6.2.3. Output Impedance

Similarly, as studied in the SSM presented in [39,40], the ISIP-OSOP converter output impedance can be found by modifying (100), such that:

$$\begin{cases} \begin{array}{l} \hat{\imath}\_{L11} + \hat{\imath}\_{L21} + \dots + \hat{\imath}\_{L\mathfrak{a}1} + \hat{\imath}\_{out} = \frac{g\underline{\mathcal{C}}}{s\underline{\mathcal{R}}\underline{\mathcal{C}} + \Gamma} \boldsymbol{\vartheta}\_{out1} + \frac{\underline{\mathcal{C}}\_{out}}{R} \\ \hat{\imath}\_{L12} + \hat{\imath}\_{L22} + \dots + \hat{\imath}\_{L\mathfrak{a}2} + \hat{\imath}\_{out} = \frac{g\underline{\mathcal{C}}}{s\underline{\mathcal{R}}\underline{\mathcal{C}} + \Gamma} \boldsymbol{\vartheta}\_{out2} + \frac{\underline{\mathcal{C}}\_{out}}{R} \\ \vdots \\ \hat{\imath}\_{L1b} + \hat{\imath}\_{L2b} + \dots + \hat{\imath}\_{Lab} + \hat{\imath}\_{out} = \frac{g\underline{\mathcal{C}}}{s\underline{\mathcal{R}}\underline{\mathcal{C}} + \Gamma} \boldsymbol{\vartheta}\_{outn} + \frac{\underline{\mathcal{C}}\_{out}}{R} \end{array} \tag{112}$$

Summing Equations in (112):

$$\sum\_{i=1}^{a} \sum\_{j=1}^{b} \mathfrak{i}\_{Lij} = \frac{s\mathcal{C}}{s\mathcal{R}\mathcal{L} + 1} \mathfrak{d}\_{\text{out}} + \frac{b\mathfrak{d}\_{\text{out}}}{R} - b\mathfrak{i}\_{\text{out}} \tag{113}$$

Accordingly, (100) is modified as follows:

$$\sum\_{i=1}^{a} \sum\_{j=1}^{b} \hat{\imath}\_{Lij} = \mathfrak{d}\_{out} \left( \frac{sRC + sbR\_iC + b}{R(1 + sR\_iC)} \right) - b\hat{\imath}\_{out} \tag{114}$$

The output voltage and the output current relationship is found by considering the same assumptions as in Section 7.2, summing Equations in (99), and substituting (95), (97), (103), (104), and (114).

$$\frac{\partial V\_{\rm ini}}{\partial \mathcal{K}} \left( -\frac{\mathcal{J} \mathcal{K} \mathcal{R}\_d}{V\_{\rm ini}} \right) \sum\_{j=1}^n \hat{i}\_{Lj} = s\mathcal{L} \sum\_{j=1}^n \hat{i}\_{Lj} + c\hat{v}\_{\rm out} \tag{115}$$

$$-R\_{\rm d} \left( \mathfrak{d}\_{\rm out} \left( \frac{sRC + sbR\_{\rm c}C + b}{R(1 + sR\_{\rm c}C)} \right) - b\hat{l}\_{\rm out} \right) = sL \left( \mathfrak{d}\_{\rm out} \left( \frac{sRC + sbR\_{\rm c}C + b}{R(1 + sR\_{\rm c}C)} \right) - b\hat{l}\_{\rm out} \right) + c\mathfrak{d}\_{\rm out} \tag{116}$$

Rearranging (116) would result in (117).

$$Z\_{\rm out} = \frac{\vartheta\_{\rm out}}{\hat{\imath}\_{\rm out}} = \frac{b(R\_d + sL)(1 + sR\_c\mathcal{C})}{s^2LC\{1 + \frac{bR\_c}{R}\} + s\left(\frac{b\mathcal{I}\_c}{R} + R\_d\mathcal{C}\{1 + \frac{bR\_c}{R}\} + cR\_c\mathcal{C}\right) + \frac{bR\_d}{R} + c} \tag{117}$$
