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A pole-zero plot shows the location in the complex plane of the poles and zeros of the transfer function of a dynamic system, such as a controller, compensator, sensor, equalizer, filter, or communications channel. By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O.
Intuitively, the minimum-phase part of a general causal system implements its amplitude response with minimal group delay, while its all-pass part corrects its phase response alone to correspond with the original system function. The analysis in terms of poles and zeros is exact only in the case of transfer functions which can be expressed as ...
In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer.
For a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative and the transient behavior will tend to zero in the limit of infinite time. The steady-state output will be:
When the transfer function method is used, attention is focused on the locations in the s-plane where the transfer function is undefined (the poles) or zero (the zeroes; see Zeroes and poles). Two different transfer functions are of interest to the designer. If the feedback loops in the system are opened (that is prevented from operating) one ...
The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter (see pole–zero plot). Evans also invented in 1948 an analog computer to compute root loci, called a "Spirule" (after "spiral" and "slide rule"); it found wide use before the advent of digital computers.
A few procedures can be followed for realizing passive two-ports with transmission zeroes. As long as transmission zeros are located at the origin or infinity, all that is needed is the application of Cauer 1 or 2 steps [clarification needed] to remove poles [clarification needed] from either the admittance or the impedance at the origin or infinity.
After applying this rule, the zero poles should be neglected, i.e. if there are no other unstable poles, then the open-loop transfer function () should be considered stable. If the open-loop transfer function G ( s ) {\displaystyle G(s)} is stable, then the closed-loop system is unstable, if and only if, the Nyquist plot encircle the point −1 ...