are caps and coils linear circuit elements?

[John Larkin]

This is true. I actually checked Wikipedia also, but didn't
understand the whole sinusoidal response definition. In my head I was
using a cap's (or coil's) iv characeristic and a randon input into
this circuit:

Vin o------/\/\/\/\-----o-----| |-------GND (output across cap..
middle node).

I was thinking to myself, if I place a random voltage waveform at Vin,
I certaintly won't be guranteed an ouput that is linear with respect
to it. For example, a triangle wave input doesn't result in a
triangle wave output.

So this is why I was thinking to myself: "why is this circuit linear?
Rs are linear, but if Cs were also linear then the response to any
input waveform should be proportional to it."

But now I get that the term "linear" is only for sine wave inputs.

Not so. If a network (or component) has any arbitrary forcing input S,
and has some corresponding output Y, then the network is linear if
increasing the amplitude of S produces an exactly increased output Y.

So if S produces Y

then if

N * S produces N * Y

for any value of N, then it's linear.

A linear network can have an output that looks very different from its
input, like your example of a triangle going in but some other
waveform coming out. A passive lowpass filter can turn a triangle into
a sine wave, but a passive lowpass filter is still a linear network.
Double the triangle input and you'll get double the sinewave output.

One of the consequences is that if you apply a sine wave to a linear
network's input, you can only get a sine wave out. A linear network
can change the phase and the amplitude of frequencies that pass
through it, but it can't generate new ones.

What's sort of cool is that if you apply a sine wave to a linear
gadget and get some output, and you plot the input and output
waveforms against each other (XY plot on a scope) the only curves you
can trace are the various/degenerate versions of an ellipse.

Phil is obviously nonlinear.


John