Hello to all,

in order to safely start selfsustained oscillations the harmonic oscillator must exhibit a conjugate-complex pole pair in the RHP of the s-plane.
The frequency f1 during the start-up phase is determined by the imaginary part of this pole.
*Here is my question: Is it always possible to find this frequency f1 also as a result of a loop gain ac analysis?
The question arises because this frequency f1 in some cases (for some circuits) is NOT identical to the cross-over frequency fc nor to the frequency fo where the loop gain phase is zero.

buddypoor

Hi Rajasekhar,

thanks for replying.
Perhaps I should be a bit more specific.
For my opinion, most of the harmonic oscillators (if not all?) will have a start-up frequency Fsu which differs from the steady-state frequency Fo.
In most cases,  Fo is larger than Fsu. More than that, Fo can be verified applying the Barkhausen criterion (loop gain ac analysis).
In particular, my question is, if there are harmonic oscillator circuits for which both frequencies are equal (Fsu=Fo).  
The background of my question is the behaviour of the well known "double-integrator oscillator" which exhibits some surprising properties.
(Perhaps one could ask: Why does it oscillate at all? What about frequencies below the cross-over frequency with loop gain Al=0? Why does the circuit not go into saturation?).

Hi Rajasekhat,
thanks again for replying. I am really happy that you are willing to discuss with me some oscillator properties.
Regarding the well-known double-integrator-oscillator (DIO), I have mentioned already some key points in my former posting (last sentence in brackets).
This oscillator is a very outstanding one since it is the only circuit I know, which fulfills the phase condition over a broad range of frequency (if, and only if the opamp is considered to be ideal!).
More than that, any additional gain within the loop changes the frequency of oscillation without touching the oscillation condition.
This also is in contrast to all other harmonic oscillator circuits.
Here are some details:
1.) For real opamps the phase will cross the 360 deg line (oscillation condition) at one frequency f1 only. The loop gain at this frequency is above unity.
2.) Question: For these conditions, every other oscillator - except the DIO - will oscillate at f1 with rising amplitudes until (severe) clipping occurs. But the DIO oscillates at another frequency f2 for which the loop gain is unity.
This oscillating frequency f2 is (very often far) above f1.
3.) More than that, the phase of the loop gain at f2 is beyond 360  deg.  (In some cases: -5...-15 deg. !!!).
4.) The oscillator amplitude reaches its maximum value (somewhat below the supply rail) with minor clipping only. In some cases practically no clipping can be observed.          
5.) For the first time period (before the maximum amplitude is reached) the frequency is somewhat below f2.
6.) From the above, I would conclude that an oscillator with two IDEAL integrators (which do not exist, fortunately) cannot oscillate at all (?).
_______________
These are some properties which I call "surprising".  
What do you think? Or somebody else in the forum?

Regards

Added somewhat later:
To summarize in short: All harmonic/linear oscillators - except the DIO - oscillate at a frequency which can be derived from the loop gain (phase=zero, gain>0 dB due to safe start-up).
In contrast, the DIO oscillates at a frequency for which the loop gain is 0 dB and the phase is some degrees below zero. However, the DIO  never has any start-up problems.

Hi love_analog, thanks for your answer.
I am surprised a bit about the small resonance within this forum about my contribution regarding some properties of this classical oscillator structure.  
Or have I described some well-known effects?

As far as details on the schematic are concerned, I spoke about the classical topology of an integrator oscillator consisting of two opamp integrating stages connected in a closed loop (one inverting, one non-inverting):
* inverting: MILLER integrator
*non-inverting: MILLER integrator plus a unity gain inverting stage or one of the known non-inverting stages using only one opamp (BTC, NIC).

Thank you.
Regards