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stability for ckts having two feedback loops (Read 25473 times)
buddypoor
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Re: stability for ckts having two feedback loops
Reply #15 - Jan 16th, 2008, 7:16am
 
Eugene and Tosei:

1.) I doubt if any method to reduce the number of loops (signal flow graph or block diagram reduction) can solve the problem.
The reason is as follows: If a system has three different loops you can define three different loop gains by introducing three different cut points. If you instead applies some rules to reduce the system having only one single loop there are exactly three different ways to do this - depending on the rules and their sequence you are using. As a result, you get again the same different loop gains exhibiting three different margins.  
(Of course, the overall transfer function does not depend on the kind of rules resp. their sequence).

Finally, starting from the overall system transfer function H(s), there are - in the above example - exact three different ways to produce by division three different denumerators of the form "1+Fo", with Fo being the negative loop gain.  

2.) Perhaps it makes sense to ask: Why are we interested in a phase or gain margin ?
My answer is: Because of design uncertainties I would like to get a rough feeling about some additional parasitic phase shift resp. additional gain which is allowed anywhere in the system without coming to close to instability.  
Of course, for a single loop system I am able to ask for specific figures of the margin (having in mind the time response).
However, if there are - for example - three loops with three different margins, does it makes sense to compute a "system margin" (like a "mean value" or something else) ?  I think the only practical way is to find the most critical margin and to define the corresponding loop as dominant. In many cases this will be an "outer" loop to be identified by simple inspection. In this case, all inner local loops can remain closed and it is not important if they are accessible or not. Having identified this "critical" loop one can do something to enhance the margin if necessary.

Lutz      







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Eugene
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Re: stability for ckts having two feedback loops
Reply #16 - Jan 16th, 2008, 8:34am
 
All,

First let me say that I am thoroughly enjoying this discussion. Thank you everyone!!

Frank is quite right about the problem when signal flows in two directions through part of a loop. I encountered that last year and I regret that I did not have time to really interpret the apparently inconsistent results I observed.

When working with multiple loops I think it sometimes comes down to which loop you can easily access. It may be moot to worry about loops buried deep in some device for example. On the other hand, if there is a single, perhaps outermost loop that you can easily affect, then perhaps it makes sense to worry only about that one loop.

I have seen a couple of analog optimization tools now that claim to compute closed loop poles and zeros of large transistor circuits. I think they are based on symbolic analysis instead of pole zero fits to frequency responses. Perhaps these new tools open the door to direct closed loop pole zero analysis of circuits so that we no longer have to use phase/gain margin. We could simply look for the pole closest to the imaginary axis.
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Re: stability for ckts having two feedback loops
Reply #17 - Jan 16th, 2008, 5:56pm
 
Eugene,

I agree with what you pointed out concerning superposition. It is true is only valid for a special cases of multi loop systems, but for the general case does not apply.

Lutz,

Concerning your item #1, I guess I'm still slightly confused. From what you are saying three different margins are possible depending on the sequence you use in your analysis. However - from a physical standpoint - the actual system (in its closed loop form) should have an actual margin which should be unique. At least that makes sense to me. So I do not find the physical meaning of having different possible margins (Obviously that margin cannot be found when the loop is closed, but the marginality for the system, either small or large, will exist). Am I interpreting something wrong? (possibly)

As for your 2nd point, I agree that -  for most practical cases - we must look for the dominant pole and perform the analysis disregarding the inner local loops by keeping them closed as you suggested. Actually that situation is a very common  in circuits (which supports your criterion), where a simple opamp in a negative feedback configuration has a clear dominant loop (the "external feedback loop") and several inner local loops (GD capacitances of MOS devices for instance) which are negligible when it comes to the overall stability.
However, this scenario is the one I pointed out at the very beginning of this discussion (see my first reply, case 2).
So I guess the question is still open (at least for me, and from what I posed in the paragraph above) for those cases when there is no clear dominant loop...

Tosei
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Re: stability for ckts having two feedback loops
Reply #18 - Jan 17th, 2008, 12:54am
 
Tosei,

to overcome your confusion, go back to the definition of – lets say –  the phase margin. This is simply the amount of phase shift to be introduced into a loop (a physical existing loop !) in order to make the closed loop system unstable. If a system exhibits two loops this definition, of course, applies also to the second loop which normally has a different loop gain and, hence, another phase response than loop No. 1. Thus, according to the above definition, two different phase margins exist.

