## 3 The Proposed Average Small Signal Modelling and Control Circuits Design
### Conventional CCM PFC Topology
\right] \tag{36}\]  
Figure 9 presents the control blocks for the outer and inner control loops of the conventional PFC using the PI controller. The optimal parameters for the PI controllers can be obtained by deriving the voltage and current systems transfer functions using the average signal modelling. For the voltage control loop; the TF of the outer voltage system \(\mathrm{G}_{\mathrm{v}}(\mathrm{s})\) in s-domain can be obtained as the output voltage to the inductor current transfer function as (37), and the analog PI controller system can be expressed using the TF \(\mathrm{G}_{\mathrm{PIv}}(\mathrm{s})\) as (38)  
\[\mathrm{G}_{\mathrm{v}}(\mathrm{s})=\frac{\mathrm{v}_{\mathrm{o}}^{*}(\mathrm{ s})}{\mathrm{i}_{\mathrm{Lb}}^{*}(\mathrm{s})}=\frac{\left|\mathrm{v}_{\mathrm{in}} \right|}{2\,\mathrm{V}_{\mathrm{o}}\,\mathrm{C}_{\mathrm{b}}\mathrm{s}} \tag{37}\]  
\[\mathrm{G}_{\mathrm{PIv}}(\mathrm{s})=\mathrm{K}_{\mathrm{PV}}+\frac{\mathrm{K }_{\mathrm{IV}}}{\mathrm{s}}\;. \tag{38}\]  
where \(\mathrm{K}_{\mathrm{PV}}\) is the proportional gain and \(\mathrm{K}_{\mathrm{IV}}\) is the integral gain of the voltage PI controller. The closed-loop TF of the outer voltage control loop system (\(\mathrm{G}_{\mathrm{CLv}}(\mathrm{s})\)) can be obtained as  
\[\mathrm{G}_{\mathrm{CLv}}(\mathrm{s})=\frac{\mathrm{V}_{\mathrm{o}}(\mathrm{ s})}{\mathrm{V}_{\mathrm{ref}}(\mathrm{s})}=\frac{\frac{\left|\mathrm{v}_{\mathrm{in}} \right|}{2\,\mathrm{V}_{\mathrm{o}}\,\mathrm{C}_{\mathrm{b}}}\mathrm{s}+\frac{ \left|\mathrm{v}_{\mathrm{in}}\right|}{2\,\mathrm{V}_{\mathrm{o}}\,\mathrm{C }_{\mathrm{b}}}\mathrm{s}+\frac{\left|\mathrm{v}_{\mathrm{in}}\right|}{2\, \mathrm{V}_{\mathrm{o}}\,\mathrm{C}_{\mathrm{b}}}\mathrm{s}} \tag{39}\]  
For the current control loop; the TF of the inner current system \(\mathrm{G}_{\mathrm{i}}(\mathrm{s})\) in s-domain can be obtained as the output voltage to the inductor current transfer function as (40), and the analog PI controller system can be expressed using the TF \(\mathrm{G}_{\mathrm{PIi}}(\mathrm{s})\) as (41)  
\[\mathrm{G}_{\mathrm{i}}(\mathrm{s})=\frac{\mathrm{i}_{\mathrm{Lb}}^{*}(\mathrm{ s})}{\mathrm{d}^{*}(\mathrm{s})}=\frac{\left(\frac{\mathrm{V}_{\mathrm{o}}}{ \mathrm{L}_{\mathrm{b}}\mathrm{C}_{\mathrm{b}}}\mathrm{R}\right)\cdot\left( \mathrm{s}\mathrm{R}\mathrm{C}_{\mathrm{b}}+2\right)}{\mathrm{s}^{2}+\frac{1}{ \mathrm{C}_{\mathrm{b}}^{1}\mathrm{R}}\mathrm{s}+\frac{\left|\mathrm{v}_{ \mathrm{in}}\right|}{2\,\mathrm{V}_{\mathrm{o}}\,\mathrm{C}_{\mathrm{b}}}(1- \mathrm{d})^{2}} \tag{40}\]  
\[\mathrm{G}_{\mathrm{PIi}}(\mathrm{s})=\mathrm{K}_{\mathrm{PIi}}+\frac{\mathrm{ K}_{\mathrm{ii}}}{\mathrm{s}} \tag{41}\]  
Since the converter switching frequency (\(\mathrm{F}_{\mathrm{sw}}\)) was selected as about 100 kHz in this work, for high frequency analysis, the capacitor can be shorted, and the open-loop TF of the inductor current system in Equation (40) can be simplified as  
\[\mathrm{G}_{\mathrm{i}}(\mathrm{s})=\frac{2\,\mathrm{V}_{\mathrm{o}}}{\mathrm{ sL}_{\mathrm{b}}} \tag{42}\]  
Figure 9: Block diagrams of the closed loop control systems: (**a**) outer voltage loop; (**b**) inner current loop. The closed loop TF of the current control loop \(\text{G}_{\text{CLI}}(\text{s})\) can be obtained as  
\[\text{G}_{\text{CLI}}(\text{s})=\frac{\text{I}_{\text{Lb}}(\text{s})}{\text{I}_{ \text{ref}}(\text{s})}=\frac{2\,\text{V}_{\text{o}}\,\text{K}_{\text{Pi}}\, \text{s}+2\,\text{V}_{\text{o}}\text{K}_{\text{li}}}{\text{s}^{2}\text{L}_{ \text{b}}+2\,\text{V}_{\text{o}}\,\text{K}_{\text{Pi}}\,\text{s}+2\,\text{V}_ {\text{o}}\text{K}_{\text{Ri}}} \tag{43}\]  
Once the transfer functions of the control loops are determined, the PI controllers' gains can be designed with the standard form of the second order system TF by selecting the proper control system bandwidth and the undamped natural frequency which enhanced the optimal stability criteria.