# What is the physical meaning of Pole and Zero?

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#### project.email

##### Member level 4
complex frequency variable

I want to know what is the physical meaning of a zero or a pole in a system?
I know by using transfer function at zero the output of the system tend to zero and
at pole it tend to infinte.
but I dont know exactly what is the quality of zero or pole?
what is the nature of them?
can everybody help me?
thanks.

#### LvW

physical meaning of pole and zero

The explanation is connected with the introduction of the complex frequency variable s=σ+jω. By replacing the "real world"-variable ω in the transfer function of a filter by s you will get the so called system function.
This function can be drawn in the complex frequency plan (see attachment).

For the simple case of second order there are two s values for which the system function goes to infinite. In most cases, these values will constitute a conjugate-complex pair (with negative real parts). This is called a pole pair.
In an overdamped system both poles will be (negative) real.

In the attached 3D-picture the poles are located at sp=-1±j1.

It is interesting to note that the cut at σ=0 gives a curve which is identical to the frequnecy response which you can measure resp. simulate (ac-analysis).

Hope this helps. LvW.

PG1995 and ferdem

Points: 2

### PG1995

Points: 2

#### LvW

pole zero definition

In order to justify the invention of the complex frequency variable s - here are some additional comments:

The pole location can be marked in the complex plane. And now one can define two very important filter parameters:

1.) The value identical to the magnitude of the vector from 0/0 to the pole location is the so called "pole frequency" ωp.

2.) The angle δ between the negative-real axis and this vector is an indication of the so called "pole quality factor" Qp. The exact relation is: Qp=1/(2cosδ)

3.) Both parameters characterize the filter response and are given in relevant textbooks. More than that, both values can be measured with a frequency generator and an oscilloscop (better: network analyzer)

4.) These definitions - together with the 3D-picture as shown - reveal the relationship between pole location and magnitude response of a filter (with amplitude peaking corresponding to the pole location).

Added after 1 hours 41 minutes:

Sorry, I forgot to say someting to the zeros.

No, it is not correct that at system zeros the output "tends to zero" . This is only true for "real zeros".
In general, the slope of the magnitude function will change in the neighbourhood of zeros - nothing else.

For example, the slope of the magnitude function (BODE- diagram) goes from 0 to -20dB/dec (or from -20 to -40 dB/Dec) at the pole frequency .
In the same way the slope increases from -20 dB/Dec to 0 (or from -40 to -20 dB/Dec) caused by a complex zero.

LvW

Added after 2 hours 57 minutes:

The end of the passband (corner frequency, 3-dB-frequency) is NOT identical to the pole frequency. However, both frequencies normally are not very far from each other.
(in practice: 5...10 %) .Exception: For a BUTTERWORTH response both frequencies are identical.

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### PG1995

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#### kiprono

physical significance of pole and zero

The zeros of the transfer function simply represent the places (frequencies) where the system has a gain of zero. (Thus, these frequencies will be totally cut out of the signal.)

The poles of the transfer function represent places (frequencies) where the system has an infinite gain. (Thus, these frequencies will send the system into resonance with zero damping.)

For any real system, an infinite gain is impossible, and the actual poles of the system will be complex rather than real. Thus, the poles will be shifted away from the frequency axis, and the gain at any particular frequency will be finite rather than infinite. The closer the pole moves to the frequency axis, the greater the resonance, and the greater the gain of the system at those frequencies.

iukhan

V
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### iukhan

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#### LvW

pole and zero

Hi Kiprono,

sorry, but you are wrong.

Quote: The zeros of the transfer function simply represent the places (frequencies) where the system has a gain of zero. (Thus, these frequencies will be totally cut out of the signal.)

No Sir, this is true only for the special case with zeros on the Im axis. However, in general, this is NOT correct . Example: Allpass has (complex) zeros, but its amplitude is constant.

Quote: The poles of the transfer function represent places (frequencies) where the system has an infinite gain. (Thus, these frequencies will send the system into resonance with zero damping.)

Also, this is wrong. Not the system but the artificial system function with the complex variable s goes (theoretically !!!) to infinite values. See the figure in the above attachement.

Thus, the poles will be shifted away from the frequency axis, and the gain at any particular frequency will be finite rather than infinite.

No, the poles are not shifted away from the axis, because they never have been there. Look at the 3D-picture ! You should improve your knowledge of the meaning of the complex frequency variable.

Regards
LvW

#### zlatkoMM

##### Full Member level 5
pole frequency definition

Do you realy great phisics expert?
Real nature of this question is simplier then you can imagine!
OK do you ever heard about carrier of electro force?
It is something in atomic structure of any material.

