S plane
2614482
221693599
2008-06-25T17:36:07Z
Foobaz
103430
syntax
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 <math>e^{-st}</math> from 0 to infinity, where s is a [[complex number]].
:<math>\int_0^\infty f(t) e^{-st}\,dt</math>
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 (i.e. the signal's energy at a point in the frequency domain). The 's' transform does the same thing, but more generally. The e<sup>-st</sup> not only catches frequencies, but also the real e<sup>-t</sup> 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 transform]]s. 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 number|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.
[[Category:Fourier analysis]]
[[ru:S-плоскость]]
{{math-stub}}
==External links==
* [http://dspcan.homestead.com/files/Ztran/zlap.htm Illustration of how the s-plane maps to the z-plane]