# FW: AW: Re: digital contents

Paolo Predonzani predo at dist.dist.unige.it
Mon Sep 23 16:36:26 CEST 1996

```> >The non-linearities are a major problem because they are present in
> >every term of the (4th order) differential equation the filter obeys to.
> >This makes the poles move in the s-plane for constant cutoff freq and Q.
> >The small signal analysis shows 4 coincident poles in -p (open loop).
> >The large signal analysis shows that the poles move between -p and the
> >origin of the s-plane. The static open loop gain also varies.
>
> This really caught my attention! So far I have only looked at the
> nonlinearities as a cause of harmonics/distortion, and not shifting
> poles as well! This would mean that the cuttof frequency changes
> with signal input strength ...

The nonlinearities can be seen in both ways so long as reactive components
(capacitors) are present in the circuit.

> Now, this makes me question: (1) How are "poles" defined for
> large signal analysis? I only know this concept for linear circuits.
> How are they defined for nonlinear circuits? Time domain
> response for a certain input signal, and then inverse fourier
> transform? If so, for which input signal ??

Poles are defined for the small signal approximation. For large signals
the operating point moves and so does the (instantaneous) small signal
approximation.
I'll make an example. Consider an envelope tracking automatic wha-wha.
It can be described as a time-varying n-pole band-pass filter. But if
x and y are signals (loose notation) and the function the whawha performs
is f() then f(x+y) != f(x) + f(y). So the system is not linear. This is
so because the transfer function is correlated to the input signal.

> (2) How *far* would these poles move? Usually, the signal amplitude
> thru the ladder is small in comparison with the bias current. (?)

It's not so small. The open loop poles almost touch (0,0) in the s-plane.
As to the closed-loop poles I never examined them in detail because I
would have to find the roots of a 4-th order polinomial. Currently I don't
have an algorithm for that. Nevertheless I tried to find a graphical
solution by means of roots-loci. The result is (I think): if the cutoff
frequency is close to the fundamental freq of the oscillator (not more
than 1 octave above it) then the closed-loop poles perform a kind of
elliptical trajectory.

> (Equations welcome (;->) )

The current (handwritten) documentation is 11 pages long and I'm studying
new integration algorithms. I'd like to make a paper out of it.

Paolo

--
+-------------------+----------------------------------+
| Paolo Predonzani  |  email: predo at dist.dist.unige.it |
+-------------------+----------------------------------+

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