Gvg derivation of Boost converter
Ming Sun / November 29, 2022
13 min read • ––– views
Step 1 - construct small-signal equations
Voltage-second balance equation
Fig. 1 shows a synchronous Boost power stage, where it contains a low side switch S1 and a high side switch S2.
For the inductor, we can write the voltage-second balance as[1]:
Where, I is the inductor current, Vg is the Boost converter's input voltage, and V is Boost converter's output voltage Vout. 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 Gvg. 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 Gvg in Matlab
The Matlab script used to derive the Gvg transfer function is as shown below:
clc; clear; close all;
syms s
syms v i vg
syms R L C V Dp
syms Gvg Gig
eqn1 = s*L*i == vg - Dp*v;
eqn2 = s*C*v == Dp*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 Gvg derived result from Matlab.
From Fig. 2, we have:
Simplis for verification of Gvg transfer function
To simulate Gvg transfer function in Simplis, the open loop Boost converter model is as shown in Fig. 3.
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.
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 Gvg very well.
Gig verification
From Fig. 2, the Gig transfer function can be written as:
To verify the Gig transfer function, the Simplis test bench can be modified as shown in Fig. 6.
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.
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 Gig.
References and downloads
[1] Fundamentals of power electronics - Chapter 2
[2] Open-loop Boost converter model for Gvg simulation in Simplis - pdf
[3] Open-loop Boost converter model for Gvg simulation in Simplis - download
[4] Open-loop Boost converter model for Gig simulation in Simplis - pdf
[5] Open-loop Boost converter model for Gig simulation in Simplis - download