Linear and non-linear 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: more to come later.

This animation illustrates just a few of the complicated superpositions that occur when multiple sources produce sound.

The pressures of sound waves are usually very much smaller than atmospheric pressure, which means that air is a highly linear medium for sound, as we saw when deriving the wave equation for sound. Our ears, however, do not respond linearly to sound, so it is interesting to compare linear and non-linear superposition. To explain the differences, we'll use some simpler systems made from electronic components.

A simple linear system

This photograph shows a linear system: voltages V1 (on the white wire) and V2 (on the yellow wire) are the two inputs and V (the red wire) is the output. (All voltages are with respect to the ground: the black wire.) This circuit 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 frequencies (1.6 and 2.0 kHz respectively). The fourth screen is that of a spectrum analyser that displays V. (The apparatus is the same as that used for Lissajous figures.) The spectrum analyser shows just two frequencies present, those of V1 and V2, which we'll call f and g respectively henceforth.

Labelling to be fixed up on this clip: For this one the DFT was 0-5 kHz. Also a circuit diagram to add here.

 

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 voltage 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 (the red wire) 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 + ...

This page is still being made. Film clip to be inserted here. Please return later.

Why 24 mV in the equations above? At room temperature, kT/e = 24 mV, where k is Boltzmann's constant, T the absolute temperature and e the electron charge. The Boltzmann distribution specifies the proportion of electrons (or atoms, molecules etc) having a given energy E: the proportion varies as E/kT  =  eV/kT. More on this later when we do thermal physics.

Comparing current and voltage

More to write here: please return later.

 
 
Creative Commons License This work is licensed under a Creative Commons License.