Mike, I think that you can use varactors in a gdo if it has some
provisions for amplitude limiting the oscillation. A Lambda diode
oscillator comes to mind as a appropriate. Almost everything else will
require that you limit the minimum control voltage seriously even if you
use two antiseries varactors.

I built my gdo using a polyethylene variable meant for small radios. It
has the same axle you described but I was able to find buttons for it in
this country. ;-) (actually a tuning scale wire wheel).

Peter

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