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My question is about interpreting a given circuit as being itself a linear system. We know that a linear system needs to satisfy the superposition principle: if \$f(ax_1+bx_2)=af(x_1)+bf(x_2)\$. This means that a zero input should result in a zero output. Circuit components like ideal resistors, capacitors, inductors, etc. are linear since their element laws obey superposition (\$ V_r = IR, \hspace{6pt} V_i = L\frac{dI}{dt}, \hspace{6 pt} V_c = \frac{1}{c}\int I\hspace{2pt} \mathrm{d}x\$, etc).

But given a circuit that contains current or voltage sources, for example, isn't this now a nonlinear system?

Take for example some Thevenin circuit equivalent like the one below with its i-v curve at its port. The i-v curve doesn't pass through the origin since the Thevenin equivalent has a voltage source inside of it and so it has a nonzero open circuit voltage, so this system has a technically nonlinear i-v relationship.

So some bottom line questions:

  1. Sources have nonlinear i-v curves; are they included as part of "linear" circuit theory since they should actually be considered inputs as opposed to components of the circuit itself?
  2. A circuit containing resistors with i-v characteristics of the form \$V = IR+V_0\$ where \$V_0\$ is some constant could not be considered a linear circuit and you could no longer employ superposition analysis or Thevenin/Norton analysis, correct?
  3. Does it even make sense to try to define an input/output of a two terminal device? I was originally going to say that the Thevenin equivalent circuit below had nonzero "output" current for zero "input" voltage but I'm not sure if that's even meaningful to say.

Thevenin circuit i-v curve

knzy
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1 Answers1

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But given a circuit that contains current or voltage sources, for example, isn't this now a nonlinear system?

It is not strictly a linear system.

But if you consider only its response to an AC stimulus, it may still behave as a linear system.

Therefore we often call something a "linear circuit" even if it is not a linear system.

In any case, we almost always apply linear circuit theory to circuits that are not in fact linear, but that behave linearly in response to perturbations of the input around an operating or bias point. (Physicists talk about perturbation theory as a general method of finding linear approximations to the behavior of nonlinear systems, and linear circuit theory is one example of this)

A circuit containing resistors with i-v characteristics of the form \$V = IR+V_0\$ where \$V_0\$ is some constant could not be considered a linear circuit and you could no longer employ superposition analysis or Thevenin/Norton analysis, correct?

You can still do Thevenin and Norton analysis because Thevenin and Norton equivalents apply to networks containing a combination of linear resistors (or impedances if we get into AC analysis) and ideal sources.

Your "resistor" is basically a series combination of what the rest of us call a resistor and a voltage source. It's actually already its own Thevenin equivalent. And so long as the \$R\$ coefficient isnt' 0, we can also find a Norton equivalent for it.

Does it even make sense to try to define an input/output of a two terminal device?

We can choose either voltage or current as an input, and the other as output.

The Photon
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  • Thanks a lot for your response; when you say it could be linear in response to an AC stimulus, are you talking about when doing small signal analysis or still large signal? Also, is my interpretation of sources as circuit inputs rather than "real" circuit components accurate? It seems consistent with how sources get treated when doing superposition analyses. – knzy May 01 '20 at 06:58
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    @knzy 1. Yes, "small signal analysis" is basically the EE's name for perturbation theory. 2. There's no problem with doing it that way. You could take any component you like (resistor or source) out of the circuit and treat it as connected to the circuit by a new port. Doing this for all the sources would make the remaining network into a true linear system. – The Photon May 01 '20 at 16:08