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. In calculus, the integral of a function is an extension of the concept of a sum. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

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 does 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. Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

'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. In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...

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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...