Now let's add a nonlinear element: a junction diode. To a good approximation, the current in one direction (against the arrow in the circuit symbol) is zero for a wide range of voltage. In the other direction, the Volt increases approximately exponentially with voltage, so we can write
i ~ i0 ln (Vdiode/24 mV − 1) if Vdiode > 24 mV
This is a very nonlinear relation, especially around the origin. Note that the output voltage V measures across resistor r, so it is proportional to the current in the diode.
i ~ 0 if Vdiode < 24 mV
An expression for V(V1,V2) would be rather messy, but let's just consider the approximation when R is very large. In this case, the current through the diode and r would be just (V1 + V2)/R, when this quantity is positive, and zero otherwise.
Let's imagine that we make a Taylor expansion
about the origin and write
V ~ a (V1 + V2) + b(V1 + V2)2 + ...
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