# Analysis and Design of a Transient-State Resonant Converter Used as a Frequency Multiplier
## 3 Design Methodology and Experimental Testing
### Development of Experimental Tests  
To verify the operation and performance of the transient resonant converter, experimental tests of the circuit were performed. Figure 9 shows the resonant converter subjected to the experimental tests. A variable resistor was used as load on the \"R\({}_{\text{L}}\)\" output, and for the wireless power transfer, the pair of coupled inductors WE76030811 from Wurth Elektronik (Waldenburg, Germany) was used. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Description** & **Value** \\ \hline \(f_{o}\) & Frequency of current in the resonant tank. & \(f_{sw}n\) & 900 kHz \\ \(\omega_{o}\) & Angular frequency in the resonant tank. & \(2\pi f_{o}\) & \(5.65410^{6}\) rad/s \\ \(\eta_{\text{rec}}\) & Rectifier efficiency. & \(1-\frac{2V_{f}}{V_{o}}\) & 0.88 \\ \(\text{R}_{\text{L}}\) & Load at the output. & \(\frac{2\pi^{3}f_{\text{L}mp}}{8g\eta_{\text{rec}}}\) & 8 \(\Omega\) \\ \(\text{R}_{\text{eq1}}\) & Rectifier equivalent resistance. & \(\frac{8R_{\text{L}}\eta_{\text{rec}}}{\tau^{2}}\) & 6.11 \(\Omega\) \\ \(\text{X}_{\text{Lds}}\) & Secondary stray reactance. & \(f_{o}L_{\text{Lds}}\) & 9.52 \(\Omega\) \\ \(\text{X}_{\text{Lmp}}\) & Reactance of magnetizing inductance. & \(f_{o}L_{mp}=g\text{R}_{\text{eq2}}\) & 61.18 \(\Omega\) \\ \(\text{R}_{\text{s}}\) & Series resistance of the equivalent circuit. & \(\frac{\left(X_{Lmp}\right)^{2}\kappa_{\text{eq1}}}{\left(X_{Lmp}\right)^{2} \left(X_{Lmp}+X_{Lds}\right)^{2}}\) & 4.54 \(\Omega\) \\ \(\text{X}_{\text{Leq1}}\) & Equivalent inductance reactance. & \(\frac{X_{Lmp}\left[X_{Lds}\left(X_{Lmp}+X_{Lds}\right)+R_{\text{eq1}}^{2} \right]}{R_{\text{eq2}}^{2}+\left(X_{Lmp}+X_{Lds}\right)^{2}}\) & 8.63 \(\Omega\) \\ \(a\) & Constant which defines the level of underdamping & \(\frac{\pi n}{-\ln(d_{f})}\) & 58 \\ \(\text{L}_{\text{s}}\) & Series inductance of the equivalent circuit. & \(\frac{DR_{s}d}{2n\eta_{\text{sw}}\pi T}\) & 23.31 \(\mu\)H \\ \(\text{L}_{\tau}\) & Inductor in the resonant tank of the converter. & \(L_{s}-L_{\text{dp}}-L_{eq1}\) & 20.26 \(\mu\)H \\ \(\text{C}_{\text{r}}\) & Capacitor in the resonant tank. & \(\frac{\omega_{o}^{2}L_{s}}{\omega_{o}^{2}L_{s}}\) & 1.34 nF \\ \(b\) & Constant that determines the RMS current in the resonant tank. & \(\frac{a\left(1-e^{-2n\epsilon/\epsilon}\right)}{4\pi n}\) & 0.426 \\ \(\text{X}_{\text{C}}\) & Capacitive reactance. & \(\frac{1}{\omega_{o}\text{C}_{s}}\) & 131 \(\Omega\) \\ \(\text{C}_{\text{odd}}\) & Constant that determines the voltage level in the resonant tank capacitor. & \(\frac{\text{X}_{\text{C}}(1+e^{-n\epsilon/\epsilon})}{2 DR_{s}a-\text{X}_{ \text{C}}\left(1+e^{-n\epsilon/\epsilon}\right)}\) & 12.28 \\ \(\text{Q}\) & Quality factor. & \(\frac{\omega_{o}L_{s}}{R_{\text{s}}}\) & 29 \\ \(\text{V}_{\text{DC}}\) & Input voltage. & \(2\left(\sqrt{\frac{P_{\text{R}}R_{\text{L}}D^{2}a}{b\left(1+e_{\text{all}} \right)^{2}}}\right)\) & 50 V \\ M & Gain in the resonant tank. & \(\frac{\left(1+e_{\text{add}}\right)\sqrt{b}}{Du}\) & 0.3 \\ \hline \end{tabular}
\end{table} 
Table 1: Circuit Design Parameters. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Description** & **Value** \\ \hline \(\text{P}_{o}\) & Average output power & 12 W \\ \(\text{V}_{\text{out}}\) & RMS voltage at the load resistor & 10 V \\ \(f_{\text{sw}}\) & Inverter switching frequency & 300 kHz \\ \(\text{D}\) & Duty cycle & 50\% \\ \(n\) & Frequency multiplier & 3 \\ \(df\) & Damping factor & 0.9 \\ \hline \end{tabular}
\end{table} 
Table 2: Design Methodology for \(n=3\). The experimental tests consisted of checking that the converter complied with the operating specifications in Table 1, in addition to checking that it was operating in the transient state and acting as a frequency multiplier in the resonant tank.