Is there a rigorous oscillation criterion?

[Winterstone]

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.

Sorry, but I cannot answer with a simple yes/no and - more than that - I cannot understand the whole question, because:

If I could now, I wouldn`t try to discuss this matter in the forum!

Regarding the term "rigorous": You know that the english language is not my mother tongue and, thus, it is very probable that I from time to time cannot find the most appropriate term.
In such a case, I am always grateful for a correction.


By the way (because I`ve got the impression that you don`t like continuing this thread):
There are already more than 500 contributions in the following thread - and it seems they didn`t come yet to an agreement if there is a current or not.
https://www.electro-tech-online.com/threads/ac-flowing-through-a-cap-what-actually-happens.116575/

 

[steveB]

If I could give my opinion about this? ...

My first words in this thread were

That's a good question, and I've never seen a good conclusive answer on this.

As I entered and continued discussing, I wasn't under any delusion that we could fully succeed where others have failed. Rather, because it is a good question without a rigorous answer, it is a subject worth talking about to see exactly what can be said and what guidelines can be made useful.

So, if the title is a problem, then I suggest that we should say the answer to the question is ... No. But a "no" answer doesn't mean there isn't much of interest to discuss on this subject. If we can't find something rigorous, then let's at least find something useful.

 

[ericgibbs]

By the way (because I`ve got the impression that you don`t like continuing this thread):

There are already more than 500 contributions in the following thread - and it seems they didn`t come yet to an agreement if there is a current or not.

hi,
I never said I did not like the Thread?

You must be aware there are some thread topics that will never have an agreed answer.

In threads like that, do we continue posting counter arguments until either of the opposing groups just gives up out of frustration.?

I would suggest a fairer and more helpful solution, that would help future readers, is if the OP posted a summary, highlighting the views of both sides.

For this thread, why not prepare a summary of the conditions that have been posted to date that are required to part answer your question,?

Hopefully, after further research, you could come back at some future date and post your findings.

E.

 

[Winterstone]

Steve,
thank you very much. I am 100% with you.
If we do not ask questions we will get no answer.

In spite of the fact that I am "already aware from all ...previous reading that no rigorous criteria has been found" up to now (!), I am a bit curious - and this gives me some motivation to make up my mind and to ask some other people. Thus, I try to collect some (perhaps novel) ideas. Should we stop to ask questions just because - up to now - no answer was given?

W.

 

[Winterstone]

[MODNOTE]Deleted, A PM reply will be sent.[/MODNOTE]

 

[Winterstone]

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?

MrAl, I don`t know if you will get an answer to this question.
Unfortunately, the objective and technique-oriented discussion has stopped. Why?
W.

 

[MrAl]

Hello Winterstone,


I just think we are all just thinking about it more and seeing how we can improve one what we've come to know here.

I think even if we dont come up with a perfect result, we can get better than what is already out there on this subject. For one, we all agree that some start up energy is required. We can look at other things too. So maybe we wont find the perfect answer, but we should be able to improve on what we already have. We can then throw it out there and wait and see if anybody ever disproves it. That's the way it works sometimes with more complicated ideas where it's hard to prove one way or the other at first but several things look like they should be as they are. Once stated, then someone else can rise and say, "I've found a counter example", and disprove it. But if that never happens, then we're good to go. That's the way it works sometimes.

 

[Winterstone]

MrAl - I see that you have some experience in treating matters like this which have no simple answer.
Thus, I agree 100% with you. However, I am not sure if we will continue...

W.

 

[MrAl]

Hello again,

I thought i would semi conclude my contribution to this thread with a simple statement about what it looks like to me that is required for oscillation. If you guys have another idea other than this that is fine too, i'll be happy to consider yours as well. I assume we are talking about a sine oscillator, because i think if we broaden that ground rule we'll have to think considerably harder and even go as far as to try to classify every type of oscillator under the sun, and i am not prepared to do that (square, triangle, chaotic oscillator are excluded for example).


