# s-plane

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In mathematics and engineering, the s-plane is the name for the complex plane on which Laplace transforms are graphed. It 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. It is used as a graphical analysis tool in engineering and physics.

A real function (f) in time 't' is translated into the s-plane by taking the integral of the function multiplied by ${\displaystyle e^{-st}}$ from ${\displaystyle 0}$ to ${\displaystyle \infty }$ where s is a complex number with the form ${\displaystyle s=\sigma +j\omega }$. Coordinates in the s-plane use 'j' to designate the imaginary component, in order to distinguish it from the 'i' used in the normal complex plane.

${\displaystyle \int _{0}^{\infty }f(t)e^{-st}\,dt\;|\;s\;\in \mathbb {C} }$

This transformation from the t-domain into the s-domain is known as a Laplace transform. 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 Laplace transform does the same thing, but more generally. The est not only catches frequencies, but also the real et effects as well. Laplace transforms therefore cater not only for frequency response, but decay effects as well. For instance, a damped sine wave can be modeled correctly using Laplace transforms.

Analysing the complex roots of an s-plane equation and plotting them on an Argand diagram can reveal information about the frequency response and stability of a real time system. This process is called root locus analysis.