# Analysis and Design of a Transient-State Resonant Converter Used as a Frequency Multiplier
## 2 Resonant Converter Transient  
### Determination of Equivalent Circuits for Transient Design of the Resonant Converter  
The design methodology of the transient resonant converter in Figure 2a consisted of determining the equivalent circuits of each stage of the converter until reaching the equivalent RLC circuit in series, as shown in Figure 2b, where the transient analysis shown in [17] is applied.  
The analysis begins by replacing the output voltage of the full bridge inverter with a bipolar square voltage source. The rectifier stage and the load at the output are replaced by its equivalent resistive value \"\(\mathrm{R_{eq1}}\)\" which is defined by (1). The procedure to determine \(\mathrm{R_{eq1}}\) is shown in detail in [2].  
The equivalent circuit replacing the rectifier by its resistive value is shown in Figure 3, where the primary stray inductance \(\mathrm{L_{dp}}\) and secondary stray inductance \(\mathrm{L_{ds}}\) are also considered to determine the following equivalent circuits. These parasitic inductances must be measured in the circuit, considering the operating frequency and the power transmission distance [18].  
\[R_{eq1}=\frac{8R_{L}\eta_{rec}}{\pi^{2}} \tag{1}\]  
where \(\mathrm{R_{L}}\) is the load resistance at the converter output, \(\eta_{\mathrm{rec}}\) is the efficiency of the full bridge rectifier.  
The following equivalent circuit is determined considering the magnetizing inductance \(\mathrm{Lmp}\) and the secondary distortion inductance, as shown in Figure 4a. Performing the reactance analysis between the components \(\mathrm{L_{mp}}\), \(\mathrm{L_{ds}}\), and \(\mathrm{R_{eq2}}\), two new equivalent components \(\mathrm{L_{eq1}}\) and \(\mathrm{R_{s}}\) are obtained, resulting in a series circuit as shown in Figure 4b. Equations (4) and (5) determine the reactances of the equivalent components \(\mathrm{X_{leq}}\) and  
Figure 3: Replacement of the rectifier stage and load \(\mathrm{R_{L}}\) by its equivalent resistive value \(\mathrm{R_{eq1}}\).  
Figure 2: Transient design methodology applied to a resonant converter, (**a**) Full bridge resonant converter, (**b**) Equivalent series RLC circuit.  
\(\mathrm{R_{s}}\), where in each equation the operating frequency \(f_{o}\) is considered to be the switching frequency \(f_{sw}\) multiplied by the multiplier \"\(n\)\", as shown in (2) and (3).  
\[f_{o}=nf_{sw} \tag{2}\]  
\[\omega_{o}=2\pi f_{o} \tag{3}\]  
\[X_{Leq}=\frac{X_{Lmp}\left[X_{Lds}\left(X_{Lmp}+X_{Lds}\right)+R_{eq2}^{2} \right]}{R_{eq2}^{2}+\left(X_{Lmp}+X_{Lds}\right)^{2}} \tag{4}\]  
\[R_{s}=\frac{jX_{Lmp}^{2}R_{eq2}}{R_{eq2}^{2}+\left(X_{Lmp}+X_{Lds}\right)^{2}} \tag{5}\]  
where \(X_{\mathrm{Imp}}\) is the reactance of the magnetizing inductance, \(X_{\mathrm{Lds}}\) is the reactance of the secondary stray inductance, and \(\mathrm{R_{eq2}}\) is the equivalent resistance of the rectifier reflexed to the primary side, which is of the same value as \(\mathrm{R_{eq1}}\approx\mathrm{R_{eq2}}\).  
Considering the operating frequency \(f_{o}\) in the resonant tank of the converter, the value in henries of the equivalent inductor \(\mathrm{L_{eq1}}\) is determined by (6).  
\[L_{eq1}=\frac{X_{Leq}}{2\pi f_{0}} \tag{6}\]