## 2 Coupler Structure and Circuit Topology  
For the frame-type underwater vehicle, Figure 1a shows the overall structure of the WPT system. The transmitter is fixed on the underwater base station, while the receiver is carried by an AUV. Figure 1b shows the WPT coupler structure suitable for framed underwater vehicles, the ferrite core is located at the bottom of the coil, and the coupler shell is made of non-metallic material. The epoxy resin potted inside is used to fix the coil and the iron core and to ensure that the coupler does not leak water.  
For wireless power transfer systems, different circuit topologies have different output characteristics. For example, series and series (S-S) compensation topology has constant current output characteristic, inductor-capacitor-capacitor and series (LCC-S) compensation topology has constant voltage output characteristic. Most secondary batteries (Li-ion, LiPo) need a constant-current, constant-voltage charging scheme. In practical applications, the battery can be charged directly using the constant-voltage mode [21]. Considering the limitation of the internal space of AUV, this paper is based on the LCC-S compensation topology. The topology and the equivalent circuit of a wireless transfer system with LCC-S compensation are shown in Figure 2.  
In Figure 2, \(V_{in}\) and \(V_{p}\) are DC input and AC output voltages of the inverter, respectively. \(Q_{1}\)-\(Q_{4}\) are four power MOSFETs of the inverter. \(L_{f1}\), \(C_{f1}\) and \(C_{p}\) are the primary-side compensation inductor and capacitors, respectively. \(C_{s}\) is the secondary-side compensation capacitor. \(L_{p}\), \(L_{s}\) and \(M\) denote the self-inductances and mutual inductance of the coils. \(D_{1}\)-\(D_{4}\) are four diodes of the full-bridge rectifier, and \(C_{D}\) is the filter capacitor. \(R_{p}\) and \(R_{s}\) are the AC resistance of the coils. \(R\) and \(R_{L}\) denote the load resistance and the equivalent load resistance after the secondary side compensation, respectively.  
According to the circuit equivalent relationship, in Figure 2b, \(V_{p}\) and \(R_{L}\) can be expressed as follows:  
\[V_{p}=\frac{2\sqrt{2}}{\pi}V_{in} \tag{1}\]  
\[R_{L}=\frac{8}{\pi^{2}}R \tag{2}\]  
Figure 1: Underwater vehicle with wireless charging system. (**a**) Overall structure. (**b**) WPT coupler structure.  
Figure 2: The LCC-S-compensated WPT system. (**a**) Topology. (**b**) Equivalent circuit.  
According to KVL and KCL equations, the constraint equation of the equivalent circuit in Figure 2b can be established by  
\[\left\{\begin{array}{l}j\omega L_{f1}L_{f1}+\frac{l_{Cf1}}{j\omega C_{f1}}=V_{p }\\ I_{Cf1}+I_{p}=I_{Lf1}\\ \frac{l_{Cf1}}{j\omega C_{f1}}=\left(\frac{1}{j\omega C_{p}}+j\omega L_{p}+R_{ p}\right)I_{p}+j\omega MI_{s}\\ \left(R_{s}+\frac{1}{j\omega C_{s}}+j\omega L_{s}+R_{L}\right)I_{s}+j\omega MI_ {p}=0\end{array}\right. \tag{3}\]  
where \(\omega\) is the operating angular frequency. We defined \(\omega_{0}\) as the resonant angular frequency when the WPT system works at the resonant frequency, the output power and transmission efficiency can be calculated as  
\[P_{out}=I_{s}^{2}R_{L}=\frac{\omega_{0}^{4}M^{2}C_{f1}^{2}V_{p}^{2}R_{L}}{\left( R_{s}+R_{L}\right)^{2}} \tag{4}\]  
\[\eta=\frac{I_{s}^{2}R_{L}}{V_{p}I_{Lf1}}=\frac{\omega_{0}^{2}M^{2}R_{L}}{\left( R_{s}+R_{L}\right)^{2}\left(R_{p}+R_{ref}\right)} \tag{5}\]  
where \(R_{ref}\) denotes the reflection impedance from the secondary side to the primary side, it can be expressed by the following:  
\[R_{ref}=\frac{\omega_{0}^{2}M^{2}}{R_{s}+R_{L}} \tag{6}\]  
It can be seen that the output power and efficiency are proportional to the square of the mutual inductance. The loss caused by the resistance of the coupler will reduce the output power and efficiency. Therefore, the mutual inductance and loss of the coupler jointly determine the performance of the coupler.