## 2 Model of Four-Plate Parallel DCPT Coupler
### Equivalent Circuit Model of Four-Plate Parallel DCPT
By representing the elements of the \(2\times 2\) transmission matrix as \(t\) and applying the substitution in Equation (4) to simplify each element of the converted scattering matrix, it can be expressed as shown in Equation (5). To represent the scattering matrix for each circuit network, as described in Equation (3), the parameters required to express it in the form of Equation (5) are calculated and presented in Table 3. Consequently, as shown in Equation (6), the scattering matrix of the entire equivalent circuit is calculated by multiplying the scattering matrices of each circuit network. \[\mathrm{t}_{\mathrm{n}}=\mathrm{t}_{11}-\mathrm{t}_{22},\quad\mathrm{t}_{ \mathrm{m}}=\mathrm{t}_{11}+\mathrm{t}_{22},\quad\mathrm{t}_{\mathrm{i}}= \frac{\mathrm{t}_{12}}{Z_{0}}-\mathrm{t}_{21}Z_{0},\quad\mathrm{t}_{\mathrm{j} }=\frac{\mathrm{t}_{12}}{Z_{0}}+\mathrm{t}_{21}Z_{0} \tag{4}\]  
\[\left[\mathrm{S}\right]=\left[ \begin{array}{cc}\frac{\mathrm{t}_{\mathrm{n} }+\mathrm{t}_{\mathrm{i}}}{\mathrm{t}_{\mathrm{m}}+\mathrm{t}_{\mathrm{i}}}& \frac{2(\mathrm{t}_{11}\mathrm{t}_{22}-\mathrm{t}_{22}\mathrm{t}_{2})}{ \mathrm{t}_{1}+\mathrm{t}_{\mathrm{i}}}\\ \frac{2}{\mathrm{t}_{\mathrm{i}}}&\frac{2}{\mathrm{t}_{\mathrm{i}}}&\frac{ \mathrm{t}_{\mathrm{i}}}{\mathrm{t}_{\mathrm{m}}+\mathrm{t}_{\mathrm{i}}}\\ \end{array} \right] \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Matrix** & **Parameter** \\ \hline \(\left[\mathrm{S}\right]_{\mathrm{a}}\) & \(\left\{\begin{array}{l}\mathrm{t}_{\mathrm{n}}=0\\ \mathrm{t}_{\mathrm{m}}=2\\ \mathrm{t}_{\mathrm{i}}=-\mathrm{j}\omega\mathrm{C}_{\mathrm{tm}}Z_{0}\\ \mathrm{t}_{\mathrm{j}}=j\omega\mathrm{C}_{\mathrm{tm}}Z_{0}\end{array}\right.\) \\ \hline \(\left[\mathrm{S}\right]_{\mathrm{b}}\) & \(\left\{\begin{array}{l}\mathrm{t}_{\mathrm{n}}=0\\ \mathrm{t}_{\mathrm{m}}=2\\ \mathrm{t}_{\mathrm{i}}=\frac{|\omega(\mathrm{L}_{\mathrm{rr}}+\mathrm{L}_{ \mathrm{rr}})}{Z_{0}}\\ \mathrm{t}_{\mathrm{j}}=\frac{|\omega(\mathrm{L}_{\mathrm{rr}}+\mathrm{L}_{ \mathrm{rr}})}{Z_{0}}\end{array}\right.\) \\ \hline \(\left[\mathrm{S}\right]_{\mathrm{c}}\) & \(\left\{\begin{array}{l}\mathrm{t}_{\mathrm{n}}=-\frac{\mathrm{C}_{1}+\mathrm{C} _{2}}{\mathrm{C}_{\mathrm{M}}}-2\\ \mathrm{t}_{\mathrm{m}}=\frac{\mathrm{C}_{1}-\mathrm{C}_{2}}{\mathrm{C}_{ \mathrm{M}}}+2\\ \mathrm{t}_{\mathrm{i}}=-\frac{1}{|\omega\mathrm{C}_{\mathrm{M}}Z_{0}}-Z_{0} \Big{(}i\omega\mathrm{C}_{\mathrm{M}}\Big{(}\Big{(}\frac{\mathrm{C}_{1}}{ \mathrm{C}_{\mathrm{M}}}+2\Big{)}\Big{(}\frac{\mathrm{C}_{2}}{\mathrm{C}_{ \mathrm{M}}}+2\Big{)}-1\Big{)}+\mathrm{R}_{\mathrm{TX}}+\mathrm{R}_{\mathrm{RX}} \Big{)}\\ \mathrm{t}_{\mathrm{j}}=-\frac{1}{|\omega\mathrm{C}_{\mathrm{M}}Z_{0}}+Z_{0} \Big{(}i\omega\mathrm{C}_{\mathrm{M}}\Big{(}\Big{(}\frac{\mathrm{C}_{1}}{ \mathrm{C}_{\mathrm{M}}}+2\Big{)}\Big{(}\frac{\mathrm{C}_{2}}{\mathrm{C}_{ \mathrm{M}}}+2\Big{)}-1\Big{)}+\mathrm{R}_{\mathrm{TX}}+\mathrm{R}_{\mathrm{ RX}}\Big{)}\end{array}\right.\) \\ \hline \(\left[\mathrm{S}\right]_{\mathrm{d}}\) & \(\left\{\begin{array}{l}\mathrm{t}_{\mathrm{n}}=\omega^{2}\mathrm{C}_{ \mathrm{RX}}(\mathrm{L}_{\mathrm{rm}}-\mathrm{L}_{\mathrm{rr}})\\ \mathrm{t}_{\mathrm{m}}=2-\omega^{2}\mathrm{C}_{\mathrm{RX}}(\mathrm{L}_{ \mathrm{rm}}+\mathrm{L}_{\mathrm{rr}})\\ \mathrm{t}_{\mathrm{i}}=\frac{|\omega\big{(}\mathrm{L}_{\mathrm{rr}}+\mathrm{L}_{ \mathrm{rr}}-\omega^{2}\mathrm{L}_{\mathrm{rr}}\mathrm{L}_{\mathrm{rr}}\mathrm{ C}_{\mathrm{RX}}\big{)}-|\omega\mathrm{C}_{\mathrm{RX}}Z_{0}\\ \mathrm{t}_{\mathrm{j}}=\frac{|\omega\big{(}\mathrm{L}_{\mathrm{rr}}+\mathrm{L}_{ \mathrm{rr}}-\omega^{2}\mathrm{L}_{\mathrm{rr}}\mathrm{L}_{\mathrm{rr}}\mathrm{ C}_{\mathrm{RX}}\big{)}+|\omega\mathrm{C}_{\mathrm{RX}}Z_{0}\end{array}\right.\) \\ \hline \hline \end{tabular} (6)  
\end{table} 
Table 3: Parameters of the S-matrix of each detailed circuit network.