Concept of Negative Feedback question

[LvW]

Lovely discussion.

Yes - it is a lovely, interesting and challenging discussion that has created some new questions on my side.
Perhaps you (that means: All readers of this) can help me with the analysis of the feedback loop as shown in the attached drawing.

I came up with this circuit during investigation of a composite amplifier consisting of two different opamps.
As you know it is the purpose of such a block to combine - for example - good offset properties from opamp 1 with a good slewrate from opamp 2.
In this example a closed-loop gain of 40 dB is realized. A closed-loop ac analysis revealed a gain peaking of approx. 10 dB. (correction: 1 dB)
Now - the question is simply: What is the stability margin of the system and how is it determined?
(I suppose, most of you know the real background of this question - however, at the moment I don't want to go into details).
I am awaiting some answers.

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I failed to attach the drawing. Here comes the 2nd attempt.

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For clarification: The numerator of the 2nd opamp's transfer function contains in brackets the expression (1+4E-6*s)

 

[D.A.(Tony)Stewart]

The Zeta function can be determined from any 2nd order closed loop coefficients and possibly 3rd and 4th order coefficients).

I recall seeing the formulae in my 1st PLL designs in 1975 and they have direct correlation on overshoot and loop gain margin for stability. I will see if I can dig up the formulae.

The value of the damping ratio ζ determines the behavior of the system. A damped harmonic oscillator can be:

Overdamped (ζ > 1): The system returns (exponentially decays) to equilibrium without oscillating. Larger values of the damping ratio ζ return to equilibrium more slowly.
Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors.
Underdamped (0 < ζ < 1): The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.
Undamped (ζ = 0): The system oscillates at its natural resonant frequency (ωo).

I am thinking now about the time domain signal the SQUARE wave and the TRIANGLE wave. They both have only odd harmonics and but the phases of the square wave differ by the alternating polarity with harmonic number. i.e. the phase of each harmonic significantly affects the result of the time domain signal but not the spectral density display. So why it may be possible that there are more factors to consider in correlating fc Amplitude peaking factors, overshoot % and Gain margin.

These are two signals which different significantly in the time domain shape but not the scalar spectral density.

Just as Q=0.5 for 3dB bandwidth in a band pass filter is a critically damped system where ζ = 1, the Low Pass Filter may also be correlated for gain margin.

However this may only affect the sensitivity of the margin and not the 2nd order approximation of the "gain margin value" .. just thinking out loud.. and will refine later..

 

[D.A.(Tony)Stewart]

Besides Matlab for simulation calculations, you can sweep the gain and phase plots and measure Relative Stability as below.




It may be that your Step response will have PO value that corresponds to a ζ=0.5 for 1dB peaking. Since the dominant pole rolls off at 20 dB/decade and you appear to have adequate compensation to extend the spacing between 0dB & 180 deg crossovers, you ought to get the desired minimum 15 dB. gain margin. The peaking factor alone is not enough to indicate the margin but for a simple system its not a bad indicator. You need the phase measurement too.

More complex systems use "all pass filters" with phase lead compensation that are flat frequency response.

 

[FvM]

More complex systems use "all pass filters" with phase lead compensation that are flat frequency response.

Are you sure about?

In a minimal phase system, phase lead/lag is "inseparably" linked to magnitude variation, an allpass filter can only generate additional phase lag. Arbitrary phase changes are feasible in systems that involve additional delay (e.g. FIR filters), not applicable for feedback loops.

 

[LvW]

Hi SunnySkyguy, thanks for replying.

Perhaps I should have mentioned that I am aware of the principal relations between time and frequency domain - in particular for 2nd order systems.
And I also know the relation (formula) between damping and overshoot (time domain) as well as peaking (frequency domain) for a 2nd order system.
As far as I remember, I never have seen similar formulas/graphs for higher-order systems (with zeros).
On the other hand, the fact that you didn't simply wrote: "Open the loop and simulate the gain/phase responses" was an indication to me that you knew about the real background of my question.
OK, here are the details and the reason for my inquiry:

1.) The shown circuit has three loops.
2.) Each loop has it's own loop gain - and it's own stability margin.
3.) Therefore: Does it make sense to ask for the stability margin of the whole circuit?
4.) If yes - which loop determines this "system margin"? How can this loop be defined? Is there a general rule?
5.) The example presented is a very basic one. As you know, there are much more complicated feedback arrangements with many loops. Therefore question 4.).

