*Maximum Power Transfer Theorem | Proof,Formula*

*Maximum Power Transfer Theorem | Proof,Formula*

**Maximum power transfer theorem** is a tool for determining the condition of maximum power transfer. **Maximum power transfer theorem** states that the source can deliver maximum power to load if the load resistance is equal to the source resistance. The maximum power transfer theorem is applicable for AC and DC circuits. To obtain the maximum power from the source, load resistance should be equal to the source resistance. We will derive the mathematical proof of maximum power transfer theorem and efficiency of the system under maximum power transfer.

**Explanation of Maximum Power Transfer Theorem**

In case of DC source, it has certain resistance. In case of AC source it has certain impedance. The source feeds current to the load. The voltage drop takes place in the source impedance when current flows through the circuit.The power lost in the source resistance must be minimum to get the maximum power delivery of the source.

If the power loss in the source resistance is more, the maximum power delivery from source can not be ensured. There is certain relationship between the source impedance and load impedance for which maximum power delivery from the source can be achieved. The maximum power theorem states that if the load resistance is equal to the source resistance, the maximum power will be delivered by the source.

The source resistance is the resistance viewed from the load side, and we can find its value using the Thevenin theorem. Therefore, the source resistance is also called Thevenin resistance. We take DC circuit to understand the maximum power transfer theorem.

**Thevenins Equivalent Circuit**

In the above Thevenin Equivalent circuit, the maximum power transfer theorem states that “the maximum amount of power will be delivered in the load resistance if the load resistance is equal to the the Thevenin or Norton source resistance.

The source resistance viewed from the load side is R_{th} and the Thevenin voltage is V_{0}. According to maximum power transfer theorem, the maximum power transfer occurs when the source resistance is equal to the load resistance.

## R_{th }= R_{L}

## Maximum Power Transfer Example – 1

According to maximum power transfer theorem the source delivers maximum power to load if the value of load resistance is equal to the source resistance. The value of the load resistance should be 20 Ω. Now let us verify this statement by calculating the power consumed by the load at various load settings.

## Proof of Maximum Power Transfer Theorem

_{L}is P= I

^{2}R. The current I flowing through the circuit is:

_{L}is:

The maximum power delivery is possible if;

Thus, the condition for Maximum Power Transfer is the source delivers maximum power to load, if **the source resistance is equal to the load resistance**. Now , we will calculate the system **efficiency under maximum power transfer condition. **

## Efficiency of Maximum Power Transfer

**Maximum Power Delivered to the Load**

We can calculate the maximum efficiency by Dividing equation(4) and equation(5)

Hence, at the condition of maximum power transfer the efficiency is 50%. At maximum power transfer condition, the source delivers 50 % of the generated power to the load, and at other conditions the source drivers small percentage of power to the load.