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S-plane

## Summary

In mathematics and engineering, the s-plane is the complex plane on which Laplace transforms are graphed. It is a mathematical domain where, instead of viewing processes in the time domain modeled 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 ${\displaystyle f}$ in time ${\displaystyle t}$ is translated into the s-plane by taking the integral of the function multiplied by ${\displaystyle e^{-st}}$ from 0 to ${\displaystyle \infty }$ where s is a complex number with the form ${\displaystyle s=\sigma +i\omega }$.

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

This transformation from the t-domain into the s-domain is known as a Laplace transform and the function ${\displaystyle F(s)}$ is called the Laplace transform of ${\displaystyle f}$. The Laplace transform is analogous to the process of Fourier analysis; in fact, Fourier series are a special case of the Laplace transform. 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 ${\displaystyle e^{-st}}$ not only captures the frequency response via its imaginary ${\displaystyle e^{-i\omega t}}$ component, but also decay effects via its real ${\displaystyle e^{-\sigma t}}$ component. For instance, a damped sine wave can be modeled correctly using Laplace transforms.

A function in the s-plane can be translated back into a function of time using the inverse Laplace transform

${\displaystyle f(t)={1 \over 2\pi i}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}F(s)e^{st}\,ds}$

where the real number ${\displaystyle \gamma }$ is chosen so the integration path is within the region of convergence of ${\displaystyle F(s)}$. However rather than use this complicated integral, most functions of interest are translated using tables of Laplace transform pairs, and the Cauchy residue theorem.

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.