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Is there a rigorous oscillation criterion?

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Hello again,


Steve:
I agree that a way to make sure the Barkhausen criterion is met first is desired.

ALL:
The new circuit is almost the same as the old circuit with different values so the pole pair criterion can still be calculated the same way just with different resistor numbers:
z=C1*R5*R1-C2*R6*R1+C2*R5*R2

and again z shows us where the pole pair is relatively:
z=0, on the jw axis,
z<0, in the RHP and so will saturate eventually,
z>0, in the LHP and so will damp to zero eventually.

For this new circuit, z=0.03975 so the pole pair is in the LHP which means the oscillations get damped out. We can calculate how long it takes to damp them out if needed.
 
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Hi Winterstone,

I'm just waking up and still need to read through all the posts through the night and morning. There were so many, so I'll check the circuit soon.

But, basically I agree with everything you said. It's just that now I'm questioning an assumption that I've always made previously. I generally assumed that if gain>1 sufficiently for small signals and phase is near 0, then limiting will bring the Barkhausen condition in line. But, can we be sure this will always be true? Is it possible for the limiting to cause both the gain and phase to change in a way that the Barkhausen condition in never met? Hence, might there be circuits that can't oscillate because of this?

I'm just asking because if so, this fact would tie into your question in an important way.
 
For this new circuit, z=0.03975 so the pole pair is in the LHP which means the oscillations get damped out. We can calculate how long it takes to damp them out if needed.

MrAl, thank you.
Yes - this result is in accordance with my investigations.
However, an interesting point is that the loop gain for the given values is LG>1 (in fact app. LG=1.01). Nevertheless, we have a damped sinus.

Now the other way round:

If we increase R6 from 9.75 to 10.2 kohms we increase the negative feedback - thereby reducing the gain to LG<1 - the circuit starts oscillation.
However, it cannot provide sustained oscillations. Instead it will exhibit latch-up.

This is exactly one of the situations Steve has described in his last post. The circuit is not able to reduce the gain for rising amplitudes.
Instead, we have to rise the loop gain in order to come closer to the first dimensioning with a damped signal. This sounds somewhat contradictory.

In summary: There is certainly a value for R6 (between 9.75 and 10.2 kohms) which theoretically allows to meet Barkhausen`s condition.
Nevertheless, the circuit has no practical use. This becomes evident because - for a REAL amplifier model - we have a positive real pole which inhibits any oscillatory behavior.

This example again underlines my desire to add something to Barkhausen`s criterion in order to arrive at a more rigorous oscillation condition.
 
I'm just waking up and still need to read through all the posts through the night and morning.

But, basically I agree with everything you said. It's just that now I'm questioning an assumption that I've always made previously. I generally assumed that if gain>1 sufficiently for small signals and phase is near 0, then limiting will bring the Barkhausen condition in line. But, can we be sure this will always be true? Is it possible for the limiting to cause both the gain and phase to change in a way that the Barkhausen condition in never met?

Yes - exactly this can be the case. See my reply above (related to the last circuit example).
And this situation is the background of my basic question as given here in this thread.
(Mike, do you understand now? No balancing on a razors edge).

W.
 
Hello again Winterstone,

Yes, i agree, and i am seeing similar results with this new circuit. It's also making it easier to analyze too which is really nice. It makes it easier to state the Barkhausen criterion in terms of the component values:

"The loop gain must equal 1":
(2*(R6+R5))/(3*R6)=1

"The phase shift must be zero" ie the w which causes this is:
w1=1/sqrt(C1*C2*R1*R2)=1/sqrt(C^2*R^2)=1/(R*C), with:{R=R1=R2,C=C1=C2}

So the phase shift is zero at w1 above, and the gain is 1 when the first equation is true. So we see this happen with R6=10k.

And i will add my own equation to this for use unless somebody comes up with something better:
C1*R5*R1-C2*R6*R1+C2*R5*R2=0

and with C2=C1 and R2=R1 (again) this simplifies even more to the very simple:
0=2*R5-R6

or:
R6=2*R5

to place the pole pair on the jw axis.

Thanks for posting the circuit with equal resistors and caps as this helps enormously to simplify. We can see how the two gain resistor control everything.
 
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Winterstone,

I havent' given up thinking about this, but I've been tied up with several things over the weekend and my time is short over the next couple of days. I'd like to make sure I understand what you are looking for, so can I ask a couple of questions?

My response plot shows the gain of your last circuit is 3.5 dB across the band with 0 phase, except there is a notch near 1 kHz and the gain is 0.1 dB there. Is this correct?

I seem to show a LHP complex pole pair of -157±j6289 for the closed loop system which indicates stability. Is this correct?

You are saying that you have stability for startup even though you are nearly meeting the Barkhausen criteria, and you would have expected limiting to give an exact match. Is this correct?

Most importantly, I interpret your main question to be, "can we identify the stability of the closed loop system (at startup) by looking at the open loop response? Is this correct?

If this last question is the main thing, I'll spend time thinking about that. If this is not quite the full extent of what you are looking for, please provide more guidance.

At this point, without answering that main question, it seems we need to specify the necessity for unstable RHP pole(s) for the closed loop startup condition.
 
Hi Steve,
thank you for replying.
Here are my answers:

1.) My response plot shows the gain of your last circuit is 3.5 dB across the band with 0 phase, except there is a notch near 1 kHz and the gain is 0.1 dB there. Is this correct?

Yes, it is correct. In detail: For R6=9.75 k the loop gain is app. +0.1 dB and for R6=10.25 k the loop gain is app. -0.1 dB.

2.)I seem to show a LHP complex pole pair of -157±j6289 for the closed loop system which indicates stability. Is this correct?

Yes - also correct, for R6=9.75 k. For a larger value R6=10.25k the poles ar slightly in the RHP (+126 +- j6290)

3.) You are saying that you have stability for startup even though you are nearly meeting the Barkhausen criteria, and you would have expected limiting to give an exact match. Is this correct?

For R6=9.75k oscillation starts, but with decreasing amplitudes - indicating stability. For 10.2k oscillation also start - but with rising amplitudes until limiting (supply rail) with latch-up effect.
(However - this applies for the idealized opamp model only. Real models cause a real pole in the RHP causing instability for all values of R6. This is logical because the real opamp cannot find a stable bias point - we have 100% positive dc feedback).

4.) Most importantly, I interpret your main question to be, "can we identify the stability of the closed loop system (at startup) by looking at the open loop response? Is this correct?

Yes - correct. Because this is in accordance with Barkhausens condition, which also is based on loop gain response. However, I doubt if this will be possible - perhaps we must use also information on pole locations (root locus).

5.) If this last question is the main thing, I'll spend time thinking about that. If this is not quite the full extent of what you are looking for, please provide more guidance.
At this point, without answering that main question, it seems we need to specify the necessity for unstable RHP pole(s) for the closed loop startup condition.

The core of my problem (the main question) is the question which I have asked in post#70 already:

"Under which conditions a circuit with feedback is able to work as an oscillator?" If possible, the answer should be based on information that can be found easily (loop gain only).
However, perhaps some additional parameters must be specified (pole location, Nyquist properties,...)

Thank you
W.
 
Hi again Winterstone,

I'm surprised nobody is continuing on here because I think you can do what you want to do, and I think we are getting closer to the answers. I'm still busy for another day, but I'll just mention an idea that might get the discussion going.

It seems part of what you are asking is very similar to the situation we have with negative feedback. We often want to know if closing the loop will give us a stable system. There are criteria that can be applied to the open loop response when we use negative feedback. I would imagine that very similar criteria could be derived for positive feedback.

I would start with the Nyquist Stability Criteria and then derive for the case of positive feedback. There is a version of this that looks at the number of encirclements around -1+j0. Perhaps for positive feedback, we can look at encirclements of +1+j0.

https://www.electro-tech-online.com/custompdfs/2013/06/nyquist.pdf

In your example, we can almost visualize the Nyquist plot. You have stability when the magnitude does not dip below 1, and instability when the magnitude does dip below 1. (the phase is 0 there). Clearly the number of encirclements is changing here. Also, one can probably identify rules to apply to the open loop Bode plot that would make doing a formal Nyquist plot unnecessary in typical cases.
 
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I'm surprised nobody is continuing on here because I think you can do what you want to do, and I think we are getting closer to the answers.

Yes - it is a bit disappointing to me also.
In particular, because I have formulated a clear question (to MrAl and to MikeMi and all other readers)
What is your answer if somebody asks you "under which conditions a circuit with feedback is able to work as an oscillator?".

