Power stage transfer function derivations

Ming Sun

Ming Sun / October 27, 2022

6 min read––– views

Step 1 - construct small-signal equations

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

Voltage-second balance equation

Fig. 1 shows a non-synchronous Buck power stage, where it contains a switch S1 and a free wheeling diode D1.

For the inductor, we can write the voltage-second balance as[1]:

`L{dI}/{dt} = D(V_g-V) + D^'(-V) = D*V_g - V`

Where, I is the inductor current and V is Buck converter's output voltage Vout. Next, let us perturb and linearize Eq. 1 by introducting the small signal perturbation as:

`L{d(I+hat(i))}/{dt} = (D+hat(d))*V_g -V-hat(v)`

Here we are trying to derive the transfer function of Gvd. As a result, we can assume Vg is constant. Removing the DC terms from Eq. 2, we have:

`L{dhat(i)}/{dt} = hat(d)*V_g - hat(v)`

Eq. 3 can be written in s domain as:

`sL*hat(i) = hat(d)*V_g - hat(v)`

charge balance equation

For the capacitor, we can write the charge balance as[1]:

`C{dV}/{dt} = I-V/R`

Next, let us perturb and linearize Eq. 5 by introducting the small signal perturbation as:

`C{d(V+hat(v))}/{dt} = I+hat(i)-(V+hat(v))/R`

Removing the DC terms from Eq. 6, we have:

`C{dhat(v)}/{dt} = hat(i)- hat(v)/R`

Eq. 7 can be written in s domain as:

`sC*dhat(v) = hat(i)- hat(v)/R`

Step 2 - solve the Gvd in Matlab

The Matlab script used to derive the Gvd transfer function is as shown below:

clc; clear; close all;

syms s
syms v i d
syms R L C Vg
syms Gvd

eqn1 = s*L*i == d*Vg - v;
eqn2 = s*C*v == i - v/R;
eqn3 = Gvd == v/d;

results = solve(eqn1, eqn2, eqn3, [v i Gvd])

Gvd = simplify(results.Gvd)

Fig. 2 shows the Gvd derived result from Matlab.

Gvd derived result from Matlab
Fig. 2Gvd derived result from Matlab

From Fig. 2, we have:

`G_{vd} = V_g * 1/{1+s*L/R + LC*s^2}`

Compared with the Gvd transfer function we previously derived in the averaged switch model blog post[3], the result matches with each other.

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

Gvd and Gvg summary table
Fig. 3Gvd and Gvg summary table[3]

References and downloads

[1] Fundamentals of power electronics - Chapter 2

[2] Popular converters and the conversion ratio derivation

[3] Average Switch Model of Buck Power Stage

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

[5] Gvd derivation script in Matlab - download

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