2.2.1. Control-To-Output Voltage Transfer Function

The output voltage and the duty cycle relationship is found by summing up the Kirchhoff's voltage law (KVL) Equations in (6), assuming *v*ˆ*in* = 0, and ˆ *dk* = 0, where *k* = 1, 2, and 3, and *k j*, and substituting (2), (4), (9)–(11).

$$\left(\frac{V\_{in}}{3K}\hat{d}\_1 - R\_d\left(\hat{v}\_{out}\left(\mathbf{sC} + \frac{\mathfrak{Z}}{R}\right)\right) = \mathbf{s}L\left(\hat{v}\_{out}\left(\mathbf{sC} + \frac{\mathfrak{Z}}{R}\right)\right) + \hat{v}\_{out} \tag{12}$$

Simplifying (12) would result in (13):

$$G\_{\rm vd} = \frac{\hat{v}\_{\rm out}}{\hat{d}\_{\dot{\rangle}}} = \frac{\frac{V\_{\rm in}}{3K}}{\left(s^2LC + s\left(\frac{3L}{R} + R\_dC\right) + \frac{3R\_d}{R} + 1\right)}\tag{13}$$
