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 ~ i_{0} ln (V_{diode}/24 mV − 1) if V_{diode} > 24 mV
i ~ 0 if V_{diode} < 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.
An expression for V(V_{1},V_{2}) 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 (V_{1} + V_{2})/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 (V_{1} + V_{2}) + b(V_{1} + V_{2})^{2} + ...
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