## 2 Analysis and Modeling of Radiation Sources of PWM Fan
### Detailed Model of Fan Radiation Based on FIT  
In order to obtain the electromagnetic radiation characteristics of the axial flow fan, a numerical calculation of the fan is carried out by the time-domain finite integration method. According to the finite integral theory, the Maxwell grid equation is obtained as  
\[\left\{ \begin{array}{l}C\mathbf{e}=-\frac{d}{dt}\mathbf{b}\\ \widetilde{C}\mathbf{h}=\mathbf{i}_{C}+\mathbf{i}_{S}+\frac{d}{dt}\mathbf{d} \end{array} \right. \tag{1}\]  
where \(C\) and \(\widetilde{C}\) are the discrete curl matrices of the main grid and the dual grid, \(\mathbf{e}\) is the voltage along the edge, and \(\mathbf{b}\) is the magnetic flux through the face elements in the main grid. \(\mathbf{h}\) is the magnetic field along the edge of the dual grid. \(\mathbf{d}\), \(\mathbf{i}_{C}\), and \(\mathbf{i}_{S}\) are the electric flux, loss current, and external current, respectively. A central difference discretization and a leap-frog strategy are adopted for the time derivative terms of the two spinodal equations in the above equation. Then, the explicit time-domain recursive equation for the finite integration method is  
\[\left\{ \begin{array}{l}\mathbf{e}^{n+1}=C_{AE}\mathbf{e}^{n}+C_{AH}\left( \widetilde{C}\mathbf{h}^{n+\frac{1}{2}}-\mathbf{i}_{S}^{n+\frac{1}{2}}\right) \\ \mathbf{h}^{n+\frac{1}{2}}=\mathbf{h}^{n-\frac{1}{2}}-\Delta tD_{\mu}^{-1}C \mathbf{e}^{n}\end{array} \right. \tag{2}\]  
where  
\[\left\{ \begin{array}{l}C_{AE}=\left(\frac{D_{e}}{\Delta t}+\frac{D_{e}}{2} \right)^{-1}\left(\frac{D_{e}}{\Delta t}-\frac{D_{e}}{2}\right)\\ C_{AH}=\left(\frac{D_{e}}{\Delta t}+\frac{D_{e}}{2}\right)^{-1}\end{array} \right. \tag{3}\]  
where \(D_{e}\), \(D_{e}\), and \(D_{\mu}\) are the diagonal matrices of dielectric constant, electrical conductivity, and magnetic permeability, respectively. Figure 3: Multiple electromagnetic radiation sources inside the fan. The transient electromagnetic process of the axial flow fan can be calculated according to (2) and (3), which is accomplished by CST Studio Suite 2023. The 3D model of the fan radiation calculation based on the FIT method is shown in Figure 4. The model mainly consists of the winding, the motor core, the driving circuit PCB, and a shell consisting of the fan frame and impeller, in which the stator winding and drive circuit PCB are directly integrated. The shell has a certain shielding effect from radiation. The PCB model, which carries circuit components, was obtained in the universal ODB++ format. In CST, the network containing the interference input path (power path) and output path (winding path) was selected, as indicated by the blue boxes in Figure 5. By extracting the parasitic parameters of the network, a field-circuit coupling model of the PCB was established. The winding model was constructed using copper enameled wire with a diameter of 0.71 mm, with 25 turns wound around each stator slot. SolidWorks 2021 was used to create the 3D models of the slots, windings, and permanent magnets. The three-phase windings were connected in a star configuration, and the resulting 3D winding model is shown in Figure 6. The permanent magnets were made of neodymium iron boron (NdFeB) material, and the iron core was formed by laminating silicon steel sheets. Figure 4: Three-dimensional detailed model of fan radiation calculation. Figure 5: PCB 3D model. (**a**) Power input network; (**b**) inverter circuit output network. Inside the fan, there are numerous circuit components and complex structures that cannot be characterized by 3D field models. Therefore, the components in the circuit are linked to the field model through an external circuit, where the FIT method in the time domain with field-circuit coupling is used to solve the problem. As shown in Figure 7, an external driving circuit is used to achieve the PWM control of MOSFET to calculate the excitation current of the electromagnetic field. The excitation current is transferred to the field model through a coupled interface between the circuit model and the field model.