## 2 Mathematical Modeling of Class-E Series-Tuned RFPAs
### Class-E Load Network Topology with Finite DC Feed  
The schematic of a Class-E-tuned switched-mode power amplifier with a finite DC feed is shown in Figure 1(a). The load network consists of a feed inductance \(L\), which supplies current to the transistor. The series reactance \(jX\) is connected to a series resonant circuit \((L_{s}-C_{s})\) that is tuned to the design frequency \(f_{o}\). This configuration is then connected to the optimal load resistance \(R\). Due to the finite inductance \(L\), the supply current becomes time-varying, denoted as \(i_{L}(\omega t)\), in contrast to the previous case where it was represented by a constant value \(I_{0}\). The equivalent circuit is shown in Figure 1(b). The load current is given by Equation (11). \[i_{R}(\omega t)=I_{R}\sin{(\omega t+\phi)} \tag{11}\]  
where \(\phi\) is the initial phase shift and \(I_{R}\) is the peak amplitude of the current. For the time interval \(0\leq\omega t<\pi\), the switch is on and the current flowing through the switch can be given by Equation (12). \[i_{s}(\omega t)= \begin{cases}\frac{V_{DD}}{\omega L}\omega t+I_{R}[\sin( \omega t+\phi)-\sin(\phi)]&0\leq\omega t\leq\pi\\ 0&\pi<\omega t\leq 2\pi\end{cases}  \tag{12}\]  
Figure 2: Class-E with finite DC feed (**a**) schematic and (**b**) its equivalent circuit, where the arrows show the direction of current flow. When the switch is off, in the interval \(\pi<\omega t\leq 2\pi\), the current through the switch becomes zero and the current through the capacitor can be given by \(i_{c}(\omega t)=\ i_{L}(\omega t)+i_{R}(\omega t)\), with Equation (13) denoting the initial condition for the feed inductance \(L\). \[i_{L}(\pi)=\frac{V_{DD}\pi-\omega L\ I_{R}\sin\left(\phi\right)}{\omega L} \tag{13}\]  
The switch voltage can be found by solving the differential equation as reported in [22]. The solution is provided by Equation (14), as follows:  
\[v_{s}(\omega t)= \begin{cases}0&0\leq\omega t\leq\pi\\ V_{DD}[1+C_{1}\cos(q\omega t)+C_{2}\sin(q\omega t)-\frac{q^{2}p}{1-q^{2}}\cos( \omega t+\phi)]&\pi<\omega t\leq 2\pi\end{cases}  \tag{14}\]  
Without loss of generality, and using the initial off-state conditions, Equations (15)-(18) are obtained. \[C_{1}=-(\cos q\pi+q\pi\ \sin q\pi)-\frac{qp}{1-q^{2}}\left[q\cos\left(\phi \right)\cos\left(q\phi\right)-(1-2q^{2})\sin\left(\phi\right)\sin\left(q\phi \right)\right] \tag{15}\]  
\[C_{1}=-(\sin q\pi-q\pi\ \cos q\pi)-\frac{qp}{1-q^{2}}\left[q\cos\left(\phi \right)\cos\left(q\phi\right)+(1-2q^{2})\sin\left(\phi\right)\sin\left(q\phi \right)\right] \tag{16}\]  
where \(q\) and \(p\) are given by Equations (17) and (18), respectively. \[q=\frac{1}{\omega\sqrt{LC}} \tag{17}\]  
\[p=\frac{\omega L\ I_{R}}{V_{DD}} \tag{18}\]  
With variable \(q\) as a free parameter, the values for the unknown parameters, \(\phi\), and \(p\), can be calculated, applying the ZVS and ZVDS conditions. Two equations with two unknowns can be solved using Equations (19) and (20) to obtain the required values, as follows:  
\[-C_{1}q\sin\left(2q\pi\right)+C_{2}q\cos\left(2q\pi\right)+\frac{q^{2}p\sin \phi}{1-q^{2}}=0 \tag{19}\]  
\[-C_{1}q\sin\left(2q\pi\right)q^{2}+C_{2}q\cos\left(2q\pi\right)q^{2}+\frac{q^ {2}p\cos\phi}{1-q^{2}}=0 \tag{20}\]  
The parameter, \(q\), imposes limitations owing to the extensive design space and complex design equations. A comprehensive tutorial addressing these considerations was conducted by [PERSON] et al. [23]. This is simplified in this work by evaluating the following two scenarios: when \(jX=L_{x}\ (q=0.5)\) and when \(jX=0\ (q=1.412)\). * Case a: \(jX=L_{x}\), known as the _sub-harmonic_ with \(q=0.5\)[22]. Equations (19) and (20) can be solved to obtain the values of \(p=21.3\) and \(\phi=-30.2\). This configuration is referred to as Topology 2.1 in this work.