butter and pole locations

[Don Tillman]

   From: Haible_Juergen#Tel2743
   Date: Mon, 20 May 96 17:39:00 PDT

   (BTW, has anybody yet calculated the exact locations / pole spread
   for the diode ladders?)
   The Roland cascaded SV 4p-filters might be different ... anybody
   knows if *they* start on an arc (as in a butterworth)?

Derriving the transfer function isn't all that difficult, but it's a
very mistake-prone process.

The generalized version (any values for the Rs and Cs) is too long for
me to conveniently write out, but if you set all the Rs equal and all
the Cs equal, the transfer function becomes:

  Vout              1
  ---- = ----------------------------
   Vin   s^4 + 7s^3 + 15s^2 + 10s + 1

(I *think* this is correct.  If anybody cares they can do it out
independently and we'll compare notes.)

Which means the poles are at (normalized to 1Hz for clarity):
   2.35Hz, 3.53Hz, 1.00 Hz, 0.12 Hz

As expected we have four real poles.

Adding a feedback factor simply increments the 0th coefficient by that
amount, soooo, with a feedback gain of 1 the poles are:
   2.62Hz, 3.41Hz, 0.59 Hz,  0.38 Hz

At a feedback gain of about 1.07, the lower two poles meet.

Feedback gain of 2:
   3.00Hz,  3.15Hz, double pole at 0.56 Hz (Q= 0.66)

Not too interesting. 

Feedback gain of 3:
   3.14 Hz (Q= 0.50), 0.64 Hz (Q= 0.84)

(Note the higher two poles have met.)
	
At a feedback gain of 4:
   3.20 Hz (Q= 0.51), 0.70 Hz (Q= 1.04)

The lower pole pair is starting to show some gain.

At a feedback gain of 5:
   3.25 Hz (Q= 0.51), 0.75 Hz (Q= 1.25)

At a feedback gain of 10:
   3.45 Hz (Q= 0.52), 0.96 Hz (Q= 2.98)

At a feedback gain of 15:
   3.60 Hz (Q= 0.52), 1.11 Hz (Q= 9.57)

Here's some serious resonance.  And it starts oscillating with a
feedback gain of about 18.3.  The higher pole pair never really gets
off the ground

Executive summary: not too exciting.

And yes, in answer to the upcoming question, I have some software I
wrote to do the pole analysis.

  -- Don