#### project.email

##### Member level 4
define pole & zero

zlatkoMM said:
Do you realy great phisics expert?
Real nature of this question is simplier then you can imagine!
OK do you ever heard about carrier of electro force?
It is something in atomic structure of any material.
I'm familar to carrier frequency & I know what's it.
but what's the realation between carrier electro force and the nature of zero and pole?
I dont understand what dose happen at a pole or zero exactly.
:roll:

#### kiprono

physical meaning of poles and zeros

Hi LvW,
My answer is for someone who is new to control theory. Your answer is correct but starters cannot easily see the physical meaning of poles and zeros.

#### FvM

##### Super Moderator
Staff member
poles of a control system physical meaning

starters cannot easily see the physical meaning of poles and zeros
But they don't get insights from wrong simplifications. You can start e. g. with real zeros as a special case, but you certainly have to mention that there are complex zeros as well. You can't bypass complex numbers when introducing poles and zeros.

#### LvW

definition of pole and zero

I totally agree with FvM.

#### LvW

what are poles and zeros physically

Gd888888888 Discussion!!!!!!!!!

What does this mean ? Is this an SMS-shortage ? (Sorry, perhaps I am to old)

#### vivek_raj_verma

##### Junior Member level 2
physical significance of zeros and poles

this has been always a matter of confusion to me whenever i sit and think about poles and zeros. And still after reading all above discussions......i think somewhere, something is missing.....

Is it just mathematics...

#### Petre Petrov

##### Member level 3
physical implication of pole and zero

“ .. zeros… and ..poles...”
In my opinion: Inappropriate and misleading terms.
Obviously should be replaced with terms which show accurately the physical effects.

#### LvW

complex frequency variable s

Petre Petrov said:
“ .. zeros… and ..poles...”
In my opinion: Inappropriate and misleading terms.
Obviously should be replaced with terms which show accurately the physical effects.

I don´t agree with you, Petre Petrov !

Question: To be consistent, you should fight also against the concept of complex frequencies, do you really ?

Indeed, poles and zeros reflect physical effects - and both terms are the best method to formulate the transfer function for different filter approximations.
This could be discussed in detail, but I think this is not the right place.
Regards
LvW

#### Petre Petrov

##### Member level 3
transfer function zero physical situation

Hello LvW and the others!

A lot of people disagree with me all the time. So you are in good company. Many more will join you.

But as you can see from the writings above some people have real difficulties to understand and remember terms which are not easy to decode. The simple reason is that the terms do not reflect the nature of the events.

In my opinion people have rights to learn appropriate terms. Also they do not deserve to learn meaningless or misleading combination of inaccurate terms. Please note that I am not for 100.000% purity. But the the terms could be more suitable.

Also please note that I do not "fight" against anythings. There are “fighters” for that.

I do not have intentions and possibilities to correct everything which is wrong in my opinion. Let the glory and the profit go to the people who will do that. In all cases someone will correct the the mistakes.

LvW, you have a good intuition. I am against the complex numbers.
And also I am against a a lot of complex things, but I do not care very much of that. So I will not “fight” them.

LvW, sorry if you do not like the text above. My English is not good enough and there are more important things to do.

Also, sorry for the people who will study EE, Analog and Digital electronics, DSP and related subjects.

LvW, thank you for paying attention and good luck with "complex" subjects.
I have enough troubles with the simplest matters.

BR

Hello again!

Please find more topics for disagreement if you like that.

" Reevaluation and replacement of terms in the sampling theory"

http://www.pueron.org/pueron/nauchnakritika/Th_Re.pdf

There are obvious mistakes in the paper but I hope you will forgive me.

BR

#### LvW

poles and zeros definition

Hello Petre Petrow !
Thank you for your reply. More or less I am with you as far as the "philosophical" parts of your contribution are concened. (And as english is not my mother tongue I certainly have used an unappropriate word "fight" - it was my intention to assume that you do not "support" complex frequencies).
Anyway, back to electronics and some definitions.

1.) A first order lowpass has a corner frequency which is defined at 50% of the maximum phase excursion (that means -90/2=-45 deg). This sounds reasonable for my opinion.
2.) A second order lowpass has a max. frequency shift of -180 deg. For my opinion, it is quite logical to define a "characeristic" frequency at the point where the phase shift again is -180/2=90 deg. This frequency "shows physical effects"(your words) and can be measured easily. As this frequency is a good indication of the border between the pass region and the attenuation region: Why not give it a name ?
3.)But it would be not a good iedea to call it "cut-off" frequency because of the several characteristics the transfer function can have in this region (Butterworth, Chebyshev,...). So, why not call it "pole frequency" ?
That´s all.
(If some peaple like, we can justify the name using the complex frequency plane, but it is not necessary at this point of discussion).
4.) And, of course, we can find by calculation a relationship between this pole frequ. and the -3 dB-cut-off.
And as a next step, we can see some advantages of this definition using the complex frequency plane. But I repeat: This step is not necessary, we can stick at real frequencies !
__________________
5.) Question: What are your arguments against this definition of a characteristic frequency for a second order lowpass which we have decided to call "pole frequency" ?
With regards to you
LvW

#### Petre Petrov

##### Member level 3
infinite gain pole

Hello LvW

I am happy that I have some support from your side to the as you say "philosophical" parts of my contributions.

