Dear Lutz Von Wangenheim,

In contrary, stable "sinusoidal" oscillations require a smooth non-linearity of the loop gain function

Any non-linearity will lead to harmonics. The non-linearity can be observed when the loop gain is greater than unity. For example, square wave oscillations. 

How to realize smooth non-linearity of the loop gain function ?

Stable "sinusoidal" oscillations are realized when the loop gain is unity. This shows that amplifier is amplifying signal exactly to the extent of loss provided by oscillator circuit. This I believe is a linear function. Otherwise, oscillator will exhibit harmonics.

AR Reddy

Stable "sinusoidal" oscillations are realized when the loop gain is unity.

Yes - that is the theoretical aspect. However, because it is not possible to ensure unity loop gain by design we always make the loop gain >1 and introduce a "smooth" non-linearity into the circuit (diodes, FET as a resistance, tanh curce of diff. amplifiers,...). That is the common design strategy. Hence, working oscillators are non-linear circuits (some people even say: Strong nonlinear). As a logical consequence, we have to live with harmonics. No surprise.

Yes sir theoretically it is applicable on linear circuits only

"Yes sir theoretically it is applicable on linear circuits only"

However, remember: Since several decades we are desiging oscillator circuits with NON-LINEAR elements based on Barkhausens criterion, don`t we?

For this reason, I prefer to say: All "linear (sinusoidal)" oscillators must contain a small amount of non-linearity, but during all design steps (excluding the last one) we treat the elements as being linear. And - as the last step - we introduce a suitable amount of non-linearity allowing a smooth amplitude limiting. 

i agree with Lutz! Erik Lindbergh has some fine papers on aspects of this topic, including the topic of frozen eigenvalues which should be read with interest. I have investigated many chaotic oscillators (which have to have a nonlinear element to work.....which gets back to Lutz's point) and have tried to apply the barkhausen criterion. Another very interesting phenomenon I came across recently at a conference is the concept called "stability loss delay,(SLD)", where the bifurcation point is delayed by the rate of change of the parameter variation causing the bifurcation.  The delay also changes the shape of the normal pitchfork  bifurcation which I called Tangential distortion. This was pointed out by one of the attendees at the UCD conference,  Prof. Vassili Gelfreich, that it was the result of SLD occurring in the circuit. Further research on this proved that Vasilli was correct. If  anyone is interested I can upload papers on this topics.

Regards

Paul

Paul Sir please upload paper

Dear Lutz Von Wangenheim,

As a logical consequence, we have to live with harmonics. No surprise.

Since practical oscillators do have harmonics, non-linearity is inherent in the oscillator.

Therefore, the number and amplitude of harmonics is directly related to the amount of non-linearity.

We may call this as "smooth non-linearity" if harmonics are very small in amplitude and number.

AR Reddy

Yes - that´s what I was trying to say.

Dear all of you,

Linear oscillators are damped circuits.

Steady state oscillators are nonlinear circuits.

The output signal from a steady state oscillator

is the step response of the power source,

i.e. a steady state oscillator is a circuit

for which a bias point can't be found.

If you investigate your circuit as an active

lowpass filter with a bias point it becomes unstable

if the group delay (phase margin) is negative at the 3dB frequency.

Together with the modified Barkhausen criterion this is a necessary

but not sufficient criterion for steady state oscillations.

Question concerning the modified Barkhausen criterion:

How far out in RHP should the poles be placed ???

"for a circuit to safely act as an oscillator, it

must be required that ALL its equilibrium points be instable"

cite from:

M. H. W. Hoffmann, ”A novel criterion for safe start-up of oscillation and

avoidance of latch-up in oscillator circuits”, Proceedings 2012 German

Microwave Conference (GeMiC 2012) pp. 1-4, 2012.

It is not possible to tell the whole story about a nonlinear

circuit by means of a few linear approximations.

Hope to see you at ECCTD 2015 next week in Trondheim, Norway

Best regards

ERIK

Erik Lindberg

Erik - what is your answer to my question (head line) ? Yes or no?

New questions:

1.) "The output signal from a steady state oscillator is the step response of the power source".  Which power source? DC power source?

2,)"...if the group delay (phase margin) is negative at the 3dB frequency."  I don`t understand the relation between group delay, phase margin and 3dB frequency. Can you explain?

