## 2 Model of Four-Plate Parallel DCPT Coupler
### Equivalent Circuit Model of Four-Plate Parallel DCPT  
Figure 5 illustrates the equivalent circuit, including the CPT impedance-matching circuit. Figure 5a represents the equivalent circuit that incorporates the parasitic components \begin{table}
\begin{tabular}{c c c} \hline \hline
**Components** & **Notation** & **Value** \\ \hline Width of Tx and Rx substrate & L & 90 mm \\ Length of Tx substrate & TW & 165 mm \\ Length of Rx substrate & RW & 75 mm \\ Width of P1-P4 & a & 30 mm \\ Length of P1, P2 & b & 120 mm \\ Distance between metal plates of P1 and P2 & c & 10 mm \\ X-axis length from the metal plate to the origin & d & 35 mm \\ Distance between Tx and Rx & gap & 10 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of 4-plate parallel DCPT coupler. Figure 4: Configuration of 4-plate parallel DCPT coupler. (**a**) Overall structure; (**b**) lumped port and matching circuit. of the elements and the inherent loss resistance of the coupler. The parasitic capacitances \(\rm C_{tmp}\), \(\rm C_{tmp}\), \(\rm C_{tmp}\), and \(\rm C_{rrp}\) are negligible. Similarly, the parasitic resistances \(\rm R_{tm}\), \(\rm R_{tr}\), and \(\rm R_{rr}\) are extremely small and are, therefore, combined with the loss resistance of the coupler for calculation. Consequently, the equivalent circuit shown in Figure 5b is used for further calculations. The self-capacitances \(\rm C_{TX}\) and \(\rm C_{RX}\) and the mutual capacitance \(\rm C_{M}\) of the four-plate parallel CPT can be calculated using Equation (1) [44]. Descriptions and values for each parameter in Figure 5b are provided in Table 2. The values of the resonant inductors and matching circuit components are fixed to a single value for use in electromagnetic analysis. The values in Table 2 are the results of matching when the receiver is in its original position, as shown in Figure 4.  
When the receiver moves along the x-axis or y-axis, changes in the impedance and coupling degree may cause variations in self-capacitance and mutual capacitance. The self-capacitance of the coupler is calculated based on the impedance obtained from electromagnetic analysis. This is determined using formulas related to the resonant frequency and LC circuit characteristics. The capacitance values are derived by substituting the imaginary part of the Z-parameters in the target frequency range, and these values are applied to the equivalent parameters. Since the obtained values include the effects of mutual capacitance,  
Figure 5: Practical equivalent circuit model of four-plate parallel DCPT coupler. (**a**) Equivalent circuit indicating parasitic components; (**b**) simplified practical equivalent circuit (PEC). a separate process to exclude mutual capacitance, as described in the calculation using Equation (1), is necessary. The PEC model considers the loss resistance of the coupler structure. R\({}_{\text{TX}}\) and R\({}_{\text{RX}}\) are determined through an optimization process by comparing the power transfer efficiency of the equivalent circuit model with the electromagnetic simulation. These values are then applied to the circuit analysis for cases where the receiver moves. \[\left\{ \begin{aligned} \text{C}_{\text{M}}&=\frac{ \text{C}_{\text{24}}\text{C}_{\text{13}}-\text{C}_{\text{14}}\text{C}_{\text{2 3}}}{\text{C}_{\text{13}}+\text{C}_{\text{14}}+\text{C}_{\text{23}}+\text{C}_ {\text{24}}}\\ \text{C}_{\text{TX}}&=\text{C}_{\text{1}}-\text{C}_{ \text{M}}=\text{C}_{\text{12}}+\frac{(\text{C}_{\text{13}}+\text{C}_{\text{14 }})-(\text{C}_{\text{23}}+\text{C}_{\text{24}})}{\text{C}_{\text{13}}+\text{C} _{\text{14}}+\text{C}_{\text{23}}+\text{C}_{\text{24}}}-\text{C}_{\text{M}}\\ \text{C}_{\text{RX}}&=\text{C}_{\text{2}}-\text{C}_{ \text{M}}=\text{C}_{\text{34}}+\frac{(\text{C}_{\text{13}}+\text{C}_{\text{23} })-(\text{C}_{\text{14}}+\text{C}_{\text{24}})}{\text{C}_{\text{13}}+\text{C} _{\text{14}}+\text{C}_{\text{23}}+\text{C}_{\text{24}}}-\text{C}_{\text{M}} \end{aligned} \right.