# Gvg derivation of Buck converter

Ming Sun / November 29, 2022

13 min read • ––– views

## Step 1 - construct small-signal equations

^{[1~2]}

### Voltage-second balance equation

**Fig. 1** shows a synchronous Buck power stage, where it contains a high side switch **S _{1}** and a low side switch

**S**.

_{2}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**. Next, let us perturb and linearize

_{out}**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

_{vg}

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.

^{}

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**transfer function, the updated Simplis test bench is as shown in

_{vg}**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 **G _{vg}**.

^{}

## 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**.

^{}

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 **G _{ig}**.

^{}

## 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