Thank you

Lutz

Hello Lutz,

Concerning: "Erik - what is your answer to my question (Does the Barkhausen criterion for oscillation apply for linear circuits only?) ? Yes or no?"

My answer is YES. Barkhausens criterion simply says that the complex pole-pair of the linearized circuit must be on the imaginary axis in order to have an ideal linear oscillator. We modify the criterion by moving the poles to RHP so the circuit becomes unstable.

Question: How far out in RHP should we move the poles ?

Concerning: "1.) "The output signal from a steady state oscillator is the step response of the power source". Which power source? DC power source?"

Normally we need a DC power source (battery) to create a DC bias point for our signal processing circuits (active filters). Only passive filters need no battery.

Concerning: "2.)"...if the group delay (phase margin) is negative at the 3dB frequency." I don't understand the relation between group delay, phase margin and 3dB frequency. Can you explain?"

Many many years ago - when I was taught about active filters - I was warned about the risk of instability of active filters. The concepts of group delay, phase margin and bandwidth were discussed. Unfortunately I can't find my old textbooks. The same problems are faced in connection with compensation of

operational amplifiers.

Please see also: Igor Balaz et.al. "Barkhausen Criterion and Another Necessary Condition for Steady State Oscillations Existence", Proceedings 23th Conference Radioelektronika 2013, April 16-17, Pardubice, Czech Republic pp.151-155, 2013.

Best regards

ERIK

Hi Erik, here are my answers:

1.) No circuits - in particular: active BJT-based circuits - are linear. Nevertheless, since many decades we succesfully design oscillator circuits for these non-linear devices - based on Barkhausens criterion.

For example, we are using the tanh transfer curve of a diff. amplifier (long-tailed pair) for working oscillator circuits and exploit the non-linearity for rising amplitudes to allow stable oscillations. Therefore, I don`t think it is correct to limit Barkhausens criterion to linear circuits. Whether we are able (or not) to place the poles on the imaginary axis (due to tolerances) is another question which, for my opinion, does not limit the applicability of Barkhausens rule (observation).

2.) Let me ask again: The oscillator signal is the step response of the DC power source???    

3.) I cannot agree that for lowpass-based phase shift oscillators the 3dB frequency should have any relevance. The only point (frequency) of interest is the frequency where the phase shift is -180deg. 

4.) Furthermore, for a working oscillator the group delay at the intended oscillation frequency must be, of course,  positive. In contrary, negative group delays result in latch-up. This was explained in  the referenced paper from I Balaz. Of course, I know this paper because I even was involved in producing it (see references).

Best regards

Lutz

Barkhausen criterion Loop gain = ! is only a Criical criterion for Oscillations

But for Feedback Oscillators Loop gain slightly > ! as stated by Prof. Lutz. for Sustained Oscillations, because of the Presence of Inherent Damping in the Circuit. Active Devices are Non Linear. But for high Non Linearity,like Switched Thyristor Circuits with Motor Loads etc. What Prof. A.R. Reddy says holds good.But they are supplied Sinusoidal current. Oscillators are having Tuning Facility(Resonant Tuning) for the Required Frequency.

1.   Feedback Oscillators

Conditions for oscillation are (Barkhausen criteria)

– Loop gain greater than one , so Axβ>1 (sum>0 dB)

• Often want 3-4 dB margin, or controlled for waveform

– Net phase shift around loop = zero degrees (really Nx360)

• Oscillations build up from noise/transient into saturation at

the frequency that satisfies the Barkhausen conditions

Link:   http://www.qsl.net/hiarc/files/Oscillators-TheoryAndPractice_EMesser_2009-10.pdf

The Following Article will also be interesting because the Loop Gain Method produces in some cases wrong solutions because some necessary conditions have not been fulfilled..

Comments and Remarks over Classic Linear Loop-Gain Method for Oscillator Design and Analysis. New Proposed Method Based on NDF/RRT

Joś e Luis JIḾENEZ-MART́IN,  Vicente GONŹALEZ-POSADAS, Angel PARRA-ERRADA

, Luis Enrique GARĆIA-MŨNOZ, Daniel SEGOVIA-VARGAS

Abstract.

This paper describes a new and original method for designing oscillators based on the Normalized Determinant Function (NDF ) and Return Relations (RRT). Firstly,

a review of the loop-gain method will be performed. The loop-gain method pros, cons and some examples for exploring wrong solutions provided by this method will be shown. This method produces in some cases wrong solutions because some necessary conditions have not been fulfilled.