Briefly:
1. Any particular point of any characteristic has some meaning and importance.
2. Any particular point of any characteristic could have a proper suitable name closely related to that point (name of the discoverer, physical effect,etc)
3. As you know there are a lot of characteristics of the filters and amplifiers:
* amplitude to frequency
* phase to frequency
* all sorts of distortions to frequency,
* all sorts of parameters to all sorts of parameters, etc
4. Particular points at 0, 45, 90, 135, 180 etc degrees are interesting in most of the cases and could have name and symbol related with the corresponding number of degrees of clearly defined and measurable physical effect.
The abbreviations:
F0D, F45F, F90D, F135D, F180 D and similar seem clear and easy to remember for me.(Indexed notation could be used if necessary).
5. My terminology about filters is a bit different. I am sorry but at the moment I cannot discuss the topic due to other urgent tasks.
6. If you mean that “cut-off” is inappropriate term I agree with you. In all real cases noting is “cut – off” or at least I don't know similar case. There are things that are attenuated/modified at some level.
7. “pole frequency “ - Sorry I cannot associated that term with the physical effect, discoverer, particular point, some basic application, mathematical effect, etc.
8. “complex numbers” and “complex numbers” are too complicated field. I am not able to discuss it now.
9. I will support every development and proof with real value which is testable.
10. From linguistic pint of view “ pole” is associated with Poland, politics, Polymerase , Earth, etc not with “transfer functions”.
11. Human language is reach enough for words Also new appropriate words and term could be created.
12. If you are still interested in my opinion on several basic terms you could visit
A note about the definitions of "sine/cosine wave", "sinusoidal/co-sinusoidal signal” and the "simplest band limited signals".
There is a short paper about several basic terms and definitions in EE. I am not sure that you will like them, but I think that they deserve some attention

Thank you again.
Sorry but I have some urgent tasks to do.
Good luck!
BR

#### LvW

physical meaning of complex plane

Hello B.R.

Nevertheless, some short comments from my side. Perhaps anywhen within the next few days you will have some time to respond.

5. My terminology about filters is a bit different. I am sorry but at the moment I cannot discuss the topic due to other urgent tasks.

It would be very interesting for me to learn something about your view in this respect. I hope you will have time to explain your philosophy behind this subject.

6. If you mean that “cut-off” is inappropriate term I agree with you. In all real cases noting is “cut – off” or at least I don't know similar case. There are things that are attenuated/modified at some level.

I am afraid, that in many many cases our language is not "reach enough" (your words) to express technical terms exactly by 100 %. Think of such basic terms like resistor, capacitor, sinusoidal freqiencies,...... Each word/term is not exact but only an approximation to the wanted behaviour. But I think, an engineer knows that a resistor is not purely resistive and that no technical device can produce a real sinewave, for example. And is a linear circuit really linear ?

7. “pole frequency “ - Sorry I cannot associated that term with the physical effect, discoverer, particular point, some basic application, mathematical effect, etc.

Is the fact, that the phase shift at a certain frequency reaches -90 deg. not a kind of a physical effect ??? You easily can measure the corresponding frequency. The decision how this frequency should be named is another problem.

10. From linguistic pint of view “ pole” is associated with Poland, politics, Polymerase , Earth, etc not with “transfer functions”.

I am not a researcher for linguistic subjects, but I accept that anybody in the past has proposed to use the term "pole" for a mathematical phenomenon which is connected with the property of "infinite". Human language is always alive and creates new words or extends the meaning of "old" words to new fields. That´s normal for my opinion.
____________________

LvW

#### kiprono

Surely, we have guru's of every topic and others may not see the point.

#### erikl

##### Super Moderator
Staff member
Re: pole frequency

Nice discussion!

Hello LvW, pls. s. below!

Petre Petrov said:
7. “pole frequency “ - Sorry I cannot associated that term with the physical effect, discoverer, particular point, some basic application, mathematical effect, etc.
LvW said:
... Is the fact, that the phase shift at a certain frequency reaches -90 deg. not a kind of a physical effect ??? You easily can measure the corresponding frequency. The decision how this frequency should be named is another problem.
...
I am not a researcher for linguistic subjects, but I accept that anybody in the past has proposed to use the term "pole" for a mathematical phenomenon which is connected with the property of "infinite".

LvW
May be this is not a very scientific "memo trigger", but I am used to associate such a pole (resp. pole frequency) to one of the earth's poles, because their latitude angle is + or -90° . (Unfortunately, higher frequencies would tend to leave the earth's surface, in such imagination ;-) ).
erikl

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