# Thevenin’s Theorem

Thevenin’s theorem is suitable for linear bilateral network when it is desired to find the values of the current flowing through the resistor for its different values. The solution of the complicated electrical networks are simplified by the thevenin’s theorem.

This theorem is a mathematical technique and replaces a complex electrical network into a simplified form consisting of a voltage source V_{T} and resistance R_{T} and R_{L} as shown below.

Here, V_{T} is the Thevenin’s equivalent voltage, R_{T} is the thevenin’s equivalent resistance and R_{L} is the load resistance.

Where the current flowing into the load resistance in the Thevenin’s equivalent circuit is given by.

V_{T} is the open-circuit voltage which appears across the load terminals when the load is removed from the network and R_{T} is the equivalent resistance as seen inward from the load terminal when the source has been removed and replaced by its internal resistance.

## Statement

For an electrical network, the current in any passive circuit element (such as a load resistance R_{L}) will be same to that of a network in which the RL is supplied by a voltage source V_{T} in series with equivalent resistance R_{T}. Where V_{T} is the open-circuit voltage which appears across the load terminals when the load is removed from the network and R_{T} is the equivalent resistance as seen inward from the load terminal when the source has been removed and replaced by its internal resistance.

## Application of Thevenin’s Theorem

Let us consider an electrical circuit as shown below.

Here ‘r’ is the internal resistance of the source EMF ‘E’. R_{1}, R_{2} and R_{L} are the resistances connected in the circuit.

R_{L} is the load resistance where we want to determine the current using Thevenin’s theorem.

For the application of Thevenin’s theorem we have to determine the V_{T} (Thevenin’s equivalent voltage) and R_{T} (Thevenin’s equivalent resistance).

**To determine the V _{T}**,

__we will remove the load resistance R__

_{L}from the terminal AB and determine the open circuit voltage across it which will be the required V_{T}.As, the R_{2} and the terminal AB is in parallel so the voltage across R_{2} is equal to the V_{T}.

Then, current through R_{2} is

Therefore, voltage V_{T} is

**To determine R _{T},**

__we will remove the source form the circuit leaving behind its internal resistance only and view the circuit inwards from the open terminal AB and determine the equivalent resistance. This equivalent resistance will be equal to the R__

_{T}.__[__If the internal resistance is negligible or not stated in the problem then replace the voltage source by short circuit and replace the current source by open circuit.]

For this illustration, the equivalent resistance as seen inward from the terminal AB will be (r+R_{1})//R_{2}.

**Finally, the Thevenin’s equivalent circuit is formed**. This is shown below.

Where the load current I_{L} is given by

## Steps for Thevenin’s Theorem

The steps involved for analyzing a circuit or network using thevenin’s theorem is summarized in following points.

- Remove the load resistance R
_{L}(the resistance across which the current is to be determined) and determine the open circuit voltage across it. The open circuit voltage is equal to V_{T}. - Remove the source form the circuit leaving behind its internal resistance and determine the equivalent resistance R
_{T }of the circuit as seen inward form the load resistance R_{L}.__[__If the internal resistance is negligible or not stated in the problem then replace the voltage source by short circuit and replace the current source by open circuit.] - Draw the thevenin’s equivalent circuit and determine the current flowing in the circuit.

## Examples

Here we will determine the current flowing through 2 Ohm resistor using thevenin’s theorem.

At first, the 2 Ohm resistor is removed from the circuit and the open circuit voltage is calculated across the terminal AB as shown below.

Now, the R_{T} is calculated by replacing the source with its internal resistance. [As no internal resistance provided for the voltage source, so replace the source by short circuit].

Now thevenin’s equivalent circuit is drawn.

The current through 2 Ohm is given by.

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