The required necessary conditions to assure a right solution will be described. The necessity of using the NDF or the Transpose Return Relations (RRT), which are related with the True Loop-Gain, to test the additional conditions will be demonstrated. To conclude this paper, the steps for oscillator design and analysis, using the proposed NDF /RRTmethod, will be presented. The loop-gain wrong solutions

will be compared with the NDF /RRTand the accuracy of this method to estimate the oscillation frequency and QLwill be demonstrated. Some additional examples of plane reference oscillators (Z/Y /G), will be added and they will be analyzed with the new NDF /RRT proposed method, even these oscillators cannot be analyzed using the classic loop gain method.

Link:  http://www.radioeng.cz/fulltexts/2012/12_01_0478_0491.pdf

P.S.

To P.S. Subramanyam:

Thank you for your contribution - in particular for the linked documents. Since many years I am interested in oscillator circuits and principles - and I know how the work and why !.

In this context, two small comments:

*I think, oscillators will not start due to noise but as a result of power switch-on transient/step response. I think, this was discussed also in RG some time ago. 

* I think, the necessary Barkhausen condition was extended to a sufficient condition by adding a specific phase slope condition (see I. Balaz paper as referenced by E. Lindberg).

_____________

I will go through the 2nd document mentioned by you . Perhaps this paper creates new questions?

Thank you and regards

Lutz

Prof. Lutz,

Generally it is said that Oscillations are generated due to Stray Parameters,Capacitance./ Noise. Noise contains all frequencies.

But now I understand the Truth of what you say that Switching induces Oscillations. Of late, with our knowledge on Switching (particularly in Power Converter Circuits and all) Switching Transients induce Oscillations of Different Frequencies. It is a Quite interesting Insight.

I have to go through your Contention that the necessary condition is converted into a Sufficient Condition.

As you said the second reference given by me needs detailed study as new Concepts are introduced, particularly with distributed capacitance in antennas etc.

Regards.

P.S.

Regarding a more rigorous ("sufficient") condition for oscillation you can find some information and documents at the end of the discussion here:

https://www.researchgate.net/post/Can_someone_answer_a_question_about_oscillation_conditions_for_feedback_circuits

Dear Lutz von Wangenheim,

I agree with you. In my view the Barkhausen criterion applies to non-linear systems whenever it provides useful insight. In circuits that are linear prior to non-linear saturation effects at large amplitudes, or are approximately linear, we can use the gain at an equilibrium point or the average gain across an operating range and use the criterion to predict/explain quantitatively the genesis of self-sustained oscillations in terms of circuit components. We can also incorporate non-linear effects into an amplitude-dependent loop gain, again allowing us to employ the criterion to predict/explain oscillations. 

While the Barkhausen criterion is valid in the strictest sense in linear systems (here I include systems with some linear range of operation), applications of the criterion have presumably always been in non-linear systems. In the future, perhaps we could say that the criterion's use beyond strictly linear systems warrants careful consideration of the particular system's non-linearities.

I'm interested to know if you agree,

Best,

Scott

Dr .Sands - thank you for this comprehensive summary. Yes - you have exactly described also my position. 100% agreement.

Lutz vW

Can we try the loop gain A(beta) is less than zero if it is positive feedback so that the denominator of the system transfer function is not zero so that the hard limited of oscillator amplitude can be avoided . (the oscillation amplitude is less that supply voltage value)

To K. Tripetch:

Sorry - but I do not understnd the question. What should we "try"? Loop gain "less than zero"? Or do you mean less than unity?

I mean denominator is  1-0.5  or 1-0.4 .

And what is the question?

My opinion is that only LC circuits (or any circuits containing passive resonators) can be (theoretically) implemented by linear amplifiers. In this case, the LC resonator produces the sine wave while the amplifier only compensates the losses.

RC circuits must be implemented by a non-linear amplifier. In this case, the RC circuit does not produce the sine wave; it is formed by the non-linear amplifier. In this case, the main role of the non-linear amplifier is to model the sine curve.

Hi Cyril - I can imagine what you mean. However, I am afraid that somebody else could misunderstand your text.

