Norton’s Theorem

What is Norton’s Theorem ?

Fig.1 shows a complex network enclosed in a box with two terminal A and B brought out. The network in the box may consist of any number of resistors and energy sources connected in any manner.

Fig.1

According to Norton, the entire circuit enclosed in the box can be replaced by a single current source I_N in parallel with a  resistance R_N  as shown in Fig.2.

Fig.2

The resistance is same as Thevenin resistance R_TH .The value of I_N is determined by using Norton’s Theorem.

Once Norton’s equivalent circuit is obtained, then current thorough the load resistance R_L connected across AB can be found out.

Norton’s theorem as applied to d.c. circuit may be stated as below :

Any linear, bilateral network having two terminals can be replaced by an equivalent circuit consisting of a current source of current output I_N in parallel with a resistance R_N.

• The output I_N of the current source is equal to the current  through AB when A and B are short circuited.
• The resistance R_N is the resistance of the network measured between terminal A and B with load removed and voltage sources replaced by their internal resistances. Ideal voltage sources are replaced with short circuits and ideal current sources are replaced with open circuits.

Norton’s theorem is converse of Thevenin’s theorem in the following ways:

1. Norton equivalent circuit uses a current source instead of voltage source as in case of Thevenin equivalent circuit.
2. The resistance R_N (same as R_TH) in parallel with the source instead of being in series with it as in case of Thevenin equivalent circuit.

Let’s analyse the application of Norton’s theorem by taking an example as shown in Fig.3.

Fig.3

Step 1: Find the value of I_N

Here, the circuit behind the terminal AB can be replaced by a current source I_N in parallel with a resistance R_N by applying Norton’s theorem.

The output I_N of the current generator is equal to the current that would flow when terminal A and B are short circuited as shown in Fig.4.

Fig.4

Here, we can observe that terminal AB are shorted and resistance R₂ and R₃ are in parallel and this parallel combination is in series with the resistance R₁ .

So, the total resistance of the circuit is given by :

R’ = R₁ +[( R₂ × R₃) /( R₂ + R₃)]

= (R₁R₂+R₁R₃+R₂R₃)/( R₂ + R₃)

Hence, Source Current I’ = V / R’

= [V× ( R₂ + R₃)]/ (R₁R₂+R₁R₃+R₂R₃)

Short- Circuit Current  I_N = Current flowing through R₂ (as shown in Fig.4)

= I’ × [ R₃/( R₂ + R₃)]

=  (V R₃) / (R₁R₂+R₁R₃+R₂R₃)

Step 2 : Find the value of R_N

To find R_N, remove the load R_L and  replace the battery by  a short because its internal resistance is assumed to be zero.

This is shown in Fig. 5

Fig.5

R_N = Resistance at terminal AB in Fig.5

= R₂ + [(R₁R₃)/(R₁ + R₃)]

Now as the values of I_N and R_N are determined, we can draw the Norton equivalent circuit which is  shown in Fig.6.

Fig.6