# Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem states that the maximum power is delivered to a load when the resistance of the load is equal to the source resistance.

So what does that mean? It means that in order to deliver maximum power to a load of a circuit we should aim to match the resistance of our source which is driving the load to the load resistance. This is a very important point which should be kept in mind! Expressed mathematically, we need:

## Derivation:

We can derive the above by considering the circuit below:

To determine the maximum power, we need an expression for power in terms of resistance and then we can differentiate and equate to zero and solve for R:

From here we can differentiate with respect to Rl using the chain rule and equate to zero to find the maximum:

If we define the denominator to be u, then the above becomes:

Taking the derivative with respect to u:

For the chain rule, we then need to take the derivative of u with respect to Rl:

From here we can apply the chain rule to determine the derivate with respect to Rl:

The maximum can then be calculated by equating this to 0:

Which is what we expect! Maximum power is transferred to the load when the source resistance matches the resistance of the load.

### Example:

1. Calculate the value of Rs such that maximum power is delivered to the load in the following circuit:

Maximum power is delivered when Rs is equal to Rl i.e. when the source resistance is 10kΩ.