6.2.1. Control-to-Output Voltage Transfer Function

The output voltage and the duty cycle relationship is found by summing up Equations in (99), assuming *v*ˆ*in* = 0, and ˆ *dk* = 0, where *k* = 1, 2, ... , *n* and *k j*, and substituting (95), (97), (102), (103), and (104).

Summing Equations in (99):

$$\frac{D\_{eff}}{K} \sum\_{j=1}^{n} \vartheta\_{\rm clj} + \frac{V\_{\rm in}}{\beta \mathcal{K}} \left( \sum\_{j=1}^{n} d\_{ij} + \sum\_{j=1}^{n} d\_{vj} + \sum\_{j=1}^{n} d\_{j} \right) = \text{sL} \sum\_{j=1}^{n} \mathbf{\hat{i}}\_{\rm Lj} + \sum\_{j=1}^{n} \vartheta\_{\rm outj} \tag{105}$$

$$\frac{D\_{eff}}{K} \sum\_{j=1}^{n} \mathfrak{d}\_{cdj} + \frac{V\_{in}}{\mathfrak{PK}} \left( \sum\_{j=1}^{n} -\frac{\mathfrak{K} \text{KR}\_d}{V\_{in}} \mathfrak{i}\_{Lj} + \sum\_{j=1}^{n} \frac{\mathfrak{B} \mathfrak{b} \text{R}\_d D\_{eff}}{a \text{R} V\_{in}} \mathfrak{d}\_{cdj} + \mathring{d}\_1 \right) = \text{sL} \sum\_{j=1}^{n} \mathfrak{i}\_{Lj} + \sum\_{j=1}^{n} \mathfrak{d}\_{outj} \tag{106}$$

$$\frac{D\_{eff}}{K} \mathbf{y} \boldsymbol{\uptheta}\_{\text{in}} + \frac{V\_{\text{in}}}{\rho \mathbf{K}} \left( \sum\_{j=1}^{n} -\frac{\rho \mathbf{K} \mathbf{R}\_d}{V\_{\text{in}}} \mathbf{\hat{i}}\_{Lj} + \frac{\rho b \mathbf{R}\_d D\_{eff}}{aR V\_{\text{in}}} \mathbf{y} \boldsymbol{\uptheta}\_{\text{in}} + \hat{d}\_1 \right) = \mathbf{s} \boldsymbol{L} \sum\_{j=1}^{n} \mathbf{\hat{i}}\_{Lj} + c \boldsymbol{\uptheta}\_{\text{out}} \tag{107}$$

Simplifying (107) would result in (108).

$$\mathbf{G}\_{\rm rd} = \frac{\vartheta\_{\rm out}}{\hat{d}\_{\rm j}} = \frac{\frac{V\_{\rm in}}{\rho \mathcal{K}} \left(1 + s \mathcal{R}\_c \mathcal{C}\right)}{s^2 \mathcal{L} \mathcal{C} \left(1 + \frac{b \mathcal{R}\_c}{\mathcal{R}}\right) + s \left(\frac{b \mathcal{L}}{\mathcal{R}} + \mathcal{R}\_d \mathcal{C} \left(1 + \frac{b \mathcal{R}\_c}{\mathcal{R}}\right) + c \mathcal{R}\_c \mathcal{C}\right) + \frac{b \mathcal{R}\_d}{\mathcal{R}} + c} \tag{108}$$
