urian said:We obtain the pole and zero from the transfer function of a circuit.And many tutors said that the gain begin to rise at 6 db/octave above a zero,and fall off at a pole.
I dont know why this occurs.How to relate the pole and zero to the physical circuit? Anyone can show me some meterials about this?
yxo said:Good question indeed! I think, physicaly, its's just how phase of a signal changes. If a frequency of a signal much below poles or zeros, no phase lead or lag occurs(at least, we can ignore). You only have signal of zero degrees phase, 90 degre and so on. However, if you close to a pole, a signal starts to lag(at a pole frequency it is 45deg). If you close to zero-starts to lead.
About gain, is seems not always the truth. In a 2-stage amplifier we have RHP-zero= gm2/Cc, but it does not increase a gain, moreover it gives additional phase shifting.
LvW said:urian said:We obtain the pole and zero from the transfer function of a circuit.And many tutors said that the gain begin to rise at 6 db/octave above a zero,and fall off at a pole.
I dont know why this occurs.How to relate the pole and zero to the physical circuit? Anyone can show me some meterials about this?
All transfer functions for circuits containing real lumped elements have a denumerator of degree n which is a polynom in s (s=complex frequency).
This polynom has n zeros. For example, one simple first order case is H(s)=1/(1+sTx) which is a first order lowpass.
From this, it is clear that the magnitude of the denumerator increases (and the magnitude of the transfer function H(s) decreases) for rising s values - especially if they rise beyond |s|=1/Tx. That means, for |s|>1/Tx the transfer function H(s) falls asymptotically with 6 dB/oct . However, one interesting point is s=-1/Tx. In this case, the denumerator is zero and H(s) approaches infinite. Therefore, this particular frequency s=-1/Tx is called "pole frequency". However, in reality this can never happen, since a negative real frequency is just a mathematical fiction
(remember: s= σ+jω). Nevertheless, it is a nice description of this "corner" in the complex s-plane.
For more complicated denumerator functions the polynom solutions may lead to complex zeros of the denumerator (poles of H(s)).
For the numerator - when it also consists of an s-polynom - we get similar results, but of course with rising instead of falling magnitudes beyond the "zeros" of H(s).
I haven´t thought about physical meaning of poles and zero before. It´s just the spontaneous idea.urian said:Thanks yxo,your idea is novel that I have never seen it before from textbook.But I am not sure whether it is true indeed or not.And what's the consequence of a 90 degree phase shift? I only understand that a 180 degree phase will decrease the magnitude.
I agree with you, in this simple case it makes (perhaps) not much sense to create a term called "pole". However, particularly for complex solutions of the denumerator it makes much sense! Perhaps you have seen already a so called "p,n diagram" which is a graphic presentation of the pole location in the complex s-plane. With the help of this graphic two parameters are defined, which (a) very good describe the characteristic frequency response of the circuit and (b) which can be easily measured.
These parameters are: pole frequency wp (magnitude of the vector from the origin to the pole location) and pole quality factor Qp (defined as 1/2cosα) with α equal to the angle between the neg. real axis and the mentioned vector.
Both parameters are extensively used, for example, to define the different filter responses.
urian said:Yes I have seen lots of pole-zero plot.But I dont know what the exact meaning and usage of the two parameters you mentioned,Wp and Qp.I have never seen them before,can you point me out some detail materials about these parameters?
Furthemore,I still dont understand why we use a negative point instead of an positive point.
urian said:Wow,so the pole describes the relationships between time and frequency domain.These remind me of a book called Signals and Systems written by Oppenheim.I think I must return to it and find more details about the pole.
Thanks,LvW,for your patience and wisdom,I really appreciate it!
.........
these links enlightened me about why poles and zeros cause the shapes they do to both the magnitude and phase reponse, why resonant circuits look like they do, and may give some insight into the meaning of negative frequency.
I wonder which part of the paper (that is to be recommended, no doubt about it!) gave you "insight into the meaning of negative frequencies"? Please, can you clarify this? Thank you. LvW
The insight you can gain from that paper about negative frequency, is that the geometry for tracing the frequency response is the same whether you trace the positive jw axis or the negative jw axis (positive or negative frequency) which means that filters like these will respond the same to positive and negative frequency. not true for a Hilbert filter. I know that is not amazing insight, but it got me thinking.
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Hi NerdAlert,
Thank you for your detailed answer.
However, opposite to you, I don`t think that one can gain from the pn diagram resp. the Nyquist contour some "insight" into the "meaning" of negative frequencies. Here, the introduction of negative values for w=2*Pi*f is nothing else than a mathematical manipulation in the frequency domain (leading to a closed Nyquist contour).
However, the documents referenced by you (exception: Wikipedia) clearly show the meaning of the negative frequency concept which originate from Eulers complex formula for trigonometric functions.
And from this it is clear that negative frequencies do not exist physically. But it is rather convenient to perform calculations based on the assumption that there is something like negative frequencies. As a result, you always have two options to evaluate the spectral distribution of a signal: One-sided (with positive frequencies only) or two-sided (with pos. and neg. frequency components and 50% amplitudes).
---------- Post added at 21:04 ---------- Previous post was at 20:44 ----------
I forgot to comment the Wikipedia contribution: For my opinion, the description of the concept of neg. frequencies is not correct - at least misunderstanding. I`ve got the impression that real/imag. parts are mixed with pos./neg. frequencies.
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