1. Phase shift of zero at the operating frequency (Barkhausen directly).
2. Gain of 1 at the operating frequency (Barkhausen directly).
3. Start up energy in the form of a frequency which must be the frequency of operation, or equivalent time domain initial value stuffing.
4. No other poles in the RHP.
5. A continued supply of energy (power supply).
6. Noteworthy is the energy gained while operating temporarily in the RHP must be lost while operating in the LHP as the pole pair shifts back and forth.

As discussed with Steve, #3 can sometimes come from a noise source or the power supply being turned on if that one harmonic is present.
Of course #5 goes almost without saying but i thought i would included it in case this came up, because there are oscillators that dont have this requirement so somebody somewhere someday is going to ask, "Where's the power supply ! ?" <chuckle> so i thought i would save them the trouble

 

[Winterstone]

Hello MrAl,
[MODNOTE]Do NOT discuss Moderation[/MODNOTE]

* point 1) and 2) apply to the loop gain only, right?
* What about loop gain>1 at start-up?
* Why do you suppress the additional conditions I have found (and mentioned in post#102 and #107)?

Presently, I am trying to convert the whole set of conditions into a form that enables usage of the Nyquist diagram.
But I don`t know if I will have the opportunity (and the motivation) to present the results at this place.

Winterstone

 

[MrAl]

Hi Winterstone,

Well, if you have #3 satisfied then you dont need loop gain > 1 right?

Suppress what? You mean like a DC gain less than unity? Well if there is no DC then it doesnt matter, it only matters if there is DC. So i am not sure how you want to handle this. As i said before, i wanted to start with a purely theoretical model and go from there. We can always add imperfections once we have the ground floor plan worked out. That was always my intended route to paradise. Once we have that defined, we can then go on to see what can go wrong in the real world and then later what we can do to fix it. If we dont do it this way we'll be here forever...what happens if a resistor is out of tolerance, what happens if a capacitor is out of tolerance, what happens if there is noise on the power supply line of a frequency close to the operating frequency...we cant cover everything so we cover the most theoretical. What happens if lightening strikes the circuit..."Condition #98234: Make sure the circuit is such that when lightening strikes the gain can keep up with the abrupt change in dv/dt..."

 

[steveB]

MrAl,

That's a good list. Like Winterstone, I also miss not having the gain>1 explicit. I like having that clearly shown because most people think of this when then build an oscillator. I know there are other ways to say it (did you mean #6 and not #3?), but Winterstone did have an underlying goal of being able to relate conditions back to the open loop response, and gain>1 for the small signal linear conditions at startup is just so easy to understand and visualize, that I miss not seeing it there.

I'm satisfied with your #3 condition for startup energy. There are many ways to say this. In some sense, removing it from the list makes sense, but I like seeing it explicit because if I were introducing oscillators to a student, I would want to mention this. I remember it was clearly introduced to me when I was a student, and the idea has stuck in my mind because of that. It's a weird condition because it really doesn't ask us to do anything with our design, while all other conditions require that we do something. Even condition 5 requires that we turn the switch on. How many times have we forgotten to do that, and scratch our heads for several minutes before we realize how stupid we are.

I also like your mention of the power supply. How did we miss that one? I guess it was too obvious. So initial energy (kick) and continual source for power flow (fuel) are stressed in your list.

Winterstone,

I hope you keep motivated to pursue posting here. I have a strong feeling that you will make progress, and I would be interested to see the results. I plan to look at this more in my spare time too. If I make some progress on the theory side, I'll post it here too.

 

[MrAl]

Hi again Steve,


I mentioned #3 because if we have the oscillation frequency then we shouldnt need loop gain > 1. Barkhausen didnt need that right? But maybe this is more theoretical then Winterstone wants to be right now.

One of the other things i wanted to stress is that if the power supply turn on is too slow, the oscillator may not start. This could happen in a system like a solar system. Should we worry about this? It's interesting and it does happen so i think we should.

The question i asked you before was how did you see one oscillator work and the other didnt work. The standard worked and the inverted oscillator did not work when you did a simulation i guess. Then i mentioned the two gain resistors. Then you said you had them right.
Well what i was driving at was if you set the resistors correctly (slightly off from their ideal values) you should be able to see both oscillators either:
1. Have an amplitude that rises continuously until the output saturates.
2. Have an amplitude that falls continuously until it reaches zero output.
3. Have an amplitude that stays constant for a very long time.

