Gvg derivation of Buck converter

Step 1 - construct small-signal equations

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

Voltage-second balance equation

Fig. 1 shows a synchronous Buck power stage, where it contains a high side switch S₁ and a low side switch S₂.

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

Where, I is the inductor current, V_g is the Buck converter's input voltage, 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_vg. As a result, we can assume D 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_vg in Matlab

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

clc; clear; close all;

syms s
syms v i vg
syms R L C D
syms Gvg Gig

eqn1 = s*L*i == D*vg - v;
eqn2 = s*C*v == i - v/R;
eqn3 = Gvg == v/vg;
eqn4 = Gig == i/vg;

results = solve(eqn1, eqn2, eqn3, eqn4, [v i Gvg Gig]);

Gvg = simplify(results.Gvg)
Gig = simplify(results.Gig)

Fig. 2 shows the G_vg derived result from Matlab.

Fig. 2Gvg derived result from Matlab

From Fig. 2, we have:

Simplis for verification of Gvg transfer function

In Ref. [3], we have created an open-loop Buck converter model for G_vg simulation in Simplis. To simulate G_vg transfer function, the updated Simplis test bench is as shown in Fig. 3.

Fig. 3Updated open-loop Buck converter model for Gvg simulation

To set the property of the Laplace Transfer Function block, Eq. 9 can be re-written as:

We can plug in the inductor, capacitor, resistor and D values into Eq. 10. We have:

Based on Eq. 11, the property of the 2nd-order Laplace Transfer Function is as shown in Fig. 4.

Fig. 42nd-order Laplace Transfer Function block property

The Simplis simulation results are as shown in Fig. 5. From Fig. 5, we can see that the mathematical Laplace transfer function matches with the AC simulation results of G_vg.

Fig. 5Simulation results comparison between mathematical derivation and AC analysis

Gig verification

From Fig. 2, the G_ig transfer function can be written as:

To verify the G_ig transfer function, the Simplis test bench can be modified as shown in Fig. 6.

Fig. 6Gig test bench in Simplis for Buck converter

To set the property of the Laplace Transfer Function block, Eq. 12 can be re-written as:

We can plug in the inductor, capacitor, resistor and Vg values into Eq. 13. We have:

Based on Eq. 14, the property of the 2nd-order Laplace Transfer Function is as shown in Fig. 7.

Fig. 72nd-order Laplace Transfer Function block property for Gig simulation

The Simplis simulation results are as shown in Fig. 9. From Fig. 9, we can see that the mathematical Laplace transfer function matches with the AC simulation results of G_ig.

Fig. 8Gig comparison between mathematical derivation and AC analysis

References and downloads

[1] Fundamentals of power electronics - Chapter 2

[2] Popular converters and the conversion ratio derivation

[3] Gvd derivation of Buck converter

[4] Open-loop Buck converter model for Gvg simulation in Simplis - pdf

[5] Open-loop Buck converter model for Gvg simulation in Simplis - download

[6] Open-loop Buck converter model for Gig simulation in Simplis - pdf

[7] Open-loop Buck converter model for Gig simulation in Simplis - download