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

[Andrew Simper]

On 28 August 2015 at 00:31, Richie Burnett <rburnett at richieburnett.co.uk> wrote:
>
> > When it comes to the gain drop, shouldn't it be possible to add more
> > poles & phase shifts somewhere else in the frequency range in order to
> > avoid some of the cancellations?
>
> If you cascade more than four poles in the forward path, you get to your 180 degrees of phase lag required for
> resonance quicker, and with *less attenuation*. Hence you need less gain in the feedback path and therefore
> suffer less attenuation in the passband at the onset of self-oscillation. But of course a higher order filter will
> have a much steeper rolloff slope in the stopband too.

It's not just phase shift but also the level of drop of the filter at
the point there is 180 degrees phase shift. Adding more stages will
increase the phase shift, but it also changes the frequency that the
cascade will self oscillate compared to the regular -3dB frequency of
the one pole sections (which isn't a big deal). The gain needed for an
n stage cascade is 1/(cos(pi/n)^n), so the gain for self oscillation
is as follows:

n=3 gain=8
n=4 gain=4
n=5 gain=2.9
n=6 gain=2.4
n=7 gain=2.1
n=8 gain=1.9

Also the more stages you have then the more low passed and "nicer" the
resonance will sound, since more of the harmonics will be filtered
away. Four stages seems to be the sweet spot, and it is also nice that
the self oscillation happens at the 3dB point of the one poles to
boot, and the amount of change in gain compensation required isn't
that much. I personally think the extra poles of filtering would be
better put to use in other ways.


> Conversely a 3-pole filter like the Tb-303 requires a *LOT* more gain to get close to self-oscillation. And the
> drop in the passband gain at high resonance settings is even more pronounced than the moog ladder! You get
> about 26dB drop without any measures used to compensate it.

The TB-303 diode cascade is a 4 pole filter, but the stages are not
buffered from each other, so the gain at the cutoff isn't -3dB, it's
lower, which is why you need more gain to get it to self oscillate.
Without resonance you get the same shape putting 4 RC filters together
without buffers between them.

Cheers,

Andy