To approach the problem from the other side – when you expect only one single system phase margin “phi” the question arises: Where (that means: into which physical existing loop) do you introduce this amount of “phi” to make the system oscillate ?? This question cant be answered without knowing the loop this margins originates from. And, more than that, this can be verified by experiment in order to demonstrate the "physical meaning" of this statement.  

By the way, this exactly is the reason that every textbook on control theory contains the warning: You must not derive the loop gain expression from the transfer function by evaluating the denumerator “1+Fo” by allocating “Fo” to the negative loop gain. In a multi-loop system this procedure is ambiguous because several ways exist to produce this denumerator form.    

Finally, concerning the "dominant loop" another point of view: As commonly agreed by all contributors to this topic in a multi-loop system that loop will dominate (i.e. determine mostly the system behaviour) which is the outermost (principle of overall feedback). However, sometimes such a loop cannot be identified. To find an answer I have performed a lot of simulation runs for several systems and came to the (still preliminary) conclusion that in such cases that loop dominates which is the slowest one. This seems to be logical, does it not ? However, the problem is to determine the time constants of each loop.

Hopefully I could express myself in a clear way as my knowledge of the english language is rather limited.

Lutz
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Re: stability for ckts having two feedback loops
Reply #19 - Jan 18th, 2008, 7:12pm
 
Hi Lutz,

I thought carefully about your last post, and I think I'm with you now concerning the idea of a single margin does not make much sense if we associate one margin figure per loop and therefore we know where we can introduce the necessary phase shift in order to make the system oscillate.
However now Im inclined to think this will be valid only when the margin of each loop does NOT affect the margin of any other loop in the system and the margin of different loops are comparable.
Im not sure if this is a valid physical scenario (think it its though), but since Mason's gain rule involves cross products of loop gains, then I think margins of each loop could also affect margins of other loops. If this was valid, then would not make sense to think of a single margin for the system? (Also, for the dominant loop case you describe I think it also make sense to treat that loops margin as the only margin in the system. This would be another case where you could consider single margin)

Sorry Im so hard to convince  :)
Tosei
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Re: stability for ckts having two feedback loops
Reply #20 - Jan 19th, 2008, 8:02am
 
Hello Tosei,

Quote: However now Im inclined to think this will be valid only when the margin of each loop does NOT affect the margin of any other loop in the system and the margin of different loops are comparable.  

As far as I understood the discussion up to now we only speak about systems exhibiting more than one feedback loop which have at least one common node. Of course, two loops which are completely isolated (decoupled) do not influence each other.

Quote: Im not sure if this is a valid physical scenario (think it its though), but since Mason's gain rule involves cross products of loop gains, then I think margins of each loop could also affect margins of other loops. If this was valid, then would not make sense to think of a single margin for the system?

Of course, such a margin could be defined (more or less artificially ?), however, what would be the practical value of such a definition ?
To make the problem clear, Ill try do describe a simple system which is a rather common in electronics: A composite amplifier consisting of two opamps and four resistors.
EXAMPLE: The output A1 of opamp 1 is connected to the noninverting input of opamp 2 which has a local feedback loop with resistors R1, R2 producing a gain of 10. The output A2 of this amplifier is fed back via a resistive divider (R3:R4=1:99) to the inverting input of opamp 1 thus producing an overall positive gain of 100.
There are two realistice loop gain alternatives:
Case (1) Open the main loop at node A1 (internal loop of opamp 2 still closed)
Case (2) Open both loops at node A2.
(A third alternative opening only loop 2 with loop 1 still closed has no practical meaning).  
Using PSpice and 741 opamp macros here are the simulated phase margins (ac analyses):
(1) case 1 : 44 deg    and  (2) case 2: 41 deg.
Now, if an additional pole at 100 kHz is introduced into the model of opamp 1 the results are as follows:
(1) case 1: 13 deg   and  (2) case 2: margin negative, which means “unstable”.  
However, a TRAN analysis indicates that the system is stable (although the step response shows a remarkable ringing).
From this it is to be concluded that in the above example opening both loops at the same time (case 2) does NOT lead to a loop gain which gives a reliable information about the circuit behaviour. This seems to be in contradiction to some earlier statements within this topic (e.g. reply 2 and 13).

