# Has anyone ever come across the derivation of the frequency domain transfer function of a buck?

#### mike _Jacobs

Joined Jun 9, 2021
223
All i can find is state space ones.

Im trying to find something that is in the S domain and clearly outlines how the answer was derived.

#### WBahn

Joined Mar 31, 2012
30,303
You may have trouble finding one -- the s-domain is a Laplace Transform from a space in which the system is described by a set of linear, time-invariant differential equations. My understanding is that if you transform other types of differential equations, you simply end up with a set of non-linear differential equations in the s-domain.

So what is normally done is to linearize the original set of differential equations, which is exactly what the purpose of small-signal analysis is.

#### BobTPH

Joined Jun 5, 2013
9,300
I am not an EE and do not have a grasp of Laplace transforms and transfer functions.

I do understand how a transfer function in the frequency domain might describe the response of a filter to different input frequencies.

But can someone explain to me how an equation in the frequency domain applies to a buck converter, which operates at a single frequency?

#### crutschow

Joined Mar 14, 2008
34,855
Are you referring to the S-domain transfer function of the control feedback loop?
This helps explain that.

Last edited:

#### crutschow

Joined Mar 14, 2008
34,855
But can someone explain to me how an equation in the frequency domain applies to a buck converter, which operates at a single frequency?
It's the response of the analog control feedback loop in a linearized model where the switching frequency is not directly included.
Edit: But the ripple in the loop signal due to the output ripple from the switching frequency must be considered in the loop response, where the PWM modulator is modeled as a voltage-to-voltage gain block with an LC output filter.

Last edited:

#### Ian0

Joined Aug 7, 2020
10,277
This one is good, too (and seems to have disappeared from TI's website)

#### Attachments

• 4.8 MB Views: 5

#### mike _Jacobs

Joined Jun 9, 2021
223
You may have trouble finding one -- the s-domain is a Laplace Transform from a space in which the system is described by a set of linear, time-invariant differential equations. My understanding is that if you transform other types of differential equations, you simply end up with a set of non-linear differential equations in the s-domain.

So what is normally done is to linearize the original set of differential equations, which is exactly what the purpose of small-signal analysis is.
if this is true, im not sure how people go about getting the answer. I have seen what the S domain transfer function is. Its listed in several reference books. But i have no clue how they got it. I would like to see the derivation.

#### Ian0

Joined Aug 7, 2020
10,277
if this is true, im not sure how people go about getting the answer. I have seen what the S domain transfer function is. Its listed in several reference books. But i have no clue how they got it. I would like to see the derivation.
See Equation 15 in SLUP340.
Two poles for the LC filter
One zero for the ESR of the capacitor

#### crutschow

Joined Mar 14, 2008
34,855
But i have no clue how they got it. I would like to see the derivation.
And you won't get a clue if you don't look at the references we posted.
See Section 1.1 of my reference.

Last edited: