*4.2. IPOP Small-Signal Analysis*

The SSM for the IPOP converter shown in Figure 4 is derived using the SSM presented in [37].

**Figure 4.** IPOP DC-DC converter SSM.

Since the input current per module is *Iin* <sup>3</sup> , and the output current per module is *Io* <sup>3</sup> , accordingly, the load resistance per module is 3*R*. Therefore, ˆ *dij*, ˆ *dv*<sup>j</sup> and *Ieq* presented in Figure 4 can be expressed as follows, where the subscript *j* = 1, 2, *and* 3:

$$d\_{i\dot{j}} = -\frac{4}{K V\_{in}} \hat{\mathbf{r}}\_{L\dot{j}} \tag{41}$$

Rewriting (41) in terms of *Rd* would result in:

$$\hat{d}\_{ij} = -\frac{\mathcal{K}\mathcal{R}\_d}{V\_{in}}\hat{\mathbf{i}}\_{Lj} \tag{42}$$

$$
\hat{d}\_{vj} = \frac{4I\_{\rm lk} f\_s D\_{eff}}{3 \, k^2 R V\_{\rm in}} \theta\_{\rm in} \tag{43}
$$

Similarly, rewriting (43) in terms of *Rd* would result in:

$$
\hat{d}\_{vj} = \frac{R\_d D\_{eff}}{3 \, R V\_{in}} \hat{v}\_{in} \tag{44}
$$

$$I\_{eq} = \frac{V\_{in}}{3\,\text{K}\Omega} \tag{45}$$

The following equations are obtained from Figure 4:

$$\begin{cases} \begin{array}{c} \frac{D\_{eff}}{K} \mathfrak{v}\_{in} + \frac{V\_{in}}{K} (\hat{d}\_{i1} + \hat{d}\_{v1} + \hat{d}\_{1}) = sL\hat{I}\_{L1} + \mathfrak{v}\_{out} \\\ \frac{D\_{eff}}{K} \mathfrak{v}\_{in} + \frac{V\_{in}}{K} (\hat{d}\_{i2} + \hat{d}\_{v2} + \hat{d}\_{2}) = sL\hat{I}\_{L2} + \mathfrak{v}\_{out} \\\ \frac{D\_{eff}}{K} \mathfrak{v}\_{in} + \frac{V\_{in}}{K} (\hat{d}\_{i3} + \hat{d}\_{v3} + \hat{d}\_{3}) = sL\hat{I}\_{L3} + \mathfrak{v}\_{out} \end{array} \tag{46}$$

$$\begin{cases} \frac{K}{3D\_{eff}} \left( \mathbf{\hat{i}}\_{in} - \mathbf{s} \mathbf{C}\_{d} \mathbf{\hat{v}}\_{in} \right) = I\_{eq} \left( \mathbf{\hat{d}}\_{i1} + \mathbf{d}\_{v1} + \mathbf{d}\_{1} \right) + \mathbf{\hat{i}}\_{L1} \\\ \frac{K}{3D\_{eff}} \left( \mathbf{\hat{i}}\_{in} - \mathbf{s} \mathbf{C}\_{d} \mathbf{\hat{v}}\_{in} \right) = I\_{eq} \left( \mathbf{\hat{d}}\_{i2} + \mathbf{\hat{d}}\_{v2} + \mathbf{\hat{d}}\_{2} \right) + \mathbf{\hat{i}}\_{L2} \\\ \frac{K}{3D\_{eff}} \left( \mathbf{\hat{i}}\_{in} - \mathbf{s} \mathbf{C}\_{d} \mathbf{\hat{v}}\_{in} \right) = I\_{eq} \left( \mathbf{\hat{d}}\_{i3} + \mathbf{\hat{d}}\_{v3} + \mathbf{\hat{d}}\_{3} \right) + \mathbf{\hat{i}}\_{L3} \end{cases} \tag{47}$$

$$\sum\_{j=1}^{3} \hat{t}\_{Lj} = \hat{v}\_{out} \Big( \mathbf{sC} + \frac{1}{R} \Big) \tag{48}$$
