Hi all,

here is a more or less theoretical question to the community – but with a practical background.
Let me explain the problem which applies to the theory of linear oscillators:

1.) In order to start safely, each linear/harmonic oscillator circuit (e. g. with opamps) must exhibit a pair of RHP poles. As a  consequence, a sinusoidal voltage is created with a rising amplitude until clipping occurs due to the limited opamp supply voltages.

2.) Therefore, to avoid clipping effects some sort of nonlinear amplitude control circuitry is necessary (thermistor, diodes, FET, ….).

3.) To my knowledge, this is true for all harmonic oscillators: 4-pole-amplifier as well as 2-pole-negative-resistance structures.

4.) However, there is one exception and my question is simply: WHY ?

The two-integrator loop (inverting/noninverting) starts safely but never goes into saturation. Instead, the final amplitude approaches the saturation voltage without clipping. This has been proved both by experiment and simulation (for equal as well as for different time constants of the integrator stages).
Of course, the Barkhausen criterion is fulfilled for the nominal oscillation frequency where the total loop gain is unity; but for lower frequencies the loop gain is greater than unity – a situation which should lead to rising amplitudes as describes above under 1.).
Therefore, why does`nt build up an oscillation also with a frequency lower than nominal as the phase condition is satisfied in all cases (nominal and below) ?
(By the way, perhaps I`ve got an answer, but I am interested in some alternative  explanations)


Lutz

Tosei,

mmH - I don´t believe that the final DC gain of real opamps (and the non-ideal phase function associated with it) could be the background. However, in the near future I will start some more experiments with different topologies. Perhaps I can reveal the secret.  
Lutz

Hello Vivek,

thank you very much for your reply.
Here comes my answer concerning your three items:

To 1.
Yes, of course, in this case there is no extra limiting device. Nevertheless, the amplifier of the circuit goes into saturation and causes the limiting effect. The output than is nothing else as the filtered version of this signal.  


To 2.
Yes, formally spoken you are right again, of course. However, when I speak of 180 deg phase shift “at all frequencies” (did I really ?) I refer to the frequencies of interest in the middle of the frequency band, where the integrator function can be  regarded as semi-ideal. And as you know, it oscillates !!    

To 3.
Sorry, but I don´t get the point. The amplitude of which oscillator circuit do you mean ?

Thank you again
Lutz

Hi Lutz,

I also didnt' get the point when you claim that a 2-integrator loop will automatically settle (without clipping).
Maybe the oscillation is just too fast, so that together with the finite slew rate, the integrators never settle
to the final value.(??)

However, in general I think that the nyquist stability criterion can predict stabiltiy for any system (no like the
Barkhausen criterion which is intuitive but wrong).

Regards

On January 15th I started this topic with a question to the whole community and, at the same time, I mentioned to have a (preliminary) answer.
Up to now, there were no new explanations resp.verifications of the effect described and, therefore, I come to the conclusion that the following answer - based on the harmonic balance principle -  is the only one which is appropriate:

The amplitude of the first harmonic of a limited sinusoidal voltage is always smaller than the amplitude of the incoming sinus which is necessary to produce this harmonic. Therefore, any small amplitude increase above the threshold cannot lead to a sustaining oscillation at a lower frequency with a limiting effect. (However, note the additional comment below).

In terms of the rules of the harmonic balance this statement would read:
The value of the inverse describing function is always smaller than the “over-amplification”  which causes the limiting effect and which leads to the assumed describing function with a value smaller than unity.

Example: Let´s assume two integrator stages with max. possible amplitudes of 10 volts and – by accident - an incoming amplitude of 12 volts. In this case there would be a theoretical over-amplification of 1.2 causing a limiting effect. This value belongs to a first harmonic amplitude of 9.2 volts. Thus, to retain the loop gain at unity an incrased composite gain of both  integrator stages of 1/0.92=1.087 (of course at a reduced frequency) would result. However, this theoretical experiment has started with a gain increase factor of 1.2. Thus, the assumed amplitude disturbance will decay and the oscillation frequency will remain unchanged at the maximum unlimited amplitude.

