CR · DC analysis

Norton Equivalent and Source Transformation

If Thévenin folded a circuit into one voltage source and one series resistor, the same circuit can also be folded into one current source and one parallel resistor. This unit confirms that the two forms look identical at the terminals and covers the transformation that moves between them.

Make the two forms the same line

The gold line is the voltage-current line that the Thévenin form of the box draws at its terminals, and the light-blue line is the Norton form, built from a current source and a parallel resistor. Change the Norton current I_N and the light-blue line shifts up or down in parallel. Find the I_N at which the light-blue line lies exactly on the gold one.

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 from the terminals, and the terminal current is I = I_N − V/R_N. That is a straight line with slope −1/R_N, the same shape as the Thévenin V-I line. With the terminals open, all the current flows through R_N and the terminal voltage is V = I_N·R_N; with the terminals shorted, all the current flows out through the terminals and I = I_N. So the Norton current is simply the short-circuit current.

Source transformation: moving between the forms

For the two lines to coincide, both their slopes and their intercepts must match. Equal slopes require R_N = R_th, and an equal horizontal intercept, the open-circuit voltage, requires I_N·R_th = V_th. Put together, V_th = I_N·R_th, that is, I_N = V_th/R_th. With this pair of relations a series voltage source can be turned into a parallel current source and back again freely; this is called source transformation.

The transformation simplifies circuits

When solving a complicated circuit, gathering the voltage and current sources into one kind by source transformation makes branches easier to combine, because current sources in parallel simply add, and so do voltage sources in series. Thévenin, Norton, and source transformation are in the end three ways of looking at the same box, and for each problem you can pick whichever form is most convenient.

ObserveI_N = V_thR_th

The Norton current equals 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 = ?

When shorted, all of the source current flows out through the terminals.

Back to the first screen

As you changed the Norton current I_N, the light-blue line shifted up and down, and at I_N = V_th/R_th = 2 A it fell exactly onto the gold Thévenin line. Whether the same resistance stood with a voltage source or with a current source, at the terminals it was the same single V-I line. There were two ways of looking at 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 through V_th = I_N·R_th (source transformation), and at the terminals they draw the same V-I line and behave identically.