# Gvd derivation of Boost converter

Ming Sun / November 29, 2022

15 min read • ––– views

## Step 1 - construct small-signal equations

^{[1]}

### Voltage-second balance equation

**Fig. 1** shows a synchronous Boost power stage, where it contains a low side switch **S _{1}** and a high 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 Boost converter's input voltage, and

**V**is Boost 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 _{vd}**. As a result, we can assume

**V**is constant. Removing the DC terms from and second order small signal terms

_{g}**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 and second order small signal terms from **Eq. 6**, we have:

**Eq. 7** can be written in `s`

domain as:

## Step 2 - solve the **G**_{vd} in Matlab

_{vd}

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 V Dp I
syms Gvd Gid
eqn1 = s*L*i == -Dp*v + d*V;
eqn2 = s*C*v == Dp*i - d*I - v/R;
eqn3 = I*Dp == V/R;
eqn4 = Gvd == v/d;
eqn5 = Gid == i/d;
results = solve(eqn1, eqn2, eqn3, eqn4, eqn5, [v i I Gvd Gid]);
Gvd = simplify(results.Gvd)
Gid = simplify(results.Gid)
```

**Fig. 2** shows the **G _{vd}** derived result from Matlab.

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Next, we can use `wxMaxima`

to simplify the results as shown in **Fig. 3**.

^{}

From **Fig. 3**, we have:

## Simplis for verification of Gvg transfer function

To simulate **G _{vd}** transfer function in Simplis, the open loop Boost converter model is as shown in

**Fig. 4**.

^{}

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, V 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. 5**.

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The Simplis simulation results are as shown in **Fig. 6**. From **Fig. 6**, we can see that the mathematical Laplace transfer function matches with the `AC`

simulation results of **G _{vd}** very well at low frequency range. Magnitude and phase in the Bode plot starts to shown some differences when the frequency is close to the switching frequency. At half of the switching frequency, the magnitude between mathematical derivation and Simplis simulation results still match with each other very well while there is about 7º phase difference between the two.

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But the difference is small enough and the it will not impact the actual design when we use the mathematical equations since in most of the design the closed loop cross over frequency is much less than the switching frequency **f _{sw}**.

## Gid verification

From **Fig. 2**, the **G _{id}** transfer function can be written as:

To verify the **G _{id}** transfer function, the Simplis test bench can be modified as shown in

**Fig. 7**.

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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. 8**.

^{}

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 _{id}**.

^{}

## References and downloads

[1] Fundamentals of power electronics - Chapter 2

[2] Open-loop Boost converter model for Gvd simulation in Simplis - pdf

[3] Open-loop Boost converter model for Gvd simulation in Simplis - download

[4] Open-loop Boost converter model for Gid simulation in Simplis - pdf

[5] Open-loop Boost converter model for Gid simulation in Simplis - download