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.

Tosei,

to overcome your confusion, go back to the definition of – let´s 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 can´t 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

Hello Tosei,

Quote: However now I´m 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: I´m 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, I´ll 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    

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