## 3 The Proposed Average Small Signal Modelling and Control Circuits Design
### Conventional CCM PFC Topology  
The control loops of the conventional CCM PFC converter are implemented with two control loops as depicted in Figure 8, the outer loop which regulate the output voltage level to the desired value, and the inner loop which used to let the input current follows the reference current to reduce the inductor current distortion and provide input power with high power factor. The mathematical equations which describe the change in the inductor current (\(\mathrm{I_{Lb}}\)) and output voltage (\(\mathrm{V_{o}}\)) for the conventional CCM PFC converter shown in Figure 2a can be given as  
\[\frac{\mathrm{d}\mathrm{I_{Lb}}}{\mathrm{dt}}=\frac{\mathrm{V_{in}}}{\mathrm{I _{b}}}-\frac{\mathrm{V_{o}}(1-\mathrm{D})}{\mathrm{L_{b}}} \tag{29}\]  
\[\frac{\mathrm{d}\mathrm{V_{o}}}{\mathrm{dt}}=-\frac{\mathrm{V_{o}}}{\mathrm{R \,C_{b}}}+\frac{\mathrm{I_{Lb}}(1-\mathrm{D})}{\mathrm{C_{b}}} \tag{30}\]  
where \(\mathrm{R}\) is the load resistance in ohms. Assume all variables (\(\mathrm{I_{Lb}}\), \(\mathrm{V_{in}}\), \(\mathrm{V_{o}}\), and \(\mathrm{D}\)) are at the steady state at the selected operational point (\(\mathrm{i_{Lb}}\), \(\mathrm{v_{in}}\), \(\mathrm{v_{o}}\), and \(\mathrm{d}\) ) and the small-signal AC variation (\(\mathrm{i_{Lb}^{*}}\), \(\mathrm{v_{in}^{*}}\), \(\mathrm{v_{o}^{*}}\) and \(\mathrm{d}^{*}\)), where  
\[\mathrm{I_{Lb}}=\mathrm{i_{Lb}}+\mathrm{i_{Lb}^{*}}\;;\;\mathrm{V_{in}}= \mathrm{v_{in}}+\mathrm{v_{in}^{*}}\;;\;\mathrm{V_{o}}=\mathrm{v_{o}}+\mathrm{ v_{o}^{*}}\;;\;\mathrm{D}=\mathrm{d}+\mathrm{d}^{*} \tag{31}\]  
The designed PFC control system should track and modify the duty cycle (D) based on the small-signal AC variation to regulate \(\mathrm{V_{o}}\) and \(\mathrm{I_{Lb}}\). Substituting Equation (29) into (27) and (28), we obtain  
\[\frac{\mathrm{d}(\mathrm{i_{Lb}}+\mathrm{i_{Lb}^{*}})}{\mathrm{dt}}=\frac{( \mathrm{v_{in}}+\mathrm{v_{in}^{*}})}{\mathrm{L_{b}}}-\frac{(\mathrm{v_{o}}+ \mathrm{v_{o}^{*}})}{\mathrm{L_{b}}}\frac{(1-\mathrm{d}-\mathrm{d}^{*})}{ \mathrm{L_{b}}} \tag{32}\]  
\[\frac{\mathrm{d}(\mathrm{v_{o}}+\mathrm{v_{o}^{*}})}{\mathrm{dt}}=-\frac{( \mathrm{v_{o}}+\mathrm{v_{o}^{*}})}{\mathrm{R\,C_{b}}}+\frac{(\mathrm{i_{Lb}}+ \mathrm{i_{Lb}^{*}})(1-\mathrm{d}-\mathrm{d}^{*})}{\mathrm{C_{b}}} \tag{33}\]  
Equations (30) and (31) can be re-written again as Equations (32) and (33) using the averaging model which consider that over the switching period in the steady state, the inductor current change, and the capacitor voltage is zero. \[\frac{\mathrm{d}(\mathrm{i_{Lb}^{*}})}{\mathrm{dt}}=\frac{1}{\mathrm{L_{b}}}( \mathrm{v_{in}^{*}})+\frac{\mathrm{v_{o}}}{\mathrm{L_{b}}}(\mathrm{d}^{*})- \frac{(1-\mathrm{d})}{\mathrm{L_{b}}}\;(\mathrm{v_{o}^{*}}) \tag{34}\]  
\[\frac{\mathrm{d}(\mathrm{v_{o}^{*}})}{\mathrm{dt}}=\frac{(1-\mathrm{d})}{ \mathrm{C_{b}}}(\mathrm{i_{Lb}^{*}})+\frac{\mathrm{i_{Lb}}}{\mathrm{C_{b}}}( \mathrm{d}^{*})-\frac{1}{\mathrm{R\,C_{b}}}(\mathrm{v_{o}^{*}}) \tag{35}\]  
Figure 8: Implementation technique of the conventional CCM PFC analog control circuit. Arrange state Equations (34) and (35) in state space matrix form to obtain the state space representation of the small signal stability model for the conventional CCM PFC converter as given  
\[\left[ \begin{array}{c}\mathrm{i}_{\mathrm{Lb}}^{*}\\ \mathrm{v}_{\mathrm{o}}^{*}\end{array} \right]=\left[ \begin{array}{c}0-\left( \frac{1-\mathrm{d}}{\mathrm{L}_{\mathrm{b}}}\right)\\ \left(\frac{1-\mathrm{d}}{\mathrm{C}_{\mathrm{b}}}\right)\frac{-1}{\mathrm{R} \,\mathrm{C}_{\mathrm{b}}}\end{array} \right]\left[ \begin{array}{c}\mathrm{ i}_{\mathrm{Lb}}^{*}\\ \mathrm{v}_{\mathrm{o}}^{*}\end{array} \right]+\left[ \begin{array}{c}\frac{1 }{\mathrm{L}_{\mathrm{b}}}\frac{\mathrm{V}_{\mathrm{o}}}{\mathrm{L}_{\mathrm{b }}}\\ 0\frac{\mathrm{i}_{\mathrm{Lb}}}{\mathrm{C}_{\mathrm{b}}}\end{array} \right]\cdot \left[ \begin{array}{c}\mathrm{v}_{\mathrm{in}}^{*}\\ \mathrm{d}^{*}\end{array}