## 3 Design Methodology of the Ideal Class-E PA with FDI
### Design Approaches
#### 3.1.1 Approach A: Design Based on a Reported Optimal Point  
In this approach, the design specifications are based on the standard QiTM, which defines inductive charging guides for wireless power applications over short distances [36]. Following the design methodology that is proposed in Section 3, the design steps are:  
Step 1: the specifications of the design were defined from the power class 0 specification of standard QiTM, which are summarized in Table 2. The \(L_{o}\) value corresponds to a commercial transmitter coil in compliance with the standard. The \(D\) and \(q\) values were defined in order to maximize the output power with low complexity of the switch driver [17]. Additionally, the only restriction considered was the breakdown voltage of the switch. Therefore, the \(v_{C_{Slim}}\) value was limited to 30 V (typical maximum rating of MOSFETs for low voltage applications [37]).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(f_{o}\) [kHz] & \(V_{DD}\) [V] & \(L_{o}\) [\(\mu\)H] & \(P_{in}\) [W] & \(\eta\) & \(D\) & \(q\) \\ \hline
100 & 5 & 24 & 10 & 1 & 0.5 & 1.412 \\ \hline \end{tabular}
\end{table}
Table 2: Class-E PA specifications.  
Figure 4: Proposed Design Methodology for Class-E PA.  
Step 2: in this design, the unknown variables are \(L_{SH}\), \(C_{SH}\), \(C_{e}\), and \(R_{L}\). The process used to find these unknown values from the design set is illustrated in Figure 5. This diagram has four phases. In phase 1 is found the equation of \(R_{L}\). In phase 2, are identified the expressions for calculating \(C_{SH}\), \(L_{SH}\), \(X_{s}\), and \(K_{Q_{L}}\). Subsequently, phase 3 identifies the expression of \(C_{o}\), and finally, in phase 4 is found the \(C_{e}\) equation. In the diagram shown in Figure 5, the input parameters or design specifications are shown in silver boxes, the relations of the design set in blue boxes, and the unknown expressions in white boxes. We followed this graphical notation in the diagrams that will appear in Section 4.  
Step 3: the design set is extended by the expression of \(V_{C_{SHm}}\) described in (30).  
Step 4: this step is skipped, because \(D\) and \(q\) are specifications of this design.  
Step 5: the extended design set was implemented in Maple(tm) while using the support tools developed for this tutorial. The resulting model parameters are listed in Table  
Step 6: in Figure 6, the theoretical (Model) and simulated (Ideal sim.) waveforms of the designed Class-E were plotted. These waveforms present ZVS and ZVDS operation under the design restriction of \(V_{C_{SHm}}\) less than 30 V. Therefore, the simulation verifies the good performance of the designed Class-E PA.  
The Class-E PA is susceptible to the component values. For instance, when the components found in the design process of the last example are approximated to the closest commercial component values, the simulated waveforms have significant changes (Comm. sim. in Figure 6). Furthermore, the ZVS and ZVDS conditions are not met and, consequently, the switch stress is increased. This electric stress could be quantified by the output power capability, which is given by [38]  
\[C_{p}=\frac{P_{out}}{\max\left\{V_{p},V_{p_{m}}\right\}\cdot\max\left\{I_{p},I_ {p_{m}}\right\}} \tag{34}\]  
where \(I_{p}\) and \(V_{p}\) are the peak current and peak voltage across the switch, respectively. The \(I_{p_{m}}\) and \(V_{p_{m}}\) are the current and voltage values at the switch turn-on time, as shown in Figure 7.