CR · DC analysis

Norton Equivalent and Source Transformation

Where Thévenin folded a circuit into one voltage source with a series resistor, the same circuit also folds into one current source with a parallel resistor. See that the two forms are identical at the terminals, and learn the transformation between them.

Make the two forms the same line

Slide the Norton current I_N. The terminal line of the current source with its parallel resistor (blue) shifts. Find the I_N at which it lands exactly on the Thévenin line (gold) of the same box.

Thévenin (voltage source, series)Norton (current source, parallel)

Norton current I_NI_N = 1.0 A

Terminal intercepts of the Norton form

V_oc = I_N R_th = 6.0 V  ·  I_sc = 1.0 A

Misaligned

A current source in parallel draws the same line

Put a resistor R_N in parallel with a current source I_N and look at the terminals: the terminal current is I = I_N − V/R_N. That is a straight line of slope −1/R_N, the same shape as Thévenin’s V-I line. Open the terminals and all the current flows through R_N, giving V = I_N·R_N; short the terminals and all the current flows out, giving I = I_N. So the Norton current is exactly the short-circuit current.

Source transformation: moving between the forms

For the two lines to coincide they must share slope and intercepts. Equal slope means R_N = R_th; an equal x-intercept (the open-circuit voltage) means I_N·R_th = V_th. Rearranged, V_th = I_N·R_th, that is I_N = V_th/R_th. This single pair of relations lets you freely turn a series voltage source into a parallel current source and back. This is source transformation.

The transformation simplifies circuits

When solving a tangled circuit, using source transformation to gather voltage and current sources into one kind makes the branches easy to combine: parallel current sources simply add, and series voltage sources simply add. Thévenin, Norton and source transformation are in the end three ways of looking at the same box, and you pick whichever form is most convenient for the problem.

ObserveI_N = V_thR_th

The Norton current is the short-circuit current.

ChooseR_N = ?

The two resistances are equal.

Fill inV_th = I_N ?

It restores the open-circuit voltage.

On your ownI_sc = ?

Shorted, the whole source current goes to the terminals.

Back to the first screen

As you slid the Norton current I_N, the terminal line of the current source with its parallel resistor moved up and down, and at I_N = V_th/R_th = 2 A it lay exactly on the Thévenin line. With the same resistance, whether you stand up a voltage source or a current source, the terminals showed one and the same V-I line. There were two ways of seeing the box, but only one box.

The Norton equivalent represents a linear two-terminal circuit as a current source I_N with a parallel resistor R_N. The Norton current is the short-circuit current I_N = V_th/R_th, and the resistance equals Thévenin’s, R_N = R_th. The two forms convert into each other by V_th = I_N·R_th (source transformation) and behave identically at the terminals as the same V-I line.