Additional comment: It is interesting to note that this holds also for an arrangement modified as follows:
Let´s assume that the quadrature oscillator consists of two inverting integrator stages and one additional inverting amplifier with a gain of “-1”. Now, either the gain of this amplifier is increased or its supply voltages are reduced - driving it into saturation. (The integrator supplies remain high and unchanged). As a consequence, the inverter output will be a limited sinusoidal wave, but the output voltages of both integrators are still unlimited, however, with unequal amplitudes at a lower frequency.  
                                         _____________________________

As to my knowledge, this explanation of the behaviour of quadrature oscillators cannot be found in any textbook.
Any comments are welcome.

Lutz

Hi Lutz,

I am not a "Barkhausen criterion"-expert ... however it is easy to provide counterexamples where the  Barkhausen criterion doesn't hold.

You can find several examples on the web, e.g.:
http://web.mit.edu/klund/www/weblatex/node4.html

Regards

Hi Lutz,

If you would actually read the comments from Lundberg (I just did), he does not quote Barkhausen as proposing a
criteria for stability. Instead, Lundberg mentions that Barkhausen developed his criteria as a tool to determine
the frequency of oscillation of a linear oscillator. It is the unfortunate fact of other people having misinterpreted
this as a means of deciding whether a system is stable or not that has caused much confusion. So, I don't see a big
disagreement between your point of view and Lundberg's as far as the historical aspects are concerned.

When you speak of unphysical systems, I suppose you refer to Lundberg's example of a system which has a constant
gain at all frequencies. However, there are other important points:

1. The example system with 3 poles at zero and 2 zeros at -1 (not an unphysical system).
2. The arguments related to the apparent contradiction between root locus/Nyquist plots
and a simple application of the Barkhausen criterion are highly relevant, once more, if you
apply Barkhausen criterion blindly to a system which is only conditionally stable (a situation which
arises very commonly when designing complicated amplifier topologies).

I think the discussion has gone kind of offtrack.

Regards
Vivek

Hello Berti,
in your reply No. 6 (Jan. 30th) you have mentioned Barkhausen with reference to a document from K. Lundberg.
For Lundbergs publication see my contribution below.

Hello Vivek !

At first, I agree with you that the discussion has somewhat deviated from the starting point as the “secret” of the two-integrator oscillator seems to be revealed. But as there are some misunderstandings and misinterpretations connected with the term “stability” (perhaps also on my side ?) I consider this discussion nevertheless still as helpful.

Quote (Vivek): If you would actually read the comments from Lundberg (I just did), he does not quote Barkhausen as proposing a criteria for stability……… other people having misinterpreted this as a means of deciding whether a system is stable or not….

Sorry, but I cannot agree because Lundberg himself contributes to the confusion as the title of his paper is “Barkhausen Stability Criterion” ! (http://web.mit.edu/klund/www/weblatex/node4.html).
And his last sentence reads as : “…….knowing the value of the loop transfer function at one frequency gives us no information about stability. Down with Barkhausen!”

Quote (Vivek): Instead, Lundberg mentions that Barkhausen developed his criteria as a tool to determine the frequency of oscillation of a linear oscillator.

Agreed, this formulation is correct, and that´s exactly the reason I refuse a statement like “Barkhausen is wrong”.

Vivek, I have a question: you mentioned “Lundberg's example of a system which has a constant gain at all frequencies”. Can you please give me the corresponding reference ? I only know one single “counter-example” contained in the Lundberg paper referenced above.

Finally, of course I totally agree with you that Barkhausen must not applied “blindly to a system which is only conditionally stable”. This exactly is the reason for my former remark that the Barkhausen formula is nothing more than a necessary (!!!) condition for a system designed to function as an oscillator.

Generally spoken, in any case an electronic engineer must not apply a rule or a formula “blindly” because – to my knowledge - there is no formula in the world of analog electronics which is “correct up to 100%” under all conditions. (Perhaps a more philosophical remark: is such a formula “wrong” ?). Instead, one must carefully prove all presumptions and decide if the error resulting from this simplified view is acceptable for the specific application.  

Thanks to all contributers to this discussion.
Lutz