## 2 Analysis of MCRWPT System for RMINS  
### Principle of MCRWPT System  
Equivalent circuit theory is applied to research the MCRWPT system based on the two-coil model with SS structure in this paper [26]. The equivalent circuit model is shown in Figure 1, where the \(U_{S}\) represents the alternating current power supply. The internal resistance \(R_{S}\) of the power supply and resistance of the transmitting coil constitute equivalent resistance \(R_{1}\) of the transmitter. \(C_{1}\) and \(L_{1}\) represent the compensation capacitance and inductance of the transmitter, respectively. The compensation capacitance and inductance of the receiver are denoted by the symbols \(C_{2}\) and \(L_{2}\), respectively. The symbol \(R_{2}\) is the equivalent resistance of the receiver, and \(R_{L}\) is the equivalent load resistance. The mutual inductance between the transmitting and receiving winding is denoted as \(M\). \(i_{1}\) and \(i_{2}\) are the transmitter and receiver currents, respectively. The relationship between the mutual inductance \(M\) and the coupling coefficient \(k\) is shown in Equation (5). According to [PERSON]'s law, when the power source \(U_{S}\) outputs alternating voltage at the frequency of \(f=\omega/(2\pi)\), the equivalent circuit equation under the sinusoidal steady state of the system is obtained by Equations (1) and (2). \[Z_{1}\cdot i_{1}-j\omega M\cdot i_{2}=U_{S} \tag{1}\]  
\[Z_{2}\cdot i_{2}-j\omega M\cdot i_{1}=0 \tag{2}\]  
\[Z_{1}=R_{1}+j\omega L_{1}+\frac{1}{j\omega C_{1}} \tag{3}\]  
Figure 1: Equivalent circuit of the MCRWPT system. \[Z_{2}=R_{2}+R_{L}+j\omega L_{2}+\frac{1}{j\omega C_{2}} \tag{4}\]  
\[M=k\cdot\sqrt{L_{1}\cdot L_{2}} \tag{5}\]  
The voltage gain \(G_{V}\), output power \(P_{OUT}\), and PTE \(\eta_{WPT}\) are derived from Equations (1)-(4):  
\[G_{V}=\frac{\omega MR_{L}}{Z_{1}\Big{(}Z_{2}+\frac{\omega^{2}M^{2}}{Z_{1}} \Big{)}} \tag{6}\]  
\[P_{OUT}=\frac{\omega^{2}M^{2}R_{L}U_{S}{}^{2}}{\Big{(}Z_{1}+\frac{\omega^{2}M^ {2}}{Z_{2}}\Big{)}^{2}\Big{(}(R_{2}+R_{L})^{2}+\Big{(}\omega L_{2}-\frac{1}{ \omega C_{2}}\Big{)}^{2}\Big{)}} \tag{7}\]  
\[\eta_{WPT}=\frac{\omega^{2}M^{2}R_{L}}{Z_{2}{}^{2}\Big{(}Z_{1}+\frac{\omega^{2 }M^{2}}{Z_{2}}\Big{)}} \tag{8}\]  
When the MCRWPT system works in the resonance state, there are:  
\[\omega L_{1}-\frac{1}{\omega C_{1}}=\omega L_{2}-\frac{1}{\omega C_{2}}=0 \tag{9}\]  
Substituting Equations (3)-(5) in Equations (6) and (8) with the parameters listed in Table 1, the voltage gain and PTE as functions of equivalent load resistance, excitation frequency, and coupling coefficient are calculated. In Table 1, the coil inductance refers to the primary coil parameter of power transmitter design A10 that complies with the Qi wireless charging standard. When the input voltage \(U_{S}\) amplitude is set to 19 V, the output voltage \(U_{WPT}\) and PTE are plotted in \[Z_{2}\cdot i_{2}-j\omega M\cdot i_{1}=0 \tag{2}\]0 The formula for calculating the output voltage is \(U_{WPT}=G_{V}\cdot U_{S}\). Figure 2a shows that the amplitude of output voltage increases with the increase of the equivalent load resistance under the conditions that the excitation frequency is 100 kHz and the coupling coefficient is 0.8. It can be seen from Figure 2b that the output voltage increases first and then decreases with the increase of the excitation frequency under the conditions that \(R_{L}\) is 50 \(\Omega\) and \(k\) is 0.8. The maximum output voltage occurs at the resonant frequency of 100 kHz. The variation trend of output voltage with coupling coefficient is shown in Figure 2c. In the resonant state, the output voltage increases first and then decreases with the increase of coupling coefficient. Therefore, the output voltage of MCRWPT is affected by the equivalent load resistance, excitation frequency, and coupling coefficient. If the \(R_{L}\) changes, the output voltage will not be stabilized at a constant value. The PTE variation trend with the increasing equivalent load resistance is shown in Figure 2d.