# What is 's' in a transfer function?

knight92
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?

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. knight92 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?