Resonance Pot Scaling

[Martin Czech]

We all know that inverse log pots or exponential control circuits are
commonly used for resonance feedback control. These cicuits were simply
there.  They give somehow more resolution in the upper Q region.

(I try to add voltage controlled Q to "vintage" circuits via a parallel
control path, ie. the usuall "vintage" pot is not changed, but an ota is
added at or near the wiper summing node, the hope is that if the Iabc is
totally off the ota won't be audible at all. It would be a different thing
to REPLACE the original pot with an ota.  But this is another subject.).

There are many ways to control Q (or the damping D which is 1/Q), by
potentiometer voltage divider or series resistor etc.  This depends on
the implementation. I will not regard this, just take Q as given by some
mechanism, if I know how Q should change I can trace that back to the
question how the actuall implementation can achieve this.

Now, what is really the ideal Q control curve? What is the theoretical
correct curve?  Is there one? I don't know.

In the state variable bandpass case (the one with increasing resonance amplitude,
there are structures that have constant amplitude) T(s) is

T(s)=s/(s**2+1/Q*s+1)

for s=j (Resonance) we get |T(s)|=Q

So , if we optimise for good control over the resonance amplitude,
Q should be Q(a)=exp(a/k), with a:=control voltage (0-10V) and
K:=appropriate constant. This would give the normal audio feel. Is this
correct? It is a very narrow bandwidth case, maybe the exponential law
is not so appropriate for such a case.

Another goal of optimisation could be control over ringing time.
The poles of the above T(s) are

s1/2=-1/2/Q+-SQRT(1/4/(Q**2)-1)

You have a real part a and a imaginary part b to express
s=a+jb.

The ringing time is determined by the real part, as the impulse response
is 

h(t)=exp{a*t}(b*cos(b*t)+a*sin(b*t))/b (?)

the decay term exp{a*t} being exp{-1/2/Q*t}

So, if we want linear control about the decay time, Q should look like
Q=K*a (a now being our linear control voltage), K;=constant.

If we want very long decay times, exponential control might be nice:
Q=exp{a/k}.

So both goals , resonance amplitude and decay time give something between linear
and exponential control of Q, some positive (d/dt)**2.


In practise very often a potentiometer feedback is used.

Damping is then with a linear pot often chosen as 2*(1-a), a is our
rotation scaling  from 0..1, since Q=1/D we get Q(a)=1/(2*1-a) which
has a nasty pole at 1, so we have very steep d/da Q(a) arround a=1.

If we use a series resistor approachinstead, we could get something
like D=2/(10*a+1) (this means inverting sumation, with feed resistor
10*a+1 and feedback resitor 2), so Q(a)=5*a+0.5. There we are, a linear
euqation with offset.  A linear pot should be fine.
For very high Q the resolution gets bad in the lower range, and the feed
resistor needs very high values.

I think, I'll try an ota Q feedback path with some piecewise linear
function for controlling the ota. I don't think it will take
many segments, three should be good enough. I guess the actual
shape of the function doesn't really matter, as long as the end points
keep where they are (0 resonace and oscillation).

Comments ?


m.c.