# Stability Analysis of Buck Converter Based on Passivity-Based Stability Criterion
## 4 Improved Control Strategy  
In order to solve the problem that the total admittance of the system is not passive, a closed-loop control chart conforming to passivity is designed by improving the output admittance model. On the basis of the original voltage and current double closed-loop, the differential link is added to the voltage feedback, and the improved control block diagram is shown in Figure 6. The transfer function of the improved control is \(G(s)=k_{\rm FS}+1\).  
Figure 5: Nyquist diagram of system output admittance. (a) \(R_{\rm d}\) increases; (b) \(k_{\rm p}\) increases.  
The improved admittance model of the Buck converter is  
\[Y_{2}=-\frac{i_{\rm o}}{v_{\rm dc}}=\frac{a_{3}s^{3}+a_{2}s^{2}+a_{1}s^{1}+a_{0}} {b_{3}s^{3}+b_{2}s^{2}+b_{1}s^{1}+b_{0}} \tag{7}\]  
where  
\[\left\{\begin{array}{l}a_{3}=V_{8}k_{\rm F}k_{\rm P1}k_{\rm P2}\\ a_{2}=V_{8}k_{\rm F}k_{\rm P1}k_{\rm 12}+V_{8}k_{\rm F}k_{\rm P2}k_{\rm 12}+V_{8}k_{ \rm P1}k_{\rm P2}+1\\ a_{1}=V_{8}k_{\rm F}k_{\rm 12}k_{\rm 12}+V_{8}k_{\rm P1}k_{\rm 12}+V_{8}k_{\rm P2}k_{ \rm 12}\\ a_{0}=V_{8}k_{\rm 12}k_{\rm 12}\\ b_{3}=L\\ b_{2}=R_{\rm d}V_{8}k_{\rm P1}k_{\rm P2}+V_{8}k_{\rm P2}+r\\ b_{1}=R_{\rm d}V_{8}k_{\rm P1}k_{\rm 12}+R_{\rm d}V_{8}k_{\rm P2}k_{\rm 12}+V_{ 8}k_{\rm 12}\\ b_{0}=R_{\rm d}V_{8}k_{\rm 12}k_{\rm 12}\end{array}\right.\]  
When \(s\) in Equation (7) tends to 0 or infinity, respectively, the admittance \(Y_{2}\) of the Buck converter at low frequency and high frequency is  
\[\left\{\begin{array}{l}\lim_{s\to 0}Y_{2}=1/R_{\rm d}\\ \lim_{s\to\infty}Y_{2}=\frac{V_{s}k_{\rm F}k_{\rm P1}k_{\rm P2}}{L}\end{array}\right. \tag{8}\]  
In order to ensure the passivity of the Buck-CPL subsystem, the controller parameters need to be satisfied.  
\[\left\{\begin{array}{l}0\leq R_{\rm d}\leq\frac{V_{\rm d}^{2}}{\Phi}\\ k_{\rm F}k_{\rm P1}k_{\rm P2}\geq\frac{PL}{V_{\rm dc}^{2}V_{\rm s}}\end{array}\right. \tag{9}\]  
It can be seen from Equation (9) that the admittance of the Buck converter in the high frequency band is no longer 0 after adding the feedback differential link, thus improving the passivity of the admittance of the Buck converter in the high frequency band.  
The comparison between Figures 5 and 8 can also verify that the improved control makes the system admittance gradually shift to the right, thus improving the passivity and stability of the system. In Figure 8, the dashed black line is the coordinate axis.  
Figure 6: Improved control diagram of the Buck converter.  
Figure 7: Closed loop small-signal equivalent model for improved control.  
The step responses of traditional control and improved control are shown in Figure 9.  
It can be seen from Figure 9 that the overshoot of the unit step response of traditional control is 20% and the adjustment time is 0.4 s. In comparison, the unit step response of improved control has no overshoot, the response speed is faster and the adjustment time reduces to 0.2 s.  
Figure 8: The improved Nyquist diagram of the output admittance using the proposed method. (**a**) \(R_{\text{d}}\) increases; (**b**) \(k_{\text{p}}\) increases; (**c**) \(k_{i}\) increases.  
Figure 9: The unit step response of the output admittance.