# Poles and large signals ...

Haible_Juergen#Tel2743 HJ2743 at denbgm3xm.scnn1.msmgate.m30x.nbg.scn.de
Tue Sep 24 03:05:00 CEST 1996

```> Poles are defined for the small signal approximation. For large signals
> the operating point moves and so does the (instantaneous) small signal
> approximation.

Ok, this sounds reasonable to me. Just to be sure I've got it, you're
not speaking of some (averaged) effect over one or even several periodes,
but you divide the input signal in such small steps, that even a single
period
is divided into many steps, and then you define poles for a linearized
circuit in that very operating point.
But wouldn't this concept also imply that you don't calculate the
interaction
of a periodic (sine shaped, for example) signal with these poles anymore,
but rather the response of more or less linear input *segments* with this
momentary pole configuration ?? (The reality of course being an infinite
amount of very small segments ...) I am ready to believe that these
momentary
defined poles can do rather strange things (see below), but what would be
the
(average) effect on a whole period?

But let's think about these momentary steps, first.
I'll try to separate momentary bias points and momentary small signal
behaviour,
and I will look at one single stage of the moog cascade now to simplify
things.
I center my view to one capacitor in the middle, and I see two collectors
below, which will be two current sources (sinks, actually), providing
Ibias+Isignal
in the left branch and Ibias-Isignal in the right branch. Above me, I see
two
BE junctions that are tied to a bias voltage, and which form two current
controlled
resistors. (And I know that the voltage across these resistors will be
passed
on to the next stage as current, again with some nonlinearity, but this
process being not that much frequency dependend, I'll omit it here.)
I see that a large signal current will make one of these BE impedances lower
than it would be just from the bias current, and making the other one
higher.
Now let's go to the extremes, say we have much input overdrive, and we have
almost the double current in the left branch, and almost zero current in the
right branch. So my small signal approximation for this case would be like
this: a differential (floating) AC current source, a capacitor across this
current
source, and two very different resistors from either side of the current
source
to GND. My filtered output signal would be the voltage across the current
source (actually, with factor gm of the next stage). Now  the resulting "R"
of this
"RC" combination is the *sum* of both resistors ... and with one of these
becoming very large (Ibias-Isignal near zero ...), the sum would be
dominated
by this large resistor, forcing the pole of this stage down. Is this the
idea?

Then, if I try to make some kind of averaging over a whole periode, there
would
be times where you have the initial pole locations (s = -1, at the zero
crossing
of the input signal), and every lower poles when you cycle thru a half
period,
reaching its minimum at the input signal peak, and reaching a maximum of -1
at the next zero crossing.
So all in all, pole frequencies only go lower, eventually near zero,
so the "average" must be in between -1 and 0.
This would sound reasonable to me; but where exactly would be such an
"average" pole location be? As I said, the *momentary* pole locations would
be of academic use only, because they "don't have time" to interact with
a ore-or-less periodic input signal).

Now this whole approach should bring the same results for specific cases
that we know or we can measure. We could set a filter to f=100Hz,
and measure how far an input signal of 200Hz would be attenuated,
dependend on input level. Of course we would measure the strengh of the
fundamental only, at the output, to make sure that we just examine the
effect of pole shift on the fundamental.
I'd expect the fundamental being more attenuated in the case when the
input is overdriven. *But* what is the reason for this:
(1)The momentary pole migration, as described above?
(2) The well-known *compression* effect of an overloaded differential pair
??
Or would both effects add to each other ???
Or: Would (1) and (2) be just different descriptions of one and the same
thing
????

Thanks for listening - hope this is readable