Yes, it's all about "Arrow convention." Just do it, we must be careful.

I would answer -1 is the critical point, since it has a clear physical meaning: this open loop gain (at some specific range of freqs.) drives the closed loop system to a positive feedback, which means unstability. Of course, it is also important to previously assume/know that a "negative feedback" is needed to track the reference, reject disturbance and so on.

Anyway, this is just my point of view, maybe biased :)

Hi Francisco, thank you for replying.

However, including the sign inversion in the definition of "loop gain" would give a critical point of "+1" for the loop gain. This is identical to the oscillation condition!

Do you think such an approach (definition) has less "physical meaning"?

Another disadvantage of the "open loop transfer function": Without a corresponding drawing (feedback diagram) we cannot see if a given function belongs to a feedback system with positive or negative feedback.

Further comments are appreciated.

I am joining the discussion based on my understanding of the versions given by the learned professors earlier.

1. Is not the OLTF mostly a quantity used for analysis in control engineering unlike in electronics, and hence has to focus around '-1' , the critical point even for assessing relative stability and margins, wherein gain is just an adversary compared to stability, thus clearly ruling out positive feedback?

2. In electronics I see the construct of the TF which can have any kind of feedback since the concept and hence the definition is device specific or related to the design of the circuit.

For me control engineering has meant mathematical model based and hence mostly a 'before the fact' concept, where as other versions like electronics or loop gains for that matter are mostly sub-circuit oriented and in some cases may also be 'after the fact'.

Not being an expert, await your valued inputs.

Hi Lutz,

Just some opinions:

As both approaches give equivalent result (i.e., if unstable with method 1 ==> unstable with method 2), both ones are based on the same physical behavior. So maybe, as you point, the physical meaning is not a differential aspect.

Howerver, I would say that to me the relation between positive feedback ("because (-1)*(-1)=+1") and instability (amplitude growing oscillations) is more intuitve/straightforward to me.

Kind Regards,

F

Dear Lutz,

I think we should always come back to the basis of the theory of Nyquist (a result of the Caushy's principle). The critical point in Nyquist corresponds in fact to the situation where the feedback becomes positive. So, of course, if the open loop TF  has already -1, we use positive feedback, which when multiplied by -1 it becomes negative feedback.For some situations in structural control, we use positive feedback. A good example for that is "Positive Position Feedback" control for vibration damping of structures.

Either -1 or +1, it remains a "convention", and the convention states that "-1" is the critical point,

I think also that the best way to visualize the stability is to use the Nichols diagram.

Kind regards,

Bilal

The theory of Nyquist criterion is derived from the principle of argument .  This  states that  if N   is the number of encirclements made by  the 1+GH locus as the 's' takes different values along the closed contour which includes the whole of the right half plane in which P poles of the open loop TF and Z zeros of F(s)= !+GH lie is  N = Z - P. For the closed loop system (negative feed back) to be stable, there should not be any zeros of 1+GH  on the RHP,i.e. Z =0, or  N =  - P. If the OLTF has no poles inside RHP, then  N =  P = 0 , i.e. no encirclement about  the origin in 1+GH plane or about (-1, j0) point in the GH plane. If OLTF  has P poles ( open loop unstable) , then the number of encirclements should be numerically equal to P, but in a direction opposite to that of the chosen s-domain locus  Critical point is thus the origin (0, j0) in 1+GH plane or  ( -1, j0) in the GH plane. The whole development is based on negative feed back. In general, systems with positive feed back  are  known to be unstable and hence Nyquist theory (as far as I know) does not bother about the stability of such systems. So, the critical point for negative feed back systems is ( 0, 0) if you consider plot of 1 +GH  or  (-1, j0) if you consider the plot of GH.

Thank you for the last three replies and comments.

In the context of the question to be discussed perhaps the following is interesting:

In January 1934 the well-known Harold S. Black has published an article „Stabilized Feedback Amplifiers“ in „The Bell System Technical Journal“ .

On page 5 of this contribution we can read that the gain reduction factor due to feedback is

Gcf=20log|1/(1-µß)|

(µ:gain; ß:feedback factor; µß: loop gain)

From this, we can derive that H. Black considers µß=+1 as „critical point“ in the s-plane.

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@Bilal Mokrani:  If I understand your answer correctly, you do NOT automatically assume negative feedback for all systems (and I agree!). And exactly this is the reason I prefer the "loop gain approach" and NOT the "open loop transfer function" (which automatically assumes negative feedback).

Of course, there will be no problem when a diagram of the system is available showing the sign at the summing junction.  But this is not always the case.

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Further example (to explain the background of my question):

Knowing the poles and zeros of a loop gain function it is common practice to roughly construct the asymptotic magnitude response as a BODE diagram and to derive the corresponding phase response from the magnitude slopes. After this, it is relatively easy to find the phase margin, which is the distance to the -180deg line  at the cross-over frequency.

