Is there a rigorous oscillation criterion?

[Winterstone]

Hello to all,

this question, primarily, concerns „linear“ (harmonic) oscillator circuits in feedback topology.
(I know they are, in fact, non-linear - but which circuit is really linear?.)

And my question is NOT why some feedback circuits do oscillate, but - in contrast - why there are some other circuits with feedback that do NOT oscillate - in spite of the fact that they are able to meet the Barkhausen criterion.

Therefore, my question is:
If an active circuit with feedback shall produce self-contained oscillations, what are the requirements that must be fulfilled to operate as a harmonic oscillator?

 

[crutschow]

If a particular circuit doesn't oscillate then it likely is not actually meeting the Barkhausen criterion due to the components in the actual circuit not being exactly what was used to calculate the criterion.

 

[MikeMl]

If the loop gain is greater than 1, and the net phase shift around the feedback loop is 0 or multiples of 360 deg, it will oscillate...

My confusion from your post stems from the phrase "harmonic oscillator". I call them just "oscillators". Are you referring to an oscillator which also generates power at harmonics of a fundamental frequency?

 

[MrAl]

Hi,

Harmonic oscillator is a general phrase used to describe oscillators even of only one frequency.

But as far as creating an oscillator, my advice is to abandon the Barkie criterion altogether and place one complex pole pair on the jw axis. Since this is not only very hard to do it also makes it impossible to calculate the output amplitude from the component values...something we can usually do. Combining these two difficulties we can satisfy them both by placing the single pole pair slightly into the right half plane and also limiting the amplitude using a pseudo non linear scheme. To figure out what gain we need we could look at the root locus. Also, no other poles in the RHP.
The rigor would come in when we go to place the pole pair. We'd have to make sure that component variations do not pull the pair back to the left half plane or else we'll loose the oscillation over time.

 

[Winterstone]

Thank you all for your answers, however....

If a particular circuit doesn't oscillate then it likely is not actually meeting the Barkhausen criterion due to the components in the actual circuit not being exactly what was used to calculate the criterion.

As I have mentioned, there are circuits that meet Barkhausens criterion - and do not oscillate. That`s the point of my question.

If the loop gain is greater than 1, and the net phase shift around the feedback loop is 0 or multiples of 360 deg, it will oscillate...
My confusion from your post stems from the phrase "harmonic oscillator". I call them just "oscillators". Are you referring to an oscillator which also generates power at harmonics of a fundamental frequency?

No it does not oscillate necessarily. Most circuits do - but not all!
May be "harmonic" is a term, which primarily is used in Germany. It means: oscillation at the fundamental frequency, which means: sinusoidal

Hi,
Harmonic oscillator is a general phrase used to describe oscillators even of only one frequency.
But as far as creating an oscillator, my advice is to abandon the Barkie criterion altogether and place one complex pole pair on the jw axis. Since this is not only very hard to do it also makes it impossible to calculate the output amplitude from the component values...something we can usually do. Combining these two difficulties we can satisfy them both by placing the single pole pair slightly into the right half plane and also limiting the amplitude using a pseudo non linear scheme. To figure out what gain we need we could look at the root locus. Also, no other poles in the RHP.
The rigor would come in when we go to place the pole pair. We'd have to make sure that component variations do not pull the pair back to the left half plane or else we'll loose the oscillation over time.

Yes - everything correct. But that was not the question. I know about the problems of placing a pole pair slightly in the RHP - and therefore my remark about non-linearity.
However, it seems that this requirement is not enough (not "rigorous" enough). And that is the reason I am looking for a "rigorous oscillation criterion".

W.

 

[steveB]

... - why there are some other circuits with feedback that do NOT oscillate - in spite of the fact that they are able to meet the Barkhausen criterion.

Therefore, my question is:
If an active circuit with feedback shall produce self-contained oscillations, what are the requirements that must be fulfilled to operate as a harmonic oscillator?

