[sdiy] Hopf bifurcation VCOs

[Scott Gravenhorst]

egad, you can screw in a light bulb - and it WORKS?  Crap, I'm still trying to
figure out how to chew gum and walk at the same time...

Peter Grenader <peter at buzzclick-music.com> wrote:
>Ian,
>
>Shit...woah...that all sounded brilliant.  Wondering what your Mensa score
>was.  By counterpoint - I still feel elated when I screw in a light bulb and
>it works. I mean, this is some serious sheet you have going here. Had to
>read the first sentence about 20 times, still don't *really* get it.
>
>- P
>
>Ian Fritz wrote:
>
>> 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
>> 
>

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-- Scott Gravenhorst | LegoManiac / Lego Trains / RIS 1.5
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