poles and s-planes

[Magnus Danielson]

From: "urosuros" <urosuros at bits.net>
Subject: poles and s-planes
Date: Thu, 17 Feb 2000 15:50:27 +0100

> Hello good people of sdiy ,

Hi there!

> there are several student questions I would like to ask :
> What is s-plane position of poles and zeroes of  4pole moog and three pole
> 303 filters ?

It really depends on the Q value you have set up your filter for. The Moog
ladder for instance will have it's 4 poles all sitting on the real axis when
in low-Q mode (flat):

          |
          |
-----x----+-----
          |
          |

When you raises the Q value will this distort this picture, so halfways up you
are at

   x   x  |
          |
----------+-----
          |
   x   x  |

and finally you reach

          |
x         x
          |
          |
          |
----------+-----
          |
          |
          |
x         x
          |

which then is self resonance since one of the pairs has reached the jw-axis.

All the poles will be sitting on a circle which expands by the increasing
feedback. Depending on the feedback sign you will have a poles on the real
axis or not. The danger of having poles on the real axis is that the first
pole reaching the jw-axis will do so with w = 0, thus we get a huge gain for
DC as we go beyond that point. This means that our filter caps will be DC
saturated before we reach any interesting Q-values. I beleive that his is
what plauges the Formant 24 dB filter, which shows exactly this behaviour
(I must test this some of these days).

> I dont expect anyone to draw it and post it to list but can someone  please
> tell me where can I find it on net ?

If ASCII will do, look in a inbox near you.

> Also , root-locus diagrams ?

The above diagrams and text should give you a clear picture of how poles move
about in the Moog ladder and many similar filters as well.

> On common sense under normal conditions all poles should be imaginary, but
> what happens with high resonance settings ?

You mean to say that under normal conditions all poles should be complex, but
with negative real parts, that is, the are situated on the left half plane
(LHP).

> Selfoscilation ( instability ) should happen when poles have just imaginary
> part in ideal conditions but in real cases poles would enter ( more or less )
> right half of s-plane , right ?

Right. But here nonlinearity kicks in and limits the energy, so you can thus
stabilize the poles on the jw-axis.

> One of my professors told me that three pole transfer function is inherently
> much , much more instable that two pole ?

Well, in a sence this is true, since for feedback systems you are much more
prone to develop destructive feedback. This is a reason for why higher orders
PLL system is not too popular for instance.

> Also he told me that instability could cause sqare wave oscilations not just
> sine wave resonant oscilations . Has anyone experienced this in filters ?

Most have. When you get high gain your filter will clip and there you have
a squarewave oscillation rather than sine. It's a nonlinear property of a real
curcuit rather than the linear theory of s-plane.

> And finaly are there any other three pole filters other than 303 ?

The Formant 24 dB filter is interesting since you can select the number of
poles to be 1, 2, 3 or 4 with a simple selector. It does however have it's
flaws and I think one has to overlook the feedback sign. The root-locus stuff
I told you about is generic for filters having (s-a)^n denominator in
non-feedback case. This basically means a number of RC filters with buffering
inbetween them.

> Was this discused earlyer ( I'm just two months on list ) ?

The issue pops up every now and then but takes on diffrent shape each time.

Cheers,
Magnus