1.) Of course, LC based oscillators need a non-linearity to limit the amplitude increase - unless it is hard-limited (clipped) by the DC supply of the amplifier (which also is a non-linearity). However, due to the filtering of the LC circuit we have a good sinusoidal signal at the output of the LC tank. That means: We must not use the limited amplifier output. But we need an additional buffer at the LC output node.

2.) In principle, the same applies to RC oscillators also. However, in some cases the filtering action of the passive part (lowpass or bandpass) is not sufficient because of low Q values. Therefore, we need a "smooth" non-linearity for the amplifier gain. However, there are some RC filter circuits (bandpass based) with larger Q values which - in this respect - can be treated like LC oscillators.

3.) There is an extrordinary oscillator principle which also provides a sufficiently good sinus signal without any additional filtering or buffering: The two-integrator loop. This oscillator circuit is the only one with a phase for the loop gain that fulfills the oscillation condition over a relatively broad frequency range - but the gain condition is fulfilled for one single frequency only.

A final remark: The question why this oscillator oscillates at one single frequency only is not yet answered in the available literature.

For example: Why does it NOT oscillate at a loop gain of 1.1 or 1.2 ? (All other oscillator types do that ) 

As we can see: There are still some questions waiting for an answer.

Yes, Lutz... I see... Let me add more explanations about the difference between LC and RC oscillators.

Both they consist of two parts - passive (LC or RC circuit) and active (amplifier). But the fundamental difference between them is that in the case of an LC oscillator, the passive LC circuit can oscillate independently, while in the case of an RC oscillator the passive RC circuit cannot oscillate if we isolate it from the whole circuit. And at all, I can not imagine how a passive RC circuit would oscillate...

So, I repeat it again, in an LC oscillator, the amplifier only "helps" the resonating LC tank to keep a constant magnitude, while, in an RC oscillator, the amplifier creates and maintains the oscillations forming the specific shape of the sine curve. Here, its role is similar to the role of the Schmitt triger in relaxation oscillators...

"But the fundamental difference between them is that in the case of an LC oscillator, the passive LC circuit can oscillate independently"

but, of course, not under real conditions (losses of both parts).

Hence, I see no principle difference between RC and LC. Both can be used - however, only in conjunction with an amplifier. 

The difference of principle between the two passive circuits is that the LC circuit is a true resonator while the RC circuit is not...

What does this mean: true resonator?

We can build an LC tank with Q=10 and also an active bandpass with Q=10. In both cases, the circuit has a step response that is a damped sinus - and can be used as a frequency-determining circuit in conjunction with another amplfier which comensates the losses (ensuring loop gain of unity).

For me, a "true resonator" behaves like a pitchfork - when affected with a step input, it oscillates with the amplitude gradually decreasing to zero. Any passive RC circuit will not react in this way (by producing decaying oscillations)...

OK - now I know what you mean. Nevertheless, I do not see any fundamental difference between LC and RC oscillators. Both can oscillate - independent on its behaviour under open-loop conditions.

Yes, Lutz... For the purposes of oscillation, this difference is not so important. But for the understanding it is crucial since still it is not clear how sinusoidal oscillations arise in RC oscillators:

https://www.researchgate.net/post/How_do_sinusoidal_oscillations_arise_in_RC_oscillators

OK - Cyril, I can imagine what you mean. Of course, it is not sufficient to mention the oscillation condition only - what you want is a way to understand what`s going on. In this context, you earlier have mentioned (in the given link to another discussion) the energy exchange between L and C.

Well - I agree that such an effect cannot always be observed in pure RC circuits. Hower, sometimes.....For example, you know that it is possible to mimic the behaviour of a lossy L (with parallel R) with an active RC circuit. In this case, you can use the L analogy to see such an energy exchange. But is it magnetic enegy? No - I don`t think so. But the behaviour is identical to a coil with ohmic losses.

For other RC circuits I always use the simple rule "output=input". If we have a circuit with an amplifier that is able to produce (at a very low output resistance) a signal which is identical to the amplifier input signal at one frequency only, we have a unity loop gain. With other words: An amplifier that produces its own input signal which is just necessary to keep the output constant . I think, this can be a simple explanation for the observed osciillation effect.    

Yes, Lutz... I agree with you... I will only clarify that we can mimic an inductor by a capacitor and v.v. but only in respect of their behavior in the frequency and time domains. Regarding their accumulating abilities, they can not accumulate energy. They use the external energy of an additional power source, not their own energy.

