## 2 Design of Underwater Stacked Capacitive Power Transfer Coupler
### Theoretical Design Method Stacked CPT System Underwater
Figure 3 shows the practical equivalent circuit (PEC) model of the underwater stacked CPT system proposed in this study. For proper resonance, the LCL-matching method was applied to the transmitter and receiver. Since the transmitter and receiver of the coupler are designed with the same specifications, all matching elements and lumped elements used in this study are symmetrical. \(L_{m}\) and \(C_{m}\) represent the components of the inductor and capacitor from the matching network, and \(L_{res}\) represents the lumped inductor component to resonate the system at the desired frequency. The water conductive resistance \(R_{w}\) in the underwater stacked CPT coupler needs to be emphasized for proper PEC design. The water conduction resistance \(R_{w}\) is an additional resistance caused by the medium around the coupler, which is negligibly small for the air CPT system but relatively high for the underwater CPT system. Although \(C_{1}\) and \(C_{2}\) vary depending on the coupling coefficients of the transmitter and receiver, they are mostly composed of self-capacitance components. In the case of the stacked CPT system with air medium, the resistance component connected in series with the self-capacitance component is the resistance of the copper plate itself, and it is small enough to be ignored  
Figure 2: Circuit model of capacitive coupler. (**a**) Basic representation. (**b**) Pi model. in most cases. However, in the case of the underwater stacked CPT system, the resistance component connected in the series with self-capacitance has a very large value compared to the air medium. This is the resistance generated in the electric field generation path through the water medium formed around the back plate and front plate that generate the self-capacitance of the coupler. All the coupler equivalent circuit components used in the PEC analysis were measured by an RLC meter such as C\({}_{m}\), C\({}_{1}\), C\({}_{2}\), and \(R_{w}\). Since the coupler equivalent circuit components vary with the distance between the transmitter and receiver, RLC meter measurements were performed at 10 mm intervals up to a maximum distance of 10 to 200 mm. The values for each parameter in Figure 3 are provided in Table 1. The model parameters measured with the RLC meter only show the values when the distance between the transmitter and receiver is 20 mm. The RLC meter used was HIOKI IM3536, and measurements were made at a frequency of 6.78 MHz. The PEC model S-parameters can be obtained by transforming the three networks expressed in the transmission matrix as Equations (2) and (3). \[\left\{ \begin{array}{c}\left[T\right]_{a}=\left[T\right]_{c}=\left[1-\omega ^{2}L_{res}C_{m}\quad j\omega\left(L_{res}+L_{m}-\omega^{2}L_{res}C_{m}L_{m} \right)\right]\\ \left[T\right]_{b}=\left[\begin{array}{cc}-C_{2}/C_{M}&-1/j\omega C_{M}\\ j\omega\left(3C_{M}+2(C_{1}+C_{2})+(C_{1}C_{2}/C_{M})\right)+2R_{w}\quad 2+C_{1}/C_{M} \end{array}\right]\\ \\ \end{array} \right. \tag{2}\]  
\[\left[S\right]= \begin{bmatrix}\frac{t_{n}+t_{j}}{t_{n}+t_{j}}&\frac{2(t_{11}t_ {22}+t_{12}t_{21})}{t_{n}+t_{j}}\\ \frac{t_{i}+t_{i}}{t_{m}+t_{j}}&\frac{t_{i}+t_{j}}{t_{m}+t_{j}}\end{bmatrix}  \text{where }\left\{ \begin{array}{c}t_{n}=t_{11}-t_{22}\\ t_{m}=t_{11}+t_{22}\\ t_{i}=t_{12}/Z_{0}-t_{21}Z_{0}\\ t_{j}=t_{12}/Z_{0}+t_{21}Z_{0}\end{array} \right. \tag{3}\]  
The transmission matrix is a mathematical expression used to analyze transmission lines and circuits, which describes how voltages and currents change between the input \begin{table}
\begin{tabular}{c c c} \hline \hline
**Matching Component** & **Parameter** & **Value** \\ \hline Matching L & \(L_{m}\) & 110 nH \\ Matching C & \(C_{m}\) & 300 pF \\ Lumped L & \(L_{res}\) & 10.2 uH \\ Water conduction resistance & \(R_{w}\) (at d = 20 mm) & 14 \(\Omega\) \\ Mutual capacitance & \(C_{m}\) (at d = 20 mm) & 3.47 pF \\ Self-capacitance & \(C_{1}\), \(C_{2}\) (at d = 20 mm) & 47.6 pF \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of fabricated underwater stacked CPT system.