## 2 Theoretical Model of Loss Calculation and Thermal Analysis
### Innovative Analytical Models for Loss Estimation and Thermal Prediction  
Recent advancements in loss estimation and thermal prediction for high-speed magnetic field-modulated motors have been driven by the development of innovative analytical models that balance computational efficiency with predictive accuracy. These novel approaches transcend traditional methodologies by incorporating multiphysics interactions, machine learning techniques, and hybrid analytical-numerical framework. A particularly promising development is the frequency domain analytical model for eddy current loss prediction in permanent magnets, which decomposes the time-varying magnetic field into harmonic components:  
\[P_{eddy}=\frac{\pi^{2}\sigma d^{2}l_{m}}{6}\sum_{n=1}^{\infty}\left(\frac{n \omega B_{n}}{2}\right)^{2} \tag{8}\]  
where \(\sigma\) represents electrical conductivity, \(d\) is the magnet thickness, \(l_{m}\) is the magnet length, \(\omega\) is the angular frequency, and \(B_{n}\) is the magnitude of the nth harmonic component. This approach significantly reduces computational complexity while maintaining accuracy for rapid design iterations.  
The equivalent thermal circuit transformation (ETCT) technique represents another innovative approach wherein complex three-dimensional thermal networks are systematically reduced to equivalent two-dimensional representations through mathematical transformations:  
\[R_{eq}=\sum_{i=1}^{n}\frac{R_{i}}{\lambda_{i}}\cdot\frac{\prod\limits_{j=1}^{n }\lambda_{j}}{\sum\limits_{k=1}^{n}\prod\limits_{j\
eq k}\lambda_{j}} \tag{9}\]  
where \(R_{eq}\) denotes the equivalent thermal resistance, \(R_{i}\) is the original thermal resistance, and \(\lambda_{i}\) is the eigenvalue of the transformation matrix. This methodology enables rapid thermal analysis with minimal compromise in accuracy, achieving computational speedups of 50-100\(\times\) compared to full finite element models.  
The integration of physics-informed neural networks (PINNs) with domain knowledge has emerged as a powerful paradigm for loss estimation and thermal prediction. These models leverage deep learning architectures while enforcing physical constraints through custom loss functions:  
\[\mathcal{L}_{PINN}=\mathcal{L}_{data}+\lambda_{1}\mathcal{L}_{PDE}+\lambda_{2} \mathcal{L}_{BC}+\lambda_{3}\mathcal{L}_{JC} \tag{10}\]where \(\mathcal{L}_{data}\) represents the data fitting error, \(\mathcal{L}_{PDE}\) is the residual of governing differential equations, \(\mathcal{L}_{BC}\) is the boundary condition error, and \(\mathcal{L}_{IC}\) is the initial condition error, with \(\lambda_{i}\) denoting weighting coefficients. This hybrid approach enables accurate predictions even with limited training data, a common constraint in specialized electrical machine applications.  
Modal decomposition techniques have been adapted for thermal transient analysis, wherein the temperature response is expressed as a superposition of thermal modes:  
\[T(x,y,z,t)=T_{\infty}+\sum_{i=1}^{m}C_{i}\phi_{i}(x,y,z)e^{-t/\tau_{i}} \tag{11}\]  
where \(T_{\infty}\) represents the steady-state temperature, \(\phi_{i}\) is the spatial modes, \(\tau_{i}\) is the time constant, and \(C_{i}\) is the amplitude coefficient determined through proper orthogonal decomposition. This approach enables accurate prediction of thermal transients during variable-speed operation, particularly crucial for flywheel energy storage applications with frequent charging and discharging cycles.