## 2 Circuit Topology and Analysis of MCR-WPT System  
In Figure 2, the equivalent circuit structure of the MCR-WPT system is presented. The resonant inductances of the transmit and receive coils are denoted by \(L_{S}\) and \(L_{W}\), respectively. The compensated resonant capacitances (including parasitic and compensating capacitances) of the coil system are represented by \(C_{S}\) and \(C_{W}\). The mutual inductance coefficient between the transmit and receive coils is indicated by \(M\). The equivalent load resistance is \(R_{L}\), while the equivalent resistances of the transmit and receive coils are \(R_{S}\) and \(R_{W}\), respectively. \(U_{S}\) stands for the high-frequency AC input power supply, while \(I\)s and \(I_{W}\) denote the currents of the \(\mathrm{T_{X}}\) and \(\mathrm{R_{X}}\) coils in the circuit, respectively. Additionally, \(R_{S}\) refers to the equivalent internal resistance of the power supply, and \(\omega\) represents the driving angular frequency of the alternating power supply. Based on the equivalent circuit structure diagram depicted in Figure 2 and [PERSON]'s voltage law, the circuit equation for the MCR-WPT system can be derived, as shown in Equation (1). \[ \begin{bmatrix}\dot{U}s\\ 0\end{bmatrix} = \begin{bmatrix}Z_{S}&j\omega M\\ j\omega M&Z_{W}\end{bmatrix} \cdot \begin{bmatrix}\dot{I}_{S}\\ \dot{I}_{W}\end{bmatrix}  \tag{1}\]  
\[\dot{Z}_{s}=R_{S}+j\omega L_{S}+\frac{1}{j\omega C_{S}}\qquad\dot{Z}_{w}=R_{L }+R_{W}+j\omega L_{W}+\frac{1}{j\omega C_{W}} \tag{2}\]  
Phase \(U\)s represents the fundamental voltage harmonic of the AC supply that powers the transmitter coil. The harmonic currents of the transmitting and receiving coils are represented by \(\dot{I}_{s}\) and \(\dot{I}_{w}\), respectively. \(\dot{Z}_{s}\) and \(Z_{w}\) represent the equivalent impedances of the transmit resonant circuit and the receive resonant circuit, respectively. Figure 2: Schematic diagram of MCR-WPT circuit. Based on Equations (1) and (2), the AC power input (Pin) and load output power (Pout) can be computed as follows:  
\[P_{in}=\frac{{{U_{s}}^{2}}}{{{Re}\big{(}{Z_{S}^{\prime}}\big{)}}} =\frac{{{U_{s}}^{2}}\Big{(}R_{W}+R_{L}+j\Big{(}\omega{L_{W}}-\frac{1}{ \omega{\mathbb{C}_{W}}}\Big{)}\Big{)}}{{\Big{(}R_{S}+j\Big{(}\omega{L_{S}}- \frac{1}{\omega{\mathbb{C}_{S}}}\Big{)}\Big{)}\Big{(}R_{W}+R_{L}+j\Big{(} \omega{L_{W}}-\frac{1}{\omega{\mathbb{C}_{W}}}\Big{)}\Big{)}+\left(\omega{M} \right)^{2}}} \tag{3}\]  
\[P_{out}={{I_{W}}^{2}}{{R_{L}}}=\frac{{\omega^{2}{M}^{2}}{{M}^{2}}{{R_{L}}{U_{ s}}^{2}}}{{\Big{(}\Big{(}R_{S}+j\Big{(}\omega{L_{S}}-\frac{1}{\omega{\mathbb{C}_{S} }}\Big{)}\Big{)}\Big{(}R_{W}+R_{L}+j\Big{(}\omega{L_{W}}-\frac{1}{\omega{ \mathbb{C}_{W}}}\Big{)}\Big{)}+\left(\omega{M}\right)^{2}}}} \tag{4}\]  
Then, the transmission efficiency (n) of the system can be evaluated using Equation (5):  
\[\eta=\frac{{{P_{out}}}}{{{P_{in}}}}=\frac{{{R_{L}}}}{{{\Big{(}R_{W}+R_{L}+j \Big{(}\omega{L_{W}}-\frac{1}{\omega{\mathbb{C}_{W}}}\Big{)}\Big{)}+\Big{(}R_ {S}+j\Big{(}\omega{L_{S}}-\frac{1}{\omega{\mathbb{C}_{S}}}\Big{)}\Big{)}\Big{(} \frac{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{  
is then transmitted to the receiving load through the secondary compensation network in the receiving loop. The relevant electrical parameters in this system can be obtained by measurement and calculation. Based on the above design of the block diagram, we built an MCR-WPT experimental system using HTS coils and copper coils. An experimental setup was constructed as shown in Figure 4. This is a hybrid HTS-Cu WPT system using HTS Tx coils and copper Rx coils. In addition, a control experiment was also conducted for two copper-coil WPT systems. Based on the frequency characteristics study of the superconducting wireless power transfer (WPT) system in our laboratory and the practical requirements for the topology capacitance value in the resonant experiment circuit [10], we have chosen the operating resonant frequency as 220.5 kHz.