## 1 Introduction  
Radio-frequency power amplifiers (RFPAs) play a crucial role in various scientific applications, extending from space exploration to everyday technologies. They are integral to driving acoustic optical tunable filters (AOTFs) used in space exploration [1] and are also employed in induction cooking and medical applications, such as magnetic resonance imaging (MRI) [2]. However, their substantial power consumption contributes notably to overall power dissipation, raising concerns about energy efficiency in RF systems. Furthermore, RFPAs generate harmonics, due to the inherent non-linearity of the transistor that can significantly interfere with the desired output signal. The heat produced by these amplifiers [3] can also adversely affect the precision of measurements in instruments where optical performance is critical [4; 5]. In space-based applications, managing the thermal impact of RFPAs is crucial to maintaining the accuracy and reliability of optical measurements [6]. The RF signal used to drive these instruments generally falls within the medium frequency (MF) band, high frequency (HF) band (3-30 MHz), and very high frequency (VHF) band (30-300 MHz). However, designing RFPAs for these frequency ranges presents unique challenges, as the use of transmission lines becomes impractical. As a result, employing discrete or lumped \(R\), \(L\), and \(C\) components is crucial to ensure efficient operation and performance at these frequencies. In this context, Class-F and Class-E amplifiers theoretically offer an efficiency of up to 100%. However, achieving the desired square waveform in Class F amplifiers requires the use of multiple resonators [7], complicating the design. As a result, Class-E amplifiers emerge as a promising alternative, providing a simpler yet highly efficient solution for RF power amplification. From a technological perspective, gallium nitride (GaN) has emerged as a superior material for such high-efficiency power amplification, surpassing conventional materialslike silicon. GaN's intrinsic properties, particularly its substantial sheet charge density, facilitate the generation of high current densities [8]. This enables the reduction of transistor dimensions, leading to an exceptional watt-per-millimeter ratio of the gate periphery. Furthermore, the high saturated drift velocity of GaN contributes to achieving substantial saturation current densities and higher output power per unit gate periphery [9]. Consequently, GaN transistors exhibit lower output capacitance and drain-to-source resistance per watt, making them ideal for switch-mode amplifiers like Class-E and Class-F designs. These combined attributes make AlGaN/GaN HEMTs highly suitable for high-efficiency RF applications and deliver excellent performance even in demanding environments. Comprehensive reviews of GaN-based device capabilities, particularly in the context of high-power and high-frequency applications, are provided by [PERSON] et al. [10] and [PERSON] et al. [11]. Building on the advantages of GaN-based devices, the design of Class-E RFPAs involves two widely recognized load network topologies, one where the effect of the finite DC feed is ignored [12] and another considering these effects [13]. Nevertheless, both configurations incorporate a series resonator, operating under the assumption of a high-quality factor (\(Q_{s}\)) to force the fundamental current into the circuitry [14]. Although some studies explored \(Q_{s}\) and bandwidth in specific contexts [15], these findings lacked broader generalizations and comparisons across various topologies. Analytical expressions for Class-E amplifiers with arbitrary duty cycles were introduced in [16] and later extended by [17] to account for varying \(Q_{s}\). However, these equations often yield ambiguous values for \(Q_{s}\), indicating the need for more comprehensive methodologies.