Electrical Engineering ⇒ Topic : Nortons Theorem

NORTON'S THEOREM

This theorem is the converse of Thevenin's theorem. It consists of an equivalent current source in parallel to the internal resistance of the network.

This theorem states that

Any two terminal linear active bilateral networks can be replaced by an equivalent current source in parallel to a resistance. The current source being the short circuited current through the load terminals and the resistance being the internal resistance of the source network looking through the open circuited load terminals.

Explanation

Firstly, consider a circuit shown in Figure 1 (a). We will determine the current through r_L by using Norton's theorem. Let us apply the following procedures:

(1) The load resistance r_L is removed from the terminals A-B and the terminals A-B are short circuited [Figure 1 (b)].

                                                                         FIGURE (1)

(2) Now, let us determine I_SC from Figure 1 (b).

                                                   ............ (1)

                                                   ............ (2)

(3)To find out the internal resistance of the circuit, the short circuit across the terminals A-B is removed. The voltage source is replaced by short circuit as it has zero internal resistance. The circuit configuration is depicted in Figure 1 (c).

From Figure 1 (c), we get

                                                              ................... (3)

(4) Next, the load resistance r_L is connected across the terminals A-B. The circuit is represented in Figure 1 (d).

The current I_L through the load resistance r_L is given by

                                                                ...............    (4)

Recall that Norton's theorem for d.c. circuits allows us to replace a two-terminal linear d.c. circuit by a single equivalent d.c. current source  in parallel with a single .equivalent resistance . The a.c. version of Norton's theorem is similar and may be stated as under

A two-terminal linear a.c. circuit can be replaced by a single equivalent a.c. current source  in parallel with a single equivalent impedance .

                                                           FIGURE (A)

Figure (a) shows the Norton equivalent circuit of a two-terminal a.c. circuit. The impedance  (called Norton equivalent impedance) has exactly the same value as the Thevenin equivalent impedance  and is found in the same way. The current  (called Norton equivalent current) is the current that flows through a short circuit connected across the Norton terminals (i.e., load terminals). Note that the Thevenin and Norton circuits are alternative equivalents for a circuit.Norton's theorem is popular for analysing transistor circuits.

Norton's Theorem

Norton's theorem is similar to Thevenin's theorem. While Thevenin's theorem is based on the idea of an equivalent source of emf, Norton's theorem is based on the idea of an equivalent current source. Norton's theorem can be stated as follows.

Any arrangement of the sources of emf and the resistances can be replaced by an equivalent current source in parallel with a resistance r. The current from the source is the shortcircuit current in the original system and r is the equivalent resistance of the network between its two terminals A and B when all sources of emf are replaced by their internal resistances.

Consider the network shown in Fig. (1). Let V' be the potential across AB when load resistance R is disconnected, as shown in Fig. (1) (a).

Consider the load resistance R connected as shown in Fig. 1 (c). Then

Now, the network shown in Fig. 1 (a) can be replaced by a current source driving a current I through the load R as shown in Fig. 1 (d). Then, we have

                                     figure (1) Circuit to illustrate Norton's theorem