## 4 Model Verification  
The accuracy of an FEM model is determined by various factors including the implemented geometry, meshing, boundaries of the model, applied material models, etc. Two approaches for the validation of the developed FEM model were used: comparison with the results obtained by the analytical method and the experimental validation. Both models of the coils (circular and square topologies), without Fe backings, were validated using the analytical approach. The dimensions of the coils in this case were proper for EV charging applications. The experimental approach was used for the validation of the model with the ferrite backings; however, the size of the coils applied in the physical experiment was significantly smaller. Since the parametric approach was used in the FEM model development, it was scaled to these lower dimensions just by adjusting the corresponding parameters, without any additional changes made in the model itself. The self and mutual inductances of the circular coils (Figure 3) for the analytical verification of the FEM model were calculated using the expressions presented in [37]. The self-inductances of the circular coils were calculated using the numerical form of the following expression [37]:  
\[L=\frac{\mu_{0}}{4\pi}\iint\limits_{0}^{\Phi}\zeta((a\text{cos}\,\varphi_{1}- y_{1})(a\text{cos}\,\varphi_{2}-Y_{2})+(a\text{sin}\,\varphi_{1}+x_{1})(a \text{sin}\,\varphi_{2}+X_{2}))d\varphi_{1}d\varphi_{2}, \tag{8}\]  
where  
\[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]  
and \(a=\delta/(2\pi)\), \(\Phi=2\pi N\), \(x_{1}=(R_{in}+a\varphi_{1})\text{cos}\,\varphi_{1}\), \(y_{1}=(R_{in}+a\varphi_{1})\text{sin}\,\varphi_{1}\), \(X_{2}=(R_{in}+a\varphi_{2})\text{cos}\,\varphi_{2}\), \(Y_{2}=(R_{in}+a\varphi_{2})\text{sin}\,\varphi_{2}\), \(g=e^{-1/4}R_{cond}\), \(\mu_{0}=4\pi\cdot 10^{-7}\) H/m is magnetic constant, \(R_{in}\) is the inner radius of the coil, \(N\) is the number of turns, \(R_{cond}\) is the radius of the conductor, \(\delta=2R_{cond}+l_{gap}\) is the screw pitch, i.e., distance between the centers of the adjacent turns, and \(l_{gap}\) is the air-gap length between the adjacent turns. The mutual inductances between the coaxially positioned circular coils, having the same geometrical shape, were calculated using the numerical form of the following expression [37]:  
\[M=\frac{\mu_{0}}{4\pi}\int_{0}^{\Phi_{1}}\int_{0}^{\Phi_{2}}\chi((a\text{cos }\,\varphi_{1}-y_{1})(a\text{cos}\,\varphi_{2}-Y_{2})+(a\text{sin}\,\varphi_ {1}+x_{1})(a\text{sin}\,\varphi_{2}+X_{2}))d\varphi_{1}d\varphi_{2}, \tag{10}\]  
Figure 3: FEM model geometry of the planar circular coils: (**a**) geometry for the mutual inductance calculation; (**b**) the main parameters of the coils. where  
\[\chi=\frac{1}{\sqrt{\left(X_{2}-x_{1}\right)^{2}+\left(Y_{2}-y_{1}\right)^{2}+h_{ air}^{2}}}, \tag{11}\]  
And \(h_{air}\) is the distance between the coils. The comparisons of the results obtained by the developed numerical FEM model and by the analytical expressions (8) and (10) for varying numbers of coil turns are presented in Figure 4. In the verification analysis, the outer radius of the coils was kept constant at \(R_{out}=0.25\) m. The inner radius of the coils \(R_{in}\) was recalculated for each case considering the corresponding number of turns \(N\) as follows:  
\[R_{in}=R_{out}-\left(N2R_{cond}+(N-1)I_{gap}\right). \tag{12}\]  
where the conductor diameter \(R_{cond}=2.7\) mm, and the air-gap length between the adjacent turns \(I_{gap}=4\) mm were constant parameters. The self and mutual inductances of the square planar coils (Figure 5) were calculated analytically using the expressions presented in [23].