Is there a rigorous oscillation criterion?

[steveB]

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.

Hi MrAl,

Yes, I am aware that the values must be swapped and used that for the simulations. I'm familiar with the standard version, so I know the method, there, and Winterstone highlighted the other version (which I'm not familiar with) and it was clear the values were swapped. So the standard version works perfectly with a nice startup, but the latest circuit given (the one with DC feedback in the positive feedback path just latches up at the rails. The former case looks correct, but I'll double check the formulas and implementation for the latter case if you think that latching result is not correct.

 

[Winterstone]

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

Thank you gentlemen for the various contributions. I couldn`t answer earlier because of some private reasons.

I think, the "best attempt" from Steve is a very good approach. Perhaps some comments to the 5 points:

(1) is obvious, however, I would add that Barkhausen must be met for one single frequency only.
(2) I think, each practical oscillator needs a switch-on and , thus, always has a sufficient stimulus. Noise is not necessary (although always existent).
(3) Yes - also obvious, but replace "gain" by "loop gain". (I consider it as part of Barkhausen`s criterion).
(4) I propose to replace the term "saturate" by "decrease". I think, this point has a direct relation to the root locus of the system (towards the imag. axis for falling gain)
(5) Yes - a very important requirement. In detail: At start-up, only one single complex pole pair is allowed in the RHP - no real pole.

I agree to all 5 points as listed above by Steve.
These points are in full agreement with my theoretical investigations. However, in reality it is not a simple task to check all the points - in particular (4) and (5).

My problem, therefore, is to find a connection between (4) and(5) with the open loop response (loop gain).
I am not sure - it`s only a guess - but I have the feeling that (4) and (5) are satisfied if

* the loop gain for dc (w=0) is below unity (obvious requirement for a stable operating point in closed-loop configuration)
* the Bode plot has a loop phase response with a negative slope at the oscillating frequency (I think, this point was already mentioned by MikeMi) .

The phase slope for "normal" oscillators is always negative at w=wo, but the "problematic" oscillator circuit (Wien_invers) exhibits a positive phase slope.
But I haven`t yet a mathematical derivation of this requirement.

Thank you again.
W.

 

[steveB]

Winterstone,

Those all sound like good modifications. My term "gain saturation" I agree is not good in the circuit context, and probably caused some confusion here in this thread. I retained that term from working on lasers (optical oscillators). I've heard the term "gain compression" also with regards to circuit oscillators, so I think, gain-compression or gain-reduction are much better choices for circuits.

 

[Winterstone]

Steve - OK, fine.
I am really happy with the progress we have made during discussion of this problem.

 

[MrAl]

Hi again,

Most of my questions are for Steve...


First, im not sure what you mean by the most recent circuit latching up and the more standard version working fine. Is this in simulation? They both should work fine otherwise we've found a circuit that meets Barkhausen and doesnt oscillate
But maybe you can clear that up. What i cant see is how you got one to work and one not to work, and the one that 'works' somehow works without negative feedback?

So anyway, what i am proposing is that for point number (2) as outlined by Steve, that we drop the 'noise' requirement, and we drop the 'turn on' requirement, because that is more general than what is needed, and as i already said twice now 'noise' does not guarantee start up and neither does turn on. Noise may or may not contain the right frequency. If it is white noise then it may contain the right frequency for example. But it's still an over specification about what the heart of the matter is, and that's that ONE single individual frequency that is needed for startup. And power supply turn on does not guarantee start up either because it may rise too slow. This is a factor in some other circuits too that depend on a given dv/dt at startup, in that the slope must be above some minimum value.

So the proposal is to drop the term 'noise' and 'power supply startup' from the specification we are looking for, because one over states what is needed and the other understates what is needed. In other words, knowing that there is noise or unspecified dv/dt power supply startup does NOT tell us that the oscillator will start up.
In place of this we insert something like "some energy in the form of the normal oscillator frequency" to start up. And again in the time domain, we must specify the initial conditions for the two capacitors and their voltages must be one set of voltages that are a solution at one point in time when the oscillator is running.
So saying that noise is needed is too wide of a specification for something that is exact, and saying that power supply turn on is needed is also too non specific. We know that noise usually starts the oscillator in real life, and power supply turn on usually starts it, but we dont want to know what usually starts it if we are going through the trouble of figuring this out in the first place, we want to know exactly what starts it.

Do you see why i stress this point now?

