2.2.2. Control-To-Filter Inductor Current Transfer Function

The filter inductor current and the duty cycle relationship is found by substituting *v*ˆ*out* in (12) in terms of ˆ*iLj* using (9), and considering the same assumptions as in Section 2.2.1.

$$\frac{1}{2K} \frac{V\_{in}}{2K} \hat{d}\_1 - R\_d \sum\_{j=1}^3 \hat{i}\_{Lj} = sL \sum\_{j=1}^3 \hat{i}\_{Lj} + \frac{\sum\_{j=1}^3 \hat{i}\_{Lj}}{\left(sC + \frac{3}{K}\right)}\tag{14}$$

Simplifying (14) would result in (15):

$$G\_{\rm id} = \frac{\hat{\mathbf{f}\_L}}{\hat{d}\_j} = \frac{\frac{V\_{\rm id}}{3\mathcal{K}} \left(3 + s\mathcal{R}\mathcal{C}\right)}{R\left(s^2 L\mathcal{C} + s\left(\frac{3L}{\mathcal{K}} + R\_d \mathcal{C}\right) + \frac{3}{\mathcal{K}} + 1\right)}\tag{15}$$
