## 3 Computer-Aided Analytical Modeling of Class-E Series-Tuned RFPAs
### Maximum Value of \(Q_{s}\)
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Topology** & \(\mathbf{Q}\) & \(\mathbf{V}_{\text{peak}}\) & \(\mathbf{I}_{\text{peak}}\) & \(\mathbf{\eta}\) (\%) & \(\mathbf{c_{p}}\) & **2 nd (dBc)** & **3 rd (dBc)** \\ \hline
1 & 3 & 3.839 & 2.844 & 99.231 & 0.09726 & \(-\)18.929 & \(-\)44.67 \\  & 5 & 3.739 & 2.85 & 99.684 & 0.09769 & \(-\)21.846 & \(-\)68.545 \\  & 6 & 3.712 & 2.852 & 99.773 & 0.0977 & \(-\)23.576 & \(-\)70.202 \\  & 10 & 3.654 & 2.855 & 99.911 & 0.09796 & \(-\)28.355 & \(-\)77.482 \\  & 25 & 3.6 & 2.859 & 99.983 & 0.09803 & \(-\)36.687 & \(-\)100.472 \\  & 50 & 3.581 & 2.86 & 99.994 & 0.09812 & \(-\)42.848 & \(-\)105.068 \\
[22] & 100 & 3.571 & 2.861 & 99.996 & 0.09807 & \(-\)48.942 & \(-\)104.933 \\ \hline
2.1 & 3 & 3.843 & 2.82 & 99.158 & 0.097810 & \(-\)15.887 & \(-\)62.902 \\  & 5 & 3.743 & 2.827 & 99.653 & 0.09819 & \(-\)20.819 & \(-\)61.925 \\  & 6 & 3.716 & 2.829 & 99.75 & 0.09826 & \(-\)22.565 & \(-\)63.331 \\  & 10 & 3.659 & 2.833 & 99.902 & 0.09839 & \(-\)27.381 & \(-\)68.67 \\  & 25 & 3.605 & 2.837 & 99.981 & 0.09855 & \(-\)35.753 & \(-\)79.715 \\  & 50 & 3.586 & 2.839 & 99.993 & 0.09852 & \(-\)41.93 & \(-\)87.778 \\
[22] & 100 & 3.577 & 2.84 & 99.996 & 0.09853 & \(-\)48.032 & \(-\)95.024 \\ \hline
2.2 & 3 & 3.967 & 2.76 & 96.981 & 0.09699 & \(-\)7.428 & \(-\)52.801 \\  & 5 & 3.851 & 2.644 & 98.919 & 0.10263 & \(-\)12.166 & \(-\)41.025 \\  & 6 & 3.818 & 2.618 & 99.255 & 0.10384 & \(-\)14.172 & \(-\)41.64 \\  & 10 & 3.751 & 2.572 & 99.736 & 0.10603 & \(-\)18.749 & \(-\)44.228 \\  & 25 & 3.689 & 2.538 & 99.957 & 0.10776 & \(-\)27.214 & \(-\)51.474 \\  & 50 & 3.668 & 2.525 & 99.987 & 0.10840 & \(-\)33.435 & \(-\)57.373 \\
[14] & 100 & 3.658 & 2.512 & 99.995 & 0.10904 & \(-\)39.561 & \(-\)63.352 \\ \hline \end{tabular}
\end{table} 
Table 1: Various performance metrics for the three topologies compared to previous literature studies (references shown in square brackets). The condition of saturation arises when \(c>0\). Since the HB method solves nonlinear differential equations and only provides approximate solutions due to numerical errors, this upper bound is defined using perturbation theory with a small parameter \(\epsilon\)[29]. As the value of \(Q_{s}\) increases, \(e^{-cQ_{s}}\to 0\). For sufficiently large values of \(Q_{s}\), \(c_{p}\) will exhibit negligible variation. This condition is defined by the inequality given in Equation (34). \[e^{-cx}\leq\epsilon \tag{34}\]  
Figure 5: Efficiency vs. normalized angular frequency for various \(Q_{s}\). (**a**) Topology 1, (**b**) Topology 2.1, and (**c**) Topology 2.2. Figure 6: \(Q_{sat}\) model for (**a**) Topology 1, (**b**) Topology 2.1, and (**c**) Topology 2.2.  
where \(\epsilon\) is a small value chosen corresponding to the accuracy requirement. The saturation point \(Q_{sat}\) is defined as the point where the inequality (34) becomes an equation. \[e^{-cQ_{sat}}=\epsilon \tag{35}\]  
By rearranging Equation (35), Equation (36) is obtained, which determines the value of \(Q_{sat}\). \[Q_{sat}=-\frac{ln(\epsilon)}{c} \tag{36}\]  
Increasing \(Q_{sat}\) beyond this point does not enhance efficiency or other performance metrics significantly but instead narrows the effective bandwidth of the RFPA. The value of \(\epsilon\) was set to \(10^{-8}\) in this work. The bounds for all three topologies are given in Table 2. Additionally, time-domain waveforms obtained from the ADS simulation for all three topologies are shown in Figure 7. These waveforms verify the non-overlapping voltage and current characteristics of standard Class-E operation. This analysis illustrates the significant impact of effective harmonic suppression on the trade-offs between efficiency and bandwidth, as well as other critical performance metrics. Understanding and managing these trade-offs provides valuable insights into \begin{table}
\begin{tabular}{c c c} \hline \hline
**Topology** & \(Q_{min}\) & \(Q_{sat}\) \\ \hline
1 & 2.6207 & 18.4306 \\
2.1 & 2.7451 & 17.2098 \\
2.2 & 4.4976 & 24.6607 \\ \hline \hline \end{tabular}
\end{table}