## 2 Materials and Methods
### Compensation Network  
A compensation network is a reactive circuit interposed between the power supply, constituted by a high frequency inverter controlled with the phase-shift technique [52], and the transmission coil or between the receiving coil and the load. In the scheme of Figure 3, a general TT network, with three reactances per side, is represented. The optimization of the compensation network in static condition, i.e., for the aligned case, has been performed in previous works [53; 54]. In the dynamic WPTS, the optimal CNs must be searched considering the lumped parameters; L and M are not constant in time. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**x [mm]** & **0** & **50** & **100** & **150** & **200** & **300** & **400** & **500** & **600** \\ \hline M [\(\mu\)H] & 28.9 & 26.0 & 18.8 & 10.25 & 3.1 & \(-\)2.7 & \(-\)1.6 & \(-\)0.5 & \(-\)0.1 \\ L\({}_{\mathrm{t}}\) [\(\mu\)H] & 116.4 & 116.7 & 117.2 & 117.7 & 117.5 & 115.7 & 117.3 & 118.7 & 117.4 \\ L\({}_{\mathrm{r}}\) [\(\mu\)H] & 113.2 & 113.2 & 113.2 & 113.0 & 112.7 & 111.1 & 110.1 & 110.0 & 110.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Self and mutual inductances of the WPTS device for different positions. Figure 2: FE model coupled with the supply and load circuit for lumped parameter evaluation. The power, \(\mathrm{P_{L}}\), transferred to the load and the, \(\eta\), of the system are the functions to maximize. CNs operate on the basis of the resonance phenomenon to achieve a minimum of the reactance seen at the inverter output in correspondence with the nominal supply frequency, \(\omega_{0}\). This condition helps to minimize the voltage needed to supply the transmitting side CN-coil assembly and forces the inverter output current to be nearly sinusoidal despite the quasi-square waveform of the output voltage. From this condition, it is derived that the analysis of WPTSs can be performed by hypothesizing that all the involved electrical quantities are sinusoidal at 85 kHz. According to this hypothesis, the circuit of Figure 3 can be redrawn in the scheme of Figure 4, where phasor quantities are considered. Subscripts t and r indicate the receiving and transmitting coils, respectively. The power dissipated in \(\mathrm{R_{loss}}\) represents the conduction losses of the supply inverter, which are proportional to the current flowing in the power switches and, during the dead times, in the free-wheeling diodes. The commutation losses depend on the parasitic capacitance of the switches and diodes and on the voltage applied across them [55; 56]. The dc side voltage of the inverter is not an optimized design variable and hence the switching losses are not affected by the optimization process. For this reason, and considering that in the experimental setup taken as reference they are a small fraction of the conduction losses, they are not considered in the representation of Figure 4.  
The transferred power and efficiency have been obtained by a sequence of steps that starts by computing the equivalent impedance, \(Z_{r}\), seen by the voltage source, \(\overline{V}_{r}\), that represents the voltage induced on the receiving coil. By definition, this is:  
\[\dot{Z}_{r}\triangleq\frac{\overline{V}_{r}}{\overline{I}_{r}} \tag{5}\]  
The transferred power, \(\dot{Z}_{pu}\), depends on the values of the reactances forming \(\mathrm{CN_{r}}\) and on \(\mathrm{R_{L}}\), whose values must be known in order to evaluate power and efficiency. To this aim, we can introduce the reflected impedance, \(\dot{Z}_{ref}\), that is a function of \(\dot{Z}_{pu}\), \(M\), and the supply angular frequency, \(\omega\), but that does not depend on the topology of the CNs:  
\[\dot{Z}_{ref}\triangleq\frac{\overline{V}_{t}}{\overline{I}_{t}}=\frac{ \omega^{2}M^{2}}{\dot{Z}_{r}} \tag{6}\]  
Figure 4: Phasor representation of the WPTS. Figure 3: Scheme of the compensation networks to be optimized in the constant-speed moving condition.