## 3 Telecom AC/DC PFC Boost Converter Operation Principle and Design
### Design of the Outer Voltage Loop  
The design of the outer voltage control loop starts with obtaining the transfer function (TF) of the outer voltage circuit by modeling the small-signal stability of the boost converter, which describes the operation of the converter at the steady state operation point. For the boost converter shown in Figure 3,  
\[\frac{{\rm V_{o}}}{{\rm V_{in}}}=\frac{1}{1-{\rm D}}, \tag{19}\]  
\[1-{\rm D}=\frac{|{\rm v_{s}}|\sin(\omega{\rm t})}{{\rm V_{o}}}\;. \tag{20}\]  
The boost converter inductor current (I\({}_{\rm Lb}\)) and diode current (I\({}_{\rm Db}\)) can be expressed as Equations (21) and (22), respectively. \[{\rm I_{Lb}}=|{\rm i_{Lb}}|\sin(\omega{\rm t}) \tag{21}\]  
\[{\rm I_{Db}}={\rm I_{Lb}}\cdot(1-{\rm D}) \tag{22}\]  
Substituting from (19)-(21) into (22), the complete boost converter diode current (I\({}_{\rm Db}\)) equation can be written as  
\[{\rm I_{Db}}=|{\rm i_{Lb}}|\sin(\omega{\rm t}).\frac{|{\rm v_{s}}|\sin(\omega{ \rm t})}{{\rm V_{o}}}=\frac{|{\rm i_{Lb}}||{\rm v_{s}}|\sin^{2}(\omega{\rm t} )}{{\rm V_{o}}}=\frac{1}{2}\;\frac{|{\rm i_{Lb}}||{\rm v_{s}}|}{{\rm V_{o}}}- \frac{1}{2}\;\frac{|{\rm i_{Lb}}||{\rm v_{s}}|}{{\rm V_{o}}}\cos(2\omega{\rm t }), \tag{23}\]  
where \(\frac{1}{2}\;\frac{|{\rm i_{Lb}}||{\rm v_{s}}|}{{\rm V_{o}}}\) refers to the DC component and \(\frac{1}{2}\;\frac{|{\rm i_{Lb}}||{\rm v_{s}}|}{{\rm V_{o}}}\cos(2\omega{\rm t})\) is refers to the AC component of the diode current. Applying the averaged small signal perturbation to Equation (23) gives  
\[\mathrm{i}_{\mathrm{Db}}^{*}=\frac{1}{2}\frac{|\mathrm{v}_{\mathrm{s}}|\cdot \mathrm{i}_{\mathrm{Lb}}^{*}}{\mathrm{V}_{\mathrm{o}}}+\frac{\mathrm{v}_{ \mathrm{o}}^{*}\cdot\mathrm{I}_{\mathrm{Db}}}{\mathrm{V}_{\mathrm{o}}}. \tag{24}\]  
The diode current is the sum of two components, load current (\(\mathrm{I}_{\mathrm{load}}\)) and the output capacitor current (\(\mathrm{I}_{\mathrm{CB}}\)), which can be expressed as  
\[\mathrm{I}_{\mathrm{Db}}=\mathrm{I}_{\mathrm{load}}+\mathrm{I}_{\mathrm{CB}} =\mathrm{I}_{\mathrm{cb}}+\frac{V_{o}}{R} \tag{25}\]  
Applying the averaged small signal perturbation to Equation (25) with the resistive load case provides  
\[\mathrm{i}_{\mathrm{Db}}^{*}=\mathrm{v}_{\mathrm{o}}^{*}\mathrm{C}_{\mathrm{ b}}\mathrm{s}+\frac{\mathrm{v}_{\mathrm{o}}^{*}\cdot\mathrm{I}_{\mathrm{Db}}}{ \mathrm{V}_{\mathrm{o}}}. \tag{26}\]  
Comparing the right-hand parts of Equations (24) and (26), we obtain  
\[\frac{1}{2}\frac{|\mathrm{v}_{\mathrm{s}}|\cdot\mathrm{i}_{\mathrm{Lb}}^{*}}{ \mathrm{V}_{\mathrm{o}}}=\mathrm{v}_{\mathrm{o}}^{*}\mathrm{C}_{\mathrm{b}} \mathrm{s}. \tag{27}\]  
So, the open loop TF of the outer voltage system is  
\[\mathrm{G}_{\mathrm{v}}(\mathrm{s})=\frac{\mathrm{v}_{\mathrm{o}}^{*}(\mathrm{ s})}{\mathrm{i}_{\mathrm{Lb}}^{*}(\mathrm{s})}=\frac{|\mathrm{v}_{\mathrm{s}}|}{2 \,\mathrm{V}_{\mathrm{o}}\,\mathrm{C}_{\mathrm{b}}\mathrm{s}}\,. \tag{28}\]  
Figure 7 shows the control blocks for the outer voltage control loop of the PFC boost converter using the PI controller. The PI controller system can be described using the following TF:  
\[1-{\rm D}=\frac{|{\rm v_{s}}|\sin(\omega{\rm t})}{{\rm V_{o}}}\;. \tag{20}\]0  
where \(\mathrm{K}_{\mathrm{PV}}\) is the proportional gain and \(\mathrm{K}_{\mathrm{IV}}\) is the integral gain of the PI controller. The closed-loop TF of the outer voltage control loop system (\(\mathrm{G}_{\mathrm{CLV}}(\mathrm{s})\)) can be obtained as  
\[1-{\rm D}=\frac{|{\rm v_{s}}|\sin(\omega{\rm t})}{{\rm V_{o}}}\;. \tag{20}\]1  
Substituting Equations (28) and (29) into (30) gives the closed-loop TF of the outer voltage control loop, as expressed by  
\[1-{\rm D}=\frac{|{\rm v_{s}}|\sin(\omega{\rm t})}{{\rm V_{o}}}\;. \tag{20}\]2  
So, the PI controller gains \(\mathrm{K}_{\mathrm{PV}}\) and \(\mathrm{K}_{\mathrm{IV}}\) can be designed using Equation (31) and the standard form of the second order system TF by selecting the optimal stability criteria for the control system bandwidth and the undamped natural frequency.