state var RC mismatch

[Martin Czech]

Someone brought up the question of state variable filter RC
mismatch, ie. the first integrator has other RC constants
then the second.

I assume the numerator being:

RCSD*s^2+a1*S*D*s^1+a2*s^0

where R and S are resistors
and   C and D are caps.
Further a2=1 as usual. 
(This enshures lp=hp=bp at cutoff. Other values are allowed, but this is 
no improvement. All state var structures I've seen set a2=1).

I concentrate on the RC mismatch, because precise 1% resistors
are not critical today, whereas caps still are.
I'll not regard any other circuit errors due to R mismatch
in summing stages etc.

Some calculations and simulations show, that the effect is not very
strong.  At cutoff frequency hp and lp magnitude are always the same,
this cutoff frequency is shiftet.  The bp magnitude differs if the
two RC (SD) don't match.  It is very unlikely that this is audible,
but could effect combinations of the outputs (notch or allpass), these
would require trimming.

I tryed RC=1.1/0.9 CD, which is the worst case for 10% caps.


Very often R or S is replaced by an OTA, in this case the
transconductance tracking becomes interesting.
I'll have to look at that also (later).

Also, high frequency effects of the op-amps are not considered.

However, as a rule of thumb one can find out which RC is too low by the
following procedure:

Tune the filter to a lower frequency (eg. 200 Hz).  Tune Q to 0.5,
this is the pole splitting point, larger values of Q show ringing in
the impulse response, so take a square or saw , turn Q up and then down,
until the ringing just disappears. This Q setting is not critical.

Now take a sine as input and observe lp and hp with a two beam scope. Tune
the sine or the filter until hp=lp magnitude.  Now change one channel to
the bp. If bp > hp magnitude, this means RC of the bandpass integrator
(the one that leads from hp to bp) is too small. And vice versa.

This can be tested for different cutoff frequencys in order to see,
if the RC producut matches over the whole area and if there is any
tracking problem.

At higher frequencys the nonideal op-amp behaviour may add further
problems not covered here.

PRO: black box, you don't even have to open the filter 
CON: needs some time

I guess I should pop this up on my web site.

m.c.