[sdiy] Reverse-exponential VCAs

[David G. Dixon]

I thought I'd try to explain what I'm doing a bit more carefully, to see if
anyone here had anything they might like to add:

Basically, I figured out how to use the 2164 to subtract out a proportion of
a signal from a summer at fixed gain in order to provide very smooth,
gradual control of self-oscillation in a filter.  Based on simulation, it
looks like it should work very nicely.

Consider the following circuit:

                               V_c
                                |      |--[ 13k7 ]--|
                                |      |            |
(+) Signal --> [ 33k0 ] --> 2164 VCA ----> Summer ----> Out
                                       |
(-) Signal --> [ 33k0 ] -------------->|
      |                                |
      |------> [ 13k7 ] -------------->|


The (+) signal is fed through a 33k resistor to a 2164 VCA and then into an
inverting summer with a 13.7k feedback resistor.  Hence, when the VCA
control voltage V_c = 0V, the VCA's gain is 1, and the positive signal is
inverted and amplified by a factor of 0.415 = sqrt(2) - 1.  (This is a
critical factor in 4P filter resonance gain.)

The inverted (-) signal is also fed to the summer through two resistors in
parallel, one giving unity gain (13.7k) and the other giving a gain of 0.415
(33k), for a net gain of 1.415 = sqrt(2).

Hence, when the VCA is at unity gain (V_c = 0V), the (+) signal will cancel
with the (-) signal through the 33k resistor, leaving only the (-) signal
through the 13.7k resistor, which will be inverted at unity gain to reappear
as a replica of the (+) signal at the output.

As V_c is increased, the VCA's gain approaches zero, and the (+) signal will
be attenuated going into the summer.  At V_c >> 0V, the (+) signal input
will disappear entirely.  Hence, the (-) signal will be inverted at a gain
of 1.415.  At this point, presuming a similar gain block on every filter
stage, the filter will begin to self-oscillate.  (There is a similar but
slightly different arrangement on the fourth stage output.)

Hence, the net result is that the gain of the signal is varied from 1 to
1.415.  However, the key is that the final approach to this critical 1.415
gain value is very gradual, because the VCA's response is exponential at
33mV/dB, and the resonance increases as the (+) signal is subtracted from
the summer input.  Hence, instead of the typical situation with linear (or
even more so with forward exponential) control of resonance, where
self-oscillation begins somewhat abruptly and may be difficult to control
precisely, one should be able to dial in any oscillation magnitude over a
fairly large proportion of the resonance pot's rotation.  Also, whereas most
filters don't give significant resonance until the resonance pot is at 50%
rotation or more, with this sort of control resonance should become audible
within the first 10 to 20% of the pot's rotation.

This arrangement should also give a much more pleasing response to resonance
CV inputs.

Incidentally, regardless of the actual gain values desired, it's easier to
use two 1% resistors in parallel on the (-) signal in this application than
to try to find just the right value of a single 1% resistor which will give
the parallel value of the other two exactly.

Discuss!