So, the combination of only one kind of reactive element (inductor or capacitor) with resistors cannot be a true (accumulating energy) resonator. Only the combination of two dual reactive elements (inductor and capacitor, i.e. an LC tank) is a true accumulating resonator... and this is because both they store energy... they continuously exchange it but they treasure it...

This chain of thoughts is closely related to your question about the nature of the resonance phenomenon. We cannot observe a true resonance in a pure RC or RL circuit. We can observe it only in a mixed LC circuit (LC tank). For example, if you cascade low- and high-pass filter to obtain a band filter, it will not resonate at its resonant frequency. This is not a resonance in its true meaning... this is something else - the circuit simply changes its impedance so that it becomes maximum/minimum at the resonant ffrequency... There is no energy accumulation here... there is no continuos energy growth up to infinity... there is only an impedance change...

And finally, these thoughts are closely related to your present question. In my opinion, we can apply the Barkhausen criterion for oscillation only to LC linear circuits (built by a linear amplifier) but we cannot apply it to RC linear circuits where we obligatory need a non-linear amplifier... We need this nonlinearity to delay, stop and reverse the "motion" of the signal... to form the sine shape in the picture above...

Cyril - as you can see below, I have some problems with your last reply:

Yes, Lutz... I agree with you... I will only clarify that we can mimic an inductor by a capacitor and v.v. but only in respect of their behavior in the frequency and time domains. Regarding their accumulating abilities, they can not accumulate energy. They use the external energy of an additional power source, not their own energy.

If an inductive two-pole device is within a black box with two input nodes we have - as far as I can see - no chance to decide if it is a passive RL (coil) or an active RC device. Isn`t it always external energy that is accumulated?

So, the combination of only one kind of reactive element (inductor or capacitor) with resistors cannot be a true (accumulating energy) resonator. Only the combination of two dual reactive elements (inductor and capacitor, i.e. an LC tank) is a true accumulating resonator... and this is because both they store energy... they continuously exchange it but they treasure it...

I am sure, this also applies to „active inductors“ because they behave like passive coils in the frequency and the time domain.

This chain of thoughts is closely related to your question about the nature of the resonance phenomenon. We cannot observe a true resonance in a pure RC or RL circuit.

This depends on the definition of „resonance“. I think, the common def. is the frequency where the imag. part of the current (or the impedance) is ZERO. And this is possible for RC circuits (WIEN network).

We can observe it only in a mixed LC circuit (LC tank). For example, if you cascade low- and high-pass filter to obtain a band filter, it will not resonate at its resonant frequency. This is not a resonance in its true meaning...

This creates the question: What is the „true meaning“? (See the definition above).

And finally, these thoughts are closely related to your present question. In my opinion, we can apply the Barkhausen criterion for oscillation only to LC linear circuits (built by a linear amplifier) but we cannot apply it to RC linear circuits where we obligatory need a non-linear amplifier... We need this nonlinearity to delay, stop and reverse the "motion" of the signal... to form the sine shape in the picture above...

But - as you know - in fact, this criterion was applied in thousands of circuits with full success. Furthermore, each amplifier is non-linear. Linearity is a fiction. I`ll give you another interesting example: Two active RC integrators in a closed-loop (one of which non-inverting). Without any extra (additional) non-linear element this oscillator circuit will (a) safely start oscillations and (b) show continuos oscillations with very good quality (amplitude clipping is not observable).

Well, Lutz... I can understand your considerations. Really, the final result is important; it is not so important how a given circuit achieves this result. But, for me, it is interesting to understand things at this lowest level.

Now about the resonance... Yes, there is a definition... but this is only a kind of convention between people... Here, in this forum, we need more than a convention... So, in contrast to this "impedance resonance", where the circuit only shows maximum/minimum impedance at its resonant frequency, we have a notion (from physics and routine) about a kind of an "energy resonance", where the circuit accumulates the synchronously injected energy...

Now about the oscillating circuit of two active RC integrators in a closed-loop...

I can understand the phase condition. Shortly, we need a total 360 degrees phase shift to obtain an oscillation. The first integrator gives 90 degrees, the second integrator provides another 90 degrees, and the act of invertion gives the final 180 degrees (90 + 90 + 180 = 360). But I cannot understand what will be the amplitude of these oscillations if there is no clipping?