 

[steveB]

First, im not sure what you mean by the most recent circuit latching up and the more standard version working fine. Is this in simulation? They both should work fine otherwise we've found a circuit that meets Barkhausen and doesnt oscillate
But maybe you can clear that up. What i cant see is how you got one to work and one not to work, and the one that 'works' somehow works without negative feedback?

First, this is a simulation. To be absolutely clear, I should show the schematics and formulas and methods, but this is probably more detail than is needed.

The standard version is the usual Wien Bridge as can be seen at the top of this article. Here the negative feedback path requires R5/R6=2. When I simulate this it works as expected, so no surprises there. I claim this simulation gives the correct answer.

https://en.wikipedia.org/wiki/Wien_bridge_oscillator

The "most recent circuit" is the one from Winterstone in post #78. here the negative feedback path requires R5/R6=0.5. When I simulate this, the output latches to one of the power supply rails. It doesn't oscillate. I don't claim this is necessarily the correct answer, but it is what I got. It seemed reasonable to me, but I'm planning to double check the equations and implementation later today.

I also simulated the original topology from Winterstone in post #54. This topology uses a DC blocking cap in the positive feedback path. When I simulate this, I get the multivibrator oscillation. I also don't claim this result is necessarily correct, but it also seemed reasonable to me. I'll double check this one too later today.

So, my interpretation of what was said in this thread is that this inverted topology does have issues oscillating in a nice way, and that Winterstone put these forward as examples that seem to meet the Barkhausen Criteria, but don't oscillate. Please let me know if I'm misunderstanding anything. There is a lot said here, and sometimes I'm not sure I'm absorbing it all correctly. One of the reasons I wanted to do my own analysis and simulations was just to have direct information to make conclusions from.

So anyway, what i am proposing is that for point number (2) as outlined by Steve, that we drop the 'noise' requirement, and we drop the 'turn on' requirement, because that is more general than what is needed, and as i already said twice now 'noise' does not guarantee start up and neither does turn on. ... Do you see why i stress this point now? [

Yes, I do see why you stress the point, and basically, I'm in agreement, which is why my list attempted to state it in the way you indicated. I didn't mention noise or startup, but only the need for some small signal with the right frequency. Still, I'm not opposed to Winterstone changing it. I would even recommend listing noise, turn on and the frequency component as possible starters for the oscillation. Remember, this is a sufficient condition list. We know any oscillator needs to be turned on and it has noise, which is unavoidable. If somehow we could truly eliminate them then an initial condition on the caps or an injected signal would also work. This last point is relevant for simulations.

In my simulations, I've used two methods for startup. Initial conditions on the states, and addition of a white noise source to the state variables both work well. Personally, I like the noise method because the outputs look more like what we would see on a scope.

 

[Winterstone]

The "most recent circuit" is the one from Winterstone in post #78. here the negative feedback path requires R5/R6=0.5. When I simulate this, the output latches to one of the power supply rails. It doesn't oscillate. I don't claim this is necessarily the correct answer, but it is what I got. It seemed reasonable to me, but I'm planning to double check the equations and implementation later today.

Of course, it is the correct answer - I got the same results and we can justify this behavior using "our list": Point (4) is violated. The roots are moving into the LHP for RISING gain (as you can see via simulation). That`s the "wrong" direction. However, the whole circuit does not work at all for a real amplifier (model) because no stable dc operating point is possible (that is the reason for two additional elements as contained in the first circuit in post#54),

I also simulated the original topology from Winterstone in post #54. This topology uses a DC blocking cap in the positive feedback path. When I simulate this, I get the multivibrator oscillation. I also don't claim this result is necessarily correct, but it also seemed reasonable to me. I'll double check this one too later today.

Here we have a similar situation. The additional elements (capacitor, resistor) enable a stable dc bias point - however, again point (4) of our list remains to be violated (as mentioned above).
In addition, the real amplifier frequency pole introduces a third REAL pole in the RHP for the closed loop. Thus, we have immediate latch-up.

Summary: The behavior of the "pathological" circuits (which fulfill Barkhausen`condition) can be explained using the list of 5 points.
But I agree to MrAl`s comment to cancel condition (2) because it is (a) obvious and (b) a general condition that applies to each electronic circuit.
Thus we have only 4 conditions.

Thank you
W.

 

[MrAl]

Hi again,


I didnt mean to completely cancel condition #2, i meant to replace it with a more definitive requirement.

For an ideal loop of wire we satisfy Barkhausen but it does not oscillate. What makes it oscillate is to induce a current of some frequency that has a wavelength equal to an integer multiple of the length. So it satisfies the first two requirements (gain 1, phase shift 0 at the right frequency) but it cant oscillate because there is no energy of the right frequency. Perhaps this is getting too theoretical here, but that's part of what i am after.

