## 2 Materials and Methods  
### Lumped Parameter Computation  
The self- and mutual-inductances at different alignment conditions were computed using the 3D finite element model (FEM) described in [22]. In order to reduce the number of mesh elements, only half of the device was implemented in Flux 3D software (released by Altair Engineering, Inc., Troy, MI, USA, ref. [44]) by exploiting the system symmetry. The two-faced coils with an airgap of 140 mm exhibit 15 turns made of Litz's wire (internal diameter 150 mm, turn diameter 4.5 mm, turn pitch 8 mm) and are equipped with a magnetic field concentrator, a ferrite layer with a side of 404 mm made of 3C95 (Ferroxcube, initial relative permeability of 3000, a saturation flux density of 530 mT, [45]). The car frame has been modeled as a steel sheet made of ASTM A1008 steel (resistivity 14.2 \(\times\) 10\({}^{-8}\)\(\Omega\)m and initial relative permeability of 1000), 1000 mm \(\times\) 800 mm, and a thickness equal to 0.7 mm. An aluminum sheet (600 mm \(\times\) 600 mm) with a thickness of 0.76 mm has been positioned between the ferrite layer and the car frame to reduce the effect of the eddy currents on the steel. In order to reduce the number of mesh elements, the eddy current in the aluminum and steel sheet were computed using a \"shell formulation\" [46; 47; 48]. This way the mesh of the model has an average of 1,680,000 second-order tetrahedral elements. The device geometry is shown in Figure 1. All material properties have been considered constant, as reported in Table 1.  
The lumped parameters were computed by solving a time harmonic magnetic problem using the \(\mathrm{\bar{T}}\ -\phi\) formulation with \(\mathrm{\bar{T}}\) electric vector potential and \(\phi\) magnetic scalar potential [49]. The following equations, subject to suitable boundary conditions, were solved in the model domain [22; 50; 51]:  
\[\
abla\times\sigma^{-1}\ \
abla\times\mathrm{\bar{T}}\ +\ \mathrm{j}\omega\mu\left( \mathrm{\bar{T}}\ -\
abla\phi\right)=0 \tag{1}\]  
Then, the vector of the magnetic field, \(\mathrm{\bar{H}}\), is given by:  
\[\mathrm{\bar{H}}\ =\left(\mathrm{\bar{T}}\ -\
abla\phi\right) \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & \(\mu_{\mathrm{r}}\) & \(\rho\) [\(\Omega\)m] \\ \hline Ferrite 3C95 & 3000 (saturation 530 mT) & 5 \\ \hline steel & 1000 & \(14\times 10^{-8}\) \\ \hline aluminum & 1 & \(2.6\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Electrical characteristics of the materials used in the FE model.  
Figure 1: Geometry of the implemented half FE model of the WPTS system.  
To evaluate the lumped parameters of the system, the 3D model of the device has been coupled with an electric circuit that includes a perfect current source and a resistive load, whose resistance may be varied, from the actual value of the load connected to the receiving coil to a very high value to simulate an open circuit (Figure 2) [47; 48].  
The FE model in Figure 1 is 'fed' by the circuit in Figure 2. Mutual and self-inductances have been calculated for different relative positions of the receiving and transmitting coil (step of 50 mm). The self-inductances, \(\mathrm{L_{i}}\), have been computed by supplying one coil at a time with a current, \(\mathrm{I_{s}}\), of 1 Arms at 85 kHz frequency and by measuring the voltage, \(\mathrm{V_{s}}\), at the ends of the supplied coil itself. The not-powered coil is connected to a load resistance, \(\mathrm{R_{L}}=10^{6}\)\(\Omega\), to represent an open-circuit condition:  
This way, the self-inductance is:  
\[\mathrm{L_{i}}=\frac{\mathrm{V_{S}}}{\omega\mathrm{I_{s}}} \tag{3}\]  
The mutual inductance, \(\mathrm{M}\), has been evaluated by (4), measuring the voltage, \(\mathrm{V_{L}}\), induced on the not supplied coil.  
\[\mathrm{M}=\frac{\mathrm{V_{L}}}{\omega\mathrm{I_{s}}} \tag{4}\]  
Table 2 shows the self and mutual inductances evaluated with FEAs and uses Equations (3) and (4) for different distances of the receiving coil axis with respect to the transmitting coil axis [22].