## 2 Design of Underwater Stacked Capacitive Power Transfer Coupler  
### Theoretical Design Method Stacked CPT System Underwater  
Unlike the four-plate parallel CPT structure in which the copper plates of the transmitter and receiver are horizontally arranged, the stacked CPT structure has them vertically stacked for each coupler [9]. When the parallel CPT coupler and the stacked CPT coupler have the same copper plate area, the stacked CPT coupler has a relatively lower mutual capacitance under the same transfer distance, and the ratio of self-capacitance to the total capacitance is high. When the parallel and stacked CPT structure have the same copper plate size, the stacked CPT coupler has a relatively lower mutual capacitance under the same transfer distance since the self-capacitance accounts for a high proportion from the total capacitance generated in the system. However, the stacked CPT coupler enables a more compact and space-efficient system designing compared with a parallel CPT coupler when transmitting the same amount of power. It can be appropriately used in applications where the transmitter and receiver are fixed close enough. Figure 1 shows the configuration of a stacked CPT for wireless power transfer underwater. Copper plates \(P_{1}\), \(P_{2}\), \(P_{3}\), and \(P_{4}\) are vertically stacked to implement a stacked structure, and the copper plates are connected to the transmitter and receiver sides, respectively. Generally, the large area plates (\(P_{1}\), \(P_{3}\)) are called back plates, and the small area plates (\(P_{2}\), \(P_{4}\)) are called front plates. The current generated by AC voltage passes through the matching network and the resonant inductor \(L_{res}\) and generates an electric field in the water. The transmitter and receiver are electrically connected through electric field coupling, and power transfer between the transmitter and receiver is possible. The red dotted line shows the direction of the current for the generated electric field. The matching network is essential for implementing a high efficiency wireless power transfer system because it can minimize the power reflected from the receiver by matching the impedances of the transmitter and receiver. Figure 1: Underwater stacked CPT system. In order to design an appropriate matching network, it is necessary to obtain an equivalent circuit to the represent coupler. As shown in Figure 2a, the general capacitive coupler can be expressed by mutual capacitance \(C_{M}\) and equivalent capacitances \(C_{1}\) and \(C_{2}\) on both sides. Since a total of six electric field couplings can be generated in a four-plate CPT coupler, the equivalent capacitance components and mutual capacitance can be represented as six electric field couplings using Equation (1). \[ \begin{cases}C_{M}=\frac{C_{13}C_{24}-C_{14}C_{23}}{C_{13}+C_{14}+C_{23}+C_{24 }}\\ C_{1}=C_{12}+\frac{(C_{12}+C_{14})(C_{23}+C_{24})}{(C_{13}+C_{14}+C_{23}+C_{24 })}\\ C_{2}=C_{34}+\frac{(C_{12})(C_{14}+C_{23})}{(C_{13}+C_{14}+C_{23}+C_{24})}\\ C_{TX}=C_{1}-C_{M}\\ C_{RX}=C_{2}-C_{M}\end{cases}  \tag{1}\]  
where \(C_{13}\) and \(C_{24}\) represent the capacitance generated between \(P_{1}\) and \(P_{3}\) and the capacitance generated between \(P_{2}\) and \(P_{4}\), respectively, and it is called by main coupling capacitance. Also, \(C_{14}\) and \(C_{23}\) represent the capacitance generated between \(P_{1}\) and \(P_{4}\) and the capacitance generated between \(P_{2}\) and \(P_{3}\), respectively, and it is called by cross-coupling capacitance. Figure 2b shows how the stacked CPT coupler can be expressed in circuit form in the form of a Pi model using the calculated mutual capacitance and equivalent capacitance [10]. The capacitance components of a Pi model vary depending on the distance and misalignment of the transmitter and receiver. Since the stacked CPT system proposed in this study operates underwater, considering the electrical properties of the water medium are essential for calculating the equivalent circuit components.