 

[steveB]

Hi again,


I didnt mean to completely cancel condition #2, i meant to replace it with a more definitive requirement.

For an ideal loop of wire we satisfy Barkhausen but it does not oscillate. What makes it oscillate is to induce a current of some frequency that has a wavelength equal to an integer multiple of the length. So it satisfies the first two requirements (gain 1, phase shift 0 at the right frequency) but it cant oscillate because there is no energy of the right frequency. Perhaps this is getting too theoretical here, but that's part of what i am after.

I think we agree with what you are saying. My view is that what you are really driving at here is identifying an important necessary condition. If we say noise is necessary, you can argue that theoretically (for example in a simulation) you can have no noise, but still have an oscillator. Hence, noise is not necessary. If we say a startup transient is necessary, we get the same situation, and so on ... But, if we use your guidance, we can form a clear necessary condition that bypasses all of this.

I think what Winterstone and I are getting at is that this list we formed is a bit simpler because it is a sufficient condition list. If i list "noise" as a sufficient condition, then my list is complete because noise is sufficient to do the job. Then, I can put several things in the sufficient condition (noise, startup transient, initial condition or injected signal etc.). Sure, you can make another list that uses your necessary condition, and your list will be more general. Still, both lists are sufficient condition lists. But, we can take this a bit further if we are talking about sufficient conditions for a real oscillator, in the real world. Since noise and startup transients are unavoidable in the real world, then there is no need to list #2 explicitly. The fact that oscillator is real is sufficient to ensure the startup energy will be there. So, the "real" condition is given as part of the question and the "startup energy" specification is already implied when we ask that question.

However, as you say, if we want to talk theory, then we should at least give a sufficient condition for #2, and if we are going to do that, your argument that it might as well be a more general necessary condition, rather than one of many possible sufficient conditions, is a good one. The proof that this is important in theory is that I can make a simulation program that has all other requirements, but if I don't give the system a (1) kick, (2) noise, (3) a startup signal, or (4) an initial condition for at least one state variable, then it won't oscillate.

 

[MrAl]

Hello again Steve,


Well, i assume we are after a 'rigorous' definition, not a willy nilly one

Lets concentrate on noise for now to reduce the complexity of discussion a little...

So, what if we have noise but are missing the required frequency?

But my stronger point is like this...

Say we are given the simple algebraic equation:
y=x-10

and we need to solve for the x that produces y=0. This is a simple equation that's all with maybe the added feature that x may have more than one solution, or it may have only one solution (before we actually try to solve it of course).

Now would it make sense to say that the set {1,10,100} is 'sufficient' to solve this equation? Sure, that set is definitely sufficient, but only because it contains one particular element, the number 10. All the other numbers (1 and 100) are a waste of time.
But what about the set {1,20,100}, is that sufficient? No of course not, because it does not contain the number 10.

So what should we say about this? Well, one set was sufficient and the other was not, and in both cases it was because the set either contained the right number or was missing the right number. So what this means is that if we want to specify a set rather than a single number as solution, then we have to make sure that when we specify this set that it contains the right number.
But all that is a lot of extra work that we dont need because we can do it much simpler by specifying one single number and be done with it. Then we leave the 'set' up to the end user, so that if they want to use a set then they are responsible for making sure that the right number is contained within that set. It is up to them to figure out if the noise in their system contains the right harmonic.

I believe this is the better way to state the requirement, and then we dont have to go through the trouble of defining what knid of noise works and what doesnt, and of course the amplitude also. But if you would like to offer a counter example i'd like to read it.

But by now you must also see how silly it would be to specify a whole set when one number would do fine.

 

[steveB]

As I've said, I understand and agree. Still, I can agree and contemplate other viewpoints that I also find acceptable.

Please state your preferred wording for condition #2. I'm sure it will be very suitable, but it would be good to see it explicitly written.

 

[Winterstone]

Hello MrAl and Steve,

I must confess, I don`t see the necessity to discuss about noise and/or other possibilities to start oscillations.
Real oscillators (hardware) will start always for two reasons: (1) There always will be a switch-on transient and (2) real amplifiers always will cause a loading effect of one or more capacitors - thus, causing a kind of "movement" within the circuit (disturbed equilibrium).
I think, there will be only one exception where an artificial "kick-off" is required: If during circuit simulation an ideal opamp model without any delay and without any output offset is used.
In all other cases (simulation with real opamps or real hardware) the simulation will automatically start.

