# 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 IN in parallel with a  resistance Ras shown in Fig.2.

Fig.2

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

Once Norton’s equivalent circuit is obtained, then current thorough the load resistance RL 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 IN in parallel with a resistance RN.

• The output IN of the current source is equal to the current  through AB when A and B are short circuited.
• The resistance RN 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(same as RTH) 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 IN

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

The output IN 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  IN = Current flowing through R2 (as shown in Fig.4)

= I’ × [ R3/( R2 + R3)]

=  (V R3) / (R1R2+R1R3+R2R3)

##### Step 2 : Find the value of RN

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

This is shown in Fig. 5

Fig.5

RN = Resistance at terminal AB in Fig.5

= R2 + [(R1R3)/(R1 + R3)]

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

Fig.6