Hello Cyril - thank you for your reply. If I remember well, the question about the definition of "resonance" was discuused already here in R&G. And I am not sure if a definition can be more than "a convention between people". Nevertheless, as far as I can see - this subject does not help to clarify the original problem problem (condition of linearity for the Barkhausen criterion).

Regarding your last contribution (RC integrator oscillator), I fully understand your doubts. In this context, may I remind you on my contribution (some days ago):

"The question why this oscillator oscillates at one single frequency only is not yet answered in the available literature. For example: Why does it NOT oscillate at a loop gain of 1.1 or 1.2 ? (All other oscillator types do that )"

As you can see, I also have some problems. However, I think and hope to be on the "right track" now. But my inverstigations and considerations are not yet finished. 

As you know, it is the only oscillator circuit that fulfilles the phase condition for many frequencies and the amplitude condition for one single frequency only - in contrast to all other known oscillator circuits.

In this context, one of the most important question is: Assuming that the oscillation condition is fulfilled at a frequency fo (and the circuit oscillates at fo) - why does it NOT oscillate at a somewhat lower frequency (let`s say 0.95fo) ?

The problem is that I cannot realize how this circuit produces sine waves without clipping... Maybe it still oscillates near the supply rails?

Cyril - shame on me. I have made a severe error:

Some time ago I wrote:

"If an inductive two-pole device is within a black box with two input nodes we have - as far as I can see - no chance to decide if it is a passive RL (coil) or an active RC device.

Sorry - that`s wrong. This statement applies for sinusoidal signals only. Both circuits (passive RL and active RC) exhibit, for example, different step responses.

And what is the difference? I am not familiar with this topic...

The difference is in the time domain if the input signal is not sinusoidal.

Hence - in contrast to my former assumption - I am able to decide if in the black box is a passive LR or an active RC circcuit.

Maybe because the LR circuit will show a self inductance?

At this point, some additional explanations are necessary:

* Up to now, my comments and statements - as far as active RC simulations of an inductor are concerned - apply to a "lossy" active inductor circuit realized with one opamp only (L||R). In this case, both realizations (active vs. passive) are identical in the frequency domain only. That means: Not identical in the time domain (i.e. step response).

* However, the situation changes if we realize an "ideal" active inductor using two opamps (GIC principle). In this case, both alternatives (active vs. passive) exhibit identical properties in the frequency domain as well as time domain. 

To all readers of this discussion:

Recently I wrote about the well-known "two-integrator oscillator" (one inverting and one non-inverting):

"The question why this oscillator oscillates at one single frequency only is not yet answered in the available literature. For example: Why does it NOT oscillate at a loop gain of 1.1 or 1.2 ? (All other oscillator types do that )"

I must admit that I have expected some comments or answers to this basic question. Hence, I am a bit surprised (disappointed). Is there really nobody who - at least - has a "feeling" or a "guess" for a possible reason?

'

All oscillators intended for generating sinusoids - however realized -

as well as many other types, start out as linear circuits as they begin

to oscillate. In the best designs, this occurs by nothing more than a

high-Q resonance being excited by the active circuit's intrinsic noise.

.

Any analysis that neglects the role of noise in oscillator behaviour

is incomplete.

.

As the amplitude of an oscillation increases exponentially over

time, a point is reached at which some process intervenes to

slow this ongoing rise and eventually limit the amplitude. Again,

in the best designs, this is implemented by a sub-circuit which

measures the amplitude, before the oscillator's own, inherent 

nonlinearities are asserted. (Incidentally, very few oscillators

apply this common-sense design approach, leaving control of

amplitude to some arbitrary, and poorly-controlled, process). 

'

Thus, it is not simply 'academic' that a well-designed oscillator is

quintessentially a linear circuit: it is a matter of necessity that it is

constrained to remain operating in this linear mode by suitably-

designed, extrinsic amplitude-control circuitry. 

'

Of course, such rigours are not necessary in oscillators that do

not have to meet high performance objectives.

.

In any event, the mindless appeal to some generalized "criterion" 

is always a poor start to a design challenge. As professionals we

have a duty - and obligation - to understand our circuits from the

most fundamental considerations, certainly not by applying some

criteria found in a text-book.

.

To B. Gilbert:

You nicely have summarized the basic considerations for designing a "linear" oscillator - and, I suppose, everybody knows why I have used quotation marks since there are no linear circuits in reality. But we assume linearity - and this works. And we also know that oscillator circuits do oscillate if the non-linearity is even visible (example: tanh characteristic of the diff. amplifier).

