## 2 Models and Methods
\begin{array}{l}u_{\rm inv}=j\omega L_{\rm P1}i_{\rm Lp1}+\Big{(}j \omega L_{\rm L1}+\frac{1}{j\omega C_{\rm P2}}+R_{1}\Big{)}i_{\rm L1}+j\omega Mi _{\rm L2}\\ u_{\rm inv}=j\omega L_{\rm P1}i_{\rm Lp1}+\frac{1}{j\omega C_{\rm P1}^{2}} \big{(}i_{\rm Lp1}-i_{\rm L1}\big{)}\\ -j\omega Mi_{\rm L1}=\Big{(}j\omega L_{\rm 2}+\frac{1}{j\omega C_{\rm s1}}+R_{ \rm e}+R_{\rm 2}\Big{)}i_{\rm L2}\\ \end{array} \right. \tag{3}\]  
In the LCC-S compensated topology of the WPT system, the primary side and secondary side are typically in the resonant states, which means \(\omega L_{\rm P1}=1/\omega C_{\rm P1}=\omega L_{\rm 1}-1/\omega C_{\rm P2}\) and \(\omega L_{\rm 2}=1/\omega C_{\rm S}\). Under the resonant conditions, the inverter output current \(i_{\rm Lp1}\), the rectifier input voltage \(u_{\rm rec}\), and the current in the transmit coil \(i_{\rm L1}\) can be calculated based on the KVL equation and given by (4). \[\left\{ \begin{array}{l}i_{\rm Lp1}=\frac{\big{(}\omega^{2}M^{2}+R_{1}R_{2}+R _{1}R_{\rm e}\big{)}u_{\rm inv}}{\omega^{2}l_{\rm P1}^{2}(R_{2}+R_{\rm e})}\\ u_{\rm rec}=\frac{\omega Mi_{\rm inv}}{\omega L_{\rm p1}(R_{2}+R_{\rm e})}\\ i_{\rm L1}=\frac{u_{\rm inv}}{j\omega L_{\rm P1}}\\ \end{array} \right. \tag{4}\]  
Moreover, according to the above equations, the output power \(P_{\rm out}\) and the input power \(P_{\rm in}\) can be given by (5). \[\left\{ \begin{array}{l}P_{\rm in}=\frac{\big{(}\omega^{2}M^{2}+R_{1}R_{2}+R _{1}R_{\rm e}\big{)}}{(R_{2}+R_{\rm e})\omega^{2}l_{\rm P1}^{2}}U_{\rm inv}^{2 }\\ P_{\rm out}=\frac{R_{\rm e}M^{2}}{l_{\rm P1}^{2}(R_{\rm e}+R_{2})^{2}}U_{\rm inv }^{2}\\ \end{array} \right. \tag{5}\]  
Therefore, the expression of the system transmission efficiency \(\eta\) can be derived as  
\[\eta=\frac{P_{\rm out}}{P_{\rm in}}=\frac{\omega^{2}M^{2}R_{\rm e}}{(R_{\rm e }+R_{\rm 2})(\omega^{2}M^{2}+R_{1}R_{\rm 2}+R_{1}R_{\rm e})} \tag{6}\]  
By solving the derivative of \(\eta\) over \(R_{\rm e}\), the optimal load condition can be obtained from (7) in order to achieve the maximum transmission efficiency. \[\left\{ \begin{array}{l}\frac{{\rm d}\eta}{{\rm d}R_{\rm e}}=0\\ \frac{{\rm d}^{2}\eta}{{\rm d}R_{\rm e}^{2}}<0\end{array} \right.\to R_{\rm e \_\eta max}=\sqrt{\omega^{2}M^{2}\frac{R_{\rm 2}}{R_{\rm 1}}+R_{\rm 2}^{2}} \tag{7}\]  
Furthermore, the transmission efficiency is related to the mutual inductance \(M\) as shown in (6) and (7). In practice, mutual inductance \(M\) will change due to factors such as the coil misalignment during the charging process. Therefore, real-time estimation of the mutual inductance is crucial for improving system efficiency. In order to estimate the  
Figure 2: WPT equivalent circuit model with LCC-S compensation network. mutual inductance, the expression of the rectifier output voltage can be derived from (1) and (4) and given by (8). \[U_{\rm f}=\frac{U_{\rm dc}\cos\left(\frac{\varphi}{2}\right)MR_{\rm e}}{(R_{\rm e }+R_{2})L_{\rm p1}} \tag{8}\]  
According to (8), the expression for the mutual inductance \(M\) can be obtained as  
\[M=\frac{U_{\rm f}(R_{\rm e}+R_{2})L_{\rm p1}}{U_{\rm dc}\cos\left(\frac{\varphi} {2}\right)R_{\rm e}} \tag{9}\]  
Substituting (2) into (9), the relationship between the mutual inductance \(M\) and the system load resistance \(R_{\rm L}\) can be given by  
\[M=\frac{U_{\rm f}L_{\rm p1}\left(8U_{\rm f}^{2}R_{\rm L}+\pi^{2}U_{\rm out}^{2 }R_{2}\right)}{8U_{\rm dc}U_{\rm f}^{2}R_{\rm L}\cos\left(\frac{\varphi}{2} \right)} \tag{10}\]  
Equation (10) indicates that the mutual inductance \(M\) is related to the DC input voltage \(U_{\rm dc}\), the rectifier output voltage \(U_{\rm f}\), the inverter phase shift angle \(\varphi\), etc. In practical applications, the primary side compensation inductance \(L_{\rm p1}\) and the receive coil resistance \(R_{2}\) are usually constant. Meanwhile, the parameter values of the DC voltage \(U_{\rm dc}\), load resistance \(R_{\rm L}\), and the inverter phase shift angle \(\varphi\) are also known.