[sdiy] Hopf bifurcation VCOs

[yves usson]

Looks promising...

Cheers

Ian Fritz a écrit :
> Hello folks --
> 
> In nonlinear dynamics a bifurcation is a change in the nature of a 
> phase-space orbit as some parameter is varied.  In two dimensions, a 
> Hopf bifurcation is a transition from a fixed steady state to a circular 
> orbit.  A circular orbit projected onto a pair of axes gives sine and 
> cosine oscillations, i.e., it represents a quadrature oscillator.
> 
> A simple prototype for a circular limit cycle derived from a Hopf 
> bifurcation is given in polar coordinates as:
> r' = r(1-r^2)   th' = w,
> where primes indicate time derivatives, r is the radial coordinate, th 
> is the angular coordinate, and w is the angular frequency.  For r = 1 we 
> see that r' is zero, so r remains constant at r = 1, while the angle 
> increases linearly in time.  So the system is a spinor with unit length 
> and frequency w.  Orbits with larger or smaller r spiral into this r = 1 
> orbit.
> 
> To proceed with designing an oscillator based on this system, we write 
> the equations in rectangular coordinates.  The math is a bit messy, and 
> I have simplified it slightly by dropping the nonlinear cross terms.  
> The system is then
> x' = x(1-x^2) + wy
> y' = -wx + y(1-y^2).
> 
> For an electronic-circuit implementation of this system we need two 
> basic building blocks.  First we need integrators to relate x and x'  
> (input = x', output = x) and y and y'.  Then we need a pair of nonlinear 
> circuits to generate the functions x - x^3 and y-y^3.  It turns out that 
> we do not need to generate exactly this function, just a function having 
> the same general shape.  This is easily done by adding a linear slope to 
> the response of a pair of back-to-back zener diodes.  The circuit 
> requires just the zeners, an opamp and four resistors.  It is an 
> important circuit, as it can be used in a variety of nonlinear circuit 
> applications including chaos generation and waveform folding.
> 
> To build this quadrature oscillator I started by cross-coupling two 
> voltage-controlled integrators, similar to what we all use for filters, 
> phasors, etc.  These represent the linear terms in the coupled 
> equations. Then I added the nonlinear circuit described above around 
> each integrator, to implement the nonlinear terms in the equations. This 
> may sound complicated, but it only requires two chips: a dual OTA 
> (LM3700) and a quad opamp.
> 
> The result is a very nice circuit.  As far as I can tell it always 
> starts up without any latchup problems.  The oscillations are not at all 
> sensitive to circuit parameters.  In fact the coupling of the nonlinear 
> elements can be varied over a wide range, with only a small change in 
> distortion.  I recorded the oscillations into Sound Forge and did a 
> spectral analysis.  Distortion is very small, with all harmonics  down 
> by 50 dB or more.
> 
> An attractive feature of this system is that it can be extended to 
> higher orders in the obvious way of coupling several subsystems in a 
> ring.  In a slightly different system I am working on for high-order 
> chaos generation, I was able to easily make a five-phase oscillator 
> (decature oscillator?) with low distortion and non-critical circuit 
> parameters.
> 
> More on that later.  :-)
> 
>   Ian
> 


-- 
Yves Usson

http://yusynth.net