S-plane

The S plane is a mathematical domain, where instead of viewing processes in the time domain, modelled with time based functions they are viewed, as equations, in the frequency domain.

A real time function is translated into the 's' plane by taking the Integral of the function, multiplied by e^-st from 0 to infinity, where s is a complex number.

$\int _{0}^{\infty }f(t).e^{-s.t}\,dt$

One way to understand what this equation is doing is to remember how fourier analysis works. In fourier analysis harmonic sine and cosine waves are multiplied into the signal, and the resultant integration provides indication of a signal present at that frequency. The 's' transform is doing the same thing, but more generally. The e^-st not only catches frequencies, but also the real e^-t effects as well. 's' transforms therefore cater not only for frequency response, but decay effects as well. For instance a damped sine wave can be modelled correctly using 's' transforms.

's' transforms are commonly known as laplace transforms In the 's' plane multiplying by s has the effect of differentiating in the corresponding real time domain. Dividing by s integrates.

Analysing the complex roots of a 's' plane equation and plotting them on an argand diagram, can reveal information about the frequency response and stability of a real time system.