## 2 Efficiency Analysis and Derivation of Wireless Power Transmission
### Inductive Power Transmission Link Characteristic Analysis  
To minimize energy losses, the inductive power transmission link should attain maximum efficiency. However, various factors impact the overall efficiency of the link, such as the quality factor of the transmitter and receiver coils, coupling coefficient, and impedance matching [30; 31]. Efficiency analysis and optimization in inductive power transmission links are crucial, especially in applications where energy conservation and minimal losses are paramount. By considering these factors and utilizing the equivalent model, engineers can work toward designing an inductive power transmission link with maximum efficiency. Figure 3a depicts the inductive power transmission link schematic with series and parallel (SP) compensation, where \(C_{\mathrm{r}}\) and \(C_{\mathrm{t}}\) are the parallel- and series-compensated capacitors, \(L_{\mathrm{t}}\) and \(L_{\mathrm{r}}\) are the transmitter and receiver coils, \(r_{\mathrm{t}}\) denotes the equivalent series resistance (ESR) of the transmitter coil, \(r_{\mathrm{r}}\) represents the ESR of the receiver coil. To maximize the power transmission efficiency, the receiver and transmitter coils are operated at the resonant frequency [32]. The resonant frequency is calculated as follows. \[f_{r}=\frac{1}{2\pi\sqrt{L_{\mathrm{r}}C_{\mathrm{r}}}}=\frac{1}{2\pi\sqrt{L_{ \mathrm{t}}C_{\mathrm{t}}}} \tag{1}\]  
The power transmission efficiency of the inductive power transmission link is directly related to the mutual inductor between the receiver and transmitter coils, as well as their quality factors [32]. The quality factors of the receiver and transmitter coils are defined as \(Q_{\mathrm{t}}=\omega_{0}L_{\mathrm{t}}/r_{\mathrm{t}}\) and \(Q_{\mathrm{r}}=\omega_{0}L_{\mathrm{r}}/r_{\mathrm{r}}\), respectively, where \(\omega_{0}=2\pi f_{\mathrm{r}}\). Considering \(Q_{\mathrm{r}}{}^{2}>>1\), the receiver circuit in Figure 3a (indicated by the green, dashed box) can be simplified to the circuit illustrated in Figure 3b. Specifically, the parallel resistance of Figure 3b is defined as \(R_{\mathrm{p}}=Q_{\mathrm{r}}{}^{2}r_{\mathrm{r}}\). Figure 2: Wireless power transmission schematic. Based on Figure 3b, the efficiency of the receiver coil, \(\eta_{\rm r}\), is calculated as follows:  
\[\eta_{\rm r}=\frac{Q_{\rm r}^{2}r_{\rm r}}{Q_{\rm r}^{2}r_{\rm r}+R_{\rm L}} \tag{2}\]  
where \(R_{\rm L}\) is the AC load resistance. To facilitate the analysis, the inductive link can be regarded as a loosely coupled transformer model. Hence, Figure 3a can be converted into the transformer model depicted in Figure 3c. In Figure 3c, \(L_{1}\) refers to the transformer's leakage inductance, \(L_{\rm M}\) represents the magnetic inductance, and \(n\) indicates the turn ratio. Their expressions are depicted as follows:  
\[\left\{ \begin{array}{l}M=k\sqrt{L_{\rm t}L_{\rm\tau}}\\ L_{\rm M}=k^{2}L_{\rm t}\\ L_{1}=L_{\rm t}-k^{2}L_{\rm t}\\ n=\sqrt{L_{\rm r}/L_{\rm t}}/k\end{array} \right. \tag{3}\]  
where \(k\) is the coupling coefficient. In the loosely coupled transformer model, the value of \(k^{2}\) is considerably below one [33]. The receiver circuit needs to be reflected in the transmitter circuit to facilitate calculation, as illustrated in Figure 3d. When the receiver operates at its resonant frequency, the reflected resistance \(R_{\rm ref}\) is expressed as follows. \[R_{\rm ref}=\frac{Q_{\rm r}^{2}r_{\rm r}\times R_{\rm L}}{n^{2}(Q_{\rm r}^{2} r_{\rm r}+R_{\rm L})}=\frac{k^{2}\omega_{0}^{2}L_{\rm t}L_{\rm r}R_{\rm L}}{R_{ \rm L}r_{\rm r}+\omega_{0}^{2}L_{\rm r}^{2}} \tag{4}\]  
Based on Figure 3d, the transmitter coil's efficiency, \(\eta_{\rm t}\), is obtained by Equation (5).