It is possible to have more than one pole or zero at any given point. For
instance, the discrete-time transfer function

Summary: Explains poles and zeros of transfer functions.

Note: You are viewing an old version of this document. The latest version is available here.

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,

Find the poles and zeros for the transfer function

The first thing we recognize is that this transfer function will equal zero
whenever the top,

For poles, we must recognize that the transfer function will be infinite
whenever the bottom part is zero. That is when

Plotting this gives

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.

It is possible to have more than one pole or zero at any given point. For
instance, the discrete-time transfer function

An easy mistake to make with regards to poles and zeros is to think that a
function like