Is there a rigorous oscillation criterion?

[Winterstone]

My first example shows that the amplitude of oscillation is Constant with LG=1. In order for the amplitude to increase spontaneously, LG must be >1.

That is why I list LG>1 as a necessary condition. Absent LG>1, in a Practical oscillator, the output will never increase to a useful level.

Yes - everything in accordance with theory. Was there any doubt?

 

[steveB]

So Mike's necessary conditions for a practical oscillator:

1. Initial Perturbation
2. Loop Gain > 1 (not Gain=1)
3. Loop Phase shift multiples of 360 deg.
4. Amplitude limiter

MikeMl,

Let's look at the list I put up previously to your post, and compare to your list.

1. loop gain > 1 for small signals
2. gain saturation to allow for gain=1 (positive feedback mode)
3. frequency selector (filter) (although sometimes the gain spectral response does this automatically)
4. 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.)

Your #1 corresponds with my #4
Your #2 corresponds with my #1
Your #3 corresponds with my #2 part in parentheses (positive feedback)
Your #4 corresponds with my #2 gain saturation

We are pretty much saying the same thing. You stressed the phase condition better than I did, and I stressed the frequency selection better than you did. But, clearly we both (and indeed all of us contributing to this thread) understand all of the important factors.

The only thing I'm having trouble with is that you seem to be dismissing the gain=1 part, even though it is considered to be a necessary condition. You keep saying "not gain=1" and gain>1 because of the startup condition. But the startup is only one part of it.

Do you agree that gain=1 is important for the sustained oscillations? If not, why? If you do agree then we are all pretty much saying the same thing.

The main issue here is that the limiter (or gain saturation) is responsible for limiting the gain, and allowing gain=1 for a particular signal amplitude.

 

[MrAl]

MrAl, I wonder why it is not possible to communicate with you on another level
(Quotes: "I have a feeling you still dont understand this", "Go back and read the whole thing")
Is this your style to discuss technical matters?

I think, it is a normal procedure to quote that sentence only which needs a comment.
And a sentence like

"Note that in this view we can keep the pole pair in the RHP and dont have to move it back into the LHP"

contains a clear statement that in my view is not correct. That`s all.
If you like to explain or modify/correct your statement, it is up to you. I only gave my comment to this sentence.
Nevertheless, I am very interested to see the proof of your claim as announced by you (Quote: "If you need a proof of this i may be able to supply one ") .
I am looking forward.
W.

Winterstone:

In that one sentence "Note that...etc" even *i* might disagree with it, that's why i said you have to take it into the context of the rest of the writing.
So at the very least you are not telling me anything new (about that one sentence taken alone like that). It makes it look like a pole pair in the RHP is somehow always ok.
But to be honest i dont like being bothered about things like this. I would rather spend the time thinking about solutions to the Great Question, "Is there such a thing as a purely linear oscillator?". My best bet right now is on a linear oscillator with a linear gain control circuit where we get the pole pair so darn close to the jw axis that it can not be measured to be otherwise in any application that would use an oscillator like this. So i like to widen the required gap a little here too into the realm of necessity over perfection.

 

[Winterstone]

The main issue here is that the limiter (or gain saturation) is responsible for limiting the gain, and allowing gain=1 for a particular signal amplitude.

Hi Steve, everything OK and agreed - however some short comments from my side to the above condition.

1.) The condition gain=1 is correct for an overall control loop (FET or something else), although in reality we have a small variation around "1" due to the time constant of the rectifier circuit causing a small AM on the output signal. This is equivalent to a periodic pole movement across the jw axis.

2.) For amplitude limitations using diodes it seems to be somewhat more complicated because we have to distinguish between two cases:
(a) Does the diode influence the amplitude directly (as in Mike`s example), or
(b) Does the diode act as damping resistor (parallel to a capacitor) that influences the PHASE of the loop function. In this case, it is "problematic" to speak about unity loop gain.
The decision between (a) or (b) depends on the particular oscillator circuit as well as the place where the diodes are connected.

3.) Without any additional stabilizing circuitry we have hard-limiting due to the supply rail. I think, in this case, we are not allowed to use the term "gain" for the whole signal because we have a non-linearity (no sinusoidal signal anymore). Instead, we have the possibility to use the principle of harmonic balance and can apply the condition gain=1 on the fundamental part of the signal only.

This results in a new oscillation condition:
Hf*G*D=1
with Hf=passive transfer function, G=gain of active element and D=Describing function due to hard limiting.

Thank you
Regards
W.

 

[steveB]

Without any additional stabilizing circuitry we have hard-limiting due to the supply rail. I think, in this case, we are not allowed to use the term "gain" for the whole signal because we have a non-linearity (no sinusoidal signal anymore). Instead, we have the possibility to use the principle of harmonic balance and can apply the condition gain=1 on the fundamental part of the signal only.

Yes, that is exactly my point. An amplitude limiter can still be analyzed as a gain saturation component if done using Fourier components. The thing that oscillates is the fundamental (with some harmonics that we try to minimize). A system with limiter is set up to have gain greater than one when the output is less than the rails. In the other extreme, with large inputs, the output is a square wave which limits the output of the fundamental. So as the input goes up beyond a certain point, the gain of the fundamental gradually decreases. This ensures that there will be an amplitude that has gain=1, and that's the amplitude we will see. More importantly, this is now stable in the "nonlinear" sense. A linear analysis around this operating point would say we have marginal stability, but a nonlinear analysis would reveal very robust stable operation. Perturbations to the system might cause small variation of frequency and/or amplitude, but this is like a ball rolling at the bottom of a valley. It's still stable.

Another thing to consider is that, for OPAMPS, sometimes it is the slew rate limit (at higher frequency oscillations) that really saturates the gain. This would depend on the particular op-amp of course, but, slew rate can create reduced gain for larger amplitude signals.

So yes, there can be wide variation of the fine details in various circuit based oscillators, and then even beyond circuitry, the principles apply to lasers and mechanical systems too. But, the basic ideas are still there in all these cases, which I think is what you are trying to figure out with this question. What are those basic things we need as sufficient conditions? Gain saturation, - however you label it, describe it, analyze it, or interpret it, - seems to be important.

 

[Winterstone]

I would rather spend the time thinking about solutions to the Great Question, "Is there such a thing as a purely linear oscillator?". My best bet right now is on a linear oscillator with a linear gain control circuit where we get the pole pair so darn close to the jw axis that it can not be measured to be otherwise in any application that would use an oscillator like this. So i like to widen the required gap a little here too into the realm of necessity over perfection.

Hello, well - this sounds good. Let`s come back to pure technical things.
As you have indicated, a "purely linear oscillator" is not possible.
Yes - I agree, of course, because of two reasons:
* No amplifier is really linear
* There is no way-out: We need an amplitude stabilizing mechanism, which - by nature - is non-linear.

But there is a problem: I think, it is not possible to place the pole pair so "close to the jw axis" as we would like to do.
I think, the reason is as follows:
In order to start safely (in a reasonable time) and to cope with tolerances and other uncertainties (parasitic influences) that cannot be avoided, we must include something like a "safety margin" in the pole location for start-up (at t=0).
That means, the regulation mechanism has to "live" with a considerable difference between the conditions at t=0 and t=infinity (I hope I could express my self clear enough).
Example: If we could design a Wien oscillator with a gain of 3.001 (but we cannot!) a pair of diodes across the feedback resistor would introduce a negligible distortion only.

But of course, as you have indicated, it is the aim of each engineer to design an oscillator that comes as close as possible to the ideal device.

Thank you
Regards
W.

 

[Winterstone]

Another thing to consider is that, for OPAMPS, sometimes it is the slew rate limit (at higher frequency oscillations) that really saturates the gain. This would depend on the particular op-amp of course, but, slew rate can create reduced gain for larger amplitude signals.

Oh yes - indeed. I remember some published articles with a title like this: "Slew rate stabilized oscillators".
That means, in principle it is possible to exploit the slew rate SR to limit the amplitude. I think, in fact it is the delay (resp. phase shift) caused by the SR that stops further rising of the amplitudes.

 

[MrAl]

Hello, well - this sounds good. Let`s come back to pure technical things.
As you have indicated, a "purely linear oscillator" is not possible.
Yes - I agree, of course, because of two reasons:
* No amplifier is really linear
* There is no way-out: We need an amplitude stabilizing mechanism, which - by nature - is non-linear.

But there is a problem: I think, it is not possible to place the pole pair so "close to the jw axis" as we would like to do.
I think, the reason is as follows:
In order to start safely (in a reasonable time) and to cope with tolerances and other uncertainties (parasitic influences) that cannot be avoided, we must include something like a "safety margin" in the pole location for start-up (at t=0).
That means, the regulation mechanism has to "live" with a considerable difference between the conditions at t=0 and t=infinity (I hope I could express my self clear enough).
Example: If we could design a Wien oscillator with a gain of 3.001 (but we cannot!) a pair of diodes across the feedback resistor would introduce a negligible distortion only.

But of course, as you have indicated, it is the aim of each engineer to design an oscillator that comes as close as possible to the ideal device.

Thank you
Regards
W.

Hello there Winterstone,

I have to agree now with most of what you are saying. But i just want to take a second and point out my differences.

First, i do agree with your first point, "No amplifier is purely linear", but i want to draw a distinction between what realm we are working within. Are we working within the practical universe or the theoretical universe.

In the practical, nothing is linear, so we in effect cancel out just about every circuit that we can make. So if we want to rule out all amplifiers we effectively rule out every circuit under the sun and that does not do us much good because then we can always throw our arms up and exclaim, "Oh this is not linear, and that is not linear, and neither is this...so nothing is linear so why bother", etc. So by stating that every amplifier is non linear and thus we can not have a linear oscillator is not going to do us any good.

But in the theoretical world such a monster does exist, so that gives us a tool for study of an amplifier that could in fact meet the criterion we need to design a real life linear amplifier, where the linearity is accepted world wide as a linear design not a non linear design. In other words, no diodes or such for limiting, but we could use real world amplifiers in the final design and that would be acceptable. In the theoretical framework however, we would use a perfectly linear amplifier and this would help us study not only the effects of the theoretical circuit but also the non theoretical. So the search for the answer to the Great Question should start from a theoretical platform and proceed later to a practical one once we know what we need. It's only after that point that we begin to accept the tiny errors and they can be made exceptionally tiny.

In theory as i am sure you know, integrators are considered linear and so are simple amplifiers. In fact amplifiers are just a gain like Av with no frequency specification. So we have these theoretical tools to work with and design a purely linear oscillator. Once we have that designed we have answered the Great Question in theory, which is really all that we can do anyway, and then take it to the real world and use real world components. We could then look at optimizing parts of the circuit to work closer and closer to the ideal and thus obtain the most practical linear oscillator that could ever be built, or at the very least, one of the most practical.

So our first goal is to design a theoretical purely linear oscillator, then later we can take it to the practical. In theory i believe we can do this by using amplifiers, integrators, and passive networks.

Does this make sense to you?

 

[Winterstone]

So our first goal is to design a theoretical purely linear oscillator, then later we can take it to the practical. In theory i believe we can do this by using amplifiers, integrators, and passive networks.
Does this make sense to you?

I think, that`s what we (I mean: engineers around the world) are doing since a long time. No doubt about this.

My only concern was your sentence (which gave rise to my reply about ideal and "pure" linear circuits):

"I would rather spend the time thinking about solutions to the Great Question, "Is there such a thing as a purely linear oscillator?"."

And my answer, of course, is: NO.

 

[MrAl]

I think, that`s what we (I mean: engineers around the world) are doing since a long time. No doubt about this.

My only concern was your sentence (which gave rise to my reply about ideal and "pure" linear circuits):

"I would rather spend the time thinking about solutions to the Great Question, "Is there such a thing as a purely linear oscillator?"."

And my answer, of course, is: NO.

Hello Winterstone,

Well again you read my post and sorry but it seems that you are missing the point, at least that's what it looks like to me here again.
But let me state it again: i was drawing the distinction between the practical and the theoretical.

Your comment about the linear oscillator seems again aimed at the practical, that we all agreed on already.

But do you agree that there could be such a perfectly linear oscillator in theory? That is, a theoretical purely linear oscillator?

 

[Winterstone]

"I would rather spend the time thinking about solutions to the Great Question, "Is there such a thing as a purely linear oscillator?"."

As it seems - again a misunderstanding.
To me the sentence "is there such a thing as..." regards a practical implementation. Otherwise, I wouldn`t use the terms "is there" and "thing" (a pure product of my brain is something else).
However - now it`s clear. No problem at all. Of course, in a simulation environment I can create such an ideal circuit.

 

[MikeMl]

...I remember some published articles with a title like this: "Slew rate stabilized oscillators".
That means, in principle it is possible to exploit the slew rate SR to limit the amplitude. I think, in fact it is the delay (resp. phase shift) caused by the SR that stops further rising of the amplitudes.

Isn't a Slew rate stabilized oscillator just a Multivibrator?

 

[Winterstone]

Isn't a Slew rate stabilized oscillator just a Multivibrator?

No, I don`t think so. If the frequency is high enough to reach - at a certain amplitude - the slew rate limits there will be an additional phase shift of the signals violating the oscillation condition. Thus, amplitudes are limited. However, for my opinion, not a very practical solution because (a) the slew rate has some tolerances, (b) just a small frequency range is possible and (c) the amplitudes cannot be chosen.
But it is rather interesting from the system point of view.

 

[Winterstone]

Hello to all,

I think it`s time to come back to the main question of this topic.
In my first post I did mention that there are some circuits which fulfill Barkhausens condition - but they don`t oscillate as expected.
This confirms that this condition is a necessary one only.
In post#25 I have announced a corresponding example.
Well - here is it: (see the attached pdf document).

The lower graph shows the loop gain magnitude, which reaches 0 dB at approximately 130 Hz (loop gain slightly above 0 dB).
The other graph is, of course, the phase response.

Thus, one could expect that the circuit will oscillate at 130 Hz (app.). But instead, the tran simulation reveals a squarewave above 20 kHz (not shown here). As a consequence: Barkhausen is fulfilled, but no corresponding oscillations.

Explanation to the circuit:
It is one of the 8 possible derivates based on the classical WIEN configuration - but not with equal resistors and capacitors. But nevertheless, with equal time constants R3C3=R4C4.
In order to avoid dc positiv feedback a capacitor C44 is added.
To enable a dc bias current for the non-inv. opamp input a resistor R33 was added.

It is to be mentioned that the same behaviour could be observed for an idealized opamp (as shown in the diagram) as well as a real opamp model (TL082).

Why no oscillations at 130 Hz? Does anyone have an explanation ?

Winterstone

 

[MrAl]

Hello to all,

I think it`s time to come back to the main question of this topic.
In my first post I did mention that there are some circuits which fulfill Barkhausens condition - but they don`t oscillate as expected.
This confirms that this condition is a necessary one only.
In post#25 I have announced a corresponding example.
Well - here is it: (see the attached pdf document).

The lower graph shows the loop gain magnitude, which reaches 0 dB at approximately 130 Hz (loop gain slightly above 0 dB).
The other graph is, of course, the phase response.

Thus, one could expect that the circuit will oscillate at 130 Hz (app.). But instead, the tran simulation reveals a squarewave above 20 kHz (not shown here). As a consequence: Barkhausen is fulfilled, but no corresponding oscillations.

Explanation to the circuit:
It is one of the 8 possible derivates based on the classical WIEN configuration - but not with equal resistors and capacitors. But nevertheless, with equal time constants R3C3=R4C4.
In order to avoid dc positiv feedback a capacitor C44 is added.
To enable a dc bias current for the non-inv. opamp input a resistor R33 was added.

It is to be mentioned that the same behaviour could be observed for an idealized opamp (as shown in the diagram) as well as a real opamp model (TL082).

Why no oscillations at 130 Hz? Does anyone have an explanation ?

Winterstone

Hi,

Sorry, but the time constants are not equal. One is 2 times less than the other right?

 

[steveB]

Winterstone,

May I ask for clarification on your circuit? What is the purpose of the voltage source? Is it a startup signal to initiate the oscillations, and needed for simulation purposes only?

 

[MrAl]

Hi Steve,

Isnt that for testing the response?
I'll wait for Winterstone to clarify too though, along with the time constant difference.

 

[steveB]

Hi MrAl,

That was my other thought, but I would expect the loop to be open for the response test.

 

[steveB]

Winterstone,

So, here is my guess at the answer. I derived the open loop transfer function and did a bode plot. The general shapes of the magnitude and phase seem to match your plots very well to the eye. When I zoom in at 130 Hz, I find the following.

MAG=-0.35 db
Phase=1 degree

Hence this circuit does not meet the Barkhausen Criteria. We require a perfect match. This gets back to one of my original comments from post #6.

... 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 ...

But, now the question would be, why doesn't the nonlinearity, which saturates the gain, allow the oscillation? Well, here the gain never exceeds 0 dB in the region of 130 Hz (assuming I derived correctly), so saturation can not give 0 dB here. This is why one of the sufficient conditions (which is also a necessary condition) was to have gain>1 at startup.

However, above 20 kHz, the gain does exceed 0 dB (it's in the 3-4 dB range), so the oscillation can start, and then the saturation/limiting can bring the gain down to 0 dB. Also, the phase in this region is in the 2 to 10 degree range, so a 0 degree condition is not unexpected in the nonlinear limiting condition at some frequency in that regime.

I've attached my Bode plot, but note that my frequency scale is in rad/s while your is in Hz.

 

[MikeMl]

Hello to all,

I think it`s time to come back to the main question of this topic.
In my first post I did mention that there are some circuits which fulfill Barkhausens condition - but they don`t oscillate as expected.
This confirms that this condition is a necessary one only.
In post#25 I have announced a corresponding example.
Well - here is it: (see the attached pdf document)....

The circuit you posted has the Wein network inverted, which is a horrible mess as far as making a reliable oscillator.

You are not breaking the loop correctly to discover how horrible that network actually is. Here is your circuit with the correct way of breaking the loop to see the loop gain/phase. To turn this into the oscillator, remove V1, and make the dashed line connection.

Note that there are three places where the phase shift is zero! Which do you think it is going to oscillate at? I'm not surprised that you saw a near square wave at > 20kHz, because the loop gain at ~185kHz is 1.5 where the phase shift is ~0 deg. (Cursor 2).

Note that the loop gain at Cursor 1 is not high enough to oscillate, too.