## 3 The Proposed Average Small Signal Modelling and Control Circuits Design
### Interleaved CCM PFC Topology  
The average small signal modelling of the interleaved PFC can be derived in the same way as explained for the conventional boost PFC using state space averaging technique. The interleaved PFC contains two separates boost PFC converters operating 180 degrees out of phase. These two separate legs are assumed to operate identically to derive the transfer function of the model as depicted in Figure 11.  
The mathematical equations which describe the change in inductors currents (I\({}_{\text{L1}}\), I\({}_{\text{L2}}\)) and the output voltage (V\({}_{\text{o}}\)) can be given as  
\[\frac{\text{d}\text{I}_{\text{L1}}\ (\text{t})}{\text{dt}}=\frac{\text{V}_{ \text{in}}}{\text{L}_{1}}-\frac{\text{V}_{\text{o}}(1-\text{D})}{\text{L}_{1}} \tag{44}\]  
\[\frac{\text{d}\text{I}_{\text{L2}}\ (\text{t})}{\text{dt}}=\frac{\text{V}_{ \text{in}}}{\text{L}_{2}}-\frac{\text{V}_{\text{o}}(1-\text{D})}{\text{L}_{2}} \tag{45}\]  
\[\frac{\text{d}\text{V}_{\text{o}}(\text{t})}{\text{dt}}=\frac{\text{I}_{\text {L1}}(1-\text{D})}{\text{C}_{\text{b}}}+\frac{\text{I}_{\text{L2}}(1-\text{D} )}{\text{C}_{\text{b}}}-\frac{\text{V}_{\text{o}}}{\text{R}\,\text{C}_{\text {b}}} \tag{46}\]  
Assume all variables (I\({}_{\text{L1}}\), I\({}_{\text{L2}}\), V\({}_{\text{in}}\), V\({}_{\text{o}}\), and D ) are at the steady state at the selected operational point (i\({}_{\text{L1}}\),i\({}_{\text{L2}}\), v\({}_{\text{o}}\), and d ) and the small-signal AC variation (i\({}_{\text{L1}}^{*}\),i\({}_{\text{L2}}^{*}\),v\({}_{\text{in}}^{*}\), v\({}_{\text{o}}^{*}\) and d\({}^{*}\)), where  
\[\text{I}_{\text{L1}}=\text{i}_{\text{L1}}+\text{i}_{\text{L1}}^{*}\text{, }\text{I}_{\text{L2}}=\text{i}_{\text{L2}}+\text{i}_{\text{L2}}^{*}\text{, }\text{ }\text{V}_{\text{in}}=\text{v}_{\text{in}}+\text{v}_{\text{in}}^{*}\text{, }\text{V}_{\text{o}}=\text{v}_{\text{o}}+\text{v}_{\text{o}}^{*}\text{, }\text{D}=\text{d}+\text{d}^{*} \tag{47}\]  
Substituting from (47) in Equations (44)-(46), we obtain  
\[\frac{\text{d}(\text{i}_{\text{L1}}+\text{i}_{\text{L1}}^{*})}{\text{dt}}= \frac{(\text{v}_{\text{in}}+\text{v}_{\text{in}}^{*})}{\text{L}_{1}}-\frac{( \text{v}_{\text{o}}+\text{v}_{\text{o}}^{*})\ (1-\text{d}-\text{d}^{*})}{\text{L}_{1}} \tag{48}\]  
\[\frac{\text{d}(\text{i}_{\text{L2}}+\text{i}_{\text{L2}}^{*})}{\text{dt}}= \frac{(\text{v}_{\text{in}}+\text{v}_{\text{in}}^{*})}{\text{L}_{2}}-\frac{( \text{v}_{\text{o}}+\text{v}_{\text{o}}^{*})\ (1-\text{d}-\text{d}^{*})}{\text{L}_{2}} \tag{49}\]  
\[\frac{\text{d}(\text{v}_{\text{o}}+\text{v}_{\text{o}}^{*})}{\text{dt}}=-\frac{ (\text{v}_{\text{o}}+\text{v}_{\text{o}}^{*})}{\text{R}\,\text{C}_{\text{b}}}+ \frac{(\text{i}_{\text{L1}}+\text{i}_{\text{L1}}^{*})(1-\text{d}-\text{d}^{*} )}{\text{C}_{\text{b}}}+\frac{(\text{i}_{\text{L2}}+\text{i}_{\text{L2}}^{*}) (1-\text{d}-\text{d}^{*})}{\text{C}_{\text{b}}} \tag{50}\]  
Figure 11: Implementation technique of the interleaved CCM PFC analog control circuit. Equations (48)-(50) can be re-written again as Equations (51)-(53) using the averaging model as follow  
\[\frac{\mathrm{d}(\mathrm{i}_{\mathrm{L1}}^{*})}{\mathrm{dt}}=\frac{1}{\mathrm{L} _{\mathrm{1}}}(\mathrm{v}_{\mathrm{in}}^{*})+\frac{\mathrm{v}_{\mathrm{o}}}{ \mathrm{L}_{\mathrm{1}}}(\mathrm{d}^{*})-\frac{(1-\mathrm{d})}{\mathrm{L}_{ \mathrm{1}}}\;(\mathrm{v}_{\mathrm{o}}^{*}) \tag{51}\]  
\[\frac{\text{d}\text{I}_{\text{L1}}\ (\text{t})}{\text{dt}}=\frac{\text{V}_{ \text{in}}}{\text{L}_{1}}-\frac{\text{V}_{\text{o}}(1-\text{D})}{\text{L}_{1}} \tag{44}\]0  
\[\frac{\text{d}\text{I}_{\text{L1}}\ (\text{t})}{\text{dt}}=\frac{\text{V}_{ \text{in}}}{\text{L}_{1}}-\frac{\text{V}_{\text{o}}(1-\text{D})}{\text{L}_{1}} \tag{44}\]1  
Arrange state Equations (51)-(53) in state space matrix form to obtain the state space representation of the small signal stability model for the interleaved CCM PFC converter as given  
\[\left[ \begin{array}{c}\mathrm{i}_{\mathrm{L1}}^{*}\\ \mathrm{i}_{\mathrm{L2}}^{*}\\ \mathrm{v}_{\mathrm{o}}^{*}\end{array} \right]=\left[