pole and zero of a circuit

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?

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.

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?

Click to expand...

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).

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.

Click to expand...

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.

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?

Click to expand...

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).

Click to expand...

Hi,LvW,you here
From your explanation,now I think the most important point is 1/Tx,beyond which will cause the magnitude to decrease at 6 dB/oct,and at the point itself,the magnitude has rolled off by 3 dB.Then the pole frequency,the negative of 1/Tx, has nothing to do with us.In my option since we can use 1/Tx point directly to tell the -3 dB point,why we create a inexistent point called "pole frequency" to use for description?
Furthemore,I still dont understand why the magnitude decrease at 6 dB/oct from the pole frequency,why not 8 dB/oct or some point else? Is there a strict mathematical relationship to define the 6 dB/oct?

I would like to add that physically, negative frequency means only a phase shifting again,

Added after 7 minutes:

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.

Click to expand...

I haven´t thought about physical meaning of poles and zero before. It´s just the spontaneous idea.
90 degree is my mistake here. I mean that in linear circuits we have only inverting and non-inverting stages,i.e. zero and 180 degree shifting, if there are no poles and zeros.

From your explanation,now I think the most important point is 1/Tx,beyond which will cause the magnitude to decrease at 6 dB/oct,and at the point itself,the magnitude has rolled off by 3 dB.Then the pole frequency,the negative of 1/Tx, has nothing to do with us.In my option since we can use 1/Tx point directly to tell the -3 dB point,why we create a inexistent point called "pole frequency" to use for description?

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.

Furthemore,I still dont understand why the magnitude decrease at 6 dB/oct from the pole frequency,why not 8 dB/oct or some point else? Is there a strict mathematical relationship to define the 6 dB/oct?

The answer is simple: for frequencies far above 1/Tx the expression |(1+jwTx)| will double (i.e. increase by 6 dB) when the frequency doubles (octave).
With other words: 6 dB/oct or 20 dB/decade (factor 10).

Concerning the contribution of yxo: I am not sure if I've understood everything, but perhaps he refers to a great work of BODE, who was the originator of the Bode formula which describes the close relationship between magnitude and phase of complex transfer functions. But in any case, I dont't think that this approach is a good way to explain the meaning of poles and zeros.

Thx so much LvW,and yxo
Now I understand why it is 6 dB/oct since the transfer function is 20logfh/f,where fh is the -3dB frequency when f is larger enough than fh.Then with a decade increase in f,the 20logfh/f adds 20 dB.

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.

Click to expand...

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.

quote yxo: I would like to add that physically, negative frequency means only a phase shifting

Sorry yxo, but this is foolish. Try to understand the meaning of a phase shift.

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.

Click to expand...

I'll explain it with the example of a second order filter function.
The transfer function of a second order lowpass is

H(s)=Ao/(1+B1*s+B2*s²) with Ao=Gain for s=0.

Don't you think it would be interesting to see the influence of B1 and B2 on the transfer characteristic after introducing s=jω ?
Well, by calculating the zeros of the denumerator (the poles of H(s)) you can show that both coefficients B1 and B2 are related to the pole location (a conjugate-complex pair of poles in the left half of the s-plane) by the following relations:

B2=1/ωp² and A2=1/(ωp*Qp).

More than that, after calculating the transfer function of a real circuit - expressed by parts values instead of Ao, B1 and B2 - you can perform a coefficient comparison and , thus, find the relation between parts and the pole parameters.

Both parameters ωp and Qp are defined as mentioned in my last posting.
After introducing s=jω it is easy to show that both parameters are closely related to the Bode diagram, because
ωp=frequency with a phase shift of 90 deg
Qp=Ap/Ao with Ap=gain at ωp.

Finally, it is to be noted, that ωp is normally NOT identical with 2Pi*fc (3.dB-cutoff).
But it is rather close to it (exception: for Qp=0.7071 both are equal).

Details about this can be found in each textbook dealing with filters (*)
I hope this answers some of your questions.
PS: I don't know what you mean with "negative points".

(*) or books on system theory and control techniques..