That's a good question, and I've never seen a good conclusive answer on this. One thought I've always had is that a perfectly linear circuit would have a difficult time actually meeting both criteria for gain and phase perfectly, and if the condition is not perfectly matched, the noise never can create a coherent oscillation. Since this is a mathematical relation, we need to expect perfection. Perfect linearity makes perfect Barkhausen criteria matching difficult. So, there needs to be some "give" or imperfections (so to speak) in the system. Of course, there always are non-ideal behaviors, but there would need to be additional criteria for the non-ideal behavior.

Good oscillators always have a saturation nonlinearity which makes meeting the criteria much easier, although also not a certainty. So, I think a saturation non-linearity is a key factor in reducing the constraints enough to allow a perfect matching of the amplitude and phase portions of the Barkhausen criteria.

I realize some of the above posts may be saying something similar in a different way, but this is just my own quick take on a difficult question.

 

[steveB]

Another viewpoint to this is that an oscillation does not appear out of the thin air. It needs to build up dynamically and then reach a sustained stable level. The Barkhausen criteria specify the conditions for the final oscillation, but it's not clear that a dynamic path to reach that condition necessarily will always exist.

Also, as mentioned, a linear circuit has no clear amplitude to associate with the oscillation. If the Barkhausen criteria are validly met, they are valid for every amplitude. This is another view of why saturation nonlinearity is so important for a good oscillator. The existence of one amplitude that meets the condition and the allowance of amplification when the amplitude is small, means the dynamic path and the final condition are both possible.

 

[MikeMl]

...there are circuits that meet Barkhausens criterion - and do not oscillate. That`s the point of my question...

Could you post an example? I suspect that an actual circuit that is supposed to oscillate, and doesn't, it is because the circuit analysis was not rigorous enough, i.e. parasitics not accounted for, gains not up to spec, parts tolerances, etc.

 

[steveB]

Could you post an example? I suspect that an actual circuit that is supposed to oscillate, and doesn't, it is because the circuit analysis was not rigorous enough, i.e. parasitics not accounted for, gains not up to spec, parts tolerances, etc.

This request is a little bit problematic, if you think about it. The original question is somewhat theoretical in the sense that the criteria apply to a perfectly linear circuit, but perfect linearity is not possible.

If you could make (or even model) a perfect linear circuit with gain exactly equal to 1, and with the correct phase relation at one particular frequency, the oscillation could never build up above the low level noise in the system. If the gain is a little greater than 1, then amplification may take place across the noise band, but since no frequency now meets the criteria of gain equal to one, then the condition is not met and a stable oscillation is not formed.

Hence, we need to cheat a little bit to answer your very appropriate question. If we allow gain saturation as amplitude increases, and make the low amplitude gain go slightly greater than 1, then noise builds up, and then the gain saturates which then makes it possible for one frequency and one amplitude to form a stable oscillation that requires higher amplitude at the right frequency.

So, for your example, let's do the opposite, lets make a gain that does not saturate at higher amplitudes but instead increases slightly. Then the low amplitude gain can be set to 0.99 and there will exist a higher amplitude and a certain frequency that meets the criteria. However, the oscillation will not happen in this case because the low level noise can not find a route to the stable oscillation.

Indeed, for this circuit, even if you forced a signal to exist at the correct operating point, amplitude and frequency, it would not be stable there. Slight noise levels would cause the signal to get larger and then the gain would be greater than 1 and the Barkhausen criteria is no longer valid.

So, oscillations require additional criteria in the nonlinear domain. In effect, linear analysis is not the right tool for the job, but it does still provide some insight into the problem.

 

[MikeMl]

Therefore, my question is:
If an active circuit with feedback shall produce self-contained oscillations, what are the requirements that must be fulfilled to operate as a harmonic oscillator?

If you accidently make an oscillator, then loop Gain>1 while phase shift is multiples of 360deg. If you are trying to make an oscillator, then loop gain wasn't high enough, or the phase shift wasn't right.

 

[steveB]

If you accidently make an oscillator, then loop Gain>1 while phase shift is multiples of 360deg. If you are trying to make an oscillator, then loop gain wasn't high enough, or the phase shift wasn't right.

Actually, the requirement is loop gain=1 not gain>1. This is why I say that matching the condition in a perfectly linear circuit is not so easy. Real circuits with transistors have enough nonlinearity to find this condition without too much trouble, if the circuit is poorly designed as an amplifier or well designed as an oscillator. Also, OPAMP circuits, although linear over a range, saturate at the voltage rails. That's why they tend to oscillate at full amplitude of the rails.

EDIT: So, again we like saturable gains. Make gain>1 at low amplitude and let it saturate to exactly gain=1 at the correct amplitude.

 

[Winterstone]

Hello Mike and Steve,

thank you for your contributions.
In principle, I agree with everything you have mentioned (loop gain equal to or somewhat above unity, non-linearity, etc.)
However, that does not solve the problem, I think

Let me say it in mathematical terms:

I think, the well-known Barkhausen criterion is a necessary one only.
And I am wondering if somebody has heard about an oscillation criterion that is sufficient.

That`s the main question.
W.

 

[MikeMl]

No steve, I have built many oscillators, and I am here to state that the Gain MUST be > 1 for any practical oscillator.

It looks to me that Winterstone has sucked us (yet again) into one of these "How Many angels can dance on the head of a pin" forum exchanges. I, for one, am not going to play...

 

[steveB]

Let me say it in mathematical terms:

I think, the well-known Barkhausen criterion is a necessary one only.
And I am wondering if somebody has heard about an oscillation criterion that is sufficient.

Yes, I did understand this is your question, but it's a difficult question and all I can do is talk around it as I did.

My gut feeling is that there really isn't a sufficient criterion because of the issues we've talked about and agree upon. The linear analysis tool just is ill-suited to deal with the nonlinearity that provides the sufficient conditions.

The Barkhausen criteria is somewhat ambiguous for a linear system. We can define a linear system to select for frequency, but not for amplitude. If the criteria is valid for one amplitude, then it is valid for all amplitudes. Systems don't oscillate at all amplitudes.

I do think that the criterion is a sufficient condition in the sense that if you force a linear system (that meets the criterion) to have a stable signal for one amplitude and frequency, then it will stay that way. But this is not very useful, and anything useful forces us to consider nonlinearity, even if it is only a very tiny amount of nonlinearity. But, once we do that, the condition is no longer sufficient. We need to talk about noise, saturation, dynamics and complete trajectories of the time evolution. And, let's not forget that frequency domain theory become very questionable (and must be used with care) once the system becomes nonlinear.

 

[steveB]

No steve, I have built many oscillators, and I am here to state that the Gain MUST be > 1 for any practical oscillator.

It looks to me that Winterstone has sucked us (yet again) into one of these "How Many angels can dance on the head of a pin" forum exchanges. I, for one, am not going to play...

Well, first the Barkhausen criteria is what it is. I didn't define it and you shouldn't redefine it. It says gain=1, so we are stuck with that.

However, I think what you are really saying is that you design your oscillators with gain>1 for small amplitude signals. However, once the oscillation builds up, the signal saturates the gain and the stable oscillation has gain=1. So, in one sense I agree that a practical oscillator requires gain greater than one, but the oscillation condition is the gain=1 value.

I agree this is a difficult question to deal with, but to me it is not a "angel dancing on a pin".

 

[steveB]

And I am wondering if somebody has heard about an oscillation criterion that is sufficient.

.

Actually, it just occurred to me that I should ask for clarification here. Are you looking for a sufficient condition for a linear circuit, or are you allowing us to specify that including nonlinearity is needed in order to establish the sufficient conditions.

I ask because I have seen sufficient criteria for stable oscillations. I may be forgetting some of it, but the following comes to mind

loop gain > 1 for small signals
gain saturation to allow for gain=1 (positive feedback mode)
frequency selector (filter) (although sometimes the gain spectral response does this automatically)
Noise source or startup signal.

If we go beyond just circuits, other oscillators might require additional types of mode filters (polarization modes, spatial modes etc.)

 

[The Electrician]

After Nyquist published his stability criterion:

https://www.electro-tech-online.com/custompdfs/2013/06/bstj13-4-680.pdf

there was naturally much interest in it. A couple of years later some Bell Labs researchers did an experimental verification of the theory:

https://www.electro-tech-online.com/custompdfs/2013/06/bstj13-4-680.pdf

In their paper they demonstrate a circuit with loop gain greater than 1 and loop phase shift of 360° (at some particular frequency) which didn't oscillate when the loop was closed.

