Introduction

It 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.

Given a continuous-time transfer function in the Laplace domain, Hs H s , or a discrete-time one in the Z-domain, Hz H z , a zero is any value of ss or zz such that the transfer function is zero, and a pole is any value of ss or zz such that the transfer function is infinite. To define them precisely:

Definition 1: zeros

1. The value(s) for zz where the numerator of the transfer function equals zero

2. The complex frequencies that make the overall gain of the filter transfer function zero.

Definition 2: poles

1. The value(s) for zz where the denominator of the transfer function equals zero

2. The complex frequencies that make the overall gain of the filter transfer function infinite.

Pole/Zero Plots

When we plot these in the appropriate s- or z-plane, we represent zeros with "o" and poles with "x". Refer to this module for a detailed looking at plotting the poles and zeros of a z-transform on the Z-plane.

Example 1

Find the poles and zeros for the transfer function Hs=s2+6s+8s2+2 H s s 2 6 s 8 s 2 2 and plot the results in the s-plane.

The first thing we recognize is that this transfer function will equal zero whenever the top, s2+6s+8 s 2 6 s 8 , equals zero. To find where this equals zero, we factor this to get, s+2s+4 s 2 s 4 . This yields zeros at s=-2 s -2 and s=-4 s -4 . Had this function been more complicated, it might have been necessary to use the quadratic formula.

For poles, we must recognize that the transfer function will be infinite whenever the bottom part is zero. That is when s2+2 s 2 2 is zero. To find this, we again look to factor the equation. This yields s+ⅈ2s−ⅈ2 s 2 s 2 . This yields purely imaginary roots of +ⅈ2 2 and -ⅈ2 2

Plotting this gives Figure 1

Figure 1: Sample pole-zero plot

Pole and Zero Plot

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

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 Hz=z2 H z z 2 will have two zeros at the origin and the continuous-time function Hs=1s25 H s 1 s 25 will have 25 poles at the origin.

Pole-Zero Cancellation

An easy mistake to make with regards to poles and zeros is to think that a function like s+3s−1s−1 s 3 s 1 s 1 is the same as s+3 s 3 . In theory they are equivalent, as the pole and zero at s=1 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 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.