 

[ericgibbs]

hi all,

On taking time to look at what the online web documentation regarding the Barkhausen criteria for a circuit to oscillate.

I read that this subject has been often discussed/debated in great detail at many levels and no unequivocal resolution has ever been produced.

IMHO the original question 'Is there a rigorous oscillation criterion?' is ambiguous and no definitive provable, repeatable condition can be stated.

E.

EDIT:
I should also add that I have been reading the posts on this thread and noting the amount of time and patience our members are putting in trying to find a definitive solution and the difficulty entailed, is what prompted me to do some online reading of the subject.

 

[MrAl]

Hi Eric,


Thanks for joining the thread


So you think there is no reasonable conclusion? Some things that are more complicated take more time to resolve. Perhaps you think this is not the forum for such a discussion?

Steve:
I'll see what i can come up with, and i'll search on the web for other proof too. Usually when i am this sure of something it turns up somewhere else too...not all the time, but many times it turns up the same or similar, so i'll look around. We can also talk in PM's if you like.

 

[steveB]

... no definitive provable, repeatable condition can be stated.

Proving things like this can be difficult, and perhaps even impossible. I think what we've done is establish a prototype version of a sufficient condition list. We are now refining the wording and considering whether the list is either incomplete of over-specified. Still, once we're done with that, we haven't proved it's rigorously correct. We then have to ask if anyone can think of a circuit that meets these conditions, but still doesn't oscillate. So, we will never prove our case, but someone might easily disprove it with one example.

 

[ericgibbs]

So you think there is no reasonable conclusion? Some things that are more complicated take more time to resolve. Perhaps you think this is not the forum for such a discussion?

hi Al,

I dont see how the OP's question can be resolved, its the word 'rigorous' in the title, which my opinion makes the question ambiguous.

In my understanding of 'rigorous', means 'precise and without doubt'.

Can you honestly formulate such a condition that will encompass ALL the possible marginal combinations required in order to fully answer the question the OP has asked.?

Regards
Eric

 

[ericgibbs]

hi steve,

I agree that the group have proposed 'sufficient' conditions to answer the OP's question.

As you say proving it would be close to impossible.

As far I can see no one has considered the temperature effects on the imagined marginal circuit, ie: would a practical circuit oscillate over a range of temperatures.

Eric

 

[Winterstone]

hi all,
On taking time to look at what the online web documentation regarding the Barkhausen criteria for a circuit to oscillate.
I read that this subject has been often discussed/debated in great detail at many levels and no unequivocal resolution has ever been produced.

Eric, just because no "unequivocal resolution" has ever been produced, some further considerations make sense.
Do you suggest - because, up to now, no sufficient criterion was formulated - we shouldn`t try it all to collect some novel thoughts about the problem?
Isn`t this - among others - one of the aims of this forum? If I am wrong - please correct me.

IMHO the original question 'Is there a rigorous oscillation criterion?' is ambiguous and no definitive provable, repeatable condition can be stated.

I don`t think it is "ambiguous". It is a known fact - even stated in Wikipedia - that the Barkhausen criterion is a necessary one only.
Thus, don`t you agree that it makes sense to search for some additional requirements, which make the criterion more "rigorous"?
From where do you know that "no definitive provable, repeatable condition can be stated" ?.

I should also add that I have been reading the posts on this thread and noting the amount of time and patience our members are putting in trying to find a definitive solution and the difficulty entailed, is what prompted me to do some online reading of the subject.

You can be sure that I also did some "online reading" before. And not only "online" - I have consulted several textbooks with high reputation.
In case I had found a satisfying answer, I wouldn`t ask for some new thoughts in this forum.

Regards
Winterstone

 

[Winterstone]

As far I can see no one has considered the temperature effects on the imagined marginal circuit, ie: would a practical circuit oscillate over a range of temperatures.
Eric

As far as my question is concerned - the temperature plays no role at all. This is a pure design and tolerance problem only. Primarily, I am interested in the system aspect of oscillator topologies.

 

[ericgibbs]

You can be sure that I also did some "online reading" before. And not only "online" - I have consulted several textbooks with high reputation.
In case I had found a satisfying answer, I wouldn`t ask for some new thoughts in this forum.

hi,

This is the very point I making regarding the wording of your original question. 'Is there a rigorous oscillation criterion?

You are already aware from all your previous reading that no rigorous criteria has been found.

I will ask you the same question as I asked MrAl.

Can you honestly formulate such a condition that will encompass ALL the possible marginal combinations required in order to fully answer the question the OP has asked.?

A simple yes or no, would suffice.
E.