Linear and nonlinear superposition

Superposition of oscillations in a nonlinear medium or system introduces complications. The multimedia chapter Interference and consonance introduces linear superpostion, interference and consonance. Here we revise some those ideas and extend them to include superposition in a nonlinear system. This page is still being made. Apologies: please return later.

A simple linear system

This photograph shows a linear system: voltages V1 and V2 are the two inputs and V is the output. We know that it is linear because resistors are linear: the voltage across a resistor is proportional to the current through it. So we can write
    V  =  αV1 + βV2
where (applying Kirchoff's laws we see that) α = β = 1/(2 + R/r). The oscilloscope screens show this for a simple example in which V1 and V2 are sinusoids with different frequencies. The fourth screen is that of a spectrum analyser that displays V. It shows just two frequencies present, those of V1 and V2, which we'll call f and g respectively henceforth.

 

A simple nonlinear system

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
    i  ~  0   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.

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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