## 2 PA Model and Design Set  
### PA Model  
The Class-E is a power amplifier that reduces the switching losses by turning on the switch with zero voltage and zero current in the \(C_{SH}\) capacitor (i.e., ZVS and ZVDS operation), as illustrated in Figure 2c,f, respectively. This PA is fed by a DC source (\(V_{DD}\)) and a pulsed source (\(v_{g}\)) with a fundamental frequency \(f_{o}\). Furthermore, it supplies the power that is delivered to its load (\(R_{L}\)) in a narrow band centered in \(f_{o}\) while using a quasi-tuned LC circuit, where the natural resonant frequency is  
\[\omega_{e}=\frac{1}{\sqrt{L_{o}C_{e}}}=\frac{1}{\sqrt{L_{o}C_{o}}}+\Delta \omega=2\pi f_{o}+\Delta\omega \tag{1}\]  
where \(\Delta\omega\) is the frequency offset necessary to guarantee the ZVS and ZVDS operation. In the ideal Class-E PA model, the quality factor of the storage elements is infinite. The switch has: zero on-resistance, infinite off-resistance, and zero rise and fall commutation times. Under these assumptions, and supposing a high value of \(Q_{L}\), the Class-E PA with FDI circuit model could be simplified to the one that is shown in Figure 1b. The current \(I_{R_{L}}\) is out of phase by an angle \(\varphi\) regarding the voltage \(v_{g}\) (Figure 2d), and it is related to the load power (\(P_{out}\)) by  
\[i_{RL}(t)=I_{p}sin(2\pi f_{o}t+\varphi)=\sqrt{\frac{2P_{out}}{R_{L}}}sin( \omega_{o}t+\varphi). \tag{2}\]In the time intervals when the switch is on (\(t_{on}\)) and off (\(t_{off}\)), the main waveforms of the PA are predicted by the following equations [17,19]  
\[i_{C_{Sflen}}(t) = v_{C_{Sflen}}(t)=0 \tag{3}\] \[i_{S_{on}}(t) = \frac{V_{DD}}{L_{SH}}t-I_{p}sin(\varphi)+I_{p}sin(\omega_{o}t+\varphi)\] (4) \[i_{L_{Sflen}}(t) = \frac{V_{DD}}{L_{SH}}t-I_{p}sin(\varphi)\] (5) \[i_{S_{off}}(t) = 0\] (6) \[i_{L_{Sflpf}}(t) = \frac{V_{DD}}{L_{SH}}t-I_{p}sin(\varphi)-\frac{1}{L_{SH}}\int_{ \frac{2\pi D}{4\omega_{0}}}^{t}v_{C_{SH}}d\tau\] (7) \[i_{C_{Sflpf}}(t) = \frac{V_{DD}}{L_{SH}}t+I_{p}(sin(\omega_{o}t+\varphi)-sin(\varphi ))-\frac{1}{L_{SH}}\int_{\frac{2\pi D}{4\omega_{0}}}^{t}v_{C_{SH}}d\tau\] (8) \[v_{C_{Sflpf}}(t) = V_{DD}\bigg{(}\frac{C1}{V_{DD}}cos(q\omega_{o}t)+\frac{C2}{V_{DD }}sin(q\omega_{o}t)+1-\frac{q^{2}}{1-q^{2}}pos(\omega_{o}t+\varphi)\bigg{)} \tag{9}\]  
where  
\[q=\frac{1}{\omega_{o}\sqrt{L_{SH}C_{SH}}} \tag{10}\]  
\[D=\frac{t_{on}}{T}=t_{on}f_{o} \tag{11}\]  
\[p=\frac{\omega_{o}L_{SH}}{V_{DD}/I_{p}}=\frac{\omega_{o}L_{SH}}{\sqrt{R_{L} \cdot R_{DC}}} \tag{12}\]  
The variable \(p\) is the ratio between the impedance of \(L_{SH}\) at the operation frequency and the geometric mean of the resistance that the amplifier shows to the power supply (\(R_{DC}\)) and its load (\(R_{L}\)). The constants C1 and C2 are related to the solution of the involved differential equation. It is important to notice that the variable \(q\) cannot take the value of 1 because of the mathematical singularity that is introduced by (9). Furthermore, under ZVS and ZVDS conditions, \(\varphi\), \(p\), \(C1\), and C2 could be expressed as a function of \(D\) and \(q\) variables [17]  
\[\varphi=\arctan\left(\frac{b_{1}c_{2}-b_{2}c_{1}}{\pm\sqrt{(a_{1}c_{2}-a_{2}c _{1})^{2}+(b_{1}c_{2}-b_{2}c_{1})^{2}}},\frac{a_{1}c_{2}-a_{2  
\[a_{2}=\frac{q}{q^{2}-1}\sin\left(q2\pi(D-1)\right)-\frac{q^{2}}{q^{2}-1}\sin\left( 2\pi D\right) \tag{20}\]  
\[b_{2}=-\frac{q}{q^{2}-1}\cos\left(2\pi D\right)-\frac{q^{2}}{q^{2}-1}\cos\left( q2\pi(D-1)\right) \tag{21}\]  
\[c_{2}=1-\cos\left(q2\pi(D-1)\right). \tag{22}\]