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).

 

[oswaler]

Hello,

Thank you for sharing your thoughts on understanding poles and zeros of transfer functions. It is great that you are trying to establish a connection between the mathematics and the real-world behavior of filters.

Firstly, let's define what poles and zeros are in the context of transfer functions. Poles are the points in the complex plane where the transfer function becomes infinite, while zeros are the points where the transfer function becomes zero. These points are important because they determine the frequency response of the filter.

Now, let's address your question about the physical significance of poles and zeros. It is true that poles and zeros exist in the complex plane and do not have a physical meaning in themselves. However, they do have a physical significance in the behavior of the filter. The poles and zeros of a transfer function represent the frequencies at which the filter either amplifies or attenuates the input signal. In other words, they indicate the frequencies at which the filter has a significant effect on the input signal.

To understand this better, let's take the example you mentioned where the frequency being input into the filter is exactly the same as one of the pole frequencies. In this case, the transfer function goes to infinity, indicating that the filter amplifies the input signal at that particular frequency. This means that the filter has a significant effect on the signal at that frequency, and the output signal will be amplified accordingly.

Similarly, if the input frequency is at one of the zero frequencies, the transfer function becomes zero, indicating that the filter attenuates the input signal at that frequency. This means that the filter has little to no effect on the signal at that frequency, and the output signal will be significantly reduced.

In summary, poles and zeros of transfer functions are important because they represent the frequencies at which the filter has a significant effect on the input signal. I hope this helps in your understanding of poles and zeros and their significance in filter design. And yes, studying Laplace transforms can definitely help in gaining a better understanding of transfer functions. Keep up the good work in your studies!