# Variable Multiple Interleaved Bi-Directional DC/DC Converter with Current Ripple Optimization
## 2 The Principle of Current Ripple Optimization  
Assuming that the bidirectional DC/DC converter operates in Buck mode, the topology of the converter is shown in Figure 1.  
The relationship among load-side voltage \(U_{\text{Load}}\), DC bus voltage, and duty cycle \(D\) is shown as follows:  
\[V_{\text{Load}}=DV_{\text{DC}} \tag{1}\]  
For inductance \(L\), the slope of the current rising phase is defined as \(k_{\text{up}}\), and the slope of the current decreasing phase is defined as \(k_{\text{down}}\). \[\left\{ \begin{array}{l}k_{\text{up}}=\frac{V_{\text{DC}}-V_{\text{Load}}}{ L}\\ k_{\text{down}}=\frac{V_{\text{Load}}}{L}\end{array} \right. \tag{2}\]  
In same-phase driving mode, \(i_{\text{L1}}\), \(i_{\text{L2}}\), \(i_{\text{L3}}\), \(i_{\text{L4}}\), and so on are exactly equal, then the peak-to-peak value of the ripple of total current can be shown as:  
\[\Delta i_{\text{Ls}}=\Delta i_{\text{L1}}+\Delta i_{\text{L2}}+\ldots+\Delta i _{\text{Ln}}=nDk_{\text{up}}T \tag{3}\]  
In the phase-shift control, the relationship between the current and time is established by setting the current \(i_{\text{L1}}\) as the zero point at the initial time of a switching cycle [0,T]. The current \(i_{\text{L1}}\), \(i_{\text{L2}}\), \(i_{\text{L3}}\), \(i_{\text{L4}}\), and so on are shown in Figure 2.  
Assuming that the slope of \(n_{\text{down}}\)-fold inductor current at the initial stage is \(k_{\text{down}}\), and satisfies \(n_{\text{down}}+n_{\text{up}}=n\). Then the following relationship can be obtained. \[\left\{ \begin{array}{l}\left[(1-D)n-1\right]=n_{\text{down}}\\ \left[Dn+1\right]=n_{\text{up}}\end{array} \right. \tag{4}\]  
Then, the duty cycle has the following relationship: \(1-\frac{n_{\text{down}}+1}{n}\leq D\leq 1-\frac{n_{\text{down}}}{n}\). Assuming that \(D\) is 50%, and \(n_{\text{down}}=\frac{n}{2}\) or \(n_{\text{down}}=\frac{n-1}{2}\) can be obtained according to the parity of \(n\). Figure 1: Bidirectional DC/DC converter in Buck mode. Figure 2: Multiple inductor current and total current waveform. The expression of the first inductance current is shown as:  
\[ \begin{cases}i_{\text{L1}}=k_{\text{up}}t&t\in[0,DT]\\ i_{\text{L1}}=k_{\text{down}}(t-T)&t\in[DT,T]\end{cases}  \tag{5}\]  
The expression of the second inductance current can be shown as:  
\[ \begin{cases}i_{\text{L2}}=k_{\text{down}}\cdot\left(t-\frac{T}{n}\right)&t \in\left[0,\frac{T}{n}\right]\\ i_{\text{L2}}=k_{\text{up}}\cdot\left(t-\frac{T}{n}\right)&t\in\left[\frac{T}{n },DT+\frac{T}{n}\right]\\ i_{\text{L2}}=k_{\text{down}}\cdot\left(t-\frac{n+1}{n}T\right)&t\in\left[DT+ \frac{T}{n},T\right]\end{cases}  \tag{6}\]  
The expression of the third inductance current can be shown as:  
\[ \begin{cases}i_{\text{L3}}=k_{\text{down}}\cdot\left(t-\frac{2T}{n}\right)&t \in\left[0,\frac{2T}{n}\right]\\ i_{\text{L3}}=k_{\text{up}}\cdot\left(t-\frac{2T}{n}\right)&t\in\left[\frac{ 2T}{n},DT+\frac{2T}{n}\right]\\ i_{\text{L3}}=k_{\text{down}}\cdot\left(t-\frac{n+2}{n}T\right)&t\in\left[DT+ \frac{2T}{n},T\right]\end{cases}  \tag{7}\]  
The expression of the n-th inductance current can be shown as:  
\[ \begin{cases}i_{\text{Ln}}=k_{\text{up}}\cdot\left(t+\frac{T}{n}\right)&t\in \left[0,DT-\frac{T}{n}\right]\\ i_{\text{Ln}}=k_{\text{down}}\cdot\left(t-\frac{n-1}{n}T\right)&t\in\left[DT- \frac{T}{n},\frac{n-1}{n}T\right]\\ i_{\text{L3}}=k_{\text{up}}\cdot\left(t-\frac{n-1}{n}T\right)&t\in\left[\frac {n-1}{n}T,T\right]\end{cases}  \tag{8}\]  
According to [PERSON]'s law, the expression of the \(i_{\text{L}}\) in a switching cycle can be obtained and can be extended to the n-th cycle, which is shown in (9). \[