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Comments on Complex frequency of a pole

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Complex frequency of a pole

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If we have the transfer function of a LC high pass filter:

$H(s) = \frac{sL}{sL+\frac{1}{sC}}$

If we want to find the pole of that filter in the end we get:

$s = \frac{j}{\sqrt{LC}}$

and for a sinuisodal input signal $s = j\omega$ the pole exists at the resonant frequency $\omega_{r}$

However if we dont apply a sinuisodal signal at the input s may become a complex number->the frequency of the pole may be complex.But what physical meaning does it have?

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Your question doesn't make a lot of sense.

for a sinusoidal input signal s = jω the pole exists at the resonant frequency ωr.

Actually the pole exists regardless of what the input signal is. The transfer function is a description of what happens with any* input signal.

However if we don't apply a sinusoidal signal at the input s may become a complex number->the frequency of the pole may be complex.

This is the same misconception. The pole is part of the system. It is always there. It has nothing to do with what input might be thrown at that system. The pole doesn't change as a function of the input. The only thing that changes as a function of the input signal is the output signal.


* Any real world signal. Technically, any analytic signal, which means that it is infinitely differentiable. That in turn means the signal or any of its derivatives can't jump instantly. Real world signal always have some finite bandwidth, so can't jump instantly. Such real world signals can then be decomposed into a set of sines. Since we're talking about linear systems, contributions from multiple superimposed input signals can be analyzed separately and their results added to determine the output signal. Therefore, an input signal can be decomposed into a sum of sines, the result of each sine computed separately, then those results added to find the overall output signal. (Credit to user LvW for pointing out "any" should be qualified).

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Quote: "The transfer function is a description of what happens with any input signal." What is the ... (5 comments)
General Comments (6 comments)
General Comments
MissMulan‭ wrote over 2 years ago

For different input signals the frequency of the pole will have different values ,the pole as a pole exists for some specific values of L,C s but the frequency of the pole changes.

a concerned citizen‭ wrote over 2 years ago · edited over 2 years ago

MissMulan‭ "For different input signals the frequency of the pole will have different values ,the pole as a pole exists for some specific values of L,C s but the frequency of the pole changes." -- This is false. For a transfer function of $1/(s+1)$ the pole is fixed, at all times, at $-1$. Otherwise you're not talking about an LTI. Think of it this way: you , yourself, gave an example transfer function, and it's made of the values from the consituent elements, L and C. Unless those values will change, their numbers are fixed in stone, therefore the poles/zeroes do not change. Also, the pole is $\pm j/\sqrt{LC}$.

Olin Lathrop‭ wrote over 2 years ago

@a concerned citizen: You should make this an answer. It shouldn't be buried down here in comments. When you do, I'll delete this comment chain.

MissMulan‭ wrote over 2 years ago

@a concerned citizen if s isnt any longer equal to jω maybe with the new input signal becomes 2jω the frequency at which we have a pole changes.

a concerned citizen‭ wrote over 2 years ago

MissMulan‭ I don't understand. You have several people telling you that your view is not correct and they all bring arguments, yet you insist on your view. Now, I'm just a random nickname, but Olin is well known. If you think he isn't, take some time to read some of his answers here, or on ee.se. The way you're answering suggests you are not asking because you are looking for a question, rather because you are looking for a confirmation of the idea that you already have in your mind. And if that confirmation doesn't come, you insist, trying to make people change their views only to support your ready made conclusion. I'm not saying this based on this topic, only. If in this moment you would have to be completely honest with youself, would you say this is a thinking that will serve you well in life? I'm not talking about engineering, only. An opinionated view is likely to only make you fight with everyone that doesn't share your views and, so far, not all of them are correct (if any).

MissMulan‭ wrote over 2 years ago

@a concerned citizen I am just trying to explain to you what is inside my head.