# 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 AC Small Signal Model
The Design of Control Strategy for the Variable Multiplicity Interleaving Bidirectional DC/DC Power Converter  
The Analysis of the Open-Loop Characteristic of the Variable Multiplicity Interleaving Bidirectional DC/DC Power Converter  
In order to simplify the analysis, it is assumed that the parameters of each corresponding component in the variable multiplicity interleaving bidirectional DC/DC power converter are entirely equal. \[\left\{ \begin{array}{l}L_{1}=L_{2}=\cdots=L_{\text{n}}=L\\ L_{\text{L1}}=L_{\text{L2}}=\cdots=L_{\text{Ln}}=L_{\text{L}}\\ D_{1}=D_{2}=\cdots=D_{\text{n}}=D\\ R_{1}=R_{2}=\cdots=R_{\text{n}}=R\end{array} \right. \tag{26}\]  
The equation of the frequency domain is obtained by the Laplace transform of Equations (24) and (25). \[\left\{ \begin{array}{l}sL_{\text{L}}^{\ast}(s)=D\vartheta_{\text{DC}}(s)+V_ {\text{DC}}\hat{d}(s)-\vartheta_{\text{Load}}(s)-\hat{i}_{\text{L}}(s)R\\ \hat{i}_{\text{Load}}(s)=sC\vartheta_{\text{Load}}(s)=\hat{m_{\text{L}}^{ \ast}}(s)\end{array} \right. \tag{27}\]  
In order to control the load voltage \(V_{\text{Load}}\), the input can be assumed to be the ideal voltage source,\(\vartheta_{\text{DC}}(s)=0\). Then, the transfer function \(G_{\text{id}}\)(s) between duty cycle \(d\)(s) and inductance current \(i_{\text{L}}\)(s) can be obtained by simplifying Equation (27). \[G_{\text{id}}(s)=\left.\frac{\hat{i}_{\text{L}}(s)}{\hat{d}(s)}\right|_{ \vartheta_{\text{DC}}(s)=0}=\frac{sCV_{\text{DC}}}{s^{2}LC+sCR+n} \tag{28}\]  
The transfer function \(G_{\text{vd}}\)(s) between duty cycle \(d\)(s) and load voltage \(V_{\text{Load}}\) (s) can be obtained. \[G_{\text{vd}}(s)=\left.\frac{\vartheta_{\text{Load}}(s)}{\hat{d}(s)}\right|_{ \vartheta_{\text{DC}}(s)=0}=\frac{nV_{\text{DC}}}{s^{2}LC+sCR+nV_{\text{DC}}} \tag{29}\]  
The transfer function \(G_{\text{vi}}\)(s) between inductance current \(i_{\text{L}}\)(s) and load voltage \(V_{\text{Load}}\) (s) can be obtained according to Equations (28) and (29). \[G_{\text{vi}}(s)=\frac{G_{\text{vd}}(s)}{G_{\text{id}}(s)}=\frac{n\big{(}s^{2 }LC+sCR+nV_{\text{DC}}\big{)}}{s^{3}C^{2}L+s^{2}C^{2}R+nsCV_{\text{DC}}} \tag{30}\]  
Bring the data in Table 2 into Equation (28), and the transfer function \(G_{\text{id}}\)(s) can be obtained. \[G_{\text{id}}(s)=\frac{sCV_{\text{DC}}}{s^{2}LC+sCR+n}=\frac{6s}{1.8\times 10^{ -4}s^{2}+6\times 10^{-3}s+n} \tag{31}\]  
The Bode diagram corresponds to \(G_{\text{id}}\)(s) when the working multiples are 4, 5, and 6 respectively, as shown in Figure 9. It can be seen from the figure that the shapes of the Bode diagrams under different multiplicities are the same, so the transfer function of the \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline DC voltage \(V_{\text{DC}}\) & 30 V \\ Switching period \(T_{\text{S}}\) & 0.0001 s (10 kHz) \\ Inductance \(L\) & 3 mH \\ Inductance resistance \(R\) & 0.1 G \\ Load capacitance \(C\) & 0.06 F \\ Initial voltage of load capacitor \(V_{\text{C0}}\) & 5 V \\ Load current \(I_{\text{Load}}\) & 6 A \\ Maximum number of parallel channels \(N\) & 6 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of variable multiplicity interleaving bidirectional DC/DC converter. five multiplicities is selected as the research object, and the current compensation network is designed. As can be seen from Figure 9, the amplitude-frequency characteristic curve is rising in the low-frequency band, there is a large steady-state error in the system. In the middle-frequency band, the crossover frequency is very high, and the system may be affected by switching frequency, harmonics, parasitic oscillation, and so on, unable to work stably. The high-frequency band declines slowly, and the system is vulnerable to high-frequency noise [29].