systems. The total electromagnetic loss distribution shown in Figure 3a reveals significant spatial non-uniformity, with peak loss densities of 5.5 \(\times\) 10\({}^{5}\) W/m\({}^{3}\) occurring predominantly in the modulator region due to the intense flux modulation effect. Figure 3b illustrates the permanent magnet eddy current  
Figure 2: Finite element model of high-speed magnetic field modulated motor: (**a**) three-dimensional solid model with detailed representation of stator, rotor, permanent magnets, and flux modulators for electromagnetic–thermal coupled analysis; (**b**) cross-sectional view of finite element mesh showing adaptive element refinement in critical regions including air gap and magnetic interfaces.  
losses, which exhibit pronounced concentration at the magnet edges and corners where flux density gradients are steepest, with localized hotspots forming at regions aligned with modulator teeth. The modulator component losses depicted in Figure 3c demonstrate the highest magnitude among all components, constituting approximately 45% of total system losses at rated speed, with distinctive periodic patterns corresponding directly to the modulator pole geometry. Armature winding losses visualized in Figure 3d show a complex distribution characterized by slot-specific concentration patterns and significant axial variation, with end-region losses exceeding central-region values by 42% due to enhanced three-dimensional field effects. The end-region loss concentration illustrated in Figure 3e highlights the pronounced axial non-uniformity, with substantially higher loss densities at the axial extremities where magnetic flux fringing creates complex eddy current paths in both active and structural components. The axial cross-sectional view in Figure 3f further confirms this axial variation, demonstrating that two-dimensional analyses would substantially underestimate total losses by neglecting these end effects. Comparative analysis reveals a distinct operational speed dependency, with permanent magnet losses increasing more rapidly than other components due to their approximately quadratic relationship with frequency, suggesting that they become dominant at extreme speeds. These spatially resolved loss distributions provide essential insights for targeted thermal management strategies, particularly for the modulator components and permanent magnets where cooling pathways are most constrained, and for optimized material selection to enhance system efficiency across diverse operating conditions.  
Figure 3: Comparative visualization of loss distribution in magnetic field-modulated motor: (**a**) total electromagnetic loss distribution showing peak values in modulator region; (**b**) permanent magnet eddy current losses with concentration at magnet edges; (**c**) modulator component losses with distinctive periodic patterns; (**d**) armature winding losses showing slot-specific concentration; (**e**) end-region loss concentration illustrating axial non-uniformity; (**f**) axial cross-sectional view confirming pronounced end effects.