## 4 Model Verification
The self-inductances of the square coils were calculated according to the Wheeler formula for square coils [38]:  
\[L=2.34\mu_{0}\frac{N^{2}d_{avg}}{1+2.75\rho^{\prime}}, \tag{13}\]  
where \(d_{avg}\) is average diameter of the coil:  
\[d_{avg}=R_{in}+R_{out}, \tag{14}\]  
and \(\rho^{\prime}\) is the fill ratio:  
\[\rho^{\prime}=\frac{R_{in}-R_{out}}{R_{in}+R_{out}}. \tag{15}\]  
The mutual inductance between the coaxially positioned coils of the same geometrical shape were calculated by applying the following expression [23]:  
\[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]0  
where \(b=\left(d_{avg}{}^{2}+h^{2}\right)^{1/2}\), and \(x=\left(2d_{avg}{}^{2}+h^{2}\right)^{1/2}\). \[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]1 Comparison of the circular coil inductances obtained by the FEM model and analytically: (**a**) self-inductance vs. number of turns; (**b**) mutual inductance vs. number of turns (\(h_{air}=15\) cm). The comparison of the results obtained by the numerical FEM model and the analytical calculations obtained by applying (13) and (16) are presented in \[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]2 The results presented in Figures 4 and 6 show the adequacy of the FEM model. The slightly higher differences obtained for the case of the square coils could be explained by the fact that the applied analytical model in this case is less detailed, in comparison to the circular coil case, as the former one does not consider the diameter of the windings. The FEM model (Figure 7a) was additionally verified experimentally by applying two identical Wirth Elektronik 400 W planar circular wireless power transfer coils [39], in this case with ferrite backings. Experimentally, the values of mutual inductance between the coils were obtained by applying the following procedure. The primary and secondary coils were connected in series at first in aiding and then in opposing ways. The total inductances of the coil system in each case were measured by applying an impedance analyzer. In the aiding case, the total inductance of the coil system could be expressed as the sum of self-inductances of the coils plus double mutual inductance:  
\[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]3  
where \(L_{p}\) and \(L_{s}\) are the self-inductances of each coil. \[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]4 FEM model geometry of the planar square coils: (**a**) geometry for the mutual inductance calculation; (**b**) the main parameters of the coils. \[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]5 Comparison of square coil inductances obtained by the FEM model and analytically: (**a**) self-inductances; (**b**) mutual inductances of the coaxially positioned coils (\(h_{air}=15\) cm). In the opposing case, the total inductance of the coil system equals the sum of self-inductances minus double the mutual inductance:  
\[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]6  
The mutual inductance could be expressed by subtracting (18) from (17) and leads to the following expression:  
\[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]7  
which was applied for the experimentally obtained mutual inductance calculation. The comparison of the results obtained by the FEM model and experimentally is presented in \[\zeta=\frac{1}{\sqrt{(X_{2}-x_{1})^{2}+(Y_{2}-y_{1})^{2}+g^{2}}}, \tag{9}\]8  
The \"Wayne Kerr 6500B\" (Wayne Kerr Europe GmbH, Iserlohn, Germany) impedance analyzer was used for the measurement of the inductances. A view of the applied experimental setup is presented in Figure 7b. To assure the particular required distance between the coils, coil holders made by a 3D printer were used. The physical experiment, as well as the analytical verification, confirmed the adequacy of the FEM model.