## 3 Mathematical Modelling and Analysis  
For analysis, the system topology of Figure 2a was rearranged assuming the parasitic resistances are neglected for all components, as shown in Figure 2c. The inverter in Figure 2a produces a square wave voltage. This voltage is applied to the input terminal of the resonant circuit. The resonant  
Figure 2: The proposed system: (**a**) CPT system with a DC–DC buck converter; (**b**) circuit simplification by the sinusoidal approximation, and (**c**) equivalent circuit used for analysis.  
circuit frequency is tuned to the fundamental component of the inverter square wave, i.e., to the switching frequency \(f\). As a result, the resonant circuit current and the resonant circuit output voltage and current have essentially sinusoidal waveforms at the fundamental frequency \(f\), with negligible harmonics. Accordingly, the sinusoidal approximation was first considered where \(V_{in}\) can be obtained from Equation (1), as shown in Figure 2b.  
The two series interface capacitances were replaced by \(C_{c}\) as shown in Equation (2). The series compensation inductor \(L\) is obtained from Equation (3). The equivalent AC load \(R\)[26], assuming the rectifier and the buck converter are ideal, is described in Equation (4).  
\[V_{in}\ =\ \frac{2V_{d}}{\pi}\ sin(\omega t), \tag{1}\]  
\[C_{c}\ =\ \frac{C_{c1}\ C_{c2}}{C_{c1}+C_{c2}}, \tag{2}\]  
\[L\ =\ \frac{1}{C_{c}\ \omega^{2}}, \tag{3}\]  
\[R\ =\ \frac{8}{D^{2}\pi^{2}}\ R_{L}=\frac{R_{load}}{D^{2}}, \tag{4}\]  
where \(\omega\) is the angular switching frequency; \(R_{load}\) is the effective load resistance at the AC side; which is equal to 81% of the actual load resistance \(R_{L}\) at the DC side after the rectifier; and \(D\) is the buck converter duty cycle. The values in Table 1 were used as a design example to prove the concept.  
The equivalent circuit shown in Figure 2c would not be applicable for analysis if the cross coupling or the leakage capacitance is too high due to the complicated coupling configuration [27] or if the operating frequency is increased to a very high level. In these cases, more accurate modelling [14] for the capacitive interface should be used.  
To fairly compare the system sensitivity and the voltage stress over the capacitive coupling interface in the proposed system and the conventional system previously described, we assumed that the same amount of power was delivered to the load and the input voltage would change according to the value of the duty cycle (\(D\)) of the buck converter. When \(D\) was 100% represented the conventional system. Figure 3 shows the needed input voltage for the design example to deliver 10 W to the load depending on \(D\), as shown in Equations (5) and (6):  
\[V_{in}=\frac{V_{in\_conv}}{D}. \tag{5}\]  
From Equation (1), we obtain:  
\[V_{D}=\frac{V_{D\_conv}}{D}, \tag{6}\]  
where \(V_{D}\) and \(V_{in}\) are the DC input voltage and AC input voltage needed to deliver a specific amount of power to the load, respectively. The DC input voltage and AC input voltage for the conventional system are represented by \(V_{D\_conv}\) and \(V_{in\_conv}\), respectively.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline \(P_{load}\) & Watt & 10 \\ \(f\) & MHz & 1 \\ \(C_{c1}\)= \(\dot{C_{c2}}\) = 2\(C_{c}\) & nF & 2 \\ \(L\) & \(\mu\)H & 25 \\ \(R_{load}\) & \(\Omega\) & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters used for modelling.