Qualitative explanation of S domain

[BlueIntegral]

I'm doing LaPlace transforms for a circuits class, and I realized that I don't REALLY know what the S domain is. When you do Fourier transforms, obviously you're in the frequency domain and that's pretty easy to understand. The other domain you work in a lot is time. But up until now, I've just been sort of blindly transforming to the S domain and not asking questions about where the equation came from. What I'm really curious about is how the S domain manifests itself in physical circuits. I think it's related to tau (time constant), but that's about as far as I've gotten.

 

[madmike159]

This is what wiki had to say

'The Laplace transform is related to the Fourier transform, but whereas the Fourier transform resolves a function or signal into its modes of vibration, the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.'

I have the same problem with Laplace transforms, still don't always know what the mean.
Have you done much work on stability and control of systems? The transfer function in the Laplace domain allows poles and zeros to be found, and you can also plot Nyquist diagrams.