## 3 Electromagnetic-Thermal Bidirectional Coupling Model Formulation of TPMLG  
### Electromagnetic Finite Element Model of the TPMLG and Loss Calculation  
The TPMLG is rotationally symmetrical around the central axis. Compared with the precise electromagnetic loss calculation of a three-dimensional model, a two-dimensional model can significantly reduce overall computational time and keep the loss values within an acceptable error range. Furthermore, the establishment of a finite element model merely necessitates the involvement of relevant topologies and components within the magnetic circuit, allowing for the simplification or omission of insignificant elements and minuscule parts. On account of the structural characteristics of the LG and the above considerations, a transient two-dimensional finite element model of the TPMLG was constructed by Maxwell 2022 R1 software, as illustrated in Figure 3. The finite element model consists of the stator core (stator yoke and stator tooth), winding A (red), winding B (blue), winding C (yellow), slot wedges, permanent magnets, silicon steel sheets, and a stainless-steel sleeve. Some basic material properties are sourced from the Maxwell 2022 R1 software while the others need to be set [28]. The electromagnetic properties can be adjusted according to different temperatures by setting object temperature and enabling the thermal modifier. To study the losses in the power generation process, the model is simulated under the load condition of stable power generation, when it works at the working frequency of 30 Hz and the stroke of 110 mm. The external circuit excitation is applied in the excitation and the Dirichlet boundary with a value of zero is set for the boundary condition. The resistance of each phase winding is 3 \(\Omega\), and the load resistance of each phase is 15 \(\Omega\). The electrical energy produced by the LG is consumed by load resistance. The band area is modelled to cover the whole length of the mover's stroke and the region is modelled to guarantee continuity and integrity of the model. Losses generated in energy conversion from electrical energy to kinetic energy directly led to the temperature rise of the LG. The losses in the TPMLG mainly incorporate copper loss, core loss, and eddy current loss. The 2-D transient electromagnetic field simulation is applied to calculate the losses of the LG. The copper loss accounts for the greatest proportion of the total loss. Ignoring the skin effect and proximity effect, the copper loss is computed based on [PERSON]'s law as follows [29]:  
\[P_{\text{Cu}} =I_{\text{A}}^{2}R_{\text{A}}+I_{\text{B}}^{2}R_{\text{B}}+I_{ \text{C}}^{2}R_{\text{C}} \tag{1}\] \[R =R_{0}\cdot[1+\theta(T-T_{0})] \tag{2}\]  
where \(P_{\text{Cu}}\) represents copper loss; \(I\) represents the phase current; \(R\) represents resistance; subscripts A, B, C represent the A-phase, B-phase, and C-phase windings, respectively; \(\theta\) is the temperature coefficient of resistance; \(T\) is the actual temperature; and subscript 0 represents the basic ambient temperature. For permanent magnet motors, temperature rise can affect the permeability of the permanent magnets, and the magnetic field excitation provided by the permanent magnets affects the loss calculation results of the motor's electromagnetic field. Therefore, the linear demagnetization characteristics of NeFdB magnets are considered and the temperature coefficients of relative permeability and magnet coercivity need to be set. According to  
Figure 3: Finite element model of the TPMLG. the classical Bertotti's core loss theory, the core loss includes hysteresis loss, classical eddy current loss and excess loss [30; 31]:  
\[P_{Fe}=K_{hf}P_{m}^{2}+K_{c}(fB_{m})^{2}+K_{e}(fB_{m})^{1.5} \tag{3}\]  
where \(P_{Fe}\) denotes core loss; \(K_{h}\) denotes hysteresis core loss coefficient; \(K_{c}\) denotes classical eddy current core loss coefficient; \(K_{e}\) denotes excess core loss coefficient; \(f\) denotes frequency; and \(B_{m}\) denotes amplitude of flux density.