# 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**.