Power stage transfer function derivations

Step 1 - construct small-signal equations

Fig. 1Buck power stage^[1~2]

Voltage-second balance equation

Fig. 1 shows a non-synchronous Buck power stage, where it contains a switch S₁ and a free wheeling diode D₁.

For the inductor, we can write the voltage-second balance as^[1]:

Where, I is the inductor current and V is Buck converter's output voltage V_out. Next, let us perturb and linearize Eq. 1 by introducting the small signal perturbation as:

Here we are trying to derive the transfer function of G_vd. As a result, we can assume V_g is constant. Removing the DC terms from Eq. 2, we have:

Eq. 3 can be written in s domain as:

charge balance equation

For the capacitor, we can write the charge balance as^[1]:

Next, let us perturb and linearize Eq. 5 by introducting the small signal perturbation as:

Removing the DC terms from Eq. 6, we have:

Eq. 7 can be written in s domain as:

Step 2 - solve the G_vd in Matlab

The Matlab script used to derive the G_vd transfer function is as shown below:

clc; clear; close all;

syms s
syms v i d
syms R L C Vg
syms Gvd

eqn1 = s*L*i == d*Vg - v;
eqn2 = s*C*v == i - v/R;
eqn3 = Gvd == v/d;

results = solve(eqn1, eqn2, eqn3, [v i Gvd])

Gvd = simplify(results.Gvd)

Fig. 2 shows the G_vd derived result from Matlab.

Fig. 2Gvd derived result from Matlab

From Fig. 2, we have:

Compared with the G_vd transfer function we previously derived in the averaged switch model blog post^[3], the result matches with each other.

In Ref. [3], the small signal transfer function of G_vd and G_vg for Buck, Boost and Buck-boost are summarized as shown in Fig. 3.

Fig. 3Gvd and Gvg summary table^[3]

References and downloads

[1] Fundamentals of power electronics - Chapter 2

[2] Popular converters and the conversion ratio derivation

[3] Average Switch Model of Buck Power Stage

[4] Fundamentals of Power electronics - Chapter 8, slide 84

[5] Gvd derivation script in Matlab - download