## 3 Telecom AC/DC PFC Boost Converter Operation Principle and Design
### PFC Boost Converter Average Small-Signal Modeling  
For the PFC boost converter shown in Figure 3, the inductor voltage (V\({}_{\text{Lb}}\)) can be expressed as:  
\[\text{V}_{\text{Lb}}=\text{L}_{\text{b}}\frac{\text{d}\text{I}_{\text{Lb}}}{ \text{dt}}=\text{V}_{\text{in}}-\text{V}_{\text{o}}(1-\text{D}). \tag{11}\]  
The output capacitor current (I\({}_{\text{Cb}}\)) can be expressed as  
\[\text{I}_{\text{Cb}}=\text{C}_{\text{b}}\frac{\text{d}\text{V}_{\text{o}}}{ \text{dt}}=-\frac{\text{V}_{\text{o}}}{\text{R}}+\text{I}_{\text{L}}(1-\text{ D}). \tag{12}\]  
where D refers to the converter duty cycle, V\({}_{\text{o}}\) is the output bus voltage (V\({}_{\text{o}}\) = V\({}_{\text{bus}}\)), and R is the load resistance in ohms.  
Assume all variables (I\({}_{\text{Lb}}\), V\({}_{\text{in}}\), V\({}_{\text{o}}\), and D ) are at the steady state at the selected operational point (i\({}_{\text{Lb}}\), v\({}_{\text{in}}\), v\({}_{\text{o}}\), and d ) and the small-signal AC variation (i\({}_{\text{Lb}}^{*}\), v\({}_{\text{in}}^{*}\), v\({}_{\text{o}}^{*}\) and d\({}^{*}\)), where  
\[\text{I}_{\text{Lb}}=\text{i}_{\text{Lb}}+\text{i}_{\text{Lb}}^{*}\text{; }\text{; }\text{V}_{\text{in}}=\text{v}_{\text{in}}+\text{v}_{\text{in}}^{*}\text{; }\text{; }\text{V}_{\text{o}}=\text{v}_{\text{o}}+\text{v}_{\text{o}}^{*}\text{; }\text{D}=\text{d}+\text{d}^{*}. \tag{13}\]  
Figure 5: AC-DC telecom PFC boost converter with digitally controlled technique.  
Figure 6: Implementation technique of the controlling circuit inside the DSP.  
The PFC boost converter control circuit shown in Figure 6 should modify the duty cycle (D) based on the small-signal AC variation to control the output voltage (V\({}_{\rm o}\)) and the inductor current (I\({}_{\rm Lb}\)).  
Substituting Equation (13) into (11) and (12), we obtain  
\[{\rm L_{b}}\frac{{\rm d}({\rm i_{Lb}}+{\rm i_{Lb}^{*}})}{{\rm dt}t}=({\rm v_{in}} +{\rm v_{in}^{*}})-({\rm v_{o}}+{\rm v_{o}^{*}})\;(1-{\rm d}-{\rm d}^{*}), \tag{14}\]  
\[{\rm C_{b}}\frac{{\rm d}({\rm v_{o}}+{\rm v_{o}^{*}})}{{\rm dt}t}=-\frac{({\rm v _{o}}+{\rm v_{o}^{*}})}{{\rm R}}+{\rm I_{load}}(1-{\rm d}-{\rm d}^{*}). \tag{15}\]  
Equations (14) and (15) can be re-written again as Equations (16) and (17) neglecting the term \({\rm v_{o}^{*}}{\rm d}^{*}\), which is the product of two AC small signals.  
\[{\rm L_{b}}\frac{{\rm d}({\rm i_{Lb}^{*}})}{{\rm dt}t}=({\rm v_{in}^{*}})-({ \rm v_{o}^{*}})\;(1-{\rm d})+{\rm v_{o}}\cdot{\rm d}^{*} \tag{16}\]  
\[{\rm C_{b}}\frac{{\rm d}({\rm v_{o}^{*}})}{{\rm dt}t}=-\frac{{\rm v_{o}^{*}}}{ {\rm R}}+{\rm I_{Lb}}\cdot{\rm d}^{*}+({\rm i_{Lb}^{*}})(1-{\rm d}) \tag{17}\]  
Use the Laplace transform to obtain the small-signal model, and arrange the model in state space matrix form to obtain  
\[\left[\begin{array}{cc}{\rm sL_{b}}\;1-{\rm d}\\ 1-{\rm d}\;-{\rm sC_{b}}-\frac{1}{{\rm R}}\end{array}\right]\left[\begin{array} []{cc}{\rm i_{Lb}^{*}}({\rm s})\\ {\rm v_{o}^{*}}({\rm s})\end{array}\right]=\left[\begin{array}{cc}{\rm v_{o} }\\ {\rm i_{Lb}}\end{array}\right]\cdot{\rm d}^{*}({\rm s})+\left[\begin{array}{ cc}1\\ 0\end{array}\right]\cdot{\rm v_{in}^{*}}({\rm s}). \tag{18}\]  
With the derived model of the PFC boost converter and selecting the appropriate system stability criterions, which enhance the system stability and improve power quality, the two control loops of the PFC boost converter can be designed, as discussed in the following subsections.