## 2 Materials and Methods
### Inverse Problem  
The reactances of the CNs in Figure 3, coupled to the WPTS device, have been designed considering the movement of the receiving inductor at a constant speed. The power transferred to the load, P\({}_{\text{L}}\), has been evaluated considering nine different positions of the receiving coil with respect to the transmitting one. Different positions correspond to different self and mutual inductances, as reported in Table 2. In each position, the power transfer efficiency and the power transferred to the load have been evaluated following the procedure described in the previous section. In order to find the CNs that are the best trade-off in the design of a dynamic WPTS, an inverse problem (identification problem) has been solved considering different combinations of power and efficiency. To this aim, three different formulations of the optimization problem have been considered. The first problem is based on the following objective functions (OF) to be maximized:  
\[P_{Ls}=\sum_{i=1}^{9}P_{Li}(x_{i}) \tag{11}\]  
\[\eta_{1}=\frac{P_{\text{L,1}}(x_{1})}{P_{\text{t,1}}(x_{1})} \tag{12}\]  
where P\({}_{\text{L}}\)(x\({}_{\text{i}}\))  
The second problem considers the first four displacements (from 0 to 150 mm, see Table 2) and it is based on the following OFs:  
\[P_{\Pi}=\min(P_{Li}(x_{i})\cdot\eta_{i}(x_{i}))\;\mathrm{i}=1,\ldots,4 \tag{13}\]  
\[P_{h}=max(P_{Li}(x_{i}))-min(P_{Li}(x_{i}))\;\mathrm{i}=1,\ldots,4 \tag{14}\]  
The OF (13) is to be maximized, while function (14) is to be minimized. In particular, maximizing the \(P_{\mathrm{h}}\) function means searching for a power transfer homogeneity over the first four positions. Finally, the third problem considers the function \(P_{\mathrm{h}}\), i.e., Equation (14) and the following (15), to be maximized:  
\[P_{\Pi 4}=\sum_{i=1}^{4}P_{Li}(x_{i})\cdot\eta_{i}(x_{i}) \tag{15}\]  
The OFs (13)-(15) use only the first four displacements out of the nine used to compute the inductive parameters L and M because, as can be deduced by analysis of the second row of Table 2, the considered coils are affected by the null-power point phenomenon [22; 41], i.e., a value of displacement exists where the mutual inductance is 0 even if the coils are still partially faced. For these coils, the null-power point happens with a displacement between 200 mm and 300 mm. At the null power point, \(P_{\mathrm{L}}\) is zero irrespectively from the topology and the reactances of the CN. With displacements higher than 200 mm, M is small and consequently there is no reason to attempt to maximize \(P_{\mathrm{L}}\) in conditions where it is inherently zero or small. The three problems were solved considering the following constraint: \(Z_{\omega 0}/Z_{\omega 100}<1\) and \(Z_{\omega 0}/Z_{\omega 0.01}<1\), where \(Z_{\omega 0}\) is the impedance at the supply inverter output at the nominal supply angular frequency \(\omega_{0}\) (@ 85 kHz), and \(Z_{\omega 100}\) and \(Z_{\omega 0\text{-}01}\) are the impedances at the angular frequencies \(\omega_{100}\) (100\(\omega_{0}\)) and \(\omega_{0\text{-}001}\) (\(\omega_{0}/100\)). The first constraint is opposed to the flow of distorted currents in the CN-coils assembly, while the second constraint avoids the flowing of low frequency current in the CN-coil assembly. Together, the two constraints enforce the minimum of the inverter load impedance at \(\omega_{0}\) discussed in the previous subsection. The second and third problems have been solved by including, eventually, the following constraint: the CN at the receiving side must be composed of capacitors only and the CN at the transmitting side can be composed of inductors or capacitors. In the following, when the constraint is applied, the relevant problem is marked with the receiving capacitor \"R.C.\" label.