## 3 The Proposed Average Small Signal Modelling and Control Circuits Design
### Interleaved CCM PFC Topology
\begin{array}{c}0\;0- \frac{(1-\mathrm{d})}{\mathrm{L}_{\mathrm{1}}}\\ 0\;0-\frac{(1-\mathrm{d})}{\mathrm{L}_{\mathrm{2}}}\\ \frac{(1-\mathrm{d})}{\mathrm{C}_{\mathrm{b}}}\frac{(1-\mathrm{d})}{\mathrm{C}_{ \mathrm{b}}}-\frac{1}{\mathrm{R}\,\mathrm{C}_{\mathrm{b}}}\end{array} \right] \left[ \begin{array}{c}\mathrm{i}_{\mathrm{L1}}^{*}\\ \mathrm{i}_{\mathrm{L2}}^{*}\\ \mathrm{v}_{\mathrm{o}}^{*}\end{array} \right]+\left[ \begin{array}{c}\frac{ \mathrm{L}_{\mathrm{1}}}{\mathrm{L}_{\mathrm{1}}}\frac{\mathrm{v}_{\mathrm{o}} }{\mathrm{L}_{\mathrm{1}}}\\ \frac{\mathrm{v}_{\mathrm{o}}^{*}}{\mathrm{L}_{\mathrm{2}}}\frac{\mathrm{v}_{ \mathrm{o}}}{\mathrm{L}_{\mathrm{2}}}\\ 0\;\frac{\mathrm{i}_{\mathrm{L1}}+\mathrm{i}_{\mathrm{L1}}}{\mathrm{C}_{ \mathrm{b}}}\end{array} \right]\left[ \begin{array}{c}\mathrm{v}_{\mathrm{ in}}^{*}\\ \mathrm{d}^{*}\end{array} \right] \tag{54}\]  
The transfer function (TF) of the output voltage control system for the interleaved CCM PFC converter can be obtained as given  
\[\mathrm{G}_{\mathrm{v}}(\mathrm{s})=\frac{\mathrm{v}_{\mathrm{o}}^{*}(\mathrm{ s})}{\mathrm{i}_{\mathrm{L1},\mathrm{2}}^{*}(\mathrm{s})}=\frac{|\mathrm{v}_{ \mathrm{in}}|}{2\mathrm{V}_{\mathrm{o}}\,\mathrm{C}_{\mathrm{b}}\mathrm{s}} \tag{55}\]  
The transfer function (TF) of the inner current control loops for the inductor's currents \(\mathrm{I}_{\mathrm{L1}}\) and \(\mathrm{I}_{\mathrm{L2}}\) for the interleaved CCM PFC converter can be obtained as given  
\[\mathrm{G}_{\mathrm{i}}(\mathrm{s})=\frac{\mathrm{i}_{\mathrm{L1},\mathrm{2}}^ {*}(\mathrm{s})}{\mathrm{d}^{*}(\mathrm{s})}=\frac{2\mathrm{V}_{\mathrm{o}}}{ 2\mathrm{R}(1-\mathrm{d})^{2}}\times\frac{1+\frac{\mathrm{s}\mathrm{R} \mathrm{C}_{\mathrm{b}}}{2}}{1+\frac{\mathrm{s}\mathrm{L}_{\mathrm{1},\mathrm{2} }}{\mathrm{R}(1-\mathrm{d})^{2}}+\frac{\mathrm{s}\mathrm{I}_{\mathrm{L1}, \mathrm{2}}}{\mathrm{R}(1-\mathrm{d})^{2}}+\frac{\mathrm{s}\mathrm{I}_{\mathrm{ L1},\mathrm{2}}\mathrm{C}_{\mathrm{b}}}{(1-\mathrm{d})^{2}} \tag{56}\]  
After rearranging Equation (56), the open loop TF of the inductors currents systems can be written as  
\[\mathrm{G}_{\mathrm{1}}(\mathrm{s})=\mathrm{G}_{\mathrm{2}}(\mathrm{s})=\frac{ \left(\frac{\mathrm{V}_{\mathrm{o}}}{\mathrm{L}_{\mathrm{1}}\mathrm{L}_{ \mathrm{2}}\mathrm{C}_{\mathrm{b}}\mathrm{R}}\right).\left(\mathrm{s}\mathrm{R }\mathrm{C}_{\mathrm{b}}+2\right)}{\mathrm{s}^{2}+\frac{\mathrm{I}_{\mathrm{ 1}}\mathrm{L}_{\mathrm{2}}\mathrm{C}_{\mathrm{b}}}{\mathrm{s}}+\frac{2}{ \mathrm{L}_{\mathrm{1}}\mathrm{L}_{\mathrm{2}}\mathrm{C}_{\mathrm{b}}}(1- \mathrm{d})^{2}} \tag{57}\]  
For the interleaved CCM PFC converter, the closed loop bandwidth values were assumed to be about 150 rad/s and 18,000 rad/s for the outer voltage and inner current loops, respectively. Figure 12 depicts the bode plot of the closed-loop TF of the control loops, which shows that the current controller offers unity gain for frequencies less than 5.5 kHz. This current control system, working as a low pass filter, helps to remove the switching frequency noise. Furthermore, the voltage controller offers unity gain for frequencies less than 21 Hz. This voltage controller works as a low pass filter to remove the 60 Hz voltage ripple.