[sdiy] Non-linear properties of SSM2040

[Rutger Vlek]

Dear all,

In the sparse moments between day time job and fatherhood I'm trying to
push forward with an idea I've had for a long time. I'm trying to capture
the variation in favourable non-linear characteristics from well-known
filters (read SSM2040 and Moog Ladder) and implement them in a more modern
topology (read SSI2164). Would also be great to be able to select between
them, while using the same filter core. I'm not necessarily aiming for
perfectly cloning the response of vintage filters, but rather hope to take
inspiration from them and perhaps to discover other pleasant
non-linearities.

I've been approaching this with Spice as well as with mathematical
modelling in Python, using a multi-dimensional Newton-Raphson solver with
the system of equations needed to describe the various filters.

Right now I'm trying to understand the character of the SSM2040, and am a
bit puzzled. I am familiar with the inverting cascaded topology of this
filter. And with the typical math that describes the non-linearities of an
OTA-based filter:
Vout = g * tanh(Vin-Vout).

I have been reading across the internet about the asymmetrical saturation
of this filter, most notably in the application notes for making filters
with the SSI2164: https://www.soundsemiconductor.com/downloads/AN701.pdf.
Based on this information, I had assumed that the tanh only operates in one
direction, something like this:
y = tanh(x)    if x < 0
y =x               if x > 0

However, in Spice, when simulating the internals of the SSM2040 using
Jurgen Haible's schematic (
http://jhaible.com/legacy/tonline_stuff/jh2040.gif), I don't see that
happening. Instead, I see tanh distortion in both directions.

Only when I push the input harder (beyond 1Vpp), I see one side of the
output clipping much sooner than the other. This seems in line with the
effect described in AN701, but is the SSM2040 really driven that hard in
real world applications, and is it really *this* additional effect, on top
of the already present tanh distortion, that explains the SSM2040's
character? And if so... how would one model it mathematically?

Rutger
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