*2.2. ISOS Small-Signal Analysis*

The SSM for the ISOS converter shown in Figure 2 is derived, using the SSM presented in [37].

**Figure 2.** ISOS DC-DC converter small-signal modeling (SSM).

Since the input voltage per module is *Vin* <sup>3</sup> , and the output voltage per module is *Vo* <sup>3</sup> , accordingly, the load resistance per module is *<sup>R</sup>* <sup>3</sup> . Therefore, <sup>ˆ</sup> *dij* and ˆ *dv*j, which are the effect of changing the filter inductor current and the effect of changing the input voltage on the duty cycle modulation, as well as *Ieq* presented in Figure 2 can be expressed as follows, where the subscript *j* = 1, 2, and 3:

$$\hat{d}\_{i\rangle} = -\frac{12 \text{ L}\_{\text{lk}} f\_s}{K V\_{\text{in}}} \mathbf{\hat{i}}\_{L\text{j}} \tag{1}$$

Equation (1) can be rewritten in terms of *Rd*, where; *Rd* <sup>=</sup> <sup>4</sup>*Llk fs <sup>k</sup>*<sup>2</sup> as:

$$\hat{d}\_{i\circ} = -\frac{3\mathcal{K}\mathcal{R}\_d}{V\_{in}}\hat{\imath}\_{L\circ} \tag{2}$$

$$\delta\_{vj} = \frac{36 \, L\_{\text{lk}} f\_s D\_{eff}}{k^2 R V\_{in}} \theta\_{cdj} \tag{3}$$

Similarly, Equation (3) can be rewritten as:

⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩

$$
\hat{d}\_{vj} = \frac{9\,\text{R}\_d\text{D}\_{eff}}{\text{R}V\_{in}}\mathfrak{d}\_{cdj} \tag{4}
$$

$$I\_{\rm lq} = \frac{V\_{\rm in}}{KR} \tag{5}$$

The following equations are obtained from Figure 2:

$$\begin{array}{ll}\frac{D\_{eff}}{K}\boldsymbol{\hat{\sigma}}\_{cd1} + \frac{V\_{in}}{3K} \left(\boldsymbol{\hat{d}}\_{l1} + \boldsymbol{\hat{d}}\_{v1} + \boldsymbol{\hat{d}}\_{1}\right) = sL\boldsymbol{\hat{\zeta}}\_{L1} + \boldsymbol{\hat{\sigma}}\_{out1} \\ \frac{D\_{eff}}{K}\boldsymbol{\hat{\sigma}}\_{cd2} + \frac{V\_{in}}{3K} \left(\boldsymbol{\hat{d}}\_{l2} + \boldsymbol{\hat{d}}\_{v2} + \boldsymbol{\hat{d}}\_{2}\right) = sL\boldsymbol{\hat{\zeta}}\_{l2} + \boldsymbol{\hat{\sigma}}\_{out2} \\ \frac{D\_{eff}}{K}\boldsymbol{\hat{\sigma}}\_{cd3} + \frac{V\_{in}}{3K} \left(\boldsymbol{\hat{d}}\_{l3} + \boldsymbol{\hat{d}}\_{v3} + \boldsymbol{\hat{d}}\_{3}\right) = sL\boldsymbol{\hat{\zeta}}\_{l2} + \boldsymbol{\hat{\sigma}}\_{out3} \end{array} \tag{6}$$

$$\begin{cases} \begin{array}{c} \frac{K}{D\_{eff}} \left(\hat{\mathbf{i}}\_{\rm in} - s\mathbf{C}\_{d}\boldsymbol{\mathfrak{d}}\_{\rm cdl}\right) = I\_{eq} \left(\hat{d}\_{\rm l1} + \hat{d}\_{\rm v1} + \hat{d}\_{\rm l}\right) + \hat{\mathbf{i}}\_{\rm L1} \\\ \frac{K}{D\_{eff}} \left(\hat{\mathbf{i}}\_{\rm in} - s\mathbf{C}\_{d}\boldsymbol{\mathfrak{d}}\_{\rm cdl}\right) = I\_{eq} \left(\hat{d}\_{\rm L2} + \hat{d}\_{\rm v2} + \hat{d}\_{\rm L2}\right) + \hat{\mathbf{i}}\_{\rm L2} \\\ \frac{K}{D\_{eff}} \left(\hat{\mathbf{i}}\_{\rm in} - s\mathbf{C}\_{d}\boldsymbol{\mathfrak{d}}\_{\rm cdl}\right) = I\_{eq} \left(\hat{d}\_{\rm B} + \hat{d}\_{\rm v3} + \hat{d}\_{\rm B}\right) + \hat{\mathbf{i}}\_{\rm L3} \end{array} \tag{7}$$

$$\begin{cases} \begin{array}{l} \hat{\mathbf{i}}\_{L1} = \mathbf{s} \mathbf{C} \mathbf{t}\_{out1} + \frac{\boldsymbol{\vartheta}\_{\text{out}}}{\mathcal{R}}\\ \hat{\mathbf{i}}\_{L2} = \mathbf{s} \mathbf{C} \hat{\mathbf{i}}\_{out2} + \frac{\boldsymbol{\vartheta}\_{\text{out}}}{\mathcal{R}}\\ \hat{\mathbf{i}}\_{L3} = \mathbf{s} \mathbf{C} \mathbf{t}\_{out3} + \frac{\boldsymbol{\vartheta}\_{\text{out}}}{\mathcal{R}} \end{array} \tag{8}$$

Summing Equations in (8) would result in (9):

$$\sum\_{j=1}^{3} \hat{\imath}\_{Lj} = \imath\_{out} \left( s\mathbb{C} + \frac{3}{R} \right) \tag{9}$$

where

$$
\mathfrak{d}\_{out1} + \mathfrak{d}\_{out2} + \mathfrak{d}\_{out3} = \mathfrak{d}\_{out} \tag{10}
$$

$$
\left\|\mathfrak{d}\_{cd1} + \mathfrak{d}\_{cd2} + \mathfrak{d}\_{cd3} = \mathfrak{d}\_{\text{in}} \text{ or } \sum\_{j=1}^{3} \mathfrak{d}\_{cdj} = \mathfrak{d}\_{\text{in}} \tag{11}
$$
