SIMON FRASER UNIVERSITY
 School of Engineering Science
 ENSC 380 Linear Systems, ENSC 320 Electric Circuits II
POLES, ZEROS AND SYSTEM RESPONSE
 Jim Cavers
1. INTRODUCTION
An engineer should be able to look at any one of these system descriptions:
and imagine how the others look and how the system behaves. You will learn analytical methods for this in the course. However, visual presentations stick, so this worksheet is a set of animations showing all these system descriptions together, changing in unison as some system parameter moves through a range of values.
Each section below presents a type of system response, such as a single time constant response or a resonant response. You'll see examples of physical systems - electronic, thermal or mechanical - with that response, comments on how they are used and, of course, the animations themselves.
In the case of systems with the same number of zeros as poles, the presentation does not include the impulse response. That is because it contains an impulse itself (why?) and it's hard to put it on a numerically computed graph. In these cases, you'll have to be content with mentally differentiating the step response.
A few notes about the frequency response. First, frequency will be expressed in radians/sec, instead of Hz, because the Laplace variable s = s+jw plays such a prominent role in finite order systems. Second, the frequency responses are shown with linear scales, not as Bode plots. Finally, frequency response is defined only for stable systems. Consequently, it will be displayed only if all poles are in the left half plane.
Links to the details and the animation of the various systems are shown below as bold and underlined.
2. SINGLE POLE
This system is about as simple as they come (other than a constant gain), and it's very useful.
3. ONE POLE, ONE ZERO
If you add a zero to the single time constant (single pole) system, we usually get a high pass characteristic, although it depends on where you place the zero.
4. TWO REAL POLES
Now back to lowpass behaviour. There are plenty of systems with more than one time constant, so you will run across this form many times. I have to admit it's not very exciting, but the animation will at least demonstrate a point in modelling of systems.
5. TWO REAL POLES AND A ZERO
What do you get if you cascade a lowpass filter and a highpass filter? Answer: a bandpass filter. This is the simplest example of a bandpass. But choose the parameters badly, and you get a "no-pass." Oops.
6. TWO COMPLEX POLES - THE LOWPASS FORM
Systems with complex poles can have oscillations in their impulse and step responses, which makes them very useful in filter design. This section deals with the lowpass form, which includes one famous class of filter - the Butterworth lowpass - as a special case.
7. TWO COMPLEX POLES - THE RESONANT BANDPASS FILTER
Finally, two complex poles and a zero. This is the bandpass form, and it is widely used to extract tones or other narrowband signals from noise.
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