## 5 Results and Discussion  
The CAD-based analytical modeling demonstrates that Topology 2.2. L3, referred to as the final topology, can achieve at least a bandwidth of 30% while maintaining optimal performance in other key metrics and ensuring an efficient response across the bandwidth. Improving one performance metric will, however, result in the degradation of others. Figure 11: Efficiency-bandwidth comparison between the final topology and Topology 2.2.1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Topology** & \(V_{\text{peak}}\) & \(I_{\text{peak}}\) & \(\eta\) **(\%)** & \(c_{p}\) & **2 nd (dBc)** & **3 rd (dBc)** \\ \hline
2.2.1 & 4.071 & 3.032 & 89.676 & 0.08312 & \(-9.411\) & \(-21.292\) \\
2.2.2 & 4.593 & 3.123 & 99.05 & 0.08609 & \(-94.6\) & \(-23.937\) \\
2.2.3 & 4.095 & 2.992 & 89.798 & 0.08390 & \(-8.665\) & \(-89.68\) \\
2.2.23 & 4.626 & 3.065 & 99.934 & 0.0801 & \(-58.633\) & \(-101.285\) \\
2.2.L & 3.441 & 2.742 & 98.145 & 0.10291 & \(-314.34\) & \(-38.905\) \\
2.2.C & 4.884 & 5.639 & 88.806 & 0.03670 & \(-295.8\) & \(-19.634\) \\
2.2.L3 & 3.348 & 2.748 & 99.128 & 0.10608 & \(-314.705\) & \(-59.022\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of modified topologies. Figure 12: Time-domain normalized voltage (in blue, its corresponding axis is marked by the arrow) and current (in red) waveforms from the ADS simulation. Building on the analytical expressions, a simplified design methodology was developed to assist designers. A complete circuit of the final topology is shown in Figure 10, where the idealized values can be calculated using the equations from previous sections. Additionally, low-pass matching networks can be designed to match the load resistance \(R\) of the RFPA to the standard 50-\(\Omega\) load of [31]. Since harmonics are already suppressed due to the modified topology, higher-order filters can be avoided, thus reducing losses associated with the filter. Based on the analytical expression, the following methodology is developed to assist designers. This methodology provides a systematic approach for optimizing Class-E RFPs to achieve the best trade-off among all performance metrics. The methodology is outlined below. 1. Set the design frequency: The design frequency, set at which the RFPA is intended to operate, is a critical step that influences all subsequent parameters and choices. 2. Determine required bias voltage levels: The necessary bias voltage levels are identified for the design. These levels are essential for ensuring the proper operation of the transistor and achieving the desired output power. 3. Calculate idealized values: The ideal values for the lumped elements are then calculated using computed Equations (25)-(27), (38), (39) and (45)-(47). These calculations serve as a reference for the initial simulation and guide the optimization process. The values for \(L_{3}\) and \(C_{3}\) are selected such that the series resonator is tuned to the 3 rd harmonic. 4. Optimize all parameters: All design parameters are optimized, including a frequency range with the goals as follows: * The output power at the fundamental achieves the specified power level. * The efficiency is at least 95% (adjustable based on the requirements). The optimization is performed by searching within the range of [0.5-2] times the initial values for \(R\), \(L\), and \(C\), and [0.5-3] times for \(L_{an}\), \(L_{pn}^{L}\), \(C_{pn}^{L}\), \(C_{0}\), \(L_{3}\), and \(C_{3}\) to find the optimal design parameters. This optimization accounts for package effects, which are critical to preventing potential design issues. Optimizing the ideal values ensures that package-induced variations do not compromise the design's performance. 5. Finalize the design: After completing the optimization, the design parameters are reviewed to ensure they meet the desired specifications. Necessary adjustments are made to fine-tune the design. 6.