In many practical situations, a circuit is designed to provide power to a load. There are applications in areas such as communications where it is desirable to maximize the power delivered to a load. We now address the problem of delivering the maximum power to a load when given a system with known internal losses. It should be noted that this will result in significant internal losses greater than or equal to the power delivered to the load.

This circuit analysis theorems are classified as:

This circuit analysis theorems are classified as:

- Superposition theorem
- Source transformation
- Thevenin theorem
- Norton theorem
- Maximum power transfer

## Maximum Power Transfer Theory

The Thevenin equivalent is useful in finding the maximum power a linear circuit can deliver to a load. We assume that we can adjust the load resistance

*R*._{L}Figure 1 |

(1) |

*V*and

_{Th}*R*are fixed. By varying the load resistance

_{Th}*R*, the power delivered to the load varies as sketched in Figure.(2). We notice from Figure.(2) that the power is small for small or large values of

_{L}*R*but maximum for some value of

_{L}*R*between 0 and ∞.

_{L}Figure 2 |

*R*is equal to

_{L}*R*. This is known as the

_{Th}*maximum power theorem*.

Maximum poweris transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (R=_{L}R)_{Th}

To prove the maximum power transfer theorem, we differentiate

*p*in Equation.(1) with the respect to*R*and set the result equal to zero. We obtain_{L}(3) |

*R*equals the Thevenin resistance

_{L}*R*. We can readily confirm that Equation.(3) gives the maximum power by showing that

_{Th}*d*.

^{2}p/dR_{L}^{2}<0
The maximum power transferred is obtained by susbtituting Equations.(3) to (1), for

(4) |

*R*. When

_{L }= R_{Th}*R*, we compute the power delivered to the load using Equation.(1).

_{L }≠ R_{Th}## Maximum Power Transfer Example

For better understanding, let us review the example below :

**1.Find the value of**

**R**_{L}**for maximum power transfer in the circuit of Figure.(3)**

We need to find the Thevenin resistance

*R*and the Thevenin voltage_{Th}*V*_{Th}across the terminals*a-b*. To get*R*, we use the circuit in Figure.(4a) and obtain_{Th}
Solving for

*i*, we get_{1}*i*= -2/3. Applying KVL around the outer loop to get_{1}*V*across terminals_{Th}*a-b*, we obtain
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