## 2 PA Model and Design Set
### Design Set of the Ideal Class-E PA  
Commonly, the conventional design set \(K\) of the ideal Class-E PA is composed of four equations (i.e., \(K_{P}\), \(K_{L}\), \(K_{C}\), and \(K_{X}\)), which relate to the PA specifications and circuit components [7; 17; 23; 30; 31]. For instance, by fixing the specification values of \(P_{out}\) and \(V_{DD}\), the \(K_{P}\) equation can be used to find the \(R_{L}\) value through the expression that is shown in Figure 3. Therefore, in order to find the values of the circuit model from the design specifications, the expressions between the PA design variables (e.g., \(\omega_{o}\), \(R_{L}\), \(C_{SH}\)) that are shown in Figure 3 can be used. This design set is composed of analytical functions of the \(D\) and \(q\) variables, which are given by Equations (23)-(27) [17; 19]. \[K_{L}=\frac{\omega_{o}L_{SH}}{R_{L}}=\frac{p(D,q)}{2g_{x}(D,q)} \tag{23}\]  
\[K_{C}=\omega_{o}C_{SH}R_{L}=\frac{2g_{x}(D,q)}{q^{2}p(D,q)} \tag{24}\]  
\[K_{P}=\frac{P_{out}R_{L}}{V_{DD}^{2}}=2(g_{x}(D,q))^{2} \tag{25}\]  
\[K_{X}=\frac{X_{s}}{R_{L}}=\frac{\frac{1}{\pi}\int_{0}^{\frac{2\pi}{\omega}} \left(V_{C_{SH}}(t)cos(\omega t+\varphi)\right)dt}{\frac{1}{\pi}\int_{0}^{ \frac{2\pi}{\omega}}\left(V_{C_{SH}}(t)sin(\omega t+\varphi)\right)dt} \tag{26}\]  
\[K_{Q_{L}}=Q_{L}=\frac{\omega_{o}L_{o}}{R_{L}}=\frac{1}{\omega_{o}C_{o}R_{L}} \tag{27}\]  
where \(X_{s}\) is a capacitive or inductive reactance that must be added to the resonant network (i.e., \(L_{o}\) and \(C_{o}\)) to guarantee the ZVS and ZVDS operation. We add the \(K_{Q_{L}}\) equation into the conventional design set, because the \(Q_{L}\) defines the \(L_{o}\) and \(C_{o}\) values. The current gain of the PA under the small ripple approximation is given by [20]  
\[g_{x}=\frac{\langle i_{s}(t)\rangle_{2\pi}}{I_{p}}=\left(\frac{1-\cos\left(2 \pi D\right)}{2\pi}\right)\cos\left(\varphi\right)+\left(\frac{\sin\left(2\pi D \right)}{2\pi}-D\right)\sin\left(\varphi\right)+\frac{D^{2}\pi}{p} \tag{28}\]  
where \(\langle x\rangle_{2\pi}\) is the moving average operator over the \(v_{g}\) period. The moving average of \(i_{s}\) is given by [32]  
\[\langle i_{s}(t)\rangle_{2\pi}=\frac{\omega_{o}}{2\pi}\int_{0}^{\frac{2\pi D}{ \omega_{o}}}i_{s}(\tau)d\tau. \tag{29}\]  
In the design set that is composed of (23)-(27), the designer must fix the \(D\) and \(q\) values to design the PA from its specifications. In the literature, some \"optimal\" design equations have been proposed under particular restrictions. Table 1 shows a summary of the selected proposals, which could be used in order to design the PA according to the application. The most common design set considers a duty cycle \(D=0.5\) because it reduces the design complexity of the oscillator that controls the switch, and \(q=1.412\) because it maximizes the load power for a fixed \(V_{DD}\) and \(R_{L}\)[17]. In a particular application, the PA designer can explore the design space defined by the \(D\) and \(q\) variables. On one hand, the design space of \(q\) was analyzed in [16] with a practical range between 0 and 1.9 for \(D=0.5\). On the other hand, the design space of \(D\) was studied in [33; 34], with a practical range between 0.25 and 0.75. Additionally, the maximum current and voltage values supported by the PA components can be analyzed into the design process as specifications.