# Variable Multiple Interleaved Bi-Directional DC/DC Converter with Current Ripple Optimization
## 2 The Principle of Current Ripple Optimization
\begin{cases}i_{\text{L}}=(n_{\text{up}}k_{\text{up}}+n_{\text{down}}k_{\text {down}})t+\frac{n_{\text{up}}k_{\text{up}}T}{2n}(n_{\text{up}}-1)\\ -\frac{n_{\text{down}}k_{\text{up}}T}{2n}(n_{\text{down}}+1)&t\in\left[0,DT- \frac{n_{\text{up}}-1}{n}T\right]\\ i_{\text{L}}=\left((n_{\text{up}}-1)k_{\text{up}}+(n_{\text{down}}+1)k_{ \text{down}}\right)t\\ +\frac{k_{\text{up}}(n_{\text{up}}-1)(n_{\text{up}}-2)}{2n}T-\frac{k_{\text{ down}}(n_{\text{down}}+1)(n_{\text{down}}+2)}{2n}T&t\in\left[DT-\frac{n_{\text{up}}-1}{n}T, \frac{T}{n}\right]\\ i_{\text{L}}=\left((n_{\text{up}}-1)k_{\text{up}}+(n_{\text{down}}+1)k_{ \text{down}}\right)t\\ +\frac{k_{\text{up}}(n_{\text{up}}-1)(n_{\text{up}}-4)}{2n}T-\frac{k_{\text{ down}}(n_{\text{down}}+1)(n_{\text{down}}+4)}{2n}T&t\in\left[DT-\frac{n_{\text{up}}-2}{n}T, \frac{2T}{n}\right]\\ \vdots\\ i_{\text{L}}=(n_{\text{up}}k_{\text{up}}+n_{\text{down}}k_{\text{down}})t\\ +\frac{n_{\text{up}}k_{\text{up}}T}{2n}(n_{\text{up}}-2n+1)-\frac{n_{\text{ down}}k_{\text{down}}T}{2n}(n_{\text{down}}+2n-1)&t\in\left[\frac{n-1}{n}T,DT+ \frac{n_{\text{down}}T}{n}\right]\\ i_{\text{L}}=\left((n_{\text{up}}-1)k_{\text{up}}+(n_{\text{down}}+1)k_{ \text{down}}\right)t\\ +\frac{k_{\text{up}}(n_{\text{up}}-1)(n_{\text{up}}-2n)}{2n}T-\frac{k_{\text{ down}}(n_{\text{down}}+1)(n_{\text{down}}+2n)}{2n}T&t\in\left[DT+\frac{n_{\text{down}}T}{n},T \right]\end{cases}  \tag{9}\]  
According to (5)-(9) and Figure 2, when phase shift control is applied to the \(n\)-fold DC/DC converter, the total ripple of the current flowing through the inductor \(L\) is shown in (10):  
\[ \begin{array}{rl}\Delta i_{\text{Lm}}&=\big{(}n_{\text{up}}k_{\text{up}}+n_ {\text{down}}k_{\text{down}}\big{)}\cdot\left(DT-\frac{n_{\text{up}}-1}{n}T \right)\\ &=\frac{TV_{\text{DC}}}{L}\cdot\frac{[(1-D)(\lfloor Dn+1\rfloor)-D(\lceil(1-D) n-1\rceil)]\cdot\left[Dn-(\lfloor Dn+1\rfloor)+1\right]}{n}\end{array}  \tag{10}\]  
Define \(k_{\text{Lm}}\) as the ripple coefficient, which is shown in (11). \[k_{\text{Lm}}=\frac{[(1-D)(\lfloor Dn+1\rfloor)-D(\lceil(1-D)n-1\rceil)]\cdot \left[Dn-(\lfloor Dn+1\rfloor)+1\right]}{n} \tag{11}\]The expression of (11) can be rewritten to. \[\Delta i_{\text{Lm}}=\frac{TV_{\text{DC}}}{L}.k_{\text{Lm}} \tag{12}\]  
It can be seen from (11) that the ripple coefficient \(k_{\text{Lm}}\) is affected by the duty cycle \(D\) and the number of parallel multiplicity \(n\). When the bidirectional DC/DC converter operates in Buck mode, the relationship between the ripple coefficient \(k_{\text{Lm}}\), the duty cycle \(D\), and the number of parallel channel multiplicity \(n\) is shown in Figure 3. It can be seen from Figure 3 that there is a point where the ripple coefficient \(k_{\text{Lm}}\) is 0 when different numbers of parallel multiplicity \(n\) and different duty cycle \(D\) are taken. According to (11), the relationship between ripple coefficient \(k_{\text{Lm}}\) and duty cycle \(D\) can be obtained, shown in Figure 4, when the number of multiplicities is different. It can be seen from Figure 4 that there are \(n-1\) points where the ripple coefficient is zero for the \(n\)-multiplicity bidirectional DC/DC power converter. As can be seen from Figure 4, under the condition of different duty cycles, it is not that the more parallel multiplicity, the smaller the current ripple coefficient. Although the ripple coefficient generally decreases with the increase of parallel multiplicity, considering that too many parallel multiplicities will increase the complexity of the circuit; and more when the number of parallel multiplicities is greater than six, the increase of parallel multiplicity has no obvious effect on the reduction of current ripple. The  
Figure 4: Relationship between \(k_{\text{Lm}}\) and \(D\) with different parallel multiplicity. Figure 3: The relationship among \(k_{\text{Lm}}\), \(D\), and \(n\) of bidirectional DC/DC converter in Buck mode.