## 3 Finite Element Analysis of Loss Distribution  
### Simulation Setup and Parameter Configuration  
The finite element analysis of loss distribution in the high-speed magnetic field-modulated motor was conducted using ANSYS Maxwell 3D v2023 R1 (ANSYS Inc., Canonsburg, PA, USA) for electromagnetic simulations coupled with ANSYS Thermal for thermal analysis. Figure 2 presents the three-dimensional model (Figure 2a) and finite element mesh (Figure 2b) of the motor structure developed for electromagnetic-thermal coupled analysis. The model incorporates detailed geometrical features including precise representation of stator slots, permanent magnets, flux modulators, and rotor components to ensure accurate prediction of magnetic field distribution and associated losses. The finite element mesh was generated with adaptive refinement techniques, implementing higher mesh density in critical regions such as air gaps (0.15 mm element size), permanent magnet edges (0.25 mm), and flux modulator interfaces (0.3 mm) where magnetic field gradients are most pronounced. The total mesh consisted of 1.85 million tetrahedral elements with second-order shape functions, with mesh sensitivity studies confirming that further refinement produced less than 1.2% change in calculated losses. Material properties were implemented using manufacturer-supplied data with temperature dependence. The silicon-steel laminations (M250-35A) were modeled with nonlinear B-H characteristics and frequency-dependent core loss coefficients (kh = 132 W/m\({}^{3}\)/T\({}^{2}\)/Hz, kc = 0.76 W/m\({}^{3}\)/T\({}^{2}\)/Hz\({}^{2}\), ke = 1.85 W/m\({}^{3}\)/T\({}^{1.5}\)/Hz\({}^{1.5}\)). Permanent magnets (N42 SH) were implemented with Br = 1.28 T, Hc = \(-\)955 kA/m, and electrical conductivity of 6.25 \(\times\) 10\({}^{5}\) S/m with temperature coefficients of \(-\)0.12%/\({}^{\circ}\)C and \(-\)0.60%/\({}^{\circ}\)C for remanence and coercivity, respectively. Copper windings were modeled with temperature-dependent resistivity (\(\rho_{0}\) = 1.68 \(\times\) 10\({}^{-8}\)\(\Omega\cdot\)m, \(\alpha\) = 0.00393/\({}^{\circ}\)C) and a fill factor of 0.65. The simulation parameters were configured based on the actual prototype specifications, with nonlinear B-H characteristics applied to the silicon-steel laminations to account for magnetic saturation effects. The permanent magnets were modeled with temperature-dependent properties using a reversible demagnetization model valid for operational temperatures up to 150 \({}^{\circ}\)C. The time-stepping transient solver was employed with step sizes corresponding to 1\({}^{\circ}\) of mechanical rotation to accurately capture high-frequency harmonics in the magnetic field, particularly those generated at the modulator-air gap interface. Boundary conditions included zero magnetic vector potential at the outer boundaries and continuity conditions at component interfaces. Boundary conditions were carefully implemented to accurately represent the operating environment. A perfect magnetic boundary condition (magnetic vector potential A = 0) was applied at a distance of five times the motor diameter to minimize boundary effects. For thermal analysis, radiation boundary conditions with emissivity \(\varepsilon\) = 0.85 were applied to all external surfaces to simulate the vacuum environment, while internal interfaces between components were modeled with thermal contact resistances calibrated from experimental measurements (e.g., 5.2 \(\times\) 10\({}^{-5}\) m\({}^{2}\)K/W for the interface between permanent magnets and rotor back iron). Cooling interfaces at the stator water jacket were modeled with convection boundary conditions using experimentally determined heat transfer coefficients that varied with coolant flow rate according to h = 2350-v\({}^{0.8}\) W/m\({}^{2}\)K, where v represents coolant velocity in m/s. For electromagnetic loss calculation, the modified Bertotti model was implemented with frequency-dependent coefficients calibrated through material testing.