Average Switch Model of Buck Power Stage

Buck power stage

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

Fig. 1 shows a non-synchronous Buck power stage, where it contains a switch S₁ and a free wheeling diode D₁. The target is to derive the averaged switch model as highlighted by the dashed rectangle. The port voltage (v₁ and v₂) and current (i₁ and i₂) is defined as shown in 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_S time interval, where DT_S is defined as the time frame when S₁ is on and D'T_S is defined as the time frame when S₁ is off.

Fig. 2Buck power stage - DTs^[1~2]

During DT_S, switch S₁ is closed. Inductor current i_L flows through switch S₁. Because switch S₁ is closed, v₂ is equal to V_g. As a result, diode D₁ is reverse biased and there is no current flowing through it.

Fig. 3Buck power stage - D'Ts^[1~2]

During D'T_S, switch S₁ is open. To maintain a continuous current flow, the inductor will turn on the free wheeling diode D₁. If we ignore the forward voltage of D₁, v₂ will be 0V during 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.

Fig. 4Sketched waveforms of port voltage and current^[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.

Fig. 5DC averaged switch model^[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.

Fig. 6DC averaged switch for Buck power stage^[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*v₁ and d*i₂ are ignored and considered to be 0. Using Eq. 3, the AC averaged switch model can be sketched in Fig. 7.

Fig. 7DC + AC averaged switch model^[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.

Fig. 8AC averaged switch model^[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.

Fig. 9Buck power stage small signal model^[1~2]

Here we are interested of the transfer function G_vd, which is defined from d to v_o. As a result, the small signal term v_g can be set to 0, as shown in Fig. 10.

Fig. 10Gvd small signal model^[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.

Fig. 11Gvg small signal model^[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

In Ref. [3], the small signal transfer function of G_vd and G_vg for Buck, Boost and Buck-boost are summarized as shown in Fig. 12.

Fig. 12Gvd and Gvg summary table^[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.

References

[1] Converter Control

[2] Averaged-Switch Modeling and Simulation

[3] Fundamentals of Power electronics - Chapter 8, slide 84