So, I think what Winterstone is getting at is that Nyquist's criterion can be both necessary and sufficient to indicate the condition for oscillation, whereas Barkhausen's may fail.

 

[Winterstone]

It looks to me that Winterstone has sucked us (yet again) into one of these "How Many angels can dance on the head of a pin" forum exchanges. I, for one, am not going to play...

MikeMI, may I kindly ask you to explain the meaning of your contribution? (In particular, the part "yet again")?
Since the english language is not my mother´s tongue, I don`t understand the sentence "How many angels...".
Or didn`t you get the meaning of my question? I suppose, you know the difference between a "necessary" and a "sufficient" condition?
Perhaps you reconsider your judgement?

Regads
W.

Edit: There are several publications (books, articles) stating that Barkhausen`s condition is a necessary one only).
Even wikipedia tells this:
"Barkhausen's criterion is a necessary condition for oscillation but not a sufficient condition: some circuits satisfy the criterion but do not oscillate.[2] ...
Apparently there is not a compact formulation of an oscillation criterion that is both necessary and sufficient.[3]"

Additional reading: https://web.mit.edu/klund/www/weblatex/node4.html (K. Lundberg, MIT)

 

[Winterstone]

The linear analysis tool just is ill-suited to deal with the nonlinearity that provides the sufficient conditions.
The Barkhausen criteria is somewhat ambiguous for a linear system. We can define a linear system to select for frequency, but not for amplitude. .........

And, let's not forget that frequency domain theory become very questionable (and must be used with care) once the system becomes nonlinear.

...... Are you looking for a sufficient condition for a linear circuit, or are you allowing us to specify that including nonlinearity is needed in order to establish the sufficient conditions.
I ask because I have seen sufficient criteria for stable oscillations.

loop gain > 1 for small signals
gain saturation to allow for gain=1 (positive feedback mode)
frequency selector (filter) (although sometimes the gain spectral response does this automatically)
Noise source or startup signal.

Hello SteveB, thank you again.

I am not quite sure if the question of linear/non-linear plays a major role in this context.
As you know, there are no circuits, amplifiers or filters, which really are linear. Nevertheless, we apply - with success - the rules of the frequency domain.
As you probably know, I am german - and I have the opportunity to read Barkhausen`s book in original version.
In this book, it is even mentioned that for a safe start of oscillations a loop gain somewhat larger than unity is required - and, more than that, "between the lines" you can also read that it is only
a condition that is to be considered as necessary only.

For my feeling, there are two different phases to be investigated:
1.) Start-up phase: This is a pure linear analysis. Here, it is to be investigated if self-excitement in oscillatory form is possible (with rising amplitudes, of course)
2.) Steady-state phase: A non-linear mechanism must limit the amplitude in order to allow continuous oscillations (at the cost of degraded THD, of course).

According to several documents (text books, magazine publications) it is clear that Barkhausen`s condition is a necessary one only.
That means, a loop gain equal resp. slightly larger than unity at a single frequency does not necessarily mean that this circuit is able to oscillate.
That means - for my opinion - some other conditions must be fulfilled. And that`s what I call a "rigorous oscillation condition".
(I hope I have clarified my question now - it is by far not a problem if "angels can dance on the head of a pin")

Thank you and regards
Winterstone

 

[Winterstone]

In their paper they demonstrate a circuit with loop gain greater than 1 and loop phase shift of 360° (at some particular frequency) which didn't oscillate when the loop was closed.
So, I think what Winterstone is getting at is that Nyquist's criterion can be both necessary and sufficient to indicate the condition for oscillation, whereas Barkhausen's may fail.

Electrician, I know the referenced paper - but I must confess that I don`t remember an example (in this paper) which didn`t oscillate.
Thank you for this information. I will check it again.
Regarding Nyquist: This subject, of course, belongs to the whole question. Do you have an indication that the Nyquist criterion can be used as a sufficient oscillation condition?
W.