Volt-ampere characteristics of a circuit part with source

Ohm’s Law can be applied to a piece of a circuit with a source, and we can draw a Volt-Ampere characteristic for this piece of a circuit.

Let’s consider  piece of a circuit with a resistor $r$ and voltage source $E$ in parallel are shown. Current i flows from terminal 1 to terminal 2. Basically the current direction depends not only on the voltage source, but from the other part of a circuit, but we will assume that current direction is considered. The potentials difference in this way is:

$u_{12}=–E+ri$

The potentials difference in the direction from terminal 2 to 1 is:

$u_{21}=E–ri$

Voltage between terminals 1 and 2 is the following:

$u=E–ri$

If current has a negative sign (it flows in the opposite direction):

$u=E+ri$

There is a piece of a circuit with resistance $r$ and current source $I$ in parallel between terminals 1 and 2 (see figures in the previous post). As before, the current direction $i$ depends not only on the considered currents source, but also on other current sources in the circuit. With the mentioned current directions, current from terminal 2 to terminal 1 is $i–I$, then voltage:

$u_{12}=ri–rl$

Potentials difference in the direction from terminal 2 to terminal 1 is:

$u_{21}=rI–ri$

Volt-ampere characteristics of this piece of circuit are shown on the figure below.

Comparing these volt-ampere-characteristics we can say that resistor circuits with voltage source in series or current source in parallel are interchangable.

12a: Kirchhoff's Law