## 1 Introduction  
The Class-E power amplifier is a tuned switching Power Amplifier (PA), which is commonly used in Radio-Frequency (RF) applications. It uses a single switch element and a tuned reactive network between its power stage and its load. Furthermore, it achieves an ideal power efficiency of 100% under Zero Voltage Switching (ZVS) and Zero Voltage Derivative Switching (ZVDS) conditions. The circuit topology and its main waveforms are shown in Figures 1 and 2, respectively. This PA is used in many applications, such as wireless power transmission [1], dedicated short-range communications [2], low-power RF devices [3], induction heater systems [4], plasma generators [5], and street-lighting [6]. Figure 1: Class-E PA (**a**) Ideal model. (**b**) High \(Q_{L}\) model. The Class-E amplifier was introduced by Ewing in 1964, as a high-efficiency RF amplifier [7]. Since then, this PA has been analyzed in several works [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. In contrast to the original implementation with RF choke inductors for Direct Current (DC) feeding, the use of a finite inductance offers a higher degree of freedom, a feature that is particularly useful in applications with size constraints, because it allows for the reduction of the inductor size [16]. For instance, in [19], the authors design a Class-E with Finite DC Feed Inductance (FDI) that is based on an integrated split inductor. Additionally, their design involves the breakdown limit of the integrated switch as a design restriction and it maximizes the PA output power under ZVS and ZVDS conditions while using an analytical model of the Class-E PA with FDI. Furthermore, to the best of our knowledge, the use of this topology has been concentrated only on applications in which the conventional topology (i.e., with RF choke) cannot be implemented (e.g., VLSI and RF applications). In general, the Class-E with FDI models are more complex than the Class-E conventional models [15]. In particular, the conventional models assumed the feed current as a DC current that results in simple analytical equations [7], while in the Class-E with FDI models considered the current behavior into the circuit analysis [17]. Furthermore, some published models set the duty cycle (\(D\)) of the commutation signal to a fixed value (e.g., 0.5), which simplifies the amplifier model and design equations [12; 15]. Following another simplification approaches, in [17; 18], the authors assumed a high loaded quality factor and ideal components (i.e., ideal switch, inductors, and capacitors with infinite quality factor) to model the Class-E with FDI through analytical equations, which allows for an easy design procedure with mathematical software tools. However, the designer needs to implement the PA model in a software tool without some insight into its analytical equations expressed as explicit functions of the model variables and without a detailed procedure for obtaining these equations, because they were not reported in [17; 18]. Only after overcoming this challenge, the designer could optimize the Class-E with FDI to a particular application by an exhaustive exploration of the design space when considering the involved trade-offs between the design specifications and restrictions [19]. The design of the Class-E PA with FDI has been explored in several works [11; 12; 16; 17; 18; 19; 20; 22; 23; 24; 25]. In general, the complexity of the circuit design of the Class-E with FDI is higher than the calculation of conventional Class-E components in order to guarantee ZVS and ZVDS operation. This design process involves multivariable equations with high complexity solutions which could require iterative procedures to achieve the given specifications and restrictions [13; 15; 16; 17; 18; 19; 20; 25; 26; 27]. In [11; 12; 13], the authors design a Class-E with FDI using an iterative design approach, which involves time-consuming procedures. On one hand, these procedures usually involve a phase of tuning components to achieve the ZVS and ZVDS conditions [28; 21; 29].