Understanding Pole/Zero Plots on the Z-Planehttp://cnx.org/contenthttp://cnx.org/content/m10556/latest/m10556Understanding Pole/Zero Plots on the Z-Plane2.122002/04/092011/09/29 16:43:02.272 GMT-5MariyahPoonawalaMariyah Poonawalamariyah@rice.eduDanCalderonDan Calderonslycaldron@netscape.netMichaelHaagMichael Haagmjhaag@gmail.comPrashantSinghPrashant Singhprash@ece.rice.edumjhaagmariyah prash slycaldronmjhaagcomplexpolezeroz-planeScience and TechnologyThis module will look at the relationships between the z-transform and the complex plane. Specifically, the creation of pole/zero plots and some of their useful properties are discussed.enIntroduction to Poles and Zeros of the Z-TransformIt is quite difficult to qualitatively analyze the Laplace transform and
Z-transform, since
mappings of their magnitude and phase or real part and
imaginary part result in multiple mappings of 2-dimensional
surfaces in 3-dimensional space. For this reason, it is very
common to examine a plot of a transfer function's poles and zeros to
try to gain a qualitative idea of what a system does.
Once the Z-transform of a system has been determined, one can
use the information contained in function's polynomials to
graphically represent the function and easily observe many
defining characteristics. The Z-transform will have the below
structure, based on Rational
Functions:
XzPzQz
The two polynomials, Pz and
Qz, allow us
to find the poles and
zeros of the Z-Transform.
zerosThe value(s) for z where
Pz0.
The complex frequencies that make the overall gain of the
filter transfer function zero.polesThe value(s) for z where
Qz0.
The complex frequencies that make the overall gain of the
filter transfer function infinite.
Below is a simple transfer function with the poles and zeros
shown below it.
Hzz1z12z34
The zeros are:
1
The poles are:
1234The Z-Plane
Once the poles and zeros have been found for a given
Z-Transform, they can be plotted onto the Z-Plane. The
Z-plane is a complex plane with an imaginary and real axis
referring to the complex-valued variable
z. The position on the complex
plane is given by rθ
and the angle from the positive, real axis around the plane is
denoted by θ. When
mapping poles and zeros onto the plane, poles are denoted by
an "x" and zeros by an "o". The below figure shows the
Z-Plane, and examples of plotting zeros and poles onto the
plane can be found in the following section.
Examples of Pole/Zero Plots
This section lists several examples of finding the poles and
zeros of a transfer function and then plotting them onto the
Z-Plane.
Simple Pole/Zero PlotHzzz12z34
The zeros are:
0
The poles are:
1234Complex Pole/Zero PlotHzzzz1212z1212
The zeros are:
The poles are:
112121212Pole-Zero Cancellation
An easy mistake to make with regards to poles and zeros is to
think that a function like
s3s1s1
is the same as
s3.
In theory they are equivalent, as the pole and zero at
s1
cancel each other out in what is known as pole-zero
cancellation. However, think about what may happen
if this were a transfer function of a system that was
created with physical circuits. In this case, it is very
unlikely that the pole and zero would remain in exactly the
same place. A minor temperature change, for instance, could
cause one of them to move just slightly. If this were to
occur a tremendous amount of volatility is created in that
area, since there is a change from infinity at the pole to
zero at the zero in a very small range of signals. This is
generally a very bad way to try to eliminate a pole. A much
better way is to use control theory to move the
pole to a better place.
Repeated Poles and ZerosIt is possible to have more than one pole or zero at any given
point. For instance, the discrete-time transfer function
Hzz2
will have two zeros at the origin and the continuous-time
function
Hs1s25
will have 25 poles at the origin.
MATLAB - If access to MATLAB is readily
available, then you can use its functions to easily create
pole/zero plots. Below is a short program that plots the
poles and zeros from the above example onto the Z-Plane.
% Set up vector for zeros
z = [j ; -j];
% Set up vector for poles
p = [-1 ; .5+.5j ; .5-.5j];
figure(1);
zplane(z,p);
title('Pole/Zero Plot for Complex Pole/Zero Plot Example');
Interactive Demonstration of Poles and Zeros