## 3 Telecom AC/DC PFC Boost Converter Operation Principle and Design
### Design of the Inner Current-Control Loop  
The design procedure for the inner current-control loop was the same as for the voltage control; we use an appropriate stability criterion for the inner current-control loop. The inner current-control loop must have enough bandwidth higher than the outer voltage loop bandwidth so that it is fast enough to track the current changes (to let the inductor current track the reference current). The current-control loop bandwidth must be less than switching frequency (F\({}_{\text{sw}}\)) to reject the noise at the switching frequency. Using the small  
Figure 8: Bode plot of the closed loop outer voltage control system. signal model of the boost converter obtained in Equation (18), the open-loop TF of the inductor current control system \(\mathrm{G_{i}(s)}\) can be expressed as  
\[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]  
After rearranging Equation (34), the open loop TF of the inductor current system can be written as  
\[\mathrm{G_{i}(s)=\frac{\left(\frac{V_{o}}{L_{b}C_{b}R}\right)\cdot\left(sRC_{b} +2\right)}{s^{2}+\frac{1}{C_{b}R}s+\frac{1}{L_{b}C_{b}}\left(1-D\right)^{2}}. \tag{35}\]  
From Equation (35), we observe that the stability of the inductor current system depends on the converter duty cycle (D). In this work, the PFC boost converter was designed to obtain a constant output voltage with a wide range of the input voltage (85-265 V). Based on these operating conditions, the minimum duty cycle of the converter is about 0.40 and the maximum duty cycle is about 0.95. Before starting the design of the controller for the inner current-control loop, the inductor current system stability was checked using the root locus stability, which was plotted as shown in Figure 10 for the minimum and maximum converter duty cycles. At D\({}_{\mathrm{min}}\) = 0.40, the open loop roots r\({}_{1,2}\) = \(-\)6.97 \(\pm\) 826.82i, and at D\({}_{\mathrm{max}}\) = 0.95, r\({}_{1,2}\) = \(-\)6.97 \(+\) 68.55i. Varying the duty cycle from 0.4 to 0.95, the real part of the roots is usually negative; at constant value of about \(-\)6.97, only the imaginary parts change with the different values of the converter duty cycle, in which the current system of the CCM-PFC converter is absolutely stable in open loop with the operating range of the converter duty cycle (D). Figure 9: Root locus of the closed loop T.F of the outer voltage control system. Figure 11 shows the bode plot of the open loop of the inductor current system. The peak resonance frequency is proportional to the duty cycle value, and the converter high bandwidth available is proportional to the high duty cycle of the converter. Figure 12 depicts the block diagram of the inner current-control loop using the PI controller. The TF of the closed loop current-control loop can be expressed as  
\[\mathrm{G_{CLI}(s)=\frac{I_{Lb}(s)}{I_{ref}(s)}=\frac{G_{i}(s)\,G_{PIi}(s)}{1+G_ {i}(s)\,.G_{PIi}(s)}\;. \tag{36}\]  
Figure 11: Bode plot of the open loop current system with different duty cycle. Figure 12: Block diagram of the inner current control system with PI controller. Figure 10: Root locus plot of the open loop current system with different duty cycle. The TF of the PI current controller (\(\rm G_{PIi}(s)\)) can be expressed as  
\[\rm G_{PIi}(s)=K_{PI}+\frac{K_{H}}{s}. \tag{37}\]  
where \(\rm K_{PI}\) and \(\rm K_{H}\) are the proportional and integral gains for the PI current controller, respectively.