Thank you so much,LvW.From your explanation,I recall that I have seen these parameters in the appendix of Allen's book.But when I am reading it for the first time,I selected to ignore it since it looked so complex.Now I have to return to the book and pick it again.From the book,Allen said that we could find the phase margin and pole frequency through time domain measurement and will make the work easy.
And,the "negative points" I mentioned is just the pole frequency as it locates in left-plane of s domain.I mean I still dont know why we define a point not exists in our real-world to determint the characteristic of a real circuit?

OK, I see. I forgot to mention the relation of the poles to the time domain.
It's quite simple:
The denumerator of H(s) is identical to the characteristic polynom you get by solving the differential equation of the system in the time domain.
For example, the step response for a marginally damped second order system in the time domain shows a "ringing" which consists of a damped sinusoidal wave.
The complex exponent of the e-function is identical to the complex pole you get in the s-domain.
Because in a damped system the real part of this exponenent must be negative - identical to the real part of the pole (in the left hand side of the s-plane).
Hopefully, now you see the advantages of this approach. That's the basis of the system theory: Simple relationships between time and frequency domain - described by the same parameter: the pole !

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!

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!

Click to expand...

Yes, Oppenheim (chapter 11) is a rather good choice.Good luck.
LvW

see Engineering Circuit Analysis 6th ed by Hayt, Kemmerly, and Durbin. section 14.2 "complex frequency," section 15.7 "The complex frequency plane," and chapter 16 "Frequency Response"

and/or

"understanding poles and zeros"
https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf

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.

NerdAlert said:

.........
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.

Click to expand...

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

LvW said:

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

Click to expand...

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.

check out Understanding Digital Signal Processing 2nd ed. by Lyons ch. 8 "quadrature signals" for good detail about where negative frequency comes from (with lots of nice diagrams), and ch. 9 "the discrete Hilbert transform" gives insight about Hilbert filters.

i found this useful as well: Negative frequencies | Steve on Image Processing
and check out comment 3. The image Krish is referring to can be seen here:
File:Complex sinusoid 3D.svg - Wikipedia, the free encyclopedia
this image is referenced in the wikipedia article on negative frequency:Negative frequency - Wikipedia, the free encyclopedia

NerdAlert said:

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.
................

Click to expand...

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.

LvW said:

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.

Click to expand...

The insight I gained is that negative frequencies and positive frequencies react the same to that type of filter, because all that is really different about negative and positive frequencies is the direction they rotate. Nevertheless, the insight and meaning you gained or I gained or anyone else gained from the linked articles is not the debate. These insights and meanigns are just as relativly defined as negative frequency... and negative numbers for that matter.

negative frequency is defined with a point of reference. if positive frequency has increasingly positve phase angle with time, then negative frequency has an increasingly negative phase angle with time. positive and negative numbers go in oppositite directions on the real number line centered around zero, and positive and negative frequencies rotate in opposite directions around the origin of the complex plane.

In the time domain, negative freq does not exist because for a cosine wave, there is no way to discern the difference between which direction the phase is changing. but in the frequency domain I would say it exists with the same physicality that negative numbers exist.

another thing about this entire discussion is the imaginary number: i, and what it means... And I think I mis-spoke about Hilbert filters. I am still looking into them, which is why I am brushing up on all this stuff in the first place.

I agree that negative frequency, and negative numbers provide a extremly useful representation of real time varying numbers of like cos(wt)

The understanding DSP book has a nice explaination of all of this.


PS: I was not recommending the wikipedia article on negative freq, i was supplying an image that the mathworks commenter Krish spoke about, and gave a source where I cam across this image.

Quote NerdAlert: In the time domain, negative freq does not exist because for a cosine wave, there is no way to discern the difference between which direction the phase is changing. but in the frequency domain I would say it exists with the same physicality that negative numbers exist.

The question is if something can "physically exist" in the frequency domain that is artificially defined (Laplace).
But I think it`s more or less a philosophical matter.
LvW

For those who are interested in a good introductory tutorial dealing with negative frequencies (Existence yes/not, meaning, advantages) here are two recommended links:

http://www.dspguru.com/sites/dspguru/files/QuadSignals.pdf
**broken link removed**