# What is 's' in a transfer function?

I am trying to slowly learn control theory. I know how to find transfer function from a given differential equation of a system e.g mass/damper system. But what is the term 's'? Is that the frequency of inputs? In real life a mass/Spring damper would be a cars suspension. What will be a typical 's' value in this case?

marcusl
Gold Member
It's hard to know from the limited information you provided. s is often the output variable from a Laplace transform. It is complex and related to angular frequency ω by $s=\sigma+i\omega$. In control systems, the real part σ is intimately related to stability. There is lots of information about Laplace transforms on the web, and I imagine they are covered in every control theory book, as well.

BTW, if you are interested in in-depth treatment of Laplace transforms applied to physical systems like heat conduction or the spring/damper that you mentioned, I can recommend a lovely little book called Operational Methods in Applied Mathematics by Carslaw and Jaeger. You can buy a used copy of the Dover edition for under $10, if it's not in your school library. It's hard to know from the limited information you provided. s is often the output variable from a Laplace transform. It is complex and related to angular frequency ω by $s=\sigma+i\omega$. In control systems, the real part σ is intimately related to stability. There is lots of information about Laplace transforms on the web, and I imagine they are covered in every control theory book, as well. BTW, if you are interested in in-depth treatment of Laplace transforms applied to physical systems like heat conduction or the spring/damper that you mentioned, I can recommend a lovely little book called Operational Methods in Applied Mathematics by Carslaw and Jaeger. You can buy a used copy of the Dover edition for under$10, if it's not in your school library.

I meant for example when you Laplace transform dx/dt you get sX(s) so I was asking what is 's'? You said it is complex but are we ever given a value for it so that we can use it in a transfer function to calculate the response/output of the system and get a real number? Many times I see on the Internet the input given as another equation so you never get a number out instead it is just another equation.

If I was considering a spring/damper system what would be a typical s value input to find the deflection/displacement of the system?

Baluncore
But what is the term 's'?
It is the complex frequency plane. You can plot the poles and zeros of the transfer function on that plane.
Points on the vertical ω axis are sinewaves with stable amplitudes.
Points on the left hand side are decaying sinewaves. Damped suspension has poles on the LHS.
Points on the right hand side are exponentialy growing sinewaves. Poles on the RHS cause instability and oscillation.

https://en.wikipedia.org/wiki/S-plane

marcusl