[sdiy] Ladder filters and gain drop, that old chestnut

[Richie Burnett]

> I estimate there's a good 10dB between self-oscillation and "no feedback".

The low-frequency pass-band region of the filter response theoretically gets 
attenuated by 14dB when you go from no feedback to just below the point of 
self-oscillation.  This is for a 4-pole low-pass cascade with global 
negative feedback to induce resonance.  i.e. Moog ladder, Juno 106's IR3109 
filter, etc...  (State Variable is a different kettle of fish.)

> So I was wondering, what influences this?

It comes from the negative feedback equation:-

http://electronicdesign.com/site-files/electronicdesign.com/files/archive/electronicdesign.com/content/content/74496/74496_f1.gif

Here's how the moog ladder filter operates from a "control theory" / 
"stability" point of view...

At the cutoff frequency of each of the four cascaded poles contributes 45 
degrees of phase-lag, and reduces the amplitude of the signal passing 
through it to 70.7%.  The combined effect of all four poles is that the 
signal at the cutoff frequency is reduced to a quarter of its previous 
amplitude and gets a total phase lag of 180 degrees.  Then you put global 
negative feedback around the four cascaded poles to try to make it resonate. 
Once you increase the negative feedback gain up to 4, you perfectly 
compensate for the quartering of the amplitude in the forward path and the 
filter breaks into oscillation.  That equation above tells you that applying 
a negative feedback gain of 4 around the forward block will decrease the 
pass-band gain from unity down to 1/5th.  That's where the 14dB drop in 
low-frequency volume comes from!

> If I set the feedback level to be constant and quite high, would 
> controlling the resonance by reducing the gain in the differential amp 
> give a decent swing of resonance without the output increasing to silly 
> amounts?

The normal way to counter the loss in pass-band volume with increasing 
resonance is to either:

1. Boost the drive signal as the resonance is increased.
2. Boost the gain after the filter as the resonance is increased.

Both achieve the same compensation in pass-band volume in a linear filter, 
but they have subtly different effects when the filter itself is non-linear. 
Boosting the input signal (option 1) drives the filter harder than option 2 
when the resonance is increased, and gives more "compression" of the 
resonance which sounds "fatter" and more analogue to most people's ears.

Hope this helps?

-Richie Burnett,

PS. All of that maths is for an ideal filter with perfectly matched 
components.  If, for example, the capacitor sizes in the ladder were 
slightly different, then the poles wouldn't all have the same cutoff 
frequency.  Under these real-life conditions you need more than a gain of 
four in the negative feedback path to make it self-oscillate, and that 
results in more than 14dB drop in pass-band volume!