## 2 Classical Topologies
### Design for the Different Topologies  
With a view toward low-power devices, a rated power (\(P_{L}\)) of 1 W was chosen as a design parameter [22]. Assuming that a rechargeable lithium-ion cell has a terminal voltage between 3.5 and 3.7 V [22], the load voltage (\(V_{L}\)) was defined as 3.6 V. For the resonance frequency, respecting biocompatibility restrictions, the frequency (\(f_{r}\)) of 13.56 MHz was chosen, and a value was estimated of 0.25 as the design coupling factor (\(\kappa\)).  
In order to identify the \(L_{1}\) and \(L_{2}\) pairs that correspond to the system's maximum efficiency point, a range of inductance combinations from 0.01 \(\upmu\)H to 10 \(\upmu\)H was chosen, with intervals of 0.01 \(\upmu\)H. Based on these data, the efficiency behavior of the classical topologies is presented in Figure 2.  
As depicted in Figure 2, the choice of \(L_{2}\) is crucial to achieve good efficiency. On the other hand, the influence of \(L_{1}\) is neglectable on the efficiency, and as long as it does not imply other constraints, it is reasonable to arbitrate its value as being identical to that of \(L_{2}\). In order to obtain the system's maximum efficiency point, according to the values obtained in Figure 2, the inductances \(L_{1}\) and \(L_{2}\) were selected. The values of \(C_{1}\) and \(C_{2}\) were calculated, as well as the mutual inductance and efficiency, based on the equations in Table 1. Table 2 shows a summary of the parameters calculated for the different topologies. Loss resistances \(R_{1}\) and \(R_{2}\) were calculated as a proportionality rule in relation to the inductances.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{C_{1}}\) & \(\boldsymbol{\eta}\) & \(\mathbf{Z_{1}}\) \\ \hline SS & \(\frac{1}{\omega_{r}^{2}L_{1}}\) & \(\frac{R_{L}}{R_{1}(\frac{R_{2}R_{1}}{M\omega_{r}})^{2}+R_{2}+R_{L}}\) & \(R_{1}+j(L_{1}\omega_{r}-\frac{1}{C_{1}\omega_{r}})+\frac{M^{2}\omega_{r}^{2}} {R_{2}+R_{L}+j(L_{2}\omega_{r}-\frac{1}{C_{2}\omega_{r}})}\) \\ SP & \(\frac{L_{2}}{L_{1}L_{2}\omega_{r}^{2}-M^{2}\omega_{r}^{2}}\) & \(\frac{R_{L}}{R_{L}+R_{2}+\frac{R_{1}L_{2}^{2}}{M^{2}}+\frac{R_{1}R_{2}^{2}}{(M \omega_{r})^{2}}+R_{2}(\frac{R_{2}}{\omega_{r}L_{2}})^{2}}\) & \(R_{1}+j(L_{1}\omega_{r}-\frac{1}{C_{1}\omega_{r}})+\frac{M^{2}\omega_{r}^{2}(R _{1}\omega_{r}C_{2}+1)+R_{L}}{(R_{2}+jL_{2}\omega_{r})(jR_{L}\omega_{r}C_{2} +1)+R_{L}}\) \\ PS & \(\frac{(L_{2}C_{2})^{2}R_{L}^{2}L_{1}}{R_{1}^{2}L_{1}^{2}L_{2}C_{2}+M^{4}}\) & \(\frac{R_{L}}{R_{1}(\frac{R_{2}+R_{1}}{\omega_{r}M})^{2}+R_{2}+R_{L}}\) & \(\frac{(R_{1}+\frac{M^{2}\omega_{r}^{2}}{R_{2}+R_{1}(2\omega_{r}-\frac{1}{C_{2 }\omega_{r}})})+jL_{1}\omega_{r}}{(1-L_{1}C_{1}\omega_{r}^{2})+j(R_{1}C_{1} \omega_{r}+\frac{M^{2}\omega_{r}^{2}C_{1}}{R_{2}+R_{1}+j(L_{2}\omega_{r}-\frac{ 1}{C_{2}\omega_{r}})})}\) \\ PP & \(\frac{(L_{1}L_{2}-M^{2})C_{2}L_{2}^{2}}{(L_{1}L_{2}-M^{2})^{2}+\frac{R_{1}^{2}C _{2}M^{4}}{L_{2}}}\) & \(\frac{R_{L}}{R_{L}+R_{2}+\frac{R_{1}L_{2}^{2}}{M^{4}}+\frac{R_{1}R_{2}^{2}}{(M \omega_{r})^{2}}+R_{2}(\frac{R_{L}}{\omega_{r}L_{2}})^{2}}\) & \(\frac{\frac{D^{2}R_{L}}{L_{1}}+i\omega_{r}(L_{1}-\frac{M^{2}}{L_{1}})}{1- \omega_{r}^{2}C_{1}L_{1}+\frac{\omega_{r}^{2}M^{2}C_{1}}{L_{2}}+\frac{i\omega_ {r}C_{1}R_{2}M^{2}}{L_{2}^{2}}}\) \\ \hline \end{tabular}
\end{table}
Table 1: Equations for the topologies, considering \(M=\kappa\sqrt{L_{1}L_{2}}\) and \(C_{2}=\frac{1}{\omega_{0}^{2}L_{2}}\).