# A Method to Reduce Eddy Current Loss of Underwater Wireless Power Transmission by Current Control
## 2 Circuit Analysis in the Air
\begin{array}{cccc}&R_{in}+j\omega L_{P1}&\frac{1}{j\omega C_{P2}}&0&0 \\ 0&\frac{1}{j\omega C_{P2}}&R_{P}+j\omega L_{P}+\frac{1}{j\omega C_{P3}}&j\omega M \end{array} \right]\ \cdot\ \left[ \begin{array}{c}\mathbf{I}_{in}\\ \mathbf{I}_{C_{P2}}\\ \mathbf{I}_{P}\\ \mathbf{I}_{S}\end{array} \right]=\left[ \begin{array}{c}\mathbf{U}_{in}\\ 0\end{array} \right] \tag{2}\]  
\[I_{in}=I_{C_{P2}}+I_{P} \tag{3}\]  
Additionally, the equation of the secondary-side circuit can be expressed as  
\[\left[ \begin{array}{cccc}&R_{S}+j\omega L_{S}+\frac{1}{j\omega C_{S3}}& \frac{1}{j\omega C_{S2}}&0&j\omega M\\ 0&\frac{1}{j\omega C_{S2}}&R_{L}+R_{ot}+j\omega L_{S1}&0\end{array} \right] \ \cdot\ \left[ \begin{array}{c}\mathbf{I}_{S}\\ \mathbf{I}_{C_{S2}}\\ \mathbf{I}_{out}\\ \mathbf{I}_{P}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array} \right] \tag{4}\]  
\[I_{S}=I_{C_{S2}}+I_{out} \tag{5}\]  
The resonance condition of the system is satisfied by the following equations:  
\[\left\{ \begin{array}{c}j\omega_{0}L_{P1}+\frac{1}{j\omega_{0}C_{P2}}=0\\ j\omega_{0}L_{P}+\frac{1}{j\omega_{0}C_{P2}}+\frac{1}{j\omega_{0}C_{P3}}=0 \end{array} \right. \tag{6}\]  
\[\left\{ \begin{array}{c}j\omega_{0}L_{S1}+\frac{1}{j\omega_{0}C_{S2}}=0\\ j\omega_{0}L_{S}+\frac{1}{j\omega_{0}C_{S2}}+\frac{1}{j\omega_{0}C_{S3}}=0 \end{array} \right. \tag{7}\]  
Equation (6) is the resonance condition of the primary-side circuit, and Equation (7) is the resonance condition of the secondary-side circuit, where \(\omega_{0}\) is the angular resonant frequency, which is only relevant to the inductors and capacitors in the system. Substitute Equations (6) and (7) into Equations (2)-(5), and the total impedance of the loop on the primary side and the secondary side can be, respectively expressed as  
\[Z_{in}=R_{in}+\frac{\omega_{0}^{2}L_{P1}^{2}}{R_{P}+R_{ref}} \tag{8}\]  
\[Z_{S}=R_{S}+\frac{\omega_{0}^{2}L_{S1}}{R_{L}+R_{ot}} \tag{9}\]  
\[R_{ref}=\frac{\omega_{0}^{2}M^{2}}{Z_{S}} \tag{10}\]  
\(R_{ref}\) is defined as the reflection impedance, which refers to the parameters that describe the impact of the secondary side on the primary side by the equivalent power method when the WPT system is working. Substitute Equations (8)-(10) into Equations (2)-(5), and the current on primary side and secondary side can be expressed as  
\[I_{P}=-\frac{j\omega_{0}L_{P1}}{R_{P}+R_{ref}}I_{in} \tag{11}\]  
\[I_{S}=\frac{j\omega_{0}MI_{P}}{Z_{S}} \tag{12}\]Here, \(I_{in}\) is expressed as  
\[I_{in}=\frac{U_{in}}{Z_{in}} \tag{13}\]  
Additionally, we can obtain the expression of the output current:  
\[I_{out}=-\frac{j\omega_{0}L_{S1}}{R_{L}+R_{ot}}I_{S} \tag{14}\]  
If we take \(I_{P}\) as the reference, \(I_{P}\) and \(I_{S}\) can be expressed as  
\[\mathbf{I}_{P}=I_{P}\angle 0^{\circ} \tag{15}\]  
\[\mathbf{I}_{S}=\frac{\omega_{0}M}{Z_{s}}I_{P}\angle 90^{\circ}=\frac{\omega_{0}M(R _{L}+R_{ot})}{R_{S}(R_{L}+R_{ot})+\omega_{0}^{2}L_{S1}^{2}}I_{P}\angle 90^{\circ} \tag{16}\]  
From Equations (15) and (16), we can see \(\mathbf{I}_{P}\) lags \(\mathbf{I}_{S}\) by \(90^{\circ}\), and the transmission efficiency of system can be calculated as  
\[\eta=\frac{l_{out}^{2}R_{L}}{I_{in}U_{in}}=\frac{\omega_{0}^{2}L_{P1}^{2}}{R_{ in}(R_{P}+R_{ref})+\omega_{0}^{2}L_{P1}^{2}}\frac{R_{ref}}{R_{ref}+R_{P}}\frac{ \omega_{0}^{2}L_{S1}^{2}}{R_{S}(R_{L}+R_{ot})+\omega_{0}^{2}L_{S1}^{2}}\frac{ R_{L}}{R_{L}+R_{ot}} \tag{17}\]  
It can be seen from the calculation results above that when the system works in the air environment and the compensation circuit works in the resonant state, the system has no additional loss, except the inevitable line loss of the circuit. Moreover, because the WPT system works in the resonant state, the power factor reaches the maximum and the line loss is the minimum, which indicates that the energy transmission efficiency of the system will reach the maximum at this time.