Regarding the Nyquist theorem, I am not quite sure if it really can help because - for my opinion - it only can give an information like "stable or instable". However, instable does not automatically mean "able to oscillate".
But we are approaching a kind of solution, I think (I hope).

Regards
Winterstone
 
I thought we were done, but you seem to want to keep ignoring the nail ;)

Look at these two Bode plots. My experience tells me which one I would choose if I was trying to build an oscillator.

Which has an unambiguous single frequency where the phase shift is zero?

Which has a single frequency where the insertion loss is at a minimum?

I'm not sure if it is significant, but which network has a negative slope in the phase shift near 0 degrees?

OBTW, I simed the simplified Inverted Wein network in an oscillator. It oscillates at DC (latches up) at Loop Gains between 1.00001 and ~1.0007. For 1.0008<LG<~1.01, it oscillates with saturation at about 100kHz, but the frequency is greatly affected by LG. For LG>1.03, it oscillates as a square-wave multivibrator, like a 555... What a crappy oscillator!
 

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Hi,

Mike:
Well yes, but also the 'inverted' version requires less op amp gain to work so that tells me that it might work at a higher frequency with the same op amp part. But given lower frequencies, then yes the more 'standard' version seems the best bet.

ALL:
Well i thought we were going to try to find a circuit that does NOT oscillate yet satisfies Barkhausen? What happened to that? If we can find such a circuit we can do an analysis and find out what else is wrong.
 
I thought we were done, but you seem to want to keep ignoring the nail ;)

Mike, thank you for your response, but please take into account that my mother tongue is not english. I couldn`t find out what "ignoring the nail" means.

Look at these two Bode plots. My experience tells me which one I would choose if I was trying to build an oscillator.
Which has an unambiguous single frequency where the phase shift is zero?
Which has a single frequency where the insertion loss is at a minimum?

I am afraid, you didn`t get the main point of this thread. It is not my aim to find a suitable passive circuit to be used in an oscillatory network.
However, perhaps this is a new information for you, but even the inverted Wien circuit can be used in a feedback oscillator. However, it must be placed in the inverting loop!
An insertion loss that has a minimum at a certain frequency is NOT an important criterion for a feedback oscillator. That has been proofed. There are lot of counter examples.
[/QUOTE]

May I repeat my question?
What is your answer if somebody asks you "under which conditions a circuit with feedback is able to work as an oscillator?"."
 
Well i thought we were going to try to find a circuit that does NOT oscillate yet satisfies Barkhausen? What happened to that? If we can find such a circuit we can do an analysis and find out what else is wrong.

Yes, this is the core of the question. Thank you for clarification.
 
Hello again,

The only other thing i can think of right now is that in order to get it to oscillate we would need to introduce some energy, but it would have to be of the same frequency as the oscillator was going to oscillate at or close to it. We usually get this when we turn the power on because the fast rising power supply voltage is rich in harmonics, so it's bound to start up.
To test this theory, we could try 'turning on' the circuit we have with a 0v power supply, then very slowly and carefully turning the power supply voltage up. No noisy pots allowed :) If we are lucky and there is not enough external noise to start the circuit, we may see the circuit never start up. Might be hard to do though without some kind of shielding. This would demonstrate the failure of the Barkhausen Criterion for oscillators.
Any volunteers?

[LATER]
Yes, that's got to be it, the missing factor! If we try to start the oscillator with a sine that is not of the right frequency, it will not start because the gain is not right at that other frequency. So the 'kick' start has to include the right frequency because no other frequency will work.
Of course testing this might be hard to do too, because we would have to be able to apply and remove the stimulus without introducing any step change.
 
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I agree the noise or startup signal at the right frequency is critical. The need for instability at startup was mentioned by you and others, but Mike and I got it wrong by saying gain>1 at startup. Mike did list it as a necessary condition (we can see if that holds up), but i listed it as a sufficient condition, which isn't right. Winterstone's example shows that you can have gain>1 and have stability, so this was wrong as a sufficient condition, even though it usually holds up.

