## 4 Study Cases and Analysis Results
In the second design, the maximum \(R_{L}\) is found for a given \(f_{o}\) and \(C_{SH}\), the specified duty cycle is 0.5 and the \(q\) value is undefined. The expressions of unknown circuit components were obtained with the process that is shown in Figure 12. According to the \(R_{L}\) search space plotted in Figure 13a, the maximum load resistance for \(D=0.5\) is achieved at \(q=1.468\). Hence, the circuit parameters were calculated and summarized in Table 8 using \(q=1.468\). Finally, the design solution was verified, the switch voltage and current obtained in the simulation are shown in Figure 13b, where a soft-switching behavior is observed. Figure 11: Class-E PA waveforms of study case 1. (**a**) \(P_{out}\) search space for \(D=0.4\). (**b**) Model and simulated waveforms of the \(P_{out}\) and average \(P_{out}\). Figure 10: The process to find the unknown design values of study case 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline \(R_{L}\)\([\Omega]\) & 3.30 & 4.94 & 2.40 & 10.90 \\ \(L_{SH}\) [nH] & 492.19 & 520.09 & 31.05 & 1.47 \\ \(C_{SH}\) [nF] & 133.02 & 22.60 & 2.60 & 172.20 \\ \(L_{o}\) [uH] & 4.61 & 33.00 & 1.15 & 13.87 \\ \(C_{e}\) [nF] & 22.00 & 0.76 & 0.22 & 114.10 \\ \(V_{DD}\) [V] & 12.00 & 1.93 & 4.44 & 6.00 \\ \(P_{int}\) [W] & 50.28 & 1.00 & 8.00 & 6.00 \\ \(\eta\) & 1 & 1 & 1 & 1 \\ \(D\) & 0.40 & 0.50 & 0.55 & 0.75 \\ \(q\) & 1.244 & 1.468 & 1.771 & 2.504 \\ \(Q_{L}\) & 4.39 & 41.94 & 30.00 & 32.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Design solutions. In the study case 3, the maximum output power capability is found for a given \(P_{out}\), \(f_{o}\), \(R_{L}\), where \(Q_{L}\), the \(D\) and \(q\) values are undefined. The unknown circuit expressions were obtained following the process described in Figure 14. The \(C_{p}\) variable search space is shown in Figure 15a, the curve behavior shows that the maximum \(C_{p}\) is around \(D=0.5\), and it decreases at extreme \(D\) values (i.e., \(D=0.25\) and \(D=0.75\)). The maximum \(C_{p}\) is achieved at \(D=0.55\), for \(q=1.771\), as shown in Figure 15b. Therefore, the unknown parameters were calculated and are listed in Table 8 using these values. Finally, the design solution was verified. The simulated \(C_{p}\) was 0.1071, the waveforms of the switch current and output power are shown in Figure 15c. Figure 14: The process to find the unknown design values of study case 3. Figure 12: The process to find the unknown design values of study case 2. Figure 13: Class-E PA waveforms of study case 2. (**a**) \(R_{L}\) search space for \(D=0.5\). (**b**) Model and simulated waveforms of the switch current and voltage. In the study case 4, the maximum \(R_{L}\) value is found with the \(I_{RMS_{SW}}\) being limited to 2 A, the \(D\) and \(q\) values are undefined. The expressions for the unknown circuit components were obtained with the process shown in Figure 16. The search space of \(R_{L}\) and \(I_{RMS_{SW}}\) are plotted as a function of \(D\) and \(q\) in Figure 17a,b, respectively. The analytical expression of \(I_{RMS_{SW}}\) is taken from the Equation (31). The curve behavior of Figure 17a shows that the \(R_{L}\) value increases as \(D\) increases. Similarly, the curve behavior of Figure 17b shows that the \(I_{RMS_{SW}}\) decreases as \(D\) increases. Consequently, the duty cycle is set at 0.75, because it allows a high \(R_{L}\) value and restricts the RMS switch current under 2 A. According to Figure 17a, the maximum load resistance for \(D=0.75\) is achieved at \(q=2.504\). Therefore, the unknown parameters were calculated and are summarized in Table 8 while using \(D=0.75\) and \(q=2.504\). Finally, the design solution was verified by simulation, where the simulated switch RMS current was 1.39 A. The waveforms of the switch current and \(I_{RMS_{SW}}\) are shown in Figure 17c. With \(D\) values different than 0.5, the \(q\) range to obtain the optimal solution can be greater than 1.9, as shown in Figure 17a.