i-v characteristic

An important idea is the $i-v$ characteristic for a circuit. Two circuits are equivalent (in their port behaviour, or response, but not in their internals) if they have identical $i-v$ characteristics at a specified pair of terminals.

We determine this by sweeping the voltage (varying it across an interval) and measuring current, or applying some current and measuring voltage. For any given circuit, we can measure the respective $i$ and $v$. A key implication of this is that we can replace many complex circuits with simpler circuits, hence the motivation of Thevenin’s theorem and Norton’s theorem. A circuit composed of only resistors will have a linear $i-v$ characteristic through the origin.

A port are the two nodes that we isolate when examining $i-v$ characteristics; very useful for circuit design.

The $i-v$ characteristic is an affine function. This can be found with superposition:

1. Turn off all internal sources and keep the applied external $v$. Find $R_{eq}$ then write $i^{(1)}=\frac{v}{R_{eq}}$.
2. Short circuit the external $v$ and keep internal sources. Then solve for $i^{(2)}$.
3. By superposition: $i=v/R_{eq}+I_0$.