<< Chapter < Page Chapter >> Page >
This module will look at the relationships between the Laplace transform and the complex plane. Specifically, the creation of pole/zero plots and some of their useful properties are discussed.

Introduction to poles and zeros of the laplace-transform

It is quite difficult to qualitatively analyze the Laplace transform and Z-transform , since mappings of their magnitude and phase or real part andimaginary part result in multiple mappings of 2-dimensional surfaces in 3-dimensional space. For this reason, it is verycommon 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 Laplace-transform of a system has been determined, one can use the information contained in function's polynomials tographically represent the function and easily observe many defining characteristics. The Laplace-transform will have the belowstructure, based on Rational Functions :

H s P s Q s

The two polynomials, P s and Q s , allow us to find the poles and zeros of the Laplace-Transform.

zeros
The complex frequencies that make the overall gain of the filter transfer function zero.
poles
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. H s s 1 s 1 2 s 3 4

The zeros are: 1

The poles are: 1 2 3 4

Got questions? Get instant answers now!

The s-plane

Once the poles and zeros have been found for a given Laplace Transform, they can be plotted onto the S-Plane. TheS-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 r θ 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 byan "x" and zeros by an "o". The below figure shows the S-Plane, and examples of plotting zeros and poles onto theplane can be found in the following section.

S-plane

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 S-Plane.

Simple pole/zero plot

H s s s 1 2 s 3 4

The zeros are: 0

The poles are: 1 2 3 4

Pole/zero plot

Using the zeros and poles found from the transfer function, the one zero is mapped to zero and the two polesare placed at 1 2 and 3 4
Got questions? Get instant answers now!

Complex pole/zero plot

H s s s s 1 2 1 2 s 1 2 1 2

The zeros are:

The poles are: 1 1 2 1 2 1 2 1 2

Pole/zero plot

Using the zeros and poles found from the transfer function, the zeros are mapped to ± , and the poles are placed at 1 , 1 2 1 2 and 1 2 1 2
Got questions? Get instant answers now!

Pole-zero cancellation

An easy mistake to make with regards to poles and zeros is to think that a function like s 3 s 1 s 1 is the same as s 3 . In theory they are equivalent, as the pole and zero at s 1 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 wascreated with physical circuits. In this case, it is very unlikely that the pole and zero would remain in exactly thesame place. A minor temperature change, for instance, could cause one of them to move just slightly. If this were tooccur a tremendous amount of volatility is created in that area, since there is a change from infinity at the pole tozero at the zero in a very small range of signals. This is generally a very bad way to try to eliminate a pole. A muchbetter way is to use control theory to move the pole to a better place.

Got questions? Get instant answers now!

Repeated poles and zeros

It is possible to have more than one pole or zero at any given point. For instance, the discrete-time transfer function H z z 2 will have two zeros at the origin and the continuous-time function H s 1 s 25 will have 25 poles at the origin.

MATLAB - If access to MATLAB is readily available, then you can use its functions to easily createpole/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 zerosz = [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

Pole-ZeroDrillDemo
Interact (when online) with a Mathematica CDF demonstrating Pole/Zero Plots. To Download, right-click and save target as .cdf.

Applications for pole-zero plots

Stability and control theory

Now that we have found and plotted the poles and zeros, we must ask what it is that this plot gives us. Basically whatwe can gather from this is that the magnitude of the transfer function will be larger when it is closer to the poles andsmaller when it is closer to the zeros. This provides us with a qualitative understanding of what the system does at variousfrequencies and is crucial to the discussion of stability .

Pole/zero plots and the region of convergence

The region of convergence (ROC) for X z in the complex Z-plane can be determined from the pole/zero plot.Although several regions of convergence may be possible, where each one corresponds to a different impulse response, thereare some choices that are more practical. A ROC can be chosen to make the transfer function causal and/or stable dependingon the pole/zero plot.

    Filter properties from roc

  • If the ROC extends outward from the outermost pole, then the system is causal .
  • If the ROC includes the unit circle, then the system is stable .
Below is a pole/zero plot with a possible ROC of the Z-transform in the Simple Pole/Zero Plot discussed earlier. The shaded region indicates the ROC chosen for the filter. From this figure, wecan see that the filter will be both causal and stable since the above listed conditions are both met.

H z z z 1 2 z 3 4

Region of convergence for the pole/zero plot

The shaded area represents the chosen ROC for the transfer function.
Got questions? Get instant answers now!

Frequency response and pole/zero plots

The reason it is helpful to understand and create these pole/zero plots is due to their ability to help us easilydesign a filter. Based on the location of the poles and zeros, the magnitude response of the filter can be quicklyunderstood. Also, by starting with the pole/zero plot, one can design a filter and obtain its transfer function veryeasily.

Conclusion

Pole-Zero Plots are clearly quite useful in the study of the Laplace and Z transform, affording us a method of visualizing the at times confusing mathematical functions.

Questions & Answers

a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
f(n)= 2n + 1
Samantha Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask