...............
So if you need to know "why some circuits oscillate and some not" then the answer is one or more of these;
1. the phase shift is wrong
2. not enough energy is input to overcome losses
3. initial start condition was not met (if type B oscillator)

@RB: Thanks for your explanation.
However, I am afraid that does not explain my observations.
May I repeat:
The two examples I have presented in my first posting (and there some other) fulfill ALL three requirements mentioned by you. Nevertheless, they do not oscillate.
This is - in principle - no surprise because Barkhausen`s condition is a necessary one only (not sufficient)!
As I have mentioned already - I have some explanations for not being able to oscillate (real pole in the RHP in addition to the complex pair) and a positive loop gain phase slope at f=fo.
However, the last investigated example (t_n_k`s state-space realization) also has a loop phase with positive slope at f=fo - but it oscillates (even though with some unnormal properties).
Thus, I am not sure anymore, if the negative slope of the loop phase is really a necessary condition for a circuit to oscillate. (Until yesterday I was of the opinion that this an important completion of Barkhausen`s criterion).

 

Looking at the Twin-T equal component case with gain of 4 is interesting. The Bode plot might be the preliminary analytical tool of choice. I feel (only intuitively) the fact that the gain-frequency curve minimum of unity occurring concurrently at the 0 degree phase shift might be the salient feature. The local minimum in gain (albeit unity value) does not allow for a finite or discernible transition of the unity gain threshold at 0 degrees phase. Conversely, if one reduces the loop gain even minutely in value, the phase shift at either resulting unity gain crossing will no longer be 0 degrees.

The root locus plot for the function clearly indicates the closed loop poles never enter the right hand plane for any value of loop amplification.

 

..................
The local minimum in gain (albeit unity value) does not allow for a finite or discernible transition of the unity gain threshold at 0 degrees phase. Conversely, if one reduces the loop gain even minutely in value, the phase shift at either resulting unity gain crossing will no longer be 0 degrees.

The root locus plot for the function clearly indicates the closed loop poles never enter the right hand plane for any value of loop amplification.

I cannot confirm these results.
My analyses show that as soon as the gain falls below the value of +4 the closed-loop poles move into the RHP.
This can be confirmed using a tran analysis which shows an oscillatory behaviour with rising amplitudes. However, as soon as the amplitudes reach the upper limit (supply rail) latching occurs. This is true for an IDEAL amplifier.
In case of a real opamp model the oscillation start is overshadowed by a quick latching effect caused by a REAL positive pole (caused by the opamp pole).
Remark: For all tran analyses the opamp output must be decoupled for dc from the double-T input using a sufficiently large capacitor. Otherwise, we have 100% dc feedback which does not allow a stable bias point.

 

Thanks - I'll check where I went astray on my work.

EDIT:

Yes found a sign error in my work - plus the root locus program [part of Scilab] I use doesn't like transfer functions of the type with the Twin-T plus positive loop gain. Probably because of the unbounded low gain instability issue.

Your original findings are confirmed by my revised simulations.

 

@t_n_k :
In the mean time I have created and simulated a state-space model (using integrators) of the system as mentioned in your post #12 (paper from Fan He, Raymond Ribas).
Result: In principle, same behaviour as "your" model. It shows rising oscillations with latching effects due to its special properties: Lower gain causes pole shifting to the RHP (and vice versa).
Certainly, this is due to the positive slope of the loop phase function.

EDIT: Thanks again for suggesting the state-space approach for realizing the 3rd-order transfer function. This was very helpful and allows a deep insight in the circuits behaviour.

 

@RB: Thanks for your explanation.
However, I am afraid that does not explain my observations.
May I repeat:
The two examples I have presented in my first posting (and there some other) fulfill ALL three requirements mentioned by you. Nevertheless, they do not oscillate.
...

Sorry I'm not buying that!

If something meets those 3 conditions it WILL oscillate, whether it's an electric circuit, a pendulum or a squeaky door hinge.

So the fact oscillation is not happening means that they don't meet the 3 requirements, they just APPEAR to meet them at first glance.

My guess is they are failing on requirement 2 (assuming you have eliminated the usual suspect which is number 3).

Can you confirm that in your examples oscillation CAN be started, but then fades away and stops? That would go far to proving a problem with requirement 2 (not enough energy input to overcome losses).

 

Hello The_RB, thank you for replying.

If something meets those 3 conditions it WILL oscillate, whether it's an electric circuit, a pendulum or a squeaky door hinge.

Does this mean that those 3 conditions constitute a sufficient oscillation condition? I still have some doubts.

Can you confirm that in your examples oscillation CAN be started, but then fades away and stops? That would go far to proving a problem with requirement 2 (not enough energy input to overcome losses).

I suppose - yes, I can.
*As I have mentioned - for IDEAL opamp models oscillation starts with rising amplitudes. As soon as amplitudes reach the supply rail latching occurs. This can be explained by the circuit properties: Positive slope of the loop phase at f=fo. As a consequence, reducing the loop gain causes NOT a pole movement to the LHP (as required) but in the opposite direction.
*For real opamp models a real and positive pole is introduced. Thus, there is an immediate latch-up.

Summary: I cannot see why the 3 conditions you have formulated shouldn`t be met.

Finally, let me say that I appreciate this discussion - in particular with regard to the last state-space examples which have created some new insights on my side.

 

...
*For real opamp models a real and positive pole is introduced. Thus, there is an immediate latch-up.
...

Ahah! So there is a 4th failure mode?

An oscillator that seemingly satisfies the 3 conditions, but at the gain required to overcome losses it can run out of "headroom" and that causes a latchup (failure to continue oscillating).

But then that type of failure seems bordering on a circuit error (like a power failure) and so is the lack of headroom an actual cause of failure, or does the circuit error of lack of headroom cause a traditional error like corrupting the feedback phase (which is the actual thing that stops oscillation)?

I don't have an answer but I THINK you could build a model that runs out of headroom and hits the stops, but does NOT change the phase angle and it keeps oscillating. Lot's of oscillators have some digital component that by definition has no "headroom". However that would be hard (maybe impossible?) to build as a pure sine oscillator that has proper sines at all points of the feedback loop.

 

is the lack of headroom an actual cause of failure, or does the circuit error of lack of headroom cause a traditional error like corrupting the feedback phase (which is the actual thing that stops oscillation)?

I think we have to discriminate between
* A "classical" stop of oscillation caused by a violation (or disappearance) of one of the known necessary oscillation conditions (Barkhausen), and
* A latching effect caused by a particular property of the circuit (in our case: rising loop gain phase at f=fo).

It is my opinion that the mentioned rising phase characteristic plays a major role because it corresponds with a NEGATIVE group delay. More than that, I think there is a relation to the fact that reducing the loop gain (due to wanted or unwanted non-linearities) is equivalent to a pole movement to the right (within the RHP) causing the latch-up.

 

I doubt would one would have to change very much in the ideal to real case transition to "explain" certain of LvW's observations.

For instance take LvW's equal value component Twin-T oscillator with positive gain of 4 applied and the loop closed. In the ideal outcome, one obtains a steady amplitude oscillator. Gains of less than 4 cause unbounded oscillation growth.

Consider a small change comprising a non-ideal amplifier with gain defined by

\(A_v(s)=\frac{4}{T_o s+1}\)

where To defines a first order high frequency roll-off for the loop amplifier. In other words - change from infinite loop amplifier bandwidth to finite loop amplifier bandwidth. Dc gains remain the same.

The introduction of the loop amplifier roll-off introduces a real valued (closed loop) pole on the complex RHP which would presumably produce a similar result to that encountered by LvW in the examined physical implementation of the particular Twin-T oscillator.

 

...............
The introduction of the loop amplifier roll-off introduces a real valued (closed loop) pole on the complex RHP which would presumably produce a similar result to that encountered by LvW in the examined physical implementation of the particular Twin-T oscillator.

Yes - exactly this is the case. The first-order roll-off of the real amplifier model creates a positive real pole (as I have mentioned already in my post#23).

To summarize:
1.) We have a circuit that fulfills Barkhausen`s condition - but it cannot oscillate for real-world amplifiers with frequency-dependent gain characteristics.
2.) Thus, it is necessary to ask what is the main difference between this circuit (twin-T with pos. gain of 4) and all working oscillator circuits?
3.) My answer: Loop gain phase has a positive slope at f=fo
4.) Counter example: There are some other circuits which also show rising loop phase at f=fo (and fulfill Barkhausen) - but they can be configured as an oscillator applying appropriate non-linearities, e.g. diodes (for real amplifiers).
5.)That means: It is not the rising phase alone that inhibits oscillator applications.
6.) Final question: There are some circuits with feedback meeting the classical oscillation condition - however, in contrast to classical oscillators they exhibit a rising loop phase at f=fo. Nevertheless, some of these circuits can be configured as oscillators (e.g. using diodes), even for real amplifiers - but some others not (because of a real pos. pole).
WHY?

 

...
6.) Final question: There are some circuits with feedback meeting the classical oscillation condition - however, in contrast to classical oscillators they exhibit a rising loop phase at f=fo. Nevertheless, some of these circuits can be configured as oscillators (e.g. using diodes), even for real amplifiers - but some others not (because of a real pos. pole).
WHY?

This is likely to be an oversimplification, but if a diode is needed to restrain the gain (and so it stops issues with running out of headroom) then it's simply a matter of an oscillator requiring enough gain to overcome losses, but not so much as to run out of headroom.

OR, as another way of looking at it, the gain control needs to be within the "goldilocks" zone.

I would just change the oscillation requirements to consider that;
1. oscillator needs roughly correct feedback phase.
2. oscillator needs feedback gain limited within "goldilocks" zone.
3. (some) oscillators need a start condition.

 

This is likely to be an oversimplification, but if a diode is needed to restrain the gain ........

No - it`s just the opposite.
perhaps I was not detailed enough. I didn`t use the diode "to restrain the gain". In contrary: Of course, I used the diodes to enlarge the gain for rising amplitudes, because - as I have mentioned several times - lowering the gain leads to further amplitude increase!
There is another option: Usage of diodes (in conjunction with a resistor parallel to a capacitor) to influence the phase for rising amplitudes.

OR, as another way of looking at it, the gain control needs to be within the "goldilocks" zone.
I would just change the oscillation requirements to consider that;
1. oscillator needs roughly correct feedback phase.
2. oscillator needs feedback gain limited within "goldilocks" zone.
3. (some) oscillators need a start condition.

Excuse me if I interpret you wrongly - but I still have the feeling that you think each circuit with feedback meeting Barkhausen`s condition should somehow oscillate.
It is a known fact that the Barkhausen criterion is not a sufficient condition but only a necessary one. Thus, I am not surprised at all to observe that some circuits do not oscillate (in spite of meeting Barkhausen).
My only point is: WHY?
It is my intention to clearly identify the reason for this behaviour and to derive one or more additional conditions in order to make this criterion more rigorous.
For example, I think I have identified the phase slope of the loop gain function as one of the key parameters, but - as mentioned in my former post - there must be something else. (For example: In some circuits under consideration exhibiting a positive phase slope the real opamp model creates a positive real pole - and in some other circuits this is not the case).
I hope I could clearly describe the points of my interest.
Thank you.

 

...
Excuse me if I interpret you wrongly - but I still have the feeling that you think each circuit with feedback meeting Barkhausen`s condition should somehow oscillate.

No, I don't think you have been wrong in understanding me, it's just that I don't care what Barkhausen thinks.

I can make anything oscillate (or stop oscillating) by those three rules I posted. To me Barkhausen is more of an antique thought experiment.

...
It is a known fact that the Barkhausen criterion is not a sufficient condition but only a necessary one. Thus, I am not surprised at all to observe that some circuits do not oscillate (in spite of meeting Barkhausen).
My only point is: WHY?
...

Then your point seems to be that Barkhausen<oscillator and not Barkhausen=oscillator. I don't have a problem with Barkhausen not being ALL an oscillator is and prefer to round ALL that an oscillator is to those 3 points;

1. oscillator needs roughly correct feedback phase.
2. oscillator needs feedback gain limited within "goldilocks" zone.
3. (some) oscillators need a start condition.

And to answer your "WHY" the answer would be that if some circuit does not oscillate it runs foul of one or more of those 3 points EVEN if it satisfies Barkhausen.

...
For example, I think I have identified the phase slope of the loop gain function as one of the key parameters,

To me, (and generalising), that is just a phase error. Any oscillator might have different stages of phase or gain in feedback, but ultimately it is the overall gain and overall feedback phase that makes it oscillate.

So it is simply "having phase error" and failing to oscillate even though some or all of the circuit seems to satisfy Barkhausen.

I'm not sure how that will be of help to you, unless you want to phrase that as "it satisfies Barkhausen but in practice the overall phase is in error so it won't oscillate".

If your argument is really "Barkhausen<oscillator" as I inferred it earlier, then that equates to "Barkhausen+X=oscillator" you need to find X.

I'm satisfied that "3conditions=oscillator" so logical deduction would take you down that road to compare "Barkhausen:3conditions" as deviation there is likely to lead to your answer?

 

It's interesting that the THE_RB is disinterested (in the strict grammatical sense) in the Barkhausen criterion. In any event THE_RB's contribution has been most thought provoking. The 3-point approach suggested by THE_RB has the sense of encapsulating both a necessary and sufficient set of conditions for oscillation without any need to resort to complex analytical methods.

Yet, despite its detractors, Barkhausen remains integral to a substantial body of teaching in the area of oscillator analysis - right or wrong. While some teachers add their caveats to the practice, the idea seems widely embedded in the electronics fraternity psyche.

I'm impressed by LvW's quest for what THE_RB might call the 'x' factor or what I might call the missing piece of the jig-saw. I've found & read an interesting paper by LvW on a related topic and wish him well in his search.

Some, like Kent Lundberg, might regard the effort as pointless. I think it's a worthwhile exercise, if the missing piece of the jig-saw can be identified - if indeed it actually exists as a single identifiable entity or rule of thumb. It would possibly add a useful tool to the analytical tool-kit. Nyquist came up with a simple but elegant graphical technique for resolving the conditions for stability in a given system. Perhaps LvW will come up with something equally elegant.

Perhaps it is also of note, that only a few AAC members have expressed any opinion in this thread. That probably suggests the level of excitement on the topic is underwhelming.

In particular, I found the equal component Twin-T oscillator concept raised by LvW intriguing and thank him for arousing my interest.

 

I can make anything oscillate (or stop oscillating) by those three rules I posted. To me Barkhausen is more of an antique thought experiment.

I must confess - I cannot.
If you can "make anything oscillate" by meeting the mentioned rules - are you aware that you have found a sufficient oscillation criterion?
(Under the assumption that you are right).
On the other hand, do you agree that this "antique thought experiment" constitutes the basis for designing working sinusoidal oscillators - even today?

1. oscillator needs roughly correct feedback phase.
2. oscillator needs feedback gain limited within "goldilocks" zone.
3. (some) oscillators need a start condition.

And to answer your "WHY" the answer would be that if some circuit does not oscillate it runs foul of one or more of those 3 points EVEN if it satisfies Barkhausen.

Definitely NOT. If this is really your opinion, you should read my posts again.
But I think, now we have reached the point where you must be somewhat more exact. Otherwise, misunderstandings cannot be avoided.
What do you mean with "roughly correct feedback phase"?
Does this "rule" show any difference to Barkhausen`s phase requirement?

To me, (and generalising), that is just a phase error. Any oscillator might have different stages of phase or gain in feedback, but ultimately it is the overall gain and overall feedback phase that makes it oscillate.
So it is simply "having phase error" and failing to oscillate even though some or all of the circuit seems to satisfy Barkhausen.

I'm not sure how that will be of help to you, unless you want to phrase that as "it satisfies Barkhausen but in practice the overall phase is in error so it won't oscillate".

So - your only explanation is "phase error"? Sorry, but this explanation is a bit too simple. I can assure you that I am aware of the gain and phase conditions for working oscillator circuits.
By the way: It`s not the "overall gain and overall feedback phase", but the LOOP gain and phase that matters. The feedback phase is only a part of the loop phase.

 

Again reflecting upon the role of the "loop amplifier" I note that both the systems mentioned in post #1 .....

Examples:
1.) Symmetric modifications of the classical WIEN oscillator if it is seen as a tuned bridge (element exchange within the positive as well as the negative feedback path)
2.) Equal component double-T oscillator with positive gain of 4.

.... each produce a closed loop RHP real pole for a bandwidth limited amplifier acting in conjunction with the proposed frequency determining feedback topology.

You already know this as you indicated earlier.

With respect to the system based on a state-space implementation of Kent Lundburg's counter-example, I note (from one simulation) that a simple bandwidth limited loop amplifier / attenuator does not appear to produce a RHP real pole. Is this your finding?
If true - I wonder if Lundberg's counter-example is a valid case. In essence it only counters those statements which presume that a loop gain magnitude greater than unity will also ensure closed loop oscillation - provided the phase shift around the loop is zero at the frequency of interest.
This is embodied in the discussion developed by Fan He et .al in their Mixed Signal Letter submission - particularly at equation (8).

 

I must confess - I cannot.
If you can "make anything oscillate" by meeting the mentioned rules - are you aware that you have found a sufficient oscillation criterion?
(Under the assumption that you are right).
On the other hand, do you agree that this "antique thought experiment" constitutes the basis for designing working sinusoidal oscillators - even today?
...

Sure, if I made it oscillate then something there must be "sufficient".

As for "the basis for designing working sine oscillators today" I have never had to use this "antique math" (my words) to make working sine oscillators, nor have I ever had an oscillator design fail to oscillate (because I use those 3 rules and they work).

My background is multi decades in electronic repair (usually making faulty circuits STOP oscillating, sometimes making faulty stalled circuits oscillate) and multi decades in parallel as an electronic design engineer (making sure my own designs oscillate in the correct places, and never oscillate in the wrong places).

I think I have already exhausted anything I might have been able to offer this discussion from my practical experience, and have nothing at all to offer you from a theoretical math perspective since that is your area of expertise, so it's time for me to bow out.

 

The_RB,
OK - I understand, that you approach the problem from the practical side and, yes, you are right, the mentioned problem is more or less a theoretical one. However, it is NOT a mathematical problem (as you might think) - it is a part of circuit theory, which is the basis for all electronic design activities.
But it is a pity that you didn`t answer my question "What do you mean with "roughly correct feedback phase"?
I am sure you are using Barkhausen´s rule when designing oscillatory circuits - probably without knowing that. But it is a known fact (even mentioned by Barkhausen himself in his book) that his "oscillation criterion" is a necessary condition only, rather than a sufficient one. Believe me, I know what I am talking about.
Nevertheless (because you bow out), thank you for your effort to participate at the discussion.
LvW

 

Again reflecting upon the role of the "loop amplifier" I note that both the systems mentioned in post #1 .....
Examples:
1.) Symmetric modifications of the classical WIEN oscillator if it is seen as a tuned bridge (element exchange within the positive as well as the negative feedback path)
2.) Equal component double-T oscillator with positive gain of 4.

.... each produce a closed loop RHP real pole for a bandwidth limited amplifier acting in conjunction with the proposed frequency determining feedback topology.

Yes - that`s the case.

With respect to the system based on a state-space implementation of Kent Lundburg's counter-example, I note (from one simulation) that a simple bandwidth limited loop amplifier / attenuator does not appear to produce a RHP real pole. Is this your finding?

Yes - agreed.

If true - I wonder if Lundberg's counter-example is a valid case. In essence it only counters those statements which presume that a loop gain magnitude greater than unity will also ensure closed loop oscillation - provided the phase shift around the loop is zero at the frequency of interest.
This is embodied in the discussion developed by Fan He et .al in their Mixed Signal Letter submission - particularly at equation (8).

Yes - again agreed. I think, K. Lundberg is wrong - twofold wrong!

*At first he claims the transfer function could not be used in an oscillatory circuit. However, the state-space model designed by you allows oscillation - even continuously if a suitable non-linearity is incorporated (influencing the phase).
*Secondly, he thinks his example could serve as a "counter example" to Barkhausen`s condition - assuming this condition is a sufficient one (which is not the case).
_________
t_n_k, thank you for your help and the fruitful discussion.
I will be absent for 3 weeks (vacation, starting on Thursday).
Regards
LvW