# Variable Multiple Interleaved Bi-Directional DC/DC Converter with Current Ripple Optimization
## 3 The Model of Variable Multiplicity Interleaving Bidirectional DC/DC Power Converter
### The Design of the Current Compensation Network  
Due to the output current of variable multiplicity interleaving bidirectional DC/DC power converter being constant, the single current closed-loop control can be adopted, and the control strategy diagram is shown in Figure 10.  
Figure 10: Control strategy diagram of variable multiplicity interleaving bidirectional DC/DC power converter.  
Figure 9: The Bode diagrams when the working number of the parallel multiplicity is 4, 5, or 6.  
When the converter is operating in Buck mode, and the number of working multiplicities is 5, then the transfer function of the current-loop-controlled object can be shown as.  
\[G_{\text{id}}(s)=\frac{6s}{1.8\times 10^{-4}s^{2}+6\times 10^{-3}s+5} \tag{32}\]  
Through the analysis of the above Bode diagram, it can be seen that there are at least two integral links in the compensation link so that the amplitude-frequency characteristic curve can decrease at the slope of \(-20\) dB/dec in the low-frequency band. In order to make the system have a strong ability to suppress high-frequency noise, the high-frequency band of the amplitude-frequency characteristic curve should decrease at the slope of \(40\) dB/dec [29].  
The transfer function of the current compensation network is:  
\[G_{\text{idc}}(s)=\frac{K(s+\omega_{\text{Z1}})(s+\omega_{\text{Z2}})}{s^{2} \big{(}s+\omega_{\text{p}}\big{)}} \tag{33}\]  
The aim of introducing two integral links into the above formula is to make the amplitude-frequency characteristic curve decrease at the slope of \(-20\) dB/dec in the low-frequency band. \(\omega_{\text{Z1}}\) and \(\omega_{\text{Z2}}\) are introduced to offset the two conjugate poles in the original transfer function. The \(\omega_{\text{p}}\) can ensure that the corrected system has enough phase angle margin, at the same time, the amplitude-frequency characteristic curve can decrease at the slope of \(-40\) dB/dec in the high-frequency band to suppress the high-frequency noise [29].  
The transfer function of the current compensation link is:  
\[G_{\text{idc}}(s)=\frac{3\times 10^{6}\times\big{(}1.8\times 10^{-4}s^{2}+6 \times 10^{-3}s+5\big{)}}{s^{2}\times(s+4500)} \tag{34}\]  
The Bode diagram of the current compensation link is shown in Figure 11.  
The open-loop transfer function of the corrected system is shown as.  
\[G_{\text{A}}(s)=G_{\text{id}}(s)G_{\text{idc}}(s)=\frac{1.8\times 10^{7}}{s(s+450 0)} \tag{35}\]  
The open-loop amplitude-frequency characteristic curve of the corrected system is shown in Figure 12. It can be seen from the figure that, after correction, the amplitude-frequency characteristic curve of the corrected system decreases at the slope of \(20\) dB/dec in the low-frequency band, and the steady-state error of the system is zero; The phase angle margin in the middle-frequency band is \(54.2^{\circ}\), which meets the stability requirement.  
Figure 11: The Bode diagram of the current compensation link.  
The crossing frequency is \(3.24\times 10^{3}\)rad/s (\(f_{\rm c}\approx 515.7\)Hz). The amplitude-frequency characteristic curve of the corrected system decreases at the slope of \(-40\) dB/dec in the high-frequency band, and the system has a good ability to suppress the high-frequency noise. To sum up, the corrected system meets the design requirements.