[sdiy] Poles of a diode ladder filter

[Aaron Lanterman]

So, tomorrow's lecture is on the Moog ladder, which I think I can fake my 
way through OK.

I wanted to talk about the EMS/TB-303 style ladder that uses diodes (or 
diode connected transistors) instead of transistors - in those cases, it's 
like an _unbuffered_ RC ladder.

So, I wondered what the transfer function looked like and where the poles 
go.

Ugh.

I set up my ladder, and then did repeated Thevinin equivalents. (I'm 
praying there's an easier way but I didn't see one).

After an ungodly amount of algebra, I found (for a normalized RC=1 filter)

s^4 + 7 s^3 + 15 s^2 + 10 s + 1

Has anyone else ever tried to slog this out?

(I sorted through electronotes - figuring that would be the best place to 
find it - and didn't see it.)

I have no faith in my algebra.

Anyway, the poles then lie at (assuming my algebra is right, which who 
knows)

   -3.5321   -2.3473   -1.0000  -0.1206

and that pole at -0.12 really dominates

If you throw in feedback, say with a feedback gain of 2, the poles start 
to march together

    -3.4142   -2.6180   -0.5858   -0.3820

But just barely above that, at some point, the right two poles separate.

For k = 1.05,

    -3.4068   -2.6319   -0.5296   -0.4317

for k = 1.1,


   -3.3991
   -2.6459
   -0.4775 + 0.0743i
   -0.4775 - 0.0743i

(I'm using the numeric "roots" routine in MATLAB, which may have some 
numeric errors)

For k = 2, the left two poles start to come togehter

   -3.1479
   -3.0000
   -0.4261 + 0.3690i
   -0.4261 - 0.3690i

For k= 2.1, they split off along the imaginary axis:

roots([1 7 15 10 3.1])

   -3.0791 + 0.0919i
   -3.0791 - 0.0919i
   -0.4209 + 0.3867i
   -0.4209 - 0.3867i

Plotting the frequency response, I empirically start to see peaking for k 
around 2.4; as you crank up the resonance, the bump moves to the right and 
gets taller relative to the rest of the response, although the absolute 
height seems to get smaller - then bigger - then smaller again - it's very 
weird, and makes me think I have something not quite right.

Has, anyone, anywhere, tried a similar analysis?

Or just have some intuition as to whether I'm anywhere even close?

- Aaron

-----------------------------------------------------------------------------

Dr. Aaron Lanterman, Asst. Prof.
and Demetrius T. Paris Junior Prof.    Voice:  404-385-2548
School of Electrical and Comp. Eng.    Fax:    404-894-8363
Georgia Institute of Technology        E-mail: lanterma at ece.gatech.edu
Mail Code 0250                         Web:    users.ece.gatech.edu/~lanterma
Atlanta, GA 30332                      Office: Centergy 5212