# Influence of MOSFET Parasitic Capacitance on the Operation of Interleaved ZVS Boost Converters
## Appendix A  
A detailed derivation of Equation (13) is given in the Appendix A. By assuming the input power is equal to the output power the current source \(J\) current value may be defined as:  
\[J=\frac{V_{\mathrm{o}}^{2}}{V_{\mathrm{in}}R}. \tag{15}\]  
After integrating (11) and using (12) and (15) the following is obtained:  
\[\frac{V_{\mathrm{o}}^{2}i_{\mathrm{L}}(0)Z_{0}}{\omega_{0}V_{\mathrm{in}}R}- \frac{V_{\mathrm{o}}^{2}i_{\mathrm{L}}(t_{1})Z_{0}}{\omega_{0}V_{\mathrm{in}}R }+\frac{V_{\mathrm{o}}^{3}i_{\mathrm{L}}(t_{1})L}{V_{\mathrm{in}}R(V_{\mathrm{ o}}-V_{\mathrm{in}})}+\frac{V_{\mathrm{o}}^{2}i_{\mathrm{L}}(t_{3})Z_{0}}{ \omega_{0}V_{\mathrm{in}}R}+\frac{V_{\mathrm{o}}^{2}V_{\mathrm{in}}}{\omega_{ 0}V_{\mathrm{in}}R}\mathrm{asin}\left[\frac{i_{\mathrm{L}}(t_{3})Z_{0}}{V_{ \mathrm{in}}-V_{\mathrm{o}}}\right]=\frac{V_{\mathrm{o}}^{2}T}{R}, \tag{16}\]  
By solving differential equations for the equivalent circuits shown in Figure 3 the value of \(t_{1}\) may also be calculated as:  
\[t_{1}=\frac{1}{\omega_{0}}\mathrm{acos}\left[\frac{i_{\mathrm{L}}(t_{1})}{i_{ \mathrm{L}}(t_{0})}\right], \tag{17}\]thus:  
\[i_{\rm L}(t_{1})=i_{\rm L}(0)\sqrt{1-\frac{V_{\rm o}^{2}}{(i_{\rm L}(0))^{2}Z_{ 0}^{2}}}. \tag{10}\]  
Similarly \(i_{\rm L}(t_{3})\) can be obtained:  
\[i_{\rm L}(t_{3})=\frac{V_{\rm in}-V_{\rm o}}{Z_{0}}\sqrt{1-\frac{V_{\rm in}^{ 2}}{(V_{\rm o}-V_{\rm in})^{2}}}. \tag{11}\]  
Substituting (10) and (11) into (12) yields:  
\[\begin{array}{l}\frac{i_{\rm L}(0)Z_{0}}{\omega_{0}V_{\rm in}}-\frac{i_{\rm L }(0)Z_{0}}{\omega_{0}V_{\rm in}}\sqrt{1-\frac{V_{\rm o}^{2}}{(i_{\rm L}(0))^{2 }Z_{0}^{2}}}+\frac{V_{\rm o}i_{\rm L}(0)L}{V_{\rm in}(V_{\rm o}-V_{\rm in})} \sqrt{1-\frac{V_{\rm o}^{2}}{(i_{\rm L}(0))^{2}Z_{0}^{2}}}+\\ +\frac{V_{\rm in}-V_{\rm o}}{\omega_{0}V_{\rm in}}\sqrt{1-\frac{V_{\rm in}^{ 2}}{(V_{\rm o}-V_{\rm in})^{2}}}+\frac{1}{\omega_{0}}\text{asin}\left[\sqrt{1- \frac{V_{\rm in}^{2}}{(V_{\rm o}-V_{\rm in})^{2}}}\right]=T\end{array} \tag{12}\]  
The average current flowing to the output from each phase can be determined by analyzing the waveforms in Figure 2 and it is equal to:  
\[\frac{I_{o}}{N}=\frac{i_{\rm L}(t_{1})}{2T}(t_{2}-t_{1}), \tag{13}\]  
substituting (8) and (10) into (13) and using [PERSON]'s law yields:  
\[\frac{V_{o}}{NR}=\frac{L(i_{\rm L}(0))^{2}}{2T(V_{\rm o}-V_{\rm in})}-\frac{V _{\rm o}^{2}L}{2 TZ_{0}^{2}(V_{\rm o}-V_{\rm in})}, \tag{14}\]  
which can be further transformed into:  
\[i_{\rm L}(0)=\sqrt{\frac{2 TV_{o}(V_{\rm o}-V_{\rm in})}{RLN}+\frac{V_{\rm o}^{ 2}}{Z_{0}^{2}}}. \tag{15}\]  
By substituting (15) into (12) and further transforming (12) the following is obtained:  
\[\begin{array}{l}\sqrt{\frac{2 TV_{o}(V_{\rm o}-V_{\rm in})}{RLN}\frac{Z_{0} ^{2}}{V_{\rm in}^{2}}+\frac{V_{\rm o}^{2}}{V_{\rm in}^{2}}}+\sqrt{\frac{Z_{0} ^{2}}{(V_{\rm o}-V_{\rm in})^{2}}\frac{2 TV_{o}(V_{\rm o}-V_{\rm in})}{RLN}}+ \\ +\frac{V_{\rm in}-V_{\rm o}}{V_{\rm in}}\sqrt{1-\frac{V_{\rm in}^{ 2}}{(V_{\rm o}-V_{\rm in})^{2}}}+\text{asin}\left[\sqrt{1-\frac{V_{\rm in}^{ 2}}{(V_{\rm o}-V_{\rm in})^{2}}}\right]=\omega_{0}T\end{array} \tag{16}\]  
It can be seen that \(\frac{Z_{0}}{R}\) is equal to \(\frac{1}{R}\), \(\frac{Z_{0}}{L}\) is equal to \(\omega_{0}\) and \(\omega_{0}T\) equals \(2\pi\frac{f_{0}}{f_{\rm s}}\). By using these equalities and substituting the voltage ratio \(G\) into (16) the final form of the implicit equation given by (13) is obtained:  
\[\sqrt{\frac{4\pi f_{0}}{N\overline{