My understanding is that the Laplace transform evaluated at $s = i \omega t$ can be used to evaluate the steady-state of a function. How is this done? I can't find any information on this in my textbooks nor on the internet.
Thanks in advance.
My understanding is that the Laplace transform evaluated at $s = i \omega t$ can be used to evaluate the steady-state of a function. How is this done? I can't find any information on this in my textbooks nor on the internet.
Thanks in advance.
Let $f(t)$ denote the time-domain function, and $F(s)$ denote its Laplace transform. The final value theorem states that: $ \lim_{t \to \infty} f(t) = \lim_{s\to 0} sF(s), $ where the LHS is the steady state of $f(t).$ Since it is typically hard to solve for $f(t)$ directly, it is much easier to study the RHS where, for example, ODEs become polynomials or rational functions in $s.$
You can get more info in course notes like this one: PDF, or in control engineering/system theory books such as Kailath: Linear Systems or Ogata: Modern Control Engineering.
Also, as noted in the comments, there are Physics Stackexchange and Electrical Engineering Stackexchange