Thevenin’s Theorem: Thevenin’s theorem is suitable for a linear bilateral network where we desire to find the values of the current flowing through the resistor for its different values.  The solution of the complicated electrical networks is simplified by Thevenin’s theorem.

This theorem is a mathematical technique and replaces a complex electrical network into a simplified form consisting of a voltage source VT and resistance RT and RL as shown below. Figure: Thevenin’s Equivalent Circuit

Here, VT is Thevenin’s equivalent voltage, RT is Thevenin’s equivalent resistance and RL is the load resistance.

Where the current flowing into the load resistance in the Thevenin’s equivalent circuit is, VT is the open-circuit voltage that appears across the load terminals when we remove the load from the network. RT is the equivalent resistance seen inward from the load terminal when the source has been removed and replaced by its internal resistance.

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## Statement

For an electrical network, the current in any passive circuit element (such as a load resistance RL) will be the same as that of a network in which the RL is supplied by a voltage source VT in series with equivalent resistance RT. Where VT is the open-circuit voltage that appears across the load terminals when the load is removed from the network and RT 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. Figure: 1.1

Here ‘r’ is the internal resistance of the source EMF ‘E’. R1, R2, and RL are the resistances in the circuit.

RL 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 VT (Thevenin’s equivalent voltage) and RT (Thevenin’s equivalent resistance).

To determine the VT, we will remove the load resistance RL from the terminal AB and determine the open-circuit voltage across it which will be the required VT. Figure: 1.2

As the R2 and the terminal AB are in parallel so the voltage across R2 is equal to the VT.

Then, the current through R2 is Therefore, voltage VT is To determine RT, we will remove the source from 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 RT.

[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 with an open circuit.] Figure: 1.3

For this illustration, the equivalent resistance as seen inward from the terminal AB will be (r+R1)//R2. Finally, we form Thevenin’s equivalent circuit. Figure: Thevenin’s Equivalent Circuit

Where the load current IL is ## Steps for Thevenin’s Theorem

The following are the steps for analyzing a circuit or network using Thevenin’s theorem.

1. Firstly remove the load resistance RL (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 VT.
2. Secondly, remove the source from the circuit leaving behind its internal resistance, and determine the equivalent resistance RT of the circuit as seen inward from the load resistance RL. [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 with an open circuit.]
3. Finally draw the Thevenin’s equivalent circuit and determine the current flowing in the circuit.

## Examples

Here we will determine the current flowing through the 2 Ohm resistors using Thevenin’s theorem. Figure: 2.1

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. Figure: 2.2 Now, the RT is calculated by replacing the source with its internal resistance. [As no internal resistance is provided for the voltage source, so replace the source with a short circuit]. Figure: 2.3 Now Thevenin’s equivalent circuit is drawn. Figure: 2.4

The current through 2 Ohm is given by. ENGINEERING NOTES ONLINE: PID CONTROLLER

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