**Superposition Theorem** is one of the simplified circuit analysis techniques.

Suppose there is an active network. Since it is an active network, there may be numbers of active sources acting simultaneously on the network.

## Statement of Superposition Theorem

Superposition theorem states if each of the sources acts on the network independently then current through any branch of the network is the sum of the currents due to each source.

### Explanation of Superposition Theorem

Suppose there is a branch in the network through which current ‘I’ is flowing due to the sources connected to the network. Now if we make dead all the sources of the network except one source, there will still be a certain current through the same branch. This current is due to the live source alone.

Then we go to another source and make it live. At the same time, we keep other sources of the network including the previously lived source as dead. At that time also there is one current in the same branch due to the second source. After that, we will go to the next source and do the same thing. In this condition, there will be still a certain current through the same branch. In this way, we get as many as currents through the said branch as many the network has the sources (current sources and voltage sources).

The actual current through the said branch is the current through it when all the sources are acting on the network. According to superposition theorem, the actual current through the said branch is the sum of all those individual currents.

### Equation of Superposition Theorem

The equation of theorem is

Where ‘I’ be the current through the branch when all sources are acting on the network. On the other hand, I_{1}, I_{2}, I_{3},…. I_{n} are the currents through the same branch when each of the n numbers of sources is acting on the network individually.

### Example of Superposition Theorem

In figure 1, ‘I’ represents the current through the resistance R when both voltage sources of emf E_{1} and E_{2} act on the circuit simultaneously.

Besides the above figure, figure 2 shows the current through the resistance R when only the voltage source of emf E_{1} acts on the network. This current is I’. Here we have replaced the second source with its internal resistance, r_{2}.

Similarly, the current through the resistance R when only the voltage source of emf E_{2} acting on the network is I”. Here also we have replaced the source E_{1} with its internal resistance, r_{1}.

Now as per superposition theorem

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