Norton and Thévenin theorems

Determining \(Z_e\)

The equivalent impedance \(Z_e\) of a network is the impedance between the output nodes with all inputs set to zero. Setting an across-variable source to zero means the across-variable on both its terminals are equal, which is equivalent to treating them as the same node. Setting a through-variable source to zero means the through-variable through it is zero, which is equivalent to treating its nodes as disconnected.

Determining \(\mathcal{V}_e\)

The equivalent across-variable source \(\mathcal{V}_e\) is the across-variable at the output nodes of the network when they are left open (disconnected from a load). Determining this value typically requires some analysis with the elemental, continuity, and compatibility equations (preferably via impedance methods).

Norton’s theorem

Similarly, the following remarkable theorem has been proven.

The equivalent network has two quantities to determine: \(\mathcal{F}_e\) and \(Z_e\). The equivalent impedance \(Z_e\) is identical to that of Thévenin’s theorem, which leaves the equivalent through-variable source \(\mathcal{F}_e\) to be determined.

Converting between Thévenin and Norton equivalents

There is an equivalence between the two equivalent network models that allows one to convert from one to another with ease. The equivalent impedance \(Z_e\) is identical in each and provides the following equation for converting between the two representations: $$\begin{align} \mathcal{V}_e = Z_e \mathcal{F}_e. \end{align}$$

Example 12.5

For the circuit shown, find a Thévenin and a Norton equivalent.

The Thévenin equivalent is shown. Now to find Z_e and 𝒱_e. Setting V_s = 0, R₁ and R₂ are in parallel, combining to give $$\begin{aligned} Z_e = \frac{R_1 R_2} {R_1+R_2}. \end{aligned}$$

Now to find v_out. It’s a voltage divider: $$\begin{aligned} \mathcal{V}_e = v_\text{out} = \frac{R_2} {R_1+R_2} V_s. \end{aligned}$$

The Norton equivalent is shown. We know Z_e from the Thévenin equivalent, which also yields $$\begin{aligned} \mathcal{F}_e = \mathcal{V}_e/Z_e. \end{aligned}$$