## 3 Proposed Circuit with ASK Modulation
### Relationship between \(V_{out}\) and \(\varphi\)  
To achieve data transmission and maintain high-power transmission efficiency, the relationship between the receiver's voltage amplitude and the Class-E PA's load network is crucial. As discussed in Section 2.2, the load angle \(\varphi\) is employed to characterize the properties of the load network. However, up to now, no expression has been reported to  
Figure 8: Schematic of the proposed WPDT circuit.  
elucidate the relationship between \(V_{\text{out}}\) and \(\varphi\) within the high-efficiency region. Therefore, this section will delve into their relationship.  
The equivalent load, including the \(R_{\text{ref}}\) and \(r_{\text{t}}\), is illustrated in Figure 6. The optimal values about the \(C_{\text{P}}\) and \(C_{\text{t}}\) at \(\omega_{0}\) can be calculated by Equation (9) when the Class-E PA operates at its optimally tuned point. According to [36], the voltage of the equivalent load \(R_{\text{eq}}\) is expressed by Equation (10).  
\[V_{\text{eq}}=\frac{V_{\text{dc}}R_{\text{eq}}g}{R_{\text{dc}}} \tag{10}\]  
Where the \(V_{\text{dc}}\) is the input dc voltage, the dc resistance \(R_{\text{dc}}\) and the transfer ratio \(g\) of dc-ac voltage are expressed as follows:  
\[\left\{\begin{array}{l}R_{\text{dc}}=\frac{\pi^{2}+g[2\pi\sin(\varphi-0.5\pi )-4\sin\varphi]}{4\pi\omega_{0}C_{\text{s}}}\\ g=\frac{\pi\sin(\varphi+\varphi)+2\cos(\varphi+\varphi)}{-2\sin(\varphi-0.5\pi )\sin(\varphi+\varphi)+0.5\pi\cos\varphi}\end{array}\right. \tag{11}\]  
where \(\varphi\) denotes the load current phase, which is calculated as follows.  
\[\left\{\begin{array}{l}\varphi=\arctan(-\frac{4-0.5\pi^{2}+(2\cos\varphi+ \pi\sin\varphi)s}{\pi+(\pi\cos\varphi-2\sin\varphi)s})\\ s=\pi\omega_{0}C_{\text{s}}R_{\text{eq}}\sqrt{1+\tan^{2}\varphi}\end{array}\right. \tag{12}\]  
Based on Equation (8), the \(\tan(\varphi)\) is expressed as follows.  
\[\tan(\varphi)=\frac{\omega_{0}L_{1}-(\omega_{0}C_{\text{t}})^{-1}}{R_{\text{ eq}}} \tag{13}\]  
According to Equations (10)-(13), the \(\varphi\) is directly related to \(V_{\text{eq}}\). However, if \(\varphi\) is substituted directly into Equation (10), the expression about the relationship between \(\varphi\) and \(V_{\text{eq}}\) becomes very complicated. Thus, a mathematical fitting method is considered to obtain an equivalent expression. Using a math-fitting tool, the relationship between \(\tan(\varphi)\) and \(V_{\text{eq}\_m}\) can be denoted as follows.  
\[V_{\text{eq}\_m}=1.83V_{\text{dc}}\exp(-0.47\tan\varphi) \tag{14}\]  
The results of the comparison between \(V_{\text{eq}\_m}\) and \(V_{\text{eq}}\) with different \(\tan(\varphi)\) are illustrated in Figure 9, demonstrating excellent agreement. Following insights from Figure 7, the chosen range of \(\varphi\) values were between \(40^{\circ}\) and \(67^{\circ}\), resulting \(\tan(\varphi)\) values ranging from 0.8 to 2.4.  
Figure 9: Comparison of \(V_{\text{eq}\_m}\) and \(V_{\text{eq}}\) with different \(\tan(\varphi)\) values.  
According to the turn ratio \(n\), the voltage \(V_{\text{out}}\) of the receiver can be calculated as follows.  
\[V_{\text{out}}=\frac{1}{k}\sqrt{\frac{L_{\text{r}}}{L_{\text{t}}}}(V_{\text{eq} \_m}\frac{R_{\text{ref}}}{R_{\text{eq}}}) \tag{15}\]  
The Class-E PA enters a sub-optimal state under low data conditions. In this state, an inductor, \(L_{\text{a}}\) is added to the load network. Selecting an appropriate value for \(L_{\text{a}}\) becomes critical in achieving the desired system efficiency, load angle, modulation index, and output voltage amplitude. This choice necessitates a careful trade-off among various performance parameters to ensure effective system operation.