s-plane
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 [math]\displaystyle{ f }[/math] in time [math]\displaystyle{ t }[/math] is translated into the s-plane by taking the integral of the function multiplied by [math]\displaystyle{ e^{-st} }[/math] from 0 to [math]\displaystyle{ \infty }[/math] where s is a complex number with the form [math]\displaystyle{ s = \sigma+i\omega }[/math].
- [math]\displaystyle{ F(s) = \int_{0}^\infty f(t) e^{-st}\,dt \; | \; s \; \in \mathbb{C} }[/math]
This transformation from the t-domain into the s-domain is known as a Laplace transform and the function [math]\displaystyle{ F(s) }[/math] is called the Laplace transform of [math]\displaystyle{ f }[/math]. 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 [math]\displaystyle{ e^{-st} }[/math] not only captures the frequency response via its imaginary [math]\displaystyle{ e^{-i\omega t} }[/math] component, but also decay effects via its real [math]\displaystyle{ e^{-\sigma t} }[/math] 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
- [math]\displaystyle{ f(t) = {1 \over 2\pi i}\lim_{T \to \infty}\int_{\gamma-iT}^{\gamma+iT} F(s) e^{st}\,ds }[/math]
where the real number [math]\displaystyle{ \gamma }[/math] is chosen so the integration path is within the region of convergence of [math]\displaystyle{ F(s) }[/math]. 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.
See also
- Root locus
- State space (controls)
- z-transform
External links
- Illustration of a mapping from the s-plane to the z-plane
- Kevin Brown (2015) Laplace Transforms at Math Pages.
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