A novel harmonic oscillator circuit

[Winterstone]

A linear oscillator circuit must always contain a non-linear part – otherwise it cannot operate in its (quasi)-linear region.

I like this sentence – and it reveals the problems to exactly describe the oscillator behaviour. For this purpose, we would need a set of non-linear differential equations.
But – in practice – we would gain nothing from this.

MrAl, thank you for your contribution. I can agree to your formula in post#14 - however, it must be emphasized that this formula gives the frequency wn for the start-up phase only
and NOT the frequency for the steady-state. That means it is not the self-sustained oscillator frequency wo.

Explanation: At t=0 (start of oscillations) the system poles must be in the RHP of the s-plane (with a small positive real part sigma).
This sigma can be found also in the time domain (pos. exponent of the e-function describing the rising amplitudes).
In the literature it has been shown that during this period the frequency is

wn=SQRT(wo^2 – sigma^2) . (However, I am not sure if this formula applies to ALL oscillator principles).

Thus, wn always is somewhat smaller than the steady-state frequency wo (as demonstrated in your formula).
For example, the classical WIEN oscillator during start-up has the frequency

wn=wo*SQRT(k-k^2/4)

where k=resistor ratio (nominal k=2, real: 2.1...2.5) and wo=1/RC.
Example: For k=2.5 the frequency ratio is wn/wo=0.97 (3% deviation).

For a better understanding it is helpful to look at the location of the system poles.
During the start-up phase poles are in the RHP of the s-plane.
Due to amplitude limiting (clipping) the poles are forced back to the imaginary axis, which is equivalent to bringing the loop gain
back to unity or – as in our case – reducing the net total loss resistance of the LC tank to zero.
But this applies to the fundamental harmonic of the distorted (clipped) signal only.
And this makes things complicated – and according to my knowledge this has not yet been mathematically analyzed in detail.

But it is more easy to verify and understand this phenomenon if we have a 4-pole oscillator with a separate gain element (like the WIEN oscillator)
because only one single parameter (gain) is involved.
Nevertheless, for an exact analysis of the resulting frequency we need non-linear diff. equations also for this simple circuitry.
As an approximation, sometimes the method of „Describing Functions“ can be applied (but not in our case).

On the other hand, in practice it does not matter at all what the „theoretical“ oscillating frequency in the steady-state mode is.
Due to tolerances, parasitics and other non-idealities the resulting frequency will deviate from its nominal value much more than caused by „violating“ the theory.

In the mean time, perhaps you can point out some of those parasitics that i asked about. What is of most concern to you.

BTW the circuit i was responding to was the one in the .pdf file that was available yesterday 07/17/2012 not the one in the more recent link posted which i think is a little outdated.

Regarding the parasitics:
In practice, the frequency will deviate from the calculated (ideal) value because of
* Finite opamp input/output impedances (idealized during calculations)
* Finite and frequency dependent opamp gain (also idealized and set to infinite during calculations)
* Tolerances of all parts and parasitics (loss resistance of capacitors) as well as board parasitic influences (parasitic capacitances).

One remark to the mentioned pdf file (EDN article). I wouldn't consider it as "outdated".
Instead, it explains the circuit from the FDNR viewpoint and it uses a special design with two different capacitors (scaling factor k).
More than that, it is shown that single supply operation of the circuit can be implemented very easily.
_______________________

Sorry for the long reply.
Regards
W.

 

[MrAl]

Hello Winterstone,


You dont have to apologize for a long reply if it is a good one like that

I assumed that the reader would infer that the equations i presented where of the linearized type. I did not intend to take the nonlinear clipping action into account as if i did the equations would not be quite as elegant, and of course would not yield much better results anyway. Also, depending on the degree of clipping we could very well see extremely close matching between the linear set and the non linear set.

The intent was (as with many linearized networks) to provide a means to analyze effects due to component variations and the linearized version must certainly be good enough for that. We need an equation with all of the components in it to do that. Already the results show very close agreement with simulation using this linearized model as a base. Also, the results agree with your results to within a tiny percentage.

What i was thinking about next was to perhaps add the parasitics that you felt were most important to the model. That way the effects can be investigated and if you wanted to you could compare with your other oscillators and see what advantages there might be.
With regard to this end, resistance losses are easiest to add to the model as well as gains.

I dont intend to do a complete nonlinear analysis except perhaps with a super precise N digit numerical solution of the NDE's, but that's going to have to wait until some time later.

 

[Winterstone]

.....................
What i was thinking about next was to perhaps add the parasitics that you felt were most important to the model. That way the effects can be investigated and if you wanted to you could compare with your other oscillators and see what advantages there might be.
With regard to this end, resistance losses are easiest to add to the model as well as gains.

I dont intend to do a complete nonlinear analysis except perhaps with a super precise N digit numerical solution of the NDE's, but that's going to have to wait until some time later.

Hello MrAl,

I understand your intention.
Regarding parasitic influences, I think that the usage of real opamp models was already a good step into the envisaged direction. And as I have mentioned in my contribution this two-opamp GIC block can be regarded as an "optimum" combination (see ref [1] in the article) because some opamp non-idealities cancel each other if both blocks have identical properties (due to the crosswise connection). I think the simulation result shows that the frequency deviation between ideal (calculated) and real is already very small.
However, the influence of passive tolerances (deviations from the calculated values) is a more severe problem - as in all active circuits like filters and oscillators.
Here a Monte Carlo simulation could give some answers. On the other hand, in most cases the frequency must be tuned anyway. And in this respect the circuit under discussion allows single-element control.
As far as the non-linear analysis is concerned - I also do not intend to take further actions because I think it is a pure academical exercise without any practical value.

W.