That was the background of my question.

More than that, of course I agree that - as an engineer - we should "understand our circuits from the most fundamental considerations, certainly not by applying somecriteria found in a text-book.".

On the other hand, students use textbooks and are asking questions about the oscillation criterion. Is it really sufficient or is it only a necessary one? And what about the requirement for linearity?  Hence, in this case, one cannot simply ignore textbook contents and the criteria to be found in it.

_____________________________________

Finally, a technical question:

Quote: Any analysis that neglects the role of noise in oscillator behaviour is incomplete.

I do not fully agree to the mentioned importance of noise. Don`t you think, that the voltage step caused by the power switch-on is the main reason for starting the oscillation? I have made many inverstigations in this context. Even in low-Q circuits (WIEN oscillator) one can show that the step response after applying power to the circuit causes a safe and quick start of oscillations. - much faster than the noise can do.

Lutz:

.

In suggesting that

[It is] the voltage step caused by the power switch-on is the main reason for starting the oscillation

.

you are making an understandable error. Certainly, in many practical cases this actually will be true. But whenever it is, we have admitted that one's oscillator is sensitive to changes in the supply (voltage or current) at least at some time (before oscillations are established) and this opens the door to the possibility that it will also be sensitive to changes in the supply even AFTER start-up. That is clearly a very undesirable state of affairs, since it implies that the oscillator’s behaviour once in operation is open to being influenced by all kinds of disturbances on that supply line. The most common problem with such oscillators is that such disturbances allow various degradations to arise in the resulting output, such as phase- and amplitude noise and jumps, reflecting the whims of the supply.

.

It is a sorry fact that numerous poorly-designed oscillators in use today exhibit this weakness (it can in no sense be called a virtue, nor even less a useful “feature”). Indeed, a well-designed oscillator will refuse to start even when the supply exhibits a large step change, even when this step has a rise-time very much less than any characteristic time-constant (most notably, the oscillator’s period). In simulation studies, this step may be as fast as 1ps; but of course, one will also examine the reaction to slower changes, down to several millisecond. In such studies it is also important to apply very stringent convergence parameters, because if these are too loose, the computation of the solution to the circuit’s matrix can lead to false consequences.

‘

Thus, the art of good oscillator design includes the provision that, in its idealized condition (usually meaning the use of perfectly-matched elements in a differential topology), it WILL REFUSE TO START when a supply is stepped. This may be equated to providing an extremely high power-supply rejection ratio (PSRR), and such is a more usual way of ensuring a near-zero sensitivity to supply disturbances.

‘

In the past, I was obliged to enter into a long and strident e-mail discussion with a well-known circuit “expert” (now deceased) over this very point, and I may be able to provide some summary notes and simulations that were prepared shortly thereafter. (I have too many computers in use, and such things get mislaid). I showed definitively that a well-designed oscillator will not start when the supply (voltage or current) suddenly appears, even in the presence of significant device mismatches; and further, that such an oscillator will continue to exhibit near-zero sensitivity in its amplitude and phase when large disturbances of various sorts are present on the said supply.

‘

Such painstaking finesse is at the heart of the design of high-performance products destined to be produced and used in quantities of several million, time after time.

‘

Barrie

Lutz:

.

I found that piece about oscillator start up. It's a PPT file.

I'm going to try to attach it. After looking over the first few

slides you'll probably want to skip the section on noise so

jump to Slide 90 for the section of relevance here. As I

recall, this was also published in a sanitized form in the

ADI magazine Analog Dialogue. I'll look into finding that.

'

Best regards, as always

'

Barrie 

Barrie,

thank you for your detailed answer and - in particular - for the ppt document. Following your recommendation, I also found your contribution "Oscar stirs" (Analog Dialogue 2006). I think, this was the text you were referring to, right?

Of course, it will took some time for me to go through these documents (trying to understrand). However, my first interpretation of your contributions regarding oscillator start-up is as follows:

You recommend that intentionally oscillators should be designed so that they do NOT start oscillations automatically after power switch-on.

With other words: If oscillator circuits meet the oscillation condition (loop gain slightly larger than unity) they would start oscillating (initiated by the switch-on transient) - however, there should be some special means which can inhibit such a start until it is released. Is that correct?

Regards

Lutz