## 2 Ideal Class-E PA with FDI Model  
The Class-E PA topology is shown in Figure 1. This amplifier is fed by a DC source (\(V_{DD}\)), and by a periodical signal (\(v_{in}\)), which drives the power switch. This signal has a period \(T\) and time intervals \(DT\) and \((1-D)T\) when the switch is on and off, respectively. The main waveforms of this amplifier are shown in Figure 2. Although this topology is known for more than 50 years [10], it is still an open research topic, particularly for the Class-E PA with FDI [6]. An analytical model of this PA was proposed in [6] and extended in [7]. That model is accurate under the following assumptions: (I) it has ideal components, which means that the real power loss occurs only on \(R_{L}\); (II) it has an ideal switch, with zero rise and fall commutation times, zero on-resistance, and infinite off-resistance; (III) high loaded quality factor (\(Q_{L}\)) of the resonant circuit. Under the stationary response of the amplifier, the fundamental frequency of the \(v_{S_{w}}\) voltage is the same of the \(v_{in}\), which is given by \(\omega=2\pi/T\). Therefore and according to assumption III, the voltage across the load (i.e., \(R_{L}\)) could be considered as a pure tone if the frequency offset (produced by the impedance \(X_{s}\), which is necessary to guarantee the ZVS and ZVDS operation) between \(\omega\) and the natural resonance frequency of the LC output network (i.e., \(L_{o}\) and \(C_{e}\)) is very low. Consequently, the load current (\(i_{R_{L}}\)) is a tone too and the PA circuit can be simplified as shown in Figure 3. It is important to notice that this high \(Q_{L}\) model is a narrow-band approximation around the resonant frequency. The main waveforms of the ideal Class-E PA could be estimated by [6;7]  
Figure 3: High \(Q_{L}\) model of the ideal Class-E PA with finite DC-feed inductance.  
Figure 2: Class-E PA ideal waveforms for duty cycle = 0.5, \(V_{DD}\) = 1 V, and output power = 1 W. (a) \(v_{in}\) voltage. (b) Switch current. (c) \(C_{SH}\) current. (d) Output current. (e) Switch voltage. (f) \(L_{SH}\) current.  
\[i_{S_{W}}(t)=\begin{cases}\frac{V_{DD}}{L_{SH}}t+I_{p}sin(\omega t+\varphi)-I_{p} sin(\varphi)&0<t\leq DT\\ 0&DT<t\leq T\end{cases} \tag{2}\]  
\[v_{S_{W}}(t)=\begin{cases}0&0<t\leq DT\\ C_{1}cos(q\omega t)+C_{2}sin(q\omega t)&\\ +V_{DD}-\frac{V_{DD}q^{2}}{1-q^{2}}pos(\omega t+\varphi)&DT<t\leq T\end{cases} \tag{3}\]  
\[i_{C_{SH}}(t)=\begin{cases}0&0<t\leq DT\\ \frac{V_{DD}}{L_{SH}}t-\frac{1}{L_{SH}}\int_{DT}^{t}V_{S_{W}}(\tau)d\tau&DT<t \leq T\\ +I_{p}(sin(\omega t+\varphi)-sin(\varphi))&\\ \end{cases} \tag{4}\]  
\[i_{L_{SH}}(t)=\begin{cases}\frac{V_{DD}}{L_{SH}}t-I_{p}sin(\varphi)&0<t\leq DT \\ \frac{V_{DD}}{L_{SH}}t-I_{p}sin(\varphi)&\\ -\frac{1}{L_{SH}}\int_{DT}^{t}V_{C_{SH}}(\tau)d\tau&DT<t\leq T\\,\end{cases} \tag{5}\]  
where \(I_{p}\) is the peak output current, \(\omega\) is the fundamental angular frequency of \(v_{in}\), \(D\) is the duty cycle of \(v_{in}\), which is defined as the time ratio of the switch when it is on and the total period of \(v_{in}\). The variable \(q\) is defined in [6] as  
\[q=\frac{1}{\omega\sqrt{L_{SH}C_{SH}}}. \tag{6}\]  
The model involves the variables \(p\), \(\varphi\), \(C_{1}\), and \(C_{2}\), which are functions of \(D\) and \(q\) proposed in [6]. However, the function notation was omitted in the equations for simplicity. Under this model, an infinite number of component combinations guarantee ZVS and ZVDS conditions [6; 7; 14]. Furthermore, to explore the best combination for a given design, the designer can tune the model parameters (i.e., the \(D\) and \(q\)). The interested reader should see [8], which detailed the ideal Class-E PA with FDI model proposed in [6] and extended in [7]. The work in [8] reported the explicit analytical expressions of the model variables (e.g., \(p\), \(\varphi\), \(C_{1}\), and \(C_{2}\)) as a function of the model parameters as well as a wide variety of study cases that show the flexibility of this analytical model.