All books I have consulted for treatment of multi-loop systems have concentrated either on the transfer function or on the binary question stable/unstable without mentioning the stability margin.
That is the background of my question.

 

[FvM]

1. The shown circuit has three loops.

I think only two.

If the right loop is stable (should be for positive u), it can be transformed to a gain block and the circuit reduces to a single feedback loop.

 

[D.A.(Tony)Stewart]

I'm not sure, but you indicated overall gain was 40dB but as well the 2nd stage gain is 40dB, which implies the 1st stage is effectively unity gain. Where is the 3rd loop? Or did you cascade two 40dB stages with 40dB overall loop? or cascade 2 x 20dB stages with 40dB loop as overall using 1e-2 (1%) feedback. Normally it is done the other way around with high gain on 1st stage and unity gain for high bandwidth& slew rate for output stage.
I agree,I only see two loops.

 

[LvW]

I think only two.

For convenience, let us introduce some abbreviations:
Both forward blocks are F1 and F2, respectively.
The corresponding feedback blocks are G1 and G2.

I have three alternatives to cut the feedback(s) - so I have three different loops, I think:
a) open the feedback for F1 only
b) open the feedback for F2 only
c) open both feedbacks at the same time.

For each of these three cases I can calculate a loop gain function. All three are different.
In case of simulation I see three corresponding places for injection of the test signal:
a) output of F1
b) output of G2
c) output of F2.

If the right loop is stable (should be for positive u), it can be transformed to a gain block and the circuit reduces to a single feedback loop.

Yes, this method of block diagram reduction is equivalent to alternative a) .

However, there are exactly two other reduction methods leading to expressions identical to the remaining alternatives b) and c).
case b): Change G2 to G2/F1 and move it to the 1st summing junction
case c): Change G1 to F1G1 and move it to the 2nd summing junction.

So again, we have three ways to turn the three-loop-system into a single loop. But which is preferred (correct) - and why?

By the way: The phase margins (simulation) for cases a) and c) are 65.9 deg and 89 deg.

As to SunnySkyguy's question:

I think it is a typical composit opamp configuration with F1 (opamp 1) without internal feedback in in series with a fixed gain amplifier (F2-G2 with 40 dB).
This new amplifier with an open-loop dc gain of 140 dB (100 dB+40dB) gets a negative feedback factor (1E-2) - resulting in an overall gain 0f 40 dB.

 

[FvM]

I append an LTSpice simulation circuit that demonstrates the transformation to a single feedback loop. I din't think about it much, it's simply my intuitive way to analyze the compound structure. Obviously the overall behaviour is kept through the transformation, which in my view clarifies that the gain measured in the feedback path through F1 is the loop gain determining the compound amplifier characteristic.

Of course, there can be only one correct single feedback loop equivalent circuit.

The F2 loop gain, which must be measured with F1 cut is in contrast a property of the internal gain block. The loop gain that can be measured by cutting the path into F1 and F2 isn't an independent circuit property, I think.

P.S.: If we count feedback loops by the number of summing nodes, there are two.

 

[LvW]

Hi FvM, thanks for replying.

Quotation: I append an LTSpice simulation circuit that demonstrates the transformation to a single feedback loop. I din't think about it much, it's simply my intuitive way to analyze the compound structure. Obviously the overall behaviour is kept through the transformation, which in my view clarifies that the gain measured in the feedback path through F1 is the loop gain determining the compound amplifier characteristic.

I’m not quite sure what you mean (cut at the output of F1 or F2 ?).
However, I suppose you mean output of F1 (with feedback around F2 remains closed) – and in this case I agree that the corresponding loop „intuitively“ seems to dominate.
There are some other good reasons supporting this „feeling“ (see last part of this reply).

