## 2 Model of Four-Plate Parallel DCPT Coupler
### Equivalent Circuit Model of Four-Plate Parallel DCPT
\tag{1}\]  
L\({}_{\text{tr}}\) and L\({}_{\text{rr}}\) are resonant inductors, and significant differences in their values arise due to the disparity in the self-capacitance of the transmitter and receiver couplers. This basic impedance difference also leads to variations in the configuration of the impedance-matching circuit. For the transmitter, a series inductor is connected first, followed by a parallel capacitor, to achieve impedance matching from the perspective of input impedance. This configuration differs from a typical LC-matching network. However, other forms of impedance-matching circuits may also be employed, and this is not a strict limitation. The S-parameters of the equivalent circuit shown in Figure 5b can be calculated through the transformation of the transmission matrix. The overall S-parameters of the equivalent circuit can be obtained by multiplying the transmission matrices of each circuit network. The transmission matrix for each circuit network in the equivalent circuit is expressed as shown in Equation (2). When expressed in impedance form, it is represented as Equation (3). \[\left\{ \begin{aligned} \left[\text{T}\right]_{\text{a}}& =\left[\begin{array}{cc}1&0\\ \text{Y}&1\end{array}\right]\\ \left[\text{T}\right]_{\text{b}}&=\left[\begin{array}{cc}1&Z\\ 0&1\end{array}\right]\\ \left[\text{T}\right]_{\text{c}}&=\left[\begin{array}{cc}1+\frac{\text{Y}_{ 2}}{\text{Y}_{3}}&\frac{1}{\text{Y}_{3}}\\ \text{Y}_{1}+\text{Y}_{3}+\frac{\text{Y}_{3}\text{Y}_{2}}{\text{Y}_{3}}&1+ \frac{\text{Y}_{1}}{\text{Y}_{3}}\end{array}\right]\\ \left[\text{T}\right]_{\text{d}}&=\left[\begin{array}{cc}1+\frac{\text{Z}_{ 1}}{\text{Z}_{3}}&Z_{1}+Z_{2}+\frac{\text{Z}_{1}Z_{2}}{\text{Z}_{3}}\\ \frac{1}{\text{Z}_{3}}&1+\frac{\text{Z}_{2}}{\text{Z}_{3}}\end{array}\right] \end{aligned} \right. \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Notation** & **Value** \\ \hline Coupler self-capacitance of Tx & C\({}_{\text{1}}\) & 3.504 pF \\ Coupler self-capacitance of Rx & C\({}_{\text{2}}\) & 1.292 pF \\ Coupler intrinsic resistance & R\({}_{\text{TX}}\), R\({}_{\text{RX}}\) & 1 \(\Omega\) \\ Tx’s lumped L & L\({}_{\text{rr}}\) & 56 \(\mu\)H \\ Rx’s lumped L & L\({}_{\text{rr}}\) & 350 \(\mu\)H \\ Tx’s impedance-matching L & L\({}_{\text{tm}}\) & 4.3 \(\mu\)H \\ Rx’s impedance-matching L & L\({}_{\text{rm}}\) & 650 nH \\ Tx’s impedance-matching C & C\({}_{\text{tm}}\) & 100 pF \\ Rx’s impedance-matching C & C\({}_{\text{rm}}\) & 750 pF \\ Mutual capacitance & C\({}_{\text{M}}\) & 0.43 pF \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Equivalent parameters of 4-plate parallel DCPT. \[\left\{ \begin{array}{l}\left[\mathrm{T}\right]_{\mathrm{a}}=\left[\begin{array}{ cc}1&0\\ j\omega\mathrm{C}_{\mathrm{tm}}&1\end{array}\right]\\ \left[\mathrm{T}\right]_{\mathrm{b}}=\left[\begin{array}{cc}1&\mathrm{j} \omega(\mathrm{L}_{\mathrm{tm}}+\mathrm{L}_{\mathrm{tr}})\\ 0&1\end{array}\right]\\ \left[\mathrm{T}\right]_{\mathrm{c}}=\left[\begin{array}{cc}-\frac{\mathrm{C} _{2}}{\mathrm{C}_{\mathrm{M}}}&-\frac{1}{j\omega\mathrm{C}_{\mathrm{M}}}\\ \mathrm{j}\omega\Big{(}3\mathrm{C}_{\mathrm{M}}+2(\mathrm{C}_{1}+\mathrm{C}_{2 })+\frac{\mathrm{C}_{1}\mathrm{C}_{2}}{\mathrm{C}_{\mathrm{M}}}\Big{)}+ \mathrm{R}_{\mathrm{TX}}+\mathrm{R}_{\mathrm{RX}}&2+\frac{\mathrm{C}_{1}}{ \mathrm{C}_{\mathrm{M}}}\end{array}\right]\\ \left[\mathrm{T}\right]_{\mathrm{d}}=\left[\begin{array}{cc}1-\omega^{2} \mathrm{L}_{\mathrm{rr}}\mathrm{C}_{\mathrm{RX}}&\mathrm{j}\omega\big{(} \mathrm{L}_{\mathrm{rr}}+\mathrm{L}_{\mathrm{rm}}-\omega^{2}(\mathrm{C}_{ \mathrm{RX}}\mathrm{L}_{\mathrm{rr}}\mathrm{L}_{\mathrm{rm}})\big{)}\\ \mathrm{j}\omega\mathrm{C}_{\mathrm{RX}}&1-\omega^{2}\mathrm{C}_{\mathrm{RX}} \mathrm{L}_{\mathrm{rm}}\end{array}\right]\end{array} \right] \tag{3}\]  
According to the formula for converting a transmission matrix into a scattering matrix, the equations can be simplified using the following relationships.