Source of poles and zeros in a physical system?

From where does the poles and zeros come into the transfer function of any system? And I am not looking for an answer in terms of mathematics but want to know what is there in any physical system that turns out forming poles and zeros when represented in the transfer function form. As in terms of mathematics, we have the dynamics of the physical systems in the form of differential equations which we transform using Laplace transform for the ease of calculation. Then using the poles and zeros, which are the solutions of the denominator and numerator polynomials of the transfer function, we do the stability analysis and rest all mathematical analysis. But this does not give me an intuition of what is there in the physical system that turns out to be a zero/pole in the system transfer function or what physical phenomenon in a system ultimately ends up forming a pole and zero.

• I am curious too. That would be a nice feature, like walking arround the plant and saying: here we have another pole, down there are two zeroes,... Oct 2, 2018 at 6:53
• There are no physical poles and zeroes; it's just a mathematical convenience. Oct 2, 2018 at 6:58
• You can often see zeroes hanging on power transmission lines, and the cables of cable-supported structures. While they have a pole, they introduce a zero into the overall structure as they're in shunt to it. Oct 2, 2018 at 7:46
• Isn't a pole associated with a physical resonance in the system? Oct 2, 2018 at 12:44
• Yes - that is true..
– LvW
Oct 2, 2018 at 15:54

I'm surprised this doesn't have an adequate answer after all this time. Yes, physical components do have pole and zero behavior. We exploit this behavior build analog PID systems.nRLC circuits are used to implement simple lead and lag compensators, here is a photo of one taken from Stevens and Lewis Aircraft Control and Simulation

Here we can see that the resistance and capacitance are chosen to determine the pole and zero placement of the compensator.

• Might help if you derive the zero, pole, and zero/pole equations and what they mean in context. Otherwise, this answer doesn't help a ton. Sep 3, 2020 at 17:26

At first - the function we call "transfer function" or "system function" is a function of the COMPLEX frequeny variable (s=a+jw) which is just an artifical one. In reality, there are no "complex frequencies". However, it is very convenient to use the complex variable s for describing most of the observations and effects within the frequency domain. Therefore, in most cases there are no real ("physical") poles or zeros.

However, there are some exceptions:

(1) The complex transfer function for some filter circuits (notch, Cauer, Chebyshev invers) has one or more real zeros. That means: The output signal approches zero at the corresponding frequency.

(2) An oscillator producing self-sustained sine wave signals at a frequency fo has a transfer function with a real pole at this frequency fo.

That means: The gain approaches infinity - and a vanishing input signal can produce an output signal. Hence, we do not need any input signal for producing an output. And - because this condition is valid for one single frequency fo only the output will consist of a sine wave fo only. (Remark: This is a - more or less - theoretical consideration, in reality we are not able to place the pole exactly on the imaginary axis, but this is a practical limitation due to tolerances etc.).

• This answer does not point out the sources of zeros. You mentioned that complex transfer functions for filter circuits have zeros but what is there in the filter circuit that led to a zero in the transfer function? Oct 3, 2018 at 15:52
• It is of common knowledge that, for example, a series LC branch can provide a zero resistance at the resonant freqency.
– LvW
Oct 3, 2018 at 16:27
• @SanjayChaturvedi As the others have already hinted plenty, it's impossible to ask a universally generic and vague "where does a zero come from" and expect an exact response. Mar 13, 2020 at 22:04
• @aconcernedcitizen Yes, you are right that this question is a generic one and will always have a fuzzy answer. I was aiming to gain different perspectives that people have to understand poles and zeros which can help me to deepen my understanding. Sep 9, 2020 at 10:36

Energy storage (resonators) often are the poles (2 at a time).

In feedback systems, the cumulative phaseshifts lead to near-360 degree situations, and these often seem as resonators particularly if lightly dampened.

• Yes, they are. But in a similar manner if I want to traceback a zero into the physical system then I am not able to find any element or process which is responsible for the formation of zeros. Oct 3, 2018 at 15:43
• what are the equational constraints that result in forming a "zero"? a high-pass-filter should do that. Oct 7, 2018 at 17:16