## 2 Efficiency Analysis and Derivation of Wireless Power Transmission
### Class-E Power Amplifier Characteristic Analysis  
The Class-E PA is usually employed to drive the inductive power transmission link. The Class-E PA's schematic with the reflected resistance of the receiver circuit is demonstrated in Figure 6, where the \(L_{\rm f}\) acts as the choke inductor, \(S_{1}\) is the MOSFET, \(C_{\rm p}\) denotes the shunt capacitor, and \(L_{1}\)-\(C_{\rm r}\)-\(R_{\rm eq}\) represents the tuned network. The switching frequency of \(S_{1}\) is equal to the inductive power transmission link's resonant frequency in the Class-E PA.  
Figure 4: Changing trend of the inductive power transmission link’s efficiency with different \(k\) and \(Q_{\rm t}\).  
Figure 5: Changing trend of \(\eta_{\tau}\), \(\eta_{\tau}\), and \(\eta_{\rm IPT}\) with different \(R_{\rm L}\).  
Figure 6: Schematic of the Class-E PA.  
It is widely recognized that the Class-E PA's high efficiency performance when it satisfies the characteristics of ZVDS and ZVS [34]. However, the Class-E PA exhibits high sensitivity to load variations in this operating state, requiring a specific set value of the \(L_{1}\)-C\({}_{t}\)-Feq\({}_{\text{Q}}\)-C\({}_{\text{P}}\) network to match. To address this challenge, the concept of the load angle \(\varphi\) is introduced, which quantifies the variation in the \(L_{1}\)-C\({}_{t}\)-\(R_{\text{eq}}\) network branch. The expression for load angle \(\varphi\) is calculated by Equation (8).  
\[\left\{\begin{array}{l}X=\omega_{0}L_{1}-(\omega_{0}\text{C}_{t})^{-1}\\ \varphi=\arctan\left(\frac{X}{R_{\text{eq}}}\right)\end{array}\right. \tag{8}\]  
The efficiency of the Class-E PA by sweep load angle \(\varphi\) is shown in Figure 7. It indicates that the efficiency can reach 1 when the \(\varphi\) is about \(49^{\circ}\), which corresponds to the optimally tuned state for the Class-E PA. In this state, both the ZVS and ZVDS characteristics are satisfied, and the optimal values of \(C_{\text{P}}\) and \(C_{\text{t}}\) can be obtained by Equation (9) [35]. On the other hand, the efficiency is close to 1 when the \(\varphi\) is about \(64.5^{\circ}\), and the ZVS characteristics are satisfied.  
\[\left\{\begin{array}{l}C_{\text{P}}=\frac{8}{\pi(\pi^{2}+4)\omega_{0}R_{ \text{eq}}}\\ \frac{16}{\omega_{0}R_{\text{eq}}[16\text{C}_{\text{L}}-\pi(\pi^{2}+4)]}\end{array}\right. \tag{9}\]  
where \(Q_{\text{L}}\) denotes the load quality factor of Class-E PA, which is equal to \(\omega_{0}L_{1}/R_{\text{eq}}\).  
In Figure 7, it is valuable to note that the Class-E PA's efficiency is near its peak in the range of \(\varphi\) from \(40^{\circ}\) to \(67^{\circ}\), which is referred to as the high-efficiency region [36]. This region represents a range of load angles where the Class-E PA operates with high efficiency, even if it is not precisely at the optimal tuning point. In addition, the voltage \(V_{\text{ref}}\) also follows \(\varphi\) variation. This beneficial phenomenon can be utilized to achieve ASK modulation for transmitting data with high-power transmission efficiency.  
According to Equation (8), it is feasible to change \(\varphi\) by adjusting the values of \(R_{\text{eq}}\), C\({}_{t}\), and \(L_{1}\) for data transmission. However, introducing an additional resistor to alter \(\varphi\) will degrade the system's efficiency, which is undesirable. Compared to changing the value of \(C_{\text{t}}\), increasing \(L_{1}\) can reduce the reactive power to improve the system's efficiency further. Consequently, adding an extra inductor can enhance the system's efficiency and achieve data transmission.