# Average Switch Model of Buck Power Stage

Ming Sun / October 25, 2022

15 min read • ––– views

## Buck power stage

^{[1~2]}

**Fig. 1** shows a non-synchronous Buck power stage, where it contains a switch **S _{1}** and a free wheeling diode

**D**. The target is to derive the averaged switch model as highlighted by the dashed rectangle. The port voltage (

_{1}**v**and

_{1}**v**) and current (

_{2}**i**and

_{1}**i**) is defined as shown in

_{2}**Fig. 1**as well.

## DC averaged switch model derivation

First let us sketch the waveform before calculating the port average voltage and current. To do so, let us re-draw **Fig. 1** during **DT _{S}** and

**D'T**time interval, where

_{S}**DT**is defined as the time frame when

_{S}**S**is on and

_{1}**D'T**is defined as the time frame when

_{S}**S**is off.

_{1}^{[1~2]}

During **DT _{S}**, switch

**S**is closed. Inductor current

_{1}**i**flows through switch

_{L}**S**. Because switch

_{1}**S**is closed,

_{1}**v**is equal to

_{2}**V**. As a result, diode

_{g}**D**is reverse biased and there is no current flowing through it.

_{1}^{[1~2]}

During **D'T _{S}**, switch

**S**is open. To maintain a continuous current flow, the inductor will turn on the free wheeling diode

_{1}**D**. If we ignore the forward voltage of

_{1}**D**,

_{1}**v**will be 0V during

_{2}**D'T**.

_{S}Through the analysis of **Fig. 2** and **Fig. 3**, the port voltage and current can be sketched as shown in **Fig. 4**.

^{[1~2]}

From **Fig. 4**, we can correlate the average voltage and current information between input port and output port as follows:

From **Eq. 1**, the DC average switch model can be sketched as shown in **Fig. 5**.

^{[1~2]}

The DC averaged switch model can be used to derive the conversion ratio in CCM. For example, we can plug in the DC averaged switch model **Fig. 5** into **Fig. 1**. In DC conditions, the inductors are short and the capacitors are open. So **Fig. 1** can be re-drawn as shown in **Fig. 6**.

^{[1~2]}

It is quite obvious that the conversition ratio can be calculated as:

## AC averaged switch model derivation

Next, let us derive the AC averaged switch model. To do so, we need to add small signal terms into **Eq. 1**.

The second order terms are ignored in **Eq. (3)**. For example, **d*v _{1}** and

**d*i**are ignored and considered to be 0. Using

_{2}**Eq. 3**, the AC averaged switch model can be sketched in

**Fig. 7**.

^{[1~2]}

In **Fig. 7**, we can set the DC terms to be zero to get the AC averaged switch model as shown in **Fig. 8**.

^{[1~2]}

## Buck power stage small model derivation - G_{vd}

Plugging **Fig. 8** into **Fig. 1**, the Buck power stage AC small signal model can be easily re-drawn as shown in **Fig. 9**.

^{[1~2]}

Here we are interested of the transfer function **G _{vd}**, which is defined from

**d**to

**v**. As a result, the small signal term

_{o}**v**can be set to 0, as shown in

_{g}**Fig. 10**.

^{[1~2]}

From **Fig. 10**, we have:

**G _{vd}** is defined as:

By combining **Eq. 4** and **Eq. 5**, we have:

**Eq. 6** can be re-written as:

Where,

**Eq. 7** and **Eq. 8** matches with the equation shown in **Ref. [3], slide 84**.

## Buck power stage small model derivation - G_{vg}

To calculate transfer function of **G _{vg}**, the small signal term

**d**can be set to 0 in

**Fig. 9**.

^{[1~2]}

From **Fig. 11**, we have:

**G _{vg}** is defined as:

By combining **Eq. 9** and **Eq. 10**, we have:

By combing **Eq. 8** and **Eq. 11**, we have:

**Eq. 12** matches with the equation shown in **Ref. [3], slide 84**.

**G**_{vd} and **G**_{vg} transfer function summary

_{vd}

_{vg}

In **Ref. [3]**, the small signal transfer function of **G _{vd}** and

**G**for Buck, Boost and Buck-boost are summarized as shown in

_{vg}**Fig. 12**.

^{[3]}

## Summary

In this blog post, the derivation of averaged switch model is shown for Buck power stage. A similar derivation process can be applied for Boost and Buck-boost converter as well. The averaged switch model can be used to derive the conversition ratio or the small signal transfer function, which can be super helpful for the control loop design.