Quotation: Of course, there can be only one correct single feedback loop equivalent circuit.

Why only one? Are the other single loops false?
For the given system (comprising F1,F2,G1 and G2) there are three different ways to rearrange the blocks (as described in my last reply) based on the rules of block diagram reduction – of course, without touching the closed-loop transfer function. Thus, I have three different forms for an equivalent single-loop system – having equal rights, or not?
But I think, in most cases a dominating one can be identified.

Quotation: P.S.: If we count feedback loops by the number of summing nodes, there are two.

Question: Must the number of summing nodes always be identical with the number of possible loops?
I think it was shown that the present circuit allows to define three different loop gains (resulting from three different ways to open feedbacks).
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In summary, I think for a multi-loop system it makes no sense to ask for one single stability margin because more than only one equivalent single-loop configuration can be defined.
And for each of these single-loops a stability margin can be found.
The number of different equivalent single-loops is identical to the number of different alternatives to place a cut for injecting a test signal (for loop gain determination) into the original system.
However, in realistic applications it seems that there is one single-loop configuration, which primarily determines the closed-loop behaviour.
In most cases this is the „outer“ loop – if such an overall loop can be identified and if it can be separated from other local (short, inner) loops.
In the present example, the F2-G2 loop can be considered as a local loop – and, thus, it should remain closed to find the „dominant“ loop gain.

The following simulation results (comparison open and closed-loop for case a) support these findings:
Phase margin 66 deg (at 55 kHz), closed-loop peak (1 dB) at 32 kHz and 1dB gain drop at 56 kHz.

However, in more complicated (nested) configurations a dominant loop identification by visual inspection only is not always possible.
I am, therefore, interested to learn if a general criterion can be found.
That was the background for presenting the block diagram in post#41 – and I am interested in your views and comments. Thank you.
(Further replies from my side not before thursday).
LvW

 

[FvM]

Quotation: Of course, there can be only one correct single feedback loop equivalent circuit.

Why only one? Are the other single loops false?

I meant there may be different ways to reduce the system to a single loop, but at the end they must be functional equivalent.

I see however, that we can proceed one step further and replace the F(s) and G(s) defined single loop feedback system by H(s) = G(s)/(1 + F(s)*G(s)), eliminating the feedback loop itself. In so far there can be in fact different F(s) and G(s) pairs resulting in the same H(s).

The intuitive aspect is that F1 is mainly defining the overall closed loop gain in the original circuit. Thus it's good if we can find an equivalent circuit keeping this property.

However, in more complicated (nested) configurations a dominant loop identification by visual inspection only is not always possible.
I am, therefore, interested to learn if a general criterion can be found.

Yes, the present example is simple enough for an easy solution. It would be interesting to see less obvious cases.

 

[LvW]

I meant there may be different ways to reduce the system to a single loop, but at the end they must be functional equivalent.

..."functional equivalent" - yes, of course, as far as the closed-loop H(s) is concerned. But this is ensured as long as the rules of block diagram manipulation are kept.
For clarification, I have prepared a small (handmade) drawing of the three alternatives for loop identification with the corresponding loop gain functions.
All circuits provide the same H(s). The form Case C is derived from Case B by eliminating only one of the parallel feedback loops.
In case the original system (at the top of the page) is NOT reduced to one of the three single-loop systems the three possible loop-gain definitions are marked by the three possible openings for injection of a test signal.

It is interesting to note that the three given loop-gain functions LG can be derived also from the closed-loop transfer function H(s):
As can be seen, the denominator of H(s) in the given form is D(s)=1+LG(B).
The other two LG's can be found by dividing H(s) by proper expressions that still allow the form D(s)=1+LG.
Case A: Division by (1+F2G2)
Case C: Division by (1+F1F2G2).

Unnecessary to say that all three loop gains have different frequency response and, hence, different stability margins.

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Sorry, I forgot the "minus" sign in all loop gain expressions.