# Influence of MOSFET Parasitic Capacitance on the Operation of Interleaved ZVS Boost Converters
## 2 Converter Operation Analysis  
The converter may be analyzed by using four equivalent circuits for one phase of the converter solely. Those equivalent circuits are presented in Figure 3 and are related to the timestamps highlighted in Figure 2. It is assumed that:  
\[L_{1}=L_{2}=L, \tag{1}\]  
\[C_{1}=C_{2}=C, \tag{2}\]  
to simplify the analysis. The following variables are defined:  
\[\omega_{0}=\frac{1}{\sqrt{LC}}, \tag{3}\]  
\[Z_{0}=\sqrt{\frac{L}{C}}, \tag{4}\]  
\[\overline{R}=\frac{R}{Z_{0}}, \tag{5}\]  
\[G=\frac{V_{\rm o}}{V_{\rm in}}, \tag{6}\]  
Figure 2: Waveforms of chosen voltages and currents in the analyzed converter.  
where \(\omega_{0}\) is the resonant angular frequency, \(Z_{0}\) is the characteristic impedance of the resonant circuit, \(\overline{R}\) is the normalized load resistance, and \(G\) is the converter voltage ratio.  
Differential equations may be composed for each of the four equivalent circuits, allowing for the calculation of the time intervals. These time intervals are equal to:  
\[t_{1}=\frac{1}{\omega_{0}}\mathrm{asin}\left[\frac{V_{\mathrm{o}}}{i_{\mathrm{L }}(t_{0})Z_{0}}\right], \tag{7}\]  
\[t_{2}-t_{1}=-\frac{i_{\mathrm{L}}(t_{1})L}{V_{\mathrm{in}}-V_{\mathrm{o}}}, \tag{8}\]  
\[t_{3}-t_{2}=\frac{1}{\omega_{0}}\mathrm{acos}\left[-\frac{V_{\mathrm{in}}}{V_{ \mathrm{o}}-V_{\mathrm{in}}}\right], \tag{9}\]  
The converter may be further analyzed by assuming that the input energy is equal to the output energy and by replacing the voltage supply \(V_{\mathrm{in}}\) with a current source \(J\) where \(J\) is equal to the average value of the input current. The average voltage on the current source is equal to the average voltage on the resonant capacitor \(C\), as the average voltage on the inductor \(L\) is equal to zero. Thus the input energy is equal to:  
\[E_{\mathrm{i}}=J\int\limits_{0}^{T}v_{\mathrm{C}}\mathrm{d}t. \tag{10}\]  
During the last state of operation (\(t_{3}-T\)) the voltage \(v_{\mathrm{C}}\) is zero, thus the input energy equation can be written as:  
\[E_{\mathrm{i}}=J\left[\int\limits_{0}^{t_{1}}(i_{\mathrm{L}}(0)Z_{0}\sin( \omega_{0}t))\mathrm{d}t+\int\limits_{t_{1}}^{t_{2}}(V_{\mathrm{o}})\ \mathrm{d}t+\int\limits_{t_{2}}^{t_{3}}(V_{\mathrm{in}}+(V_{\mathrm{o}}-V_{ \mathrm{in}})\cos(\omega_{0}(t-t_{2}))\mathrm{d}t\right]. \tag{11}\]  
Figure 3: Equivalent circuits of one phase of the converter in four states of operation. (**a**) Initial resonant capacitor charging after transistor turn-off; (**b**) Current flowing to the output through the diode; (**c**) Resonant discharging of the capacitor; (**d**) Charging the inductor after transistor turn-on.  
Output energy is equal to:  
\[E_{\text{o}}=\frac{V_{\text{o}}^{2}T}{R}. \tag{12}\]  
By assuming equality of input and output energy the following implicit equation can be derived:  
\[\sqrt{\frac{4\pi f_{0}}{N\overline{R}f_{\text{s}}}G(G-1)+G^{2}}+\sqrt{\frac{4\pi f _{0}}{N\overline{R}f_{\text{s}}}\frac{G}{G-1}}+\left(1-G\right)\sqrt{1-\frac{1 }{\left(G-1\right)^{2}}}+\arcsin\left[\sqrt{1-\frac{1}{\left(G-1\right)^{2}}} \right]=2\pi\frac{f_{0}}{f_{\text{s}}}, \tag{13}\]  
where \(N\) is the number of phases of the converter, \(f_{0}\) is the resonant frequency, and \(f_{\text{s}}\) is the switching frequency. A detailed derivation of (13) is presented in Appendix A.