Now we want to verify the phase margin by simulating the loop gain with the help of a simulation program. (There are many articles and contributions describing different methods how to break the loop for this purpose). 

The results are as follows: For low frequencies, the phase starts at -180 deg (confirming the principle of negative feedback) and the phase margin now is the difference to the -360deg line. 

According to my experience, very often this causes some confusion. Of course, it is possible to explain the difference between both criteria (180deg vs. 360deg). Nevertheless, some people (newcomers) have problems with it.   

I briefly wrote it :-)) see below

Prof. M.K. Padmanabhan:

Thank you for summarizing the Nyquist criterion.

As you said - the plot (1+GH) assumes negative feedback. Hence, it is a plot based on the "Open Loop Transfer Function". However, I think Nyquist`s criterion can also be applied to the function (1-GH) resp. the point (+1,j0). 

Of course, both approaches are possible and equivalent. However, it is important to know that the first one is based on negative feedback only and that - as a consequence - the results from a circuit`s loop gain simulation will assume (+1, j0) as the critical point.  

Josef - thank you. Yes, you have clearly described both possible conventions in your paper. May I ask you: Which convention do you prefer and WHY?

Hi Lutz

I prefer the first convention (Fig.1) because I usually work with an OPA - and it is used (mostly) negative feedback.

Josef

Gentlemen, to complete the picture, here is how H. Nyquist explains his own criterion (see: Bell system Technical Jounal):

Excerpt

Regeneration Theory

by H. Nyquist

(Bell System Technical Journal, Vol. 11, Jan. 1932, Page 136)

"Rule: Plot plus and minus the imaginary part of AJ(iw) against the real part for all frequenciews from 0 to infinity. If the point (1+i0) lies completely outside this curve the system is stable; if not it is unstable."

From the whole text, it becomes clear that the product AJ(iw) is nothing else than the quantity we call today "loop gain" - and the critical point is at (+1+j0).

The critical point is (-1+j0).

http://lpsa.swarthmore.edu/Nyquist/NyquistAll.html

The critical point is (-1+j0).

Yes - that is the opinion of somebody anywhere in the internet .

However, H. Nyquist says (+1+j0).

 I think that your question is a bit tricky, and needs some concentrations and coming back to the origin of the demonstrations. My feeling is that there is a trick lost between the steps.... I will do my best to respond properly.

Anyway, i like the question, it feeds the debate about the basics !

Kind regards,

Bilal

@Bilal Mokrani, thank you and yes - it is "tricky" or (better) not easy to answer. 

I think, it cannot be answered with "yes" or "no" without knowing what we are speaking of: (a) "real loop gain" or (b) open-loop transfer function". But both approaches do exist in literature - and the beginners sometimes get confused.  For example, the question comes up when the feedback factor already contains a sign inversion so that we have to ADD two signals at the summing junction.

For my feeling, the "loop gain" followers have no problem at all because it does not matter where and how many sign inversions exist in the loop. But I am not sure how the followers of (b) will proceed ("+1" or "-1" ?). 

I think that response of Mohandas K Padmanabhan contains the entire result.

The basic Nyquist Plot based on the argument principle gives N=Z-P for the RHP poles and zeros of the CLOSED loop system  transfer function (TF) by counting the number of encirclements of the ORIGIN (0,0). In such a case, either if we write the closed loop denominator as 1+GH or as 1-GH (assuming that there is no sign error) the result is correct.

Only for convenience we prefer to only plot GH, i.e. the Open-Loop TF, and yet, we must keep in mind that, even if we plot the Open Loop, we think of the Closed Loop. Here, if the CL was 1+GH, then the way 1+GH looks relative to the origin, GH will now look relative to  -1. If people write the CL as 1-GH, then the way 1-GH looks relative to the origin, GH will now look relative to +1.

Nothing to do: different people use diffrent conventions and languages.

When you talk about minimum-phase, make sure that you don't talk with a physicist or, at least, you know that for physicists, minimum-phase means ALL zeros AND poles in LHP.

Not to mention that while could be nice to get a gift in England, may not be as nice to get a gift (i.e., poison) in Germany. :-)

@Itzhak Barkana

Yes - I fully agree with you. There will be no problem at all - as long as one knows what he is doing (as I have mentioned at the beginning).

But I am afraid (and this results from some experients with students) that this pre-condition is not always fulfilled.

And for this reason I always prefer to teach a method which is "as safe as possible". And I think this is the loop gain approach (including all signs and sign inversions within the loop).   

@Lutz von Wangenheim

Of course! We teach Control (or as we may also call, Systems).

We must just warn the students that, in other fields, although they may hear same terms, they must make sure what exactly those terms mean there, as they can have different meanings and/or uses. 

Yes Mr. Itzhak Barkana , we must simply to tell them:

check initial assumptions for deriving the transmission of the feedback system.