[sdiy] VCO - sine output - why bother?

[Ian Fritz]

Matthew --

Thanks for posting. Your comments are right on.

I was trying to emphasize more the approach to the analysis than the 
classification.  Of course the COTA oscillator can be looked at as a 
limit cycle oscillator, ie by writing down the differential equations in 
phase space and looking for stable periodic solutions.  It's just that 
it isn't normally done that way.  Three of the equations would represent 
the three free integrators and the fourth would additionally include the 
feedback and nonlinear limiting.  The limiting could be approximated by 
an appropriate function -- not necessarily simple -- and computer code 
could crank out the solutions numerically.

My approach was based on the material in the "Cyclic Chaos System" 
section of my website.  This work is an implementation of work by Rene 
Thomas and coworkers (C. R. Biologics 326, p. 205 (2003), and Chaos 14 
(3), p. 669 (2004)).  The multiphase oscillator is a slightly simplified 
version of this, using fixed damping on each stage and just a single 
nonlinearity. What's different from the COTA approach is that each stage 
has damping.  That damping is chosen so that, without the clamping, the 
system diverges weakly.  Thus, only weak clamping is needed, resulting 
in pure sinusoids.

My third order filter (the Threeler) is set up to produce a variety of 
limit-cycle and chaotic oscillations in addition to the more standard 
filtering functions.  Demos at my you tube channel.

Ian


On 8/29/2016 10:01 PM, mskala at ansuz.sooke.bc.ca wrote:
> On Mon, 29 Aug 2016, Ian Fritz wrote:
>> Yes, that's why I abandoned the COTA approach in favor of the limit-cycle
>> approach. For the four-stage version, the waves are all the same amplitude,
>
> Where can I find more information about this "limit cycle approach"?  I
> did some Google searching and all I found was academic literature from
> chaos theory, where the term seems to refer to any dynamical system whose
> behaviour converges on some kind of cycle.  There's a more precise
> mathematical definition, but it doesn't seem useful:  my reading of it is
> that almost any electronic oscillator could be called a "limit cycle"
> oscillator, and there'd be no specific reason for that to give especially
> pure sine waves unless the oscillator happens to be designed so that the
> limit cycle is a pure sine wave, which is begging the question.
>
> The specific design of your own that you're pointing to seems to differ
> from other oscillators in that it's applying gentle limiting at all
> amplitudes increasing slowly with greater amplitude instead of harder
> limiting almost zero at low amplitude and increasing quickly at high
> amplitude, and it applies the limiting at all stages instead of just one.
> Are those the important features of the "limit cycle approach"?
>


-- 
ijfritz.byethost4.com