Understanding Z-Plane Poles and Zeros: A Sketching Guide

  • Thread starter andrey21
  • Start date
  • Tags
    Poles
In summary, the conversation discusses sketching the z-plane poles and zeros for the function H(z) = ((z-1)^2 (z+1)^2) /(z-0.7)(z+0.9+0.95j)(z+0.9-0.95j). The zeros of the function are determined to be 1 and -1, and the points (-0.9,0.95) and (-0.9,-0.95) are plotted on the z-plane. The significance of the numerator having a greater order than the denominator in this case is related to the behavior of the function as z approaches infinity.
  • #1
andrey21
476
0
I have been asked to sketch the Z planes poles and zeros for the follwoing:

H(z) = ((z-1)^2 (z+1)^2) /(z-0.7)(z+0.9+0.95j)(z+0.9-0.95j)


I have the denominator points sorted. Would the numerator points be:

(z-1)^2 =0
z-1 =0
z=1

and

(z+1)^2 = 0
z+1 = 0
z = -1
 
Physics news on Phys.org
  • #2
Yes, 1 and -1 are the zeros of this function.
 
  • #3
Ok so now to sketching these values.
Do I need to get:

z = -0.9 +0.95j
and
z=-0.9 -0.95 j

in the form r< theta

So r would be:

SQRT ((-0.9)^2 + (0.95)^2))
r = 1.309

and find theta with:

tan theta = b/a

giving me theata = -46.548 and 46.548 degress?
 
  • #4
Well, if you want to plot the values, you COULD get them into polar form. It seems you have done this incorrectly: note that the real parts of each of the numbers is negative, but you are planning to plot them in the first and fourth quadrants.

I would just plot them straight from the x+yj form; the x-axis is the real axis, and the y-axis is the imaginary axis. If you do this and compare to the polar form, you'll see that they're the same, but it seems more straightforward to me to just plot the points (-0.9,0.95) and (-0.9,-0.95).
 
  • #5
Ah I see its just the programme I have been provided to sketch the zplane poles and zeros requires values for r and theta. is this correct:

so r would be:

SQRT ((-0.9)^2 + (0.95)^2))
r = 1.309

and find theta with:

tan theta = b/a

(tan^(-1) 0.95/0.9)-pi

theta = -133.452

this would plot the two points in the second and third quadrants as required
 
  • #6
Yes, seems right.
 
  • #7
Thanks I just have a final question. On my work sheet it says what is the significence that the order of numerator is greater than that of denominator??
 
  • #8
Well, it might tell you something about the behavior of the function as |z| goes to infinity...
 

What are Z-plane poles and zeros?

Z-plane poles and zeros are mathematical concepts used to describe the behavior of a system in the Z-domain. They represent the locations of the roots of the numerator and denominator polynomials in the transfer function of the system.

How do Z-plane poles and zeros affect the stability of a system?

The location of the poles and zeros in the Z-plane can determine the stability of a system. If all poles are located inside the unit circle, the system is stable. If any poles are located outside the unit circle, the system is unstable.

What is the relationship between Z-plane poles and zeros and the frequency response of a system?

The frequency response of a system can be determined by the locations of the poles and zeros in the Z-plane. The magnitude and phase response can be calculated using the locations of the poles and zeros.

How are Z-plane poles and zeros related to the time-domain response of a system?

The time-domain response of a system can be determined by the locations of the poles and zeros in the Z-plane. The positions of the poles and zeros can affect the stability, overshoot, and settling time of a system.

What is the significance of the pole-zero cancellation in the Z-plane?

Pole-zero cancellation occurs when a pole and a zero are located at the same point in the Z-plane. This can result in a simpler transfer function and improved system performance, such as reduced overshoot and settling time.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
343
  • Calculus and Beyond Homework Help
Replies
1
Views
656
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
487
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
592
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
734
  • Calculus and Beyond Homework Help
Replies
19
Views
1K
Back
Top