# 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 Mean Value Model  
In Figure 6, \(V_{\rm DC}\) is DC side voltage, \(i_{\rm DC}\) is DC side current, \(i_{\rm Lk}\) (\(1\leq\rm k\leq n\)) is inductor current, \(R_{\rm k}\) (\(1\leq\rm k\leq n\)) is the resistance of the inductor, \(V_{\rm L}\) is inductor voltage, \(V_{\rm Load}\) is load-side voltage, \(i_{\rm Load}\) is load-side current, and \(D\) is duty cycle. \begin{table}
\begin{tabular}{c c} \hline
**Range of Duty Cycles** & **Number of Parallel Multiplicities** \\ \hline
0.1000–0.1835 & 6 \\
0.1835–0.2254 & 5 \\
0.2254–0.2887 & 4 \\
0.2887–0.3675 & 6 \\
0.3675–0.4772 & 5 \\
0.4772–0.5528 & 6 \\
0.5528–0.6324 & 5 \\
0.6324–0.7113 & 6 \\
0.7113–0.7746 & 4 \\
0.7746–0.8165 & 5 \\
0.8165–0.9000 & 6 \\ \hline \end{tabular}
\end{table} 
Table 1: The number of parallel multiplicities corresponding to different duty cycles. Figure 5: Parallel multiplicity corresponding to the minimum current ripple coefficient under different duty cycles. The DC side current is shown as:  
\[i_{\text{DC}}=\sum_{k=1}^{n}D_{\text{k}}i_{\text{Lk}} \tag{13}\]  
When the lower arm is turned off and the upper arm is turned on, the voltages at both ends of each inductance can be shown as:  
\[\left\{ \begin{aligned} V_{\text{L1}}&=L_{1}\frac{d_{i 1}}{dt}=V_{\text{DC}}-V_{\text{Load}}-i_{\text{L1}}R_{1}\\ V_{\text{L2}}&=L_{2}\frac{d_{i2}}{dt}=V_{\text{DC}}-V_{ \text{Load}}-i_{\text{L2}}R_{2}\\ &\vdots\\ V_{\text{Ln}}&=L_{\text{n}}\frac{d_{i\text{n}}}{dt}=V_{ \text{DC}}-V_{\text{Load}}-i_{\text{Ln}}R_{\text{n}}\end{aligned} \right. \tag{14}\]  
When the lower arm is turned on and the upper arm is turned off, the voltages at both ends of each inductance can be shown as:  
\[\left\{ \begin{aligned} V_{\text{L1}}&=L_{1}\frac{d_{ i1}}{dt}=-V_{\text{Load}}-i_{\text{L1}}R_{1}\\ V_{\text{L2}}&=L_{2}\frac{d_{i2}}{dt}=-V_{\text{Load }}-i_{\text{L2}}R_{2}\\ &\vdots\\ V_{\text{Ln}}&=L_{\text{n}}\frac{d_{i\text{n}}}{dt}=-V_{ \text{Load}}-i_{\text{Ln}}R_{\text{n}}\end{aligned} \right. \tag{15}\]  
Because the switching period \(T_{\text{s}}\) is very small, it can be considered that \(V_{\text{DC}}\) and \(V_{\text{Load}}\) are continuous and constant in a switching period. They can be expressed by their average values in a period. \[\left\{ \begin{aligned} V_{\text{DC}}&=\left\langle V_{ \text{DC}}\right\rangle_{T_{\text{s}}}\\ V_{\text{Load}}&=\left\langle V_{\text{Load}}\right\rangle_{T_ {\text{s}}}\end{aligned} \right. \tag{16}\]  
According to the volt-ampere characteristics and volt-second balance principle of the inductor, the average voltage of the inductor shown in (17) in a switching cycle can be obtained by dealing with Equations (14)-(16). \[\left\{ \begin{aligned} \left\langle v_{\text{L1}}(t)\right\rangle_{T_{ \text{s}}}&=\frac{1}{T_{\text{s}}}\big{[}\int_{0}^{D_{\text{T}}T_ {\text{s}}}v_{\text{L1}}(t)dt+\int_{D_{\text{T}}T_{\text{s}}}^{T_{\text{s}}}v _{\text{L1}}(t)dt\big{]}=D_{1}\left\langle V_{\text{DC}}\right\rangle_{T_{ \text{s}}}-\left\langle V_{\text{Load}}\right\rangle_{T_{\text{s}}}-\left \langle i_{\text{L1}}\right\rangle_{T_{\text{s}}}R_{1}\\ \left\langle v_{\text{L2}}(t)\right\rangle_{T_{\text{s}}}& =\frac{1}{T_{\text{s}}}\big{[}\int_{0}^{D_{\text{T}}T_{\text{s} }}v_{\text{L2}}(t)dt+\int_{D_{\text{T}}T_{\text{s}}}^{T_{\text{s}}}v_{\text{L 2}}(t)dt\big{]}=D_{2}\left\langle V_{\text{DC}}\right\rangle_{T_{\text{s}}}- \left\langle V_{\text{Load}}\right\rangle_{T_{\text{s}}}-\left\langle i_{ \text{L2}}\right\rangle_{T_{\text{s}}}R_{2}\\ &\vdots\\ \left\langle v_{\text{Ln}}(t)\right\rangle_{T_{\text{s}}}& =\frac{1}{T_{\text{s}}}\big{[}\int_{0}^{D_{\text{T}}T_{\text{s}}}v_{\text{Ln }}(t)dt+\int_{D_{\text{n}}T_{\text{s}}}^{T_{\text{s}}}v_{\text{Ln}}(t)dt\big{]} =D_{\text{n}}\left\langle V_{\text{DC}}\right\rangle_{T_{\text{s}}}-\left \langle V_{\text{Load}}\right\rangle_{T_{\text{s}}}-\left\langle i_{\text{Ln}} \right\rangle_{T_{\text{s}}}R_{\text{n}}\end{aligned} \right.