So, it seems we need sufficient conditions that go beyond the open loop linear analysis. Still noise can be assumed to be present always in the real world. Also, the stability condition can be gleaned from the open loop response of the system if we use a Nyquist type of analysis assuming positive feedback. The Barkhausen criteria seem to be integrated into the Nyquist criteria. The additional thing that goes beyond the linear analysis is the nonlinear amplitude dependent gain. We've talked about gain limiting and gain that increases at higher amplitude. We know the conditions needed for gain limiting (instability and noise at low amplitudes).

It seems we are getting closer and closer to what we should answer when Winterstone asks, "under which conditions a circuit with feedback is able to work as an oscillator?". I think we are talking about "practical" oscillators and not a crazy marginal stability oscillation. We can formalize the Barkhausen criterial under the Nyquist rules for number of encirclements, and include noise and nonlinear gain limiting. Some words about frequency selection might be useful too. Then we can look for exceptions to these rules, to see if they hold up. Even if there are exceptions, we might have a sufficient condition list because sufficient conditions don't require that we cover every case. Still, we can then expand the list to include the exceptions.
 
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Hi Steve,

I dont know what you mean by gain>1 and stability. How is this so?

I also do not believe that simply stating "noise" is sufficient for start up for the following reasons:
1. If the noise does not contain the right frequency, it wont start anyway, so 'noise' is under specified.
2. If the noise does contain the right frequency, it starts, but then 'noise' is over specified.
3. Since the "right frequency" is at the heart of it, it makes more sense to simply state the the right frequency has to be present as stimulus at start up.

So what this implies is that there is no real circuit that will fail Barkhausen unless the right frequency is not present, and if the right frequency is not present then many circuits will fail even though analytically they pass the two Barkhausen tests.

So to be specific about this other new conditions, i would state it as "having at least the right frequency stimulus present at startup" (other frequencies can of course be present too, but are not necessary but the correct one *IS* necessary). It's a necessary condition and quite succinct.
 
I dont know what you mean by gain>1 and stability. How is this so?
Sorry, I'm not explaining very well. I'm mostly in a brainstorming mode here because I find this a faster way to get to answers.

What I mean is that usually we start up the oscillator with gain>1. This seems to be a necessary condition as Mike suggested (can we find exceptions?). However, it seems to me that Winterstone's latest circuit has gain>1 but is stable for small signals at startup. Then, when the gain saturates, the circuit becomes unstable. This is the opposite of usual oscillators that are unstable at startup and become stable if the gain is reduced below 1.

So, I'm questioning this gain>1 as a sufficient condition, without further qualification about the stability/instability we will find when closing a positive feedback loop. The Nyquist stability rules about "encirclement" seem relevant here to tell us which way it goes.

To me this latest circuit is not a good situation to have for an oscillator because (I think) it can't give stable and reliable oscillations. Gain is greater than one and stable will drive the signal higher, limit the gain and then make it unstable. To me this is undesirable for sufficient conditions for a practical oscillator. I would generally prefer to design a circuit that can start unstable and drive itself into a stable condition. Does that make any sense? I'm basically brainstorming here, because I haven't had time to do simulations on this yet. I have some time now, so I'll write a simulation program for Winterstone's last circuit. Maybe I'll have to change my mind after I do that.

I also do not believe that simply stating "noise" is sufficient for start up for the following reasons:
1. If the noise does not contain the right frequency, it wont start anyway, so 'noise' is under specified.
2. If the noise does contain the right frequency, it starts, but then 'noise' is over specified.
3. Since the "right frequency" is at the heart of it, it makes more sense to simply state the the right frequency has to be present as stimulus at start up.

In theory I think you are correct, but in practice noise is always present at all frequencies. If the oscillator is unstable at startup, then the tiniest noise will eventually get amplified at frequencies where gain>1, and this would generally be sufficient to start most oscillators. But, I'm not opposed to stating the condition in the way you say, since it is simple and all-inclusive and less vulnerable to debate and exceptions.

So what this implies is that there is no real circuit that will fail Barkhausen unless the right frequency is not present, and if the right frequency is not present then many circuits will fail even though analytically they pass the two Barkhausen tests.
Winterstone's circuit appears stable for small signals and it would seem the Barkhausen condtion can be met for some particular amplitudes and conditions. Yet, the instability when the gain is limited seems to be an impediment to making a good oscillation. At small signals the circuit is stable and a signal at the right frequency can get amplified since the gain>1. So, I'm not sure how we want to classify this. Is the crude oscillations we might get with this circuit something we want to classify as a "working oscillator"?.

