## 2 Model of LCC-s Converter  
The wireless resonant converter considered in this work is an LCC-S topology. The electric scheme of this converter is shown in Figure 1.  
As can be seen in Figure 1, the considered DC-DC converter is made by the combination of a DC-AC converter (VDC as DC voltage source and MOSFETs full-bridge), a LC filter (typically called resonant tank) and an AC-DC converter (diode bridge and output filter capacitance \(C_{o}\)). The inductances \(L_{p}\) and \(L_{s}\) are associated with the coils that realize the wireless power transfer. Since the coils are in air, the coupling coefficient between these inductances is typically low; for this reason, some capacitors are generally added in wireless converters to reduce the reactive power generated by the wireless coils. The simplest configuration for wireless converters adopts two capacitances in series with \(L_{p}\) and \(L_{s}\). The resulting converter topology is called Series-Series (SS) compensation. Although the simplicity of the SS topology makes it a frequently adopted converter for wireless power transfer, there are some limitations to take into account: limited freedom in the design  
Figure 1: Electric scheme of LCC-s converter. of the converter and a limited range of output voltage achievable (it can behave just like buck converter) [31; 32]. For this reason, according to specific applications, the adoption of more complex topologies could be required. The LCC-S compensation can overcome the limitations of the SS one, giving the possibility to work as a boost converter with a proper selection of the filter inductance \(L_{f}\). Concerning the modeling of resonant converters, the typical approach adopted in the literature is the so-called First Harmonic Approximation (FHA), where the input voltage is approximated with its fundamental harmonic. This approximation is valid when the converter's working frequency corresponds to its natural resonant frequency. In this condition, independently by the nature of the input voltage waveform (square, PWM, trapezoidal, \(\ldots\)), the combination of inductances and capacitances in the circuit behaves like a band-pass filter, which results in a short circuit for the fundamental harmonic and in a big impedance for high-frequency harmonics. In addition, under the FHA hypothesis, the AC-DC part of the converter can be modeled with an equivalent resistance \(R_{e}=\frac{8}{\pi^{2}}R_{L}\). This expression can be derived as the ratio between voltage and current before the rectifier, both written as functions, respectively, of the voltage and current on the load (detailed steps in [33]). With these considerations, the converter in Figure 1 can be simplified with the equivalent circuit in Figure 2.  
In this way, the converter model in steady-state conditions can be derived in frequency domain using phasors. Applying the [PERSON]' equations to the circuit in Figure 2 the model of the LCC-S circuit can be easily derived in matrix form as follows:  
\[\left[-\omega^{2}\mathbf{L}+j\omega\mathbf{R}+\mathbf{C}\right] \begin{bmatrix}\dot{I}_{f} \\ I_{Cf}\\ I_{Cp}\\ I_{Cs}\end{bmatrix} =j\omega \begin{bmatrix}\dot{V}_{i}\\ \dot{V}_{i}\\ 0\\ 0\end{bmatrix}  \tag{1}\]  
where  
\[\mathbf{L}= \begin{bmatrix}L_{f}&0&0&0\\ L_{f}&0&L_{p}&M\\ 0&0&M&L_{s}\\ 0&0&0&0\end{bmatrix} ;\;\mathbf{R}= \begin{bmatrix}r&r&0&0\\ 0&0&r&0\\ r&0&0&R_{e}+r\\ 0&0&0&0\end{bmatrix} ;\;\mathbf{C}= \begin{bmatrix}0&1\Big{/}C_{f}&0&0\\ 0&0&1\Big{/}C_{p}&0\\ 0&0&0&1\Big{/}C_{s}\\ 1&-1&-1&0\end{bmatrix}  \tag{2}\]  
\(r\) is a parasitic resistance associated with the inductive and capacitive components. \(\dot{V}_{i}\) is the rms value of the input voltage. Since the input voltage has a trapezoidal waveform, its first harmonic can be evaluated using Fourier Transform. Considering the trapezoidal  
Figure 2: Equivalent circuit of LCC-s converter after FHA. where  
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