I think the discussion is (a) fascinating and (b) at the same time rather confusing because it seems that no general rule has been found up to now to check resp. to define the stability properties of multi-loop systems in the frequency domain.

Regards
Lutz    



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Re: stability for ckts having two feedback loops
Reply #21 - Feb 14th, 2008, 2:47am
 
buddypoor wrote on Jan 19th, 2008, 8:02am:
Hello Tosei,

EXAMPLE: The output A1 of opamp 1 is connected to the noninverting input of opamp 2 which has a local feedback loop with resistors R1, R2 producing a gain of 10. The output A2 of this amplifier is fed back via a resistive divider (R3:R4=1:99) to the inverting input of opamp 1 thus producing an overall positive gain of 100.
There are two realistice loop gain alternatives:
Case (1) Open the main loop at node A1 (internal loop of opamp 2 still closed)
Case (2) Open both loops at node A2.
(A third alternative opening only loop 2 with loop 1 still closed has no practical meaning).  
Using PSpice and 741 opamp macros here are the simulated phase margins (ac analyses):
(1) case 1 : 44 deg    and  (2) case 2: 41 deg.
Now, if an additional pole at 100 kHz is introduced into the model of opamp 1 the results are as follows:
(1) case 1: 13 deg   and  (2) case 2: margin negative, which means unstable.  
However, a TRAN analysis indicates that the system is stable (although the step response shows a remarkable ringing).
From this it is to be concluded that in the above example opening both loops at the same time (case 2) does NOT lead to a loop gain which gives a reliable information about the circuit behaviour. This seems to be in contradiction to some earlier statements within this topic (e.g. reply 2 and 13).



Hi Lutz,

Please let me insist one more time. I tried to replicate your example and I built a test bench using ideal opamp models which I parametrized such that they had similar open loop characteristics to the 741.
What I found (AC analysis) when adding the extra 100KHz pole to the first opamp is that for case (1) (loop broken at A1 node) the margin was little smaller than what you got but still positive, and for the case (2) (loop broken at node A2) a large negative margin, indicating apparently the loop is not stable, and therefore in agreement with your previous statement.
However these conclusions were derived from the Bode plot of those loops, which I'm almost sure is NOT applicable for the second case. The reason for this being :

1) The phase shift characteristic (at least in my simulation) crosses the 180 point at least twice
2) If you derive the stability from the corresponding Nyquist plot the system indicates is stable (as opposed to the Bode plot)

My TRAN simulation looked similar to what you described: heavy ringing but attenuated, indicating a very small phase margin.

Conclusion:
Previous discussions in this forum concluded the Nyquist stability criterion is the general one and is always applicable while the Bode method is not. Based on this (and on my results), statements on previous replies #2 and 13 are still valid.
The result of this is that we still have valid results from both cases which would allow us to define a single margin (not sure how) for the multi-loop system, which is the primary idea we were discussing.

I might be doing something wrong and I also do not know how you concluded case (2) was unstable, but Im pretty sure Nyquist criterion is the one telling the truth here.

Regards
Lutz


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Re: stability for ckts having two feedback loops
Reply #22 - Feb 16th, 2008, 12:59am
 
Hi, Tosei !

I agree with you that my conclusion in my last reply concerning BODE-stability was not correct; the reason was simply that I have stopped the simulation to early - i.e. at a frequency which was too low to discover the second point with 180 deg phase shift. So you are right, of course, that in cases with a magnitude response without a monotonic decrease not Bode but Nyquist tells the truth about stability.
  However, this does not solve our basic problem. Since the Nyquist plot is based on the loop gain (and not the closed loop transfer function) there still remains the question as mentioned at the beginning of this topic for a multi-loop system: Which loop gives the correct information ?
  I think, in these cases there is no other choice than to use a general – and some more complicated – stability criterion which does not need the loop gain but is based on an evaluation of the characteristic system equation in the time domain (e. g. Routh, Hurwitz).  
Thank you,
Regards
Lutz
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