So to be specific about this other new conditions, i would state it as "having at least the right frequency stimulus present at startup" (other frequencies can of course be present too, but are not necessary but the correct one *IS* necessary). It's a necessary condition and quite succinct.
I agree with that.
 
Hi Steve,


Yes that makes sense :)

I think what else we need to do is separate the perfectly theoretical from the practical. For example, in theory we do need the stimulus of that one perfect frequency, but in practice we can get away with the step change that comes about when the power supplies are first applied to the circuit and with it's rich harmonics starts the circuit oscillating. Also in theory i think we can say that all we need is gain=1 and phase shift=0 along with that special frequency excitation. But yes in practice we might have a higher gain such that the oscillation starts quicker and more reliably.

I had hoped that someone else interested in this was in the mood to perform a few bench top experiments :)
 
So, I finally got around to doing my own simulations. I found the usual Wien Bridge arrangement worked beautifully as expected. The latest circuit with the inverted feedback arrangement just latched up. I think this is expected because of the DC positive feedback. When I put a large capacitor to block DC in the positive feedback path, I obtained a nice multivibrator with about 8 kHz square wave.

So, that last simulation is interesting because neither R5/R6<0.5 nor R5/R6>0.5 has all stable poles. So, doesn't that lead us to a sufficient condition for the Barkhausen condition to give oscillations? It would be that we require stability on the gain<1 side of the Barkhausen condition.

Winterstone asks, "under which conditions a circuit with feedback is able to work as an oscillator?"

I'll suggest these sufficient conditions. (Keep in mind that sufficient conditions allow that there might be other ways that don't obey these conditions)

1. Barkhausen Criteria must be met for a nonzero amplitude and frequency
2. Stimulus signal at the correct frequency must exist at small amplitude
3. Gain > 1 must exist for small amplitudes at the frequency
4. Gain must saturate as amplitude increases at the frequency
5. No RHP poles exist to cause instability when gain saturates to G<1

I think 5 will avoid the latch up and multivibrator issues.

I'm still brainstorming here, so poke holes. Can anyone think of any circuit that meets these conditions but does not oscillate in a practical way?

If some of this holds up, we could then try to relate many of these back to the open loop response. Clearly 1 relates, but it is the open loop response of the linearized system around the point with gain in saturation that matters. Still, the small signal open loop response can usually tell us what's happening. Also, 3 would be obvious and 5 might be traceable back to the small signal open loop response, along the lines of a Nyquist encirclement criterion. I would argue that since real systems have noise at all frequencies, then 2 does not need to be related back to anything. And, 4 is a nonlinear issue that just tells us the open loop response needs gain>1 so that saturation will bring it down to gain=1 at the desired operating point.

So, it might be possible to identify sufficient conditions that are identifiable from the open loop response. However, even if we can, that doesn't mean there can't be other sufficient conditions that ensure oscillations. For example, the multivibrator oscillation in my simulation did not obey this list. But, perhaps that mode of operation is separate from this question anyway.

Anyway, that's my best attempt.
 
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Hi Steve,

I am not sure what you mean when you say that the standard Wein works beautifully and the inverted Wein latches up.
Are you aware that the resistors R5 and R6 much also be swapped for the inverted version?
In other words, if the standard version has R5=10k and R6=5k, then the inverted version requires swapping R5 and R6, so R5=5k and R6=10k. It's only then that we see similar action.

Oh yes, before i forget to mention, since i was offering as another specification (in addition to Barkhausen) of an excitation by a frequency equal to the zero phase frequency and that of course is a frequency domain specification, i should have also included the time domain version of that which could read that the state of the system be equal to a valid state as when the oscillator is already in motion. In other words, the initial cap voltages can be set to equal any measured voltages of these two voltages at any time t once the system is up and running. So for example if at time t1=100 seconds we measure vC1=A volts and vC2=B volts (not random voltages) we should see the system start up smoothly. Of course if either of these voltages is not correct, we'd then be doing the equivalent of applying a step to one of the initial conditions and that would mean it would start up too, but that's cheating because that does not narrow down the start up requirement good enough (using a step means using a bunch of frequencies not all of which are useful).

I was also assuming we were going to be working with sine oscillators to start with.
 
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