The maximum power transfer theorem deals with the condition that when the maximum power transfer of the source to the load occurs.

Here we will be discussing the theorem along with its statement, proof of the condition, steps, and the example problems.

## Statement

The theorem states that:

In a d.c. circuit, the maxim power transfer from the source to load takes place when the load resistance is equal to the internal resistance of the source as viewed from the load terminals with the load removed and all the sources replaced by their internal resistances.

In other words, the maximum power transfer shall occur when the load resistance is equal to the Thevenin’s resistance of the circuit.

## Illustration

Meanwhile, let’s look at an illustration.

In our illustration, a circuit supplies power to the load R_{L}. The circuit which we are representing by using a box can be replaced by using Thevenin’s equivalent circuit consisting of a Thevenin’s voltage E (=V_{TH}) in series with the Thevenin’s resistance R_{S} (=R_{TH}). As in Thevenin’s equivalent circuit, the Thevenin’s resistance turns out to be the internal resistance of the source. This also illustrates that Thevenin’s theorem is an application of a practical voltage source.

Finally, by the maximum power transfer theorem, the circuit will transfer the maximum power when the load resistance is equal to the internal resistance of the source.

In short, the condition of maximum power transfer is load resistance (RL) = Thevenin’s resistance (RS).

Condition is R_{L} = R_{S}.

## Proof of Maximum Power Transfer Theorem

Consider the circuit below.

The power which the source delivers to the load is

In the above relation, when R_{L} equals zero, the power P also becomes zero. Similarly, when the R_{L} tends to infinity the power P is again zero.

Hence, we know that for the transfer of maximum power, the value of R_{L} lies between zero and infinity.

We can calculate the value of R_{L} when the power will be maximum, by differentiating the expression of power P concerning R_{L} and equation it with zero.

Therefore, the condition R_{S} = R_{L} is the condition for maximum power transfer.

Finally, using this condition the equation (i) to obtain the power which the source delivers to the load is

## Steps for Solving a Problem Using Maximum Power Transfer Theorem

We can follow the following steps to solve a problem using this theorem.

- Firstly, find the Thevenin’s resistance (R
_{TH}) by removing the load and replacing all the sources with its internal resistance. Meanwhile, for the problems containing ideal sources, replace the voltage source with short-circuit and the current source with an open-circuit. - According to the maximum power transfer theorem, this R
_{TH}is the system load resistance, that is to say, R_{L}=R_{TH}that allows maximum power transfer. - Secondly, find the Thevenin’s voltage (V
_{TH}). - Finally, calculate the maximum power transfer using the relation.

Similarly, if you are approaching the problem using Norton’s theorem then the maximum power transfer will be

## Examples of Maximum Power Transfer Theorem

Here, we will determine the value of resistance R such that maximum power transfer takes place in the circuit.

Firstly, we will calculate the Thevenin’s voltage which is the open-circuit voltage across the terminal XY.

At first, apply KVL on mesh I

Secondly, applying KVL on mesh II

Then, solving equation (i) and (ii) we have

I_{1}=3.25 A

I_{2} =0.25 A

Now, to find the V_{TH }we write KVL moving from Y to X. For a clear understanding, we are writing KVL on the blue portion of the circuit in a clockwise direction.

Then, we have to find the R_{TH.}

From the above circuit,

For the transfer of maximum power, R_{L} should be equal to R_{TH}.

Further, we will draw Thevenin’s equivalent circuit as

Therefore, maximum power delivered to the load is

well explained