## 3 Analytical Model Calculation and Validation
### Analytical Calculation
#### 3.1.3 Solution of Equations  
Using the separation of variables method, the TE potential in \(\Omega_{\mathrm{II}}\) can be expressed as [13]:  
\[W_{\mathrm{a}}^{\mathrm{II}}(x,y,z)=\sum_{m=-M}^{M}\sum_{n=-N}^{N}\mathrm{e}^{ \mathrm{i}\tilde{\gamma}_{m}x}\mathrm{e}^{\tilde{\gamma}_{n}x_{2}}\Big{(}C_{mn }^{\mathrm{II}}\mathrm{e}^{\tilde{\alpha}_{mn}y}+D_{mn}^{\mathrm{II}}\mathrm{ e}^{\tilde{\beta}_{mn}y}\Big{)},-h_{2}<y<0 \tag{11}\]  
where \(M\) and \(N\) are harmonics of the Fourier series, \(\tilde{\xi}_{m}=2\pi m/l_{2}\) and \(k_{n}=2\pi n/w_{2}\) are spatial frequencies, \(C_{mn}^{\mathrm{II}}\) and \(D_{mn}^{\mathrm{II}}\) are the coefficients to be solved, \(\alpha_{mn}\) and \(\beta_{mn}\) are:  
\[\alpha_{mn}=0.5\Big{(}-\mu_{0}\sigma v_{y}+\sqrt{\left(\mu_{0}\sigma v_{y} \right)^{2}+4\gamma_{mn}{}^{2}}\Big{)} \tag{12}\]  
\[\beta_{mn}=0.5\Big{(}-\mu_{0}\sigma v_{y}-\sqrt{\left(\mu_{0}\sigma v_{y} \right)^{2}+4\gamma_{mn}{}^{2}}\Big{)} \tag{13}\]  
Figure 6: Source, reflected and transmitted TE potential. where  
\[{\gamma_{mn}}^{2}={\xi_{m}}^{2}+{k_{n}}^{2}-j\mu_{0}\sigma(v_{\rm x}\xi_{m}+v_{z}k _{n}) \tag{14}\]  
Similarly, the TE potential in the region \(\Omega_{\rm I}\) and \(\Omega_{\rm II}\) can be obtained:  
\[W_{\rm a}^{\rm Lr}(x,y,z)=\sum_{m=-M}^{M}\sum_{n=-N}^{N}{\rm e}^{i\tilde{\zeta} _{m}x}{\rm e}^{i\tilde{\zeta}_{m}x}{\rm e}^{j\tilde{\zeta}_{m}x}C_{mn}^{\rm Lr} \mathrm{e}^{-\kappa_{mn}y},0\leq y\leq d \tag{15}\]  
\[W_{\rm a}^{\rm III}(x,y,z)=\sum_{m=-M}^{M}\sum_{n=-N}^{N}{\rm e}^{i\tilde{\zeta }_{m}x}{\rm e}^{i\tilde{\zeta}_{m}x}C_{mn}^{\rm III}\mathrm{e}^{\kappa_{mn}(y+ h_{2})},y\leq-h_{2} \tag{16}\]  
where  
\[\kappa_{mn}=\sqrt{{\xi_{m}}^{2}+{k_{n}}^{2}} \tag{17}\]  
The analytical expression of the magnetic field generated by the PM can be obtained by [PERSON]'s molecular current hypothesis [14]; that is, the \(y\) component magnetic flux density generated by the PM at the upper surface of the conductive plate \(B_{\rm y}^{\rm s}(x,0,z)\) is known. Then, the magnetic flux density in \(\Omega_{\rm I}\) can be expressed by a Fourier coefficient as [15]:  
\[B_{\rm y}^{\rm LS}(x,y,z)=\sum_{m=-M}^{M}\sum_{n=-N}^{N}{\rm e}^{i\tilde{\zeta }_{m}x}{\rm e}^{j\tilde{\zeta}_{m}x}{\rm e}^{j\tilde{\zeta}_{m}x}{\rm e}^{ \kappa_{mn}y} \tag{18}\]  
where \(S_{mn}^{\rm LS}\) is the Fourier coefficient, which can be calculated by Equation (19) or by the fast Fourier transform (FFT) more efficiently. \[S_{mn}^{\rm LS}=\frac{1}{l_{2}w_{2}}\int_{-v_{2}/2}^{w_{2}/2}\int_{-l_{2}/2}^{ l_{2}/2}B_{\rm y}^{\rm s}(x,0,z){\rm e}^{-j(\tilde{\zeta}_{m}x+k_{n}z)}{\rm d }x{\rm d}z \tag{19}\]  
\(W_{\rm a}^{\rm LS}\) can be expressed by \(B_{\rm y}^{\rm LS}\) as  
\[\alpha_{mn}=0.5\Big{(}-\mu_{0}\sigma v_{y}+\sqrt{\left(\mu_{0}\sigma v_{y} \right)^{2}+4\gamma_{mn}{}^{2}}\Big{)} \tag{12}\]0  
Substituting Equation (18) into Equation (20), \(W_{\rm a}^{\rm LS}\) can be solved:  
\[\alpha_{mn}=0.5\Big{(}-\mu_{0}\sigma v_{y}+\sqrt{\left(\mu_{0}\sigma v_{y} \right)^{2}+4\gamma_{mn}{}^{2}}\Big{)} \tag{12}\]1  
where  
\[\alpha_{mn}=0.5\Big{(}-\mu_{0}\sigma v_{y}+\sqrt{\left(\mu_{0}\sigma v_{y} \right)^{2}+4\gamma_{mn}{}^{2}}\Big{)} \tag{12}\]2  
By taking Equations (11), (15), (16) and (21) into the boundary conditions in Equation (10), four unknown coefficients, \(C_{mn}^{\rm II}\), \(D_{mn}^{\rm II}\), \(C_{mn}^{\rm Lr}\) and \(C_{mn}^{\rm III}\), can be solved.