[sdiy] standard filters from ladder topology?

[Don Tillman]

   > From: "JH." <jhaible at debitel.net>
   > Date: Sun, 4 Sep 2005 00:14:45 +0200
   > 
   > Is it possible to get "standard" low pass filters such as
   > Butterworth, Bessel, etc., from a Moog ladder with multiple
   > feedback (or feed forward) loops?

Hi JH!

I haven't actually built such a thing... but I believe it's completely
possible.

The math is mostly straightforward.  We have Vin, a summing point,
four 1/(s+1) stages, the output from the last stage is called Vout,
and taps from each stage with gains B1,B2,B3,B4 go back into the
summing point.  Call the voltage at the output of the summing point
Vs, find that, then find the equation for Vout.

I get:

  Vout/Vin = 1 / [ (s+1)^4 - B1(s+1)^3 - B2(s+1)^2 - B3(s+1) - B4 ]

The coefficients for the ladder filter with no feedback are:
  1, 4, 6, 4, 1

The coefficients for a Butterworth filter are:
  1.000, 2.613, 3.414, 2.613, 1.000

The s^4 term is already 1.0.  B1 then adjusts the s^3 term, as well as
the others.  B2 adjusts the s^2 term, and so on.

I just did a quick calculation that tells me that the feedback values
for a 4 pole butterworth ladder filter are:
  B1 = 1.387
  B2 = -1.575
  B3 = 0.377
  B4 = -0.189

There may be errors here; I did not check it for accuracy.

  -- Don

-- 
Don Tillman
Palo Alto, California
don at till.com
http://www.till.com