Understanding poles and zeros of transfer functions

[bitrex]

Hello again, as I was reading today still trying to grok passive filter design, I realized that I do not entirely grasp the concept of "poles" and "zeros" from in a qualitative way. I understand, for example, that a pole is a kind of singularity where the denominator of the complex-number transfer function is zero, but I'm having trouble relating the mathematics to the real world behavior of the filter. Let's say that the frequency being input into the filter is exactly the same as one of the pole frequencies and the denominator goes to zero and the transfer function goes to infinity - how can that be physical? Is it because the singularity is in the complex plane and doesn't have any physical significance? I can now derive transfer functions and plot frequency responses of some circuits using the mathematical tools I have practiced, but I feel it is important to get an intuitive sense of what is happening concurrently in the circuit and this is proving difficult. I haven't studied much about Laplace transforms yet - perhaps this is a stumbling block.

 

[MATLABdude]

At a high level, a transfer function is just a way of describing what sort of output you get for a particular type of input.

So, for a sinusoidal input which has a frequency that just happens to correspond to a zero of your system, you'd have zero output (Why? Because your transfer function is zero!)

For a sinusoidal input which has a frequency that just happens to correspond to a pole of your system, you'd theoretically have infinite output. However, real components being real, you'd drive your system to some sort of maximal (but non-infinite) response (a.k.a. resonance).