But 3 above can only happen in the perfectly ideal case and that's not possible even in simulation i dont think, without some non linear gain of course. Let me elaborate.

In the time domain response, we see a sinusoidal part and an exponential envelope:
Vout=e^(a*t)*(sinusoidal_part)

and when a=0 the exponential part becomes equal to 1 so we get:
Vout=sinusoidal_part {ideal oscillator}

and 'a' depends on the component values. In particular, when we set the gain resistors we can change 'a' to make the exponent of the exponential part either positive or negative as well as zero. If we set them to get 'a' negative, we get a decreasing output over time, and when we set them to get 'a' positive we get an increasing output over time. So one way we get saturation and the other way we get zero. It's only when a=0 that we get a perfect output that lasts theoretically forever.

But it's not really possible to get a=0 perfectly because there are limitations in the simulation environment. So what this means is that the oscillator output will be either ramping up or ramping down. But when 'a' is very very close to zero, this make still take a LOT of cycles to see happen if the resistors are chosen correctly. So what this means is that we should be looking for a condition which causes the oscillator to ramp up, and another condition for it to ramp down, and make sure it can do both. This gives us an idea that it is working right because if we did have the right values it would work in theory.

I bring this up because i see both circuit working this way and Winterstone did too, while you mentioned that you saw the inverted version only saturate. So perhaps you can see if you can get the inverted version output to ramp down by lowering the gain slightly. What i am wondering is how you saw it ramp up only while both Winterstone and myself had seen that it can be either way. I also wonder if we can get the real life inverted circuit to ramp down or is that just in simulation.

 

[steveB]

I mentioned #3 because if we have the oscillation frequency then we shouldnt need loop gain > 1. Barkhausen didnt need that right? But maybe this is more theoretical then Winterstone wants to be right now.

I guess I'm just not following your logic here. It seems that suddenly you dropped the gain>1 and gain compression criteria. While my mind can see how #6 might tie in to both of those conditions, and could possibly replace them (too subtly for my taste), I don't see why #3, which relates to startup energy, should replace those. I don't know about Winterstone, but that is too theoretical for me, even if I get to the point of understanding it. I don't see why sufficient conditions should not be very clear and obvious, and more in line with things we need to do to actually make the oscillator.

My guess is that you are trying to focus back in to the linear conditions, and not the nonlinear ones.

One of the other things i wanted to stress is that if the power supply turn on is too slow, the oscillator may not start. This could happen in a system like a solar system. Should we worry about this? It's interesting and it does happen so i think we should.

Are you talking about "planets and a sun" or "solar cells and a battery"? If this could happen to an oscillator circuit, while all our supposed sufficient conditions are in place, then yes, of course we should worry about this. It means we failed to identify all important conditions to complete the requirement of being "sufficient". Can you give an example circuit? An example would help us identify the cause and identify the missing condtion(s).

The question i asked you before was how did you see one oscillator work and the other didnt work. The standard worked and the inverted oscillator did not work when you did a simulation i guess. Then i mentioned the two gain resistors. Then you said you had them right.
Well what i was driving at was if you set the resistors correctly (slightly off from their ideal values) you should be able to see both oscillators either:
1. Have an amplitude that rises continuously until the output saturates.
2. Have an amplitude that falls continuously until it reaches zero output.
3. Have an amplitude that stays constant for a very long time.

Thank you. I understand you now. I did try different values of the gain slightly off from the ideal, but I didn't do that systematically. I don't remember seeing those effects, but then again, I wasn't looking for them. Those facts will give me a way to check and have confidence that the simulation is correct. Originally, I was trying to just see if I had oscillations or not, and it seems that even that behavior indicates no stable oscillation. But, I will check this.

Let me be clear about how I derived the simulation equations, in case this will affect the result. I used an Opamp modeled with finite gain (10000 to 100000) and a first order compensation pole at ω=10 to 100 rad/s.

By the way, in your simulations, how to you input the initial energy? Is it a starting signal at the right frequency? or noise? or initial conditions? Or, if you didn't do a simulation, will these different ways of starting affect the behavior you outlined in 1, 2 and 3 above?

 

[MrAl]

Hello Steve,


Well i find myself in between the purely theoretical and partly practical because one of the things i saw as important was the start up power supply dv/dt, but yet i would want to mention that as a side issue rather than the main issues. For the main issues, i was trying to build on Barkhausen with as little added as possible. That's because if Barkhausen could get away with just those two criteria, then to add to that theory we probably only need to add one or two more things. To get a practical list however i believe that is another story and deserves a different kind of coverage because then we have to talk about all sorts of things. But yes, i was including start up power supply dv/dt, but if we have the right frequency to start up with as excitation then we dont need to worry about that either because the power supply rise generates the correct frequency but if we have already specified that frequency as a requirement then we dont need to mention the power supply. But the power supply is a concern (with the planetary system so we dont crash into the Sun...just kidding, i read my post over and see how funny that 'solar system' must have looked) because in a system like a photovoltaic system the power supply may rise EXTREMELY slow and that could cause an oscillator to NOT start up properly.
But then there comes the additional concern of another pole in the RHP. If we happen to have that then it could cause an increased output that saturates too.

So i hope i cleared up my intent a bit more here now. I am going for the basic theory first, then the practical as a separate list, but i may get mixed up a little at times because of my great concern about the slow rising power supply, which i will try to hold off with until we get the basic theory down pat.

I have started a simulation with various techniques:
1. Initial value stuffing (caps).
2. Small DC input source which provides a step change which contains a large number of harmonics.
3. A single initial value (one cap) just to see what would happen if the oscillator was turned off and then back on again within a very unlucky time interval where one cap voltage died while the other remained higher.

The initial turn on should not affect the overall performance as in 1,2, or 3 above (in my last post) except if there is no input it should not start up at all because there is no input energy.

I was thinking of going to a more purely theoretical simulation. This means i can get rid of the gain control resistors and do away with the imperfect op amp. The hand analysis i do allows me to plot theoretical responses as well as notice things about the circuit.

As to gain > 1, i was thinking in terms of gain=1 because that's the way Barkhausen first cited it, and with a purely theoretical circuit with gain=1 and oscillation already happening we dont need to move any poles or vary the gain or both. I was thinking along these lines (for now) because there are problems that come up even in this ultra simple configuration. For example, extra poles in the RHP.

 

[steveB]

Hi MrAl,

I feel that I understand what you are saying.

So, on the simulation issue, I was not able to resolve the oscillatory behavior for the circuit in post #78. In simulation with a nonideal model for the opamp (single pole and finite gain), the response is a very quick exponential divergence to the rails. If I take this same model to linear analysis in Matlab, the response is the same, and a single pole deep into the RHP can be seen as the cause. If I use a simple linear analysis for an ideal opamp (infinite gain, infinite bandwidth), the linear analysis does show the oscillatory behavior. I presume that if I implement a proper nonlinear simulation program using an ideal opamp model, I'll get the behavior you mentioned. However, I can't resolve this with a proper opamp model that has a compensation pole included.

My conclusion is that we are using different opamp models, with yours not having the first order compensation pole. This conclusion seems consistent with Winterstone's comment in post number #87. Here he says

"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)."

This is interesting because we are seeing two different effects with these different models. The more realistic model has no hope of ever working because the deep RHP pole drives us to the rail in a microsecond. But, if we could make a circuit like the one without that deep pole, we still have a latchup problem once we do hit the rail.

This effect is what motivated me to have a condition worded as ...

"5. No RHP poles exist to cause instability when gain saturates to G<1"

I think Winterstone didn't quite get this understanding when he commented in post #102. He mentioned "only one complex pair allowed..." and "no real pole". But, even without the real pole, we see a latchup at the rails because the gain saturates and the pole pair moves from LHP to RHP, which causes the latch. We need the pole pair to move from the RHP to the LHP when gain<1. So, I think my statement is worded in a very simple and useful way. Full stability is a good sufficient condition (in conjunction with other conditions too, of course) to require for the linearized model when G<1.

This is why I think that the gain-compression and the stability for gain<1 are important for this discussion, and are good candidates to consider as the core for sufficient conditions for oscillators made in the way we typically make them.

 

[Winterstone]

I think Winterstone didn't quite get this understanding when he commented in post #102. He mentioned "only one complex pair allowed..." and "no real pole". But, even without the real pole, we see a latchup at the rails because the gain saturates and the pole pair moves from LHP to RHP, which causes the latch. We need the pole pair to move from the RHP to the LHP when gain<1. So, I think my statement is worded in a very simple and useful way. Full stability is a good sufficient condition (in conjunction with other conditions too, of course) to require for the linearized model when G<1.

This is why I think that the gain-compression and the stability for gain<1 are important for this discussion, and are good candidates to consider as the core for sufficient conditions for oscillators made in the way we typically make them.

Yes - Steve, you got the point. Even without a real pole in the RHP we have a latch-up effect. See my next reply.
W.

 

[Winterstone]

Hello MrAl and Steve,

thank you for all your „brain-storming“ and corresponding contributions. From all these considerations I have prepared something like a summary that reflects my present knowledge.

Conditions for a practical feedback oscillator (hardware):

1.) Circuit design with a loop gain LG>1 (real, with loop phase=0 for one single frequency and small-signal amplitudes).

2.) Circuit must contain a non-linear mechanism causing a movement of the pole pair from the RHP (LG>1) to the LHP (LG<1) for rising amplitudes. That means: Root locus must enter the LHP.

3.) Only one conjugate-complex pole pair, which is located slightly in the RHP of the s-plane (due to LG>1). In particular, no real pole in the RHP.

4.) The loop gain at w=0 (DC) must be below unity.

You may be surprised, but I think that`s all - as far as practically realized circuits are concerned.

Comments and explanations:

* Provision of a power supply is evident and must not be explicitely listed

* The same applies to a stimulus, which must not be provided separately. The power switch-on transient provides a sufficient start-up impuls. In addition, each real amplifier contains a dc output offset causing a disturbance of the balance conditions (loading of a capacitor at t=0). (Of course, for circuit simulations some „kick-off“ is to be provided).

* To 2.): The reason for this requirement is as follows:
For all known classical oscillators a gain reduction for rising amplitudes causes a movement of the pole pair (from the RHP) closer to the imag. axis and then into the LHP. Thus, amplitudes decrease - and the gain rises again. Thus, the poles move again back to the right. However, there are „pathological circuits“ (see post#54 and #78) which exhibit an opposite behavior. A gain decrease (or supply limiting effects) cause a further enlargement of the amplitudes. Thus, a non-linearity that decreases gain for rising amplitudes is counter-effective. Instead, to limit the amplitude increase, we must increase the gain. This is contradictory and does not work in practice.
This was the reason both circuits (post#54 and #78) did not work but exhibit latch-up.

Comment: It seems that this root locus requirement is identical to a corresponding requirement in the Nyquist plane: The loop gain Nyquist plot must cross the critical point „+1“ in a North-South direction. (This is a preliminary statement only, further investigations necessary).


To 3.): In the example circuit in post#54 the frequency dependent gain of the real opamp model has created a real pole in the RHP. This is not allowed (immediate latch-up without any oscillations).

To 4.): It is obvious that the loop gain for DC must be below unity - otherwise no stable operating point under closed-loop conditions is possible.
____________________________________

Thank you again - perhaps we meet again in another thread or another forum. I am sorry, but this was my last contribution for this thread.

Kind regards to you
Winterstone

 

[steveB]

Nice summary, and I'm on-board with this. Of course we can squabble about small details, but all of the critical information that I would want to see is in there.

I agree, looking at the Nyquist plots, and considering a rigorous derivation along those lines is the thing that requires further investigation. If it works for negative feedback, with care, an analysis for positive feedback should say something clear and useful, and even might be rigorous in a limited context.

 

[MrAl]

Hi again,


I guess i didnt mention that i was using a perfect op amp model. That is, except for the internal open loop gain which i make high. I do this so that i can investigate the properties of the theoretical oscillator better. If i include frequency characteristics of the op amp then i am forever restricted to coming up with theories based on oscillators made with op amps. So i wanted to avoid that for now, but it is certainly clear why you would want to do this.