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Laplace transfer function, poles and quality factor
Laplace transfer function, poles and quality factor
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Old 1st July 2021, 04:21 PM   #1
JustAFrog is offline JustAFrog
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Join Date: Aug 2020
Default Laplace transfer function, poles and quality factor

Hello,
I am working on my theoretical skill, and I struggle a bit understanding the geometric interpretation of s-plane regarding eigenfrequency (natural frequency ?) and quality factor.


On this document (cf screenshot), I read that poles at constant frequency move on a circle when modifying the quality factor?
Is it because of this "natural frequency" term ?

It goes strongly against my intuition that constant frequency is equivalent to constant imaginary part of the pole.


http://jahonen.kapsi.fi/Audio/Papers/Equalization.pdf

I am understanding wrong ? is the document a bit inaccurate ?

Would you have some recommendation about resources I could use to understand those thing ? (signal theory, linear stuff, DSP, and so on). I am not too scared of maths, but I really enjoy pedagogical stuff that gives intuitions.

Thank you very much :-)
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Old 1st July 2021, 05:54 PM   #2
cbdb is offline cbdb  Canada
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I had to look up "natural frequency" to find its the same as resonant frequency. Thats the freq. an object vibrates when you hit it, or the freq where a circuit has max amplitude. It doesn't normally change so in that paper for any system the natural freq. Wn stays constant. The freq. of the signal w, is not constant. Its contained in the S as jw+rho.

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Last edited by cbdb; 1st July 2021 at 06:03 PM.
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Old 1st July 2021, 06:48 PM   #3
rayma is online now rayma  United States
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That's a good one. I don't think they even teach this stuff anymore, sad.
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Old 1st July 2021, 08:01 PM   #4
Mark Johnson is offline Mark Johnson  United States
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Laplace transfer function, poles and quality factor
For me, Norman Nise's textbook is the gold standard: amazon link

It covers stability (and instability) very thoroughly.

BTW the "natural frequency" omega sub 0, is seldom observed in real world physical systems, because real world systems contain nonzero amounts of damping. Instead, what is observed in real world systems is the "damped natural frequency" omega sub d

omega_d = omega_0 * sqrt(1 - (zeta squared))

where zeta is the damping factor. If for example an LC resonator contains any resistance R at all, then zeta>0 and omega_d < omega_0.
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Old 4th July 2021, 08:32 PM   #5
JustAFrog is offline JustAFrog
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Thank you very much all.
I think I understood. So, on circle centred on 0, we have "constant damped natural frequency", and on line parallel to the imaginary axis, constant "undamped" natural frequency ?
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