## 3 Telecom AC/DC PFC Boost Converter Operation Principle and Design
### Design of the Inner Current-Control Loop
Since the converter switching frequency (\(\rm F_{sw}\)) 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 (35) can be simplified as  
\[\rm G_{i}(s)=\frac{2\ V_{o}}{sL_{b}}. \tag{38}\]  
Substituting Equations (37) and (38) into (36), the closed-loop TF for the inner current-control system can be derived as  
\[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]0  
Using the same converter operating conditions of the voltage and power used in the voltage loop controller design and with a high bandwidth of about 5000 rad/s for the inner current-control loop, the closed-loop TF of the inner current-control loop can be expressed as  
\[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]1  
Figure 13 shows the bode plot of the closed-loop TF of the current-control loop, which shows that the controller offers unity gain for frequencies less than 1610 Hz. This current control system, also working as a low pass filter, helps to remove the switching frequency noise. The optimal PI controller transfer function designed for the inner voltage control loop can be expressed as  
\[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]2  
Figure 14 shows that the Eigenvalues of the closed-loop current controller are located on the negative side of the pole zero plane, which ensures that the current control system has absolute stability. \[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]3 Bode plot of the current controller closed loop T.F with PI controller. The IP controller is characterized by a highly dynamic response and less overshoot compared with the PI controller. So, an optimal IP controller is proposed and designed in this section to replace the PI in the inner current-control loop to eliminate the current distortion around the zero-crossing point. Figure 15 depicts the block diagram of the inner current control system using the IP controller consisting of two control loops instead of the one used by the PI controller, where the integral gain feeds in forward and the proportional gain feeds backward. From the block diagram in Figure 15, Equation (42) can be derived. \[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]4  
After rearranging Equation (42), we obtain  
\[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]5  
The closed-loop TF of the inner current-control loop using the IP controller can be expressed as  
\[\mathrm{G_{i}(s)=\frac{i_{Lb}^{*}(s)}{d^{*}(s)}=\frac{2V_{o}}{R(1-D)^{2}}\cdot \frac{1+\frac{8 RC_{b}}{2}}{1+\frac{8L_{b}}{R(1-D)^{2}}+\frac{s^{2}L_{b}C_{b}}{ (1-D)^{2}}}. \tag{34}\]6  
The problem of the slow response of the inductor current to tracking the reference current can be solved using the IP controller in the inner current- control loop. Since this controller structure provides two control loops for the current control system, not just the one used in the PI controller, the reliable and exact tracking of the inductor current to the reference current is obtained. As shown in Equation (44), there is no zero in the TF using the IP controller compared with the TF in Equation (39) with the PI controller, which also provides a system response with less overshoot.