# 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  
Equations (17) and (18) obtained in the mean value model are nonlinear equations, so it is complex to solve them directly. However, there are DC components and low-frequency small signal components in the variable, which can be solved in approximate ways. The specific idea is to study the state of the DC/DC power converter near the steady-state working point, where the linearization is satisfied, and the equation can be linearized to separate the DC components and small signal components [27]. Conditions are as follows: the interleaved bidirectional DC/DC power converters should satisfy the low-frequency assumption, small ripple assumption, and small signal assumption in the process of modeling the AC small signal model [27; 28]. \[\left|\hat{x}(t)\right|\ll\left|X\right| \tag{21}\]  
Figure 7: The average value model of the variable multiplicity bidirectional DC/DC power converter in Buck mode. In Equation (21) [27], \(|\dot{x}(t)|\) is the amplitude of the AC component, while \(|X|\) is the amplitude of the DC component. In a switching cycle, any variable can be divided into DC components and AC small signal components, which is shown in (22). \[\left\{ \begin{array}{c}\left\langle{i_{\text{Lk}}}\right\rangle_{\text{T}_{ \text{s}}}=I_{\text{Lk}}+\hat{i}_{\text{Lk}}^{\ast}(t)\\ \left\langle{D_{\text{k}}}\right\rangle_{\text{T}_{\text{s}}}=D_{\text{k}}+ \hat{d}_{\text{k}}(t)\\ \left\langle{\
u_{\text{Load}}}\right\rangle_{\text{T}_{\text{s}}}=V_{\text{Load }}+\hat{\
u}_{\text{Load}}(t)\\ \left\langle{\
u_{\text{DC}}}\right\rangle_{\text{T}_{\text{s}}}=V_{\text{DC }}+\hat{\
u}_{\text{DC}}(t)\\ \left\langle{i_{\text{Load}}}\right\rangle_{\text{T}_{\text{s}}}=I_{\text{Load }}+\hat{i}_{\text{Load}}^{\ast}(t)\end{array} \right.\right. \tag{22}\]  
Equation (22) being brought into Equation (20) can get (23). \[\left\{ \begin{array}{c}L_{1}\frac{d[I_{11}+\hat{i}_{\text{L1}}(t)]}{dt}= \left[D_{1}+\hat{d}_{1}(t)\right]\left[V_{\text{DC}}+\hat{\
u}_{\text{DC}}(t )\right]-\left[V_{\text{Load}}+\hat{\
u}_{\text{Load}}(t)\right]-\left[I_{1 1}+\hat{i}_{\text{L1}}(t)\right]R_{1}\\ L_{2}\frac{d[I_{12}+\hat{i}_{\text{L2}}(t)]}{dt}=\left[D_{2}+\hat{d}_{2}(t) \right]\left[V_{\text{DC}}+\hat{\
u}_{\text{DC}}(t)\right]-\left[V_{\text{Load }}+\hat{\
u}_{\text{Load}}(t)\right]-\left[I_{12}+\hat{i}_{\text{L2}}(t) \right]R_{2}\\ \vdots\\ L_{\text{n}}\frac{d[I_{\text{n}}+\hat{i}_{\text{n}}(t)]}{dt}=\left[D_{\text{ n}}+\hat{d}_{\text{n}}(t)\right]\left[V_{\text{DC}}+\hat{\
u}_{\text{DC}}(t) \right]-\left[V_{\text{Load}}+\hat{\
u}_{\text{Load}}(t)\right]-\left[I_{ \text{Ln}}+\hat{i}_{\text{Ln}}(t)\right]R_{\text{n}}\end{array} \right.\right. \tag{23}\]  
Ignoring the DC components and the second-order AC small signal components in Equation (23), the above equation can be transformed into the following form. \[\left\{ \begin{array}{c}L_{1}\frac{d\hat{i}_{\text{L1}}(t)}{dt}=D_{1}\hat{ \
u}_{\text{DC}}(t)+V_{\text{DC}}\hat{d}_{1}(t)-\hat{\
u}_{\text{Load}}(t)- \hat{i}_{\text{L1}}(t)R_{1}\\ L_{2}\frac{d\hat{i}_{\text{L2}}(t)}{dt}=D_{2}\hat{\
u}_{\text{DC}}(t)+V_{\text{ DC}}\hat{d}_{2}(t)-\hat{\
u}_{\text{Load}}(t)-\hat{i}_{\text{L2}}(t)R_{2}\\ \vdots\\ L_{\text{n}}\frac{d\hat{i}_{\text{Ln}}(t)}{dt}=D_{\text{n}}\hat{\
u}_{\text{ DC}}(t)+V_{\text{DC}}\hat{d}_{\text{n}}(t)-\hat{\
u}_{\text{Load}}(t)-\hat{i}_{\text{Ln}}(t)R_{ \text{n}}\end{array} \right. \tag{24}\]  
Solving Equations (22) and (18) simultaneously, the following equation can be obtained. \[\hat{i}_{\text{Load}}(t)=C\frac{d\hat{i}_{\text{Load}}(t)}{dt}=\hat{i}_{ \text{L1}}(t)+\hat{i}_{\text{L2}}(t)+\cdots+\hat{i}_{\text{Ln}}(t) \tag{25}\]  
According to Equations (24) and (25), the AC small signal model of the variable multiplicity bidirectional DC/DC power converter in Buck mode can be obtained, which is shown in Figure 8.  
Figure 8: The AC small signal model of the variable multiplicity bidirectional DC/DC power converter in Buck mode.