## 2 Classical Topologies
### Bifurcation Phenomenon  
The system behavior was verified regarding deviations from a pre-established resonance, due to the practical difficulties of obtaining identical inductive and capacitive values to the designed ones. This type of analysis is necessary to evaluate the stability of the system through frequency variations. Thus, from the equations defined in Table 1 and the values presented in Table 2, the transmitter current behavior (\(I_{1}\)) was evaluated for frequency oscillations of \(\pm 30\%\) in relation to the base frequency. The results obtained are shown in Figure 3.  
Practical projects prove that the resonance frequency varies around 10% due to tolerances in the circuit parameters [9]. Taking that into consideration, as shown in Figure 3, for the SS topology, a reduction of 10% on the frequency of operation resulted in an input current 50% higher than the expected one for 13.56 MHz. A similar behavior was observed for the SP topology; the input current rose 40% for frequency variations of up to 10%. The obtained values were found to be impractical and could potentially jeopardize the integrity of the physical project. Conversely, both the PP and PS topologies demonstrated a significant reduction in the input current, achieving a decrease of approximately 70% even with oscillations reaching up to 10%. The operation of a bifurcated system can be catastrophic for the project. However, respecting some criteria, the phenomenon can be eliminated, minimizing operational risks and ensuring stability in energy transfer [9]. As shown in Table 3, for stable operation, it is necessary to meet the criteria based on the primary and secondary quality factors \(Q_{p}\) and \(Q_{s}\), respectively. \begin{table}
\begin{tabular}{c c c c c} \hline  & **SS** & **SP** & **PS** & **PP** \\ \hline \(L_{1}\) & 610.0 nH & 40.0 nH & 610.0 nH & 40.0 nH \\ \(R_{1}\) & 0.3381 \(\Omega\) & 0.0222 \(\Omega\) & 0.3381 \(\Omega\) & 0.0222 \(\Omega\) \\ \(L_{2}\) & 610.0 nH & 40.0 nH & 610.0 nH & 40.0 nH \\ \(R_{2}\) & 0.3381 \(\Omega\) & 0.0222 \(\Omega\) & 0.3381 \(\Omega\) & 0.0222 \(\Omega\) \\ \(C_{1}\) & 225.8 pF & 3.6736 pF & 212.5 pF & 3.4517 nF \\ \(C_{2}\) & 225.8 pF & 3.4440 nF & 225.8 pF & 3.4440 nF \\ \(M\) & 152.5 nH & 10.0 nH & 152.5 nH & 10.0 nH \\ \(\eta\) & 94.9\% & 94.8\% & 94.9\% & 94.8\% \\ \hline \end{tabular}
\end{table} 
Table 2: Design parameters. Figure 2: Efficiency as a function of coil inductances: (**a**) SS; (**b**) SP; (**c**) PS; (**d**) PP. To address this undesired issue, it is necessary to select new design parameters. One viable strategy involves adjusting the \(L_{2}\) values to mitigate instability, as outlined in Table 4. While this approach may require operating outside the point of maximum efficiency, careful consideration can help minimize any significant impact on energy transfer. Table 4 provides a summary of the new parameters for the input current with bifurcation reduction. From the new parameters, as presented in Figure 4, the SS (a) and SP (b) topologies became stable for frequency variations. In addition, the PS and PP topologies brought a peculiar behavior. For frequency variations of up to 10%, the input current was maintained practically constant, which is a very attractive feature for practical implementations. After the selection of the parameter values and confirming the minimization of the bifurcation \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **SS** & **SP** & **PS** & **PP** \\ \hline \(L_{1}\) & 610.0 nH & 40.0 nH & 610.0 nH & 40.0 nH \\ \(R_{1}\) & 0.3381 \(\Omega\) & 0.0222 \(\Omega\) & 0.3381 \(\Omega\) & 0.0222 \(\Omega\) \\ \(L_{2}\) & 300.0 nH & 70.0 nH & 430.0 nH & 57.0 nH \\ \(R_{2}\) & 0.1663 \(\Omega\) & 0.0388 \(\Omega\) & 0.2385 \(\Omega\) & 0.0316 \(\Omega\) \\ \(C_{1}\) & 225.8 pF & 3.6736 nF & 219.0 pF & 3.5609 nF \\ \(C_{2}\) & 459.2 pF & 1.9680 nF & 320.2 pF & 2.4168 nF \\ \(M\) & 107.0 nH & 13.2 nH & 128.1 nH & 11.9 nH \\ \(\eta\) & 94.0\% & 93.8\% & 94.7\% & 93.6\% \\ \hline \hline \end{tabular}
\end{table}