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. This method is called maximum power transfer theorem.

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.

*Make sure to read what is electric circuit first.*

These circuit analysis theorems are classified as:

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

## Maximum Power Transfer Theorem

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 |

If the entire circuit is replaced by its Thevenin equivalent except for the load, as shown in Figure.(1), the power delivered to the load is

(1) |

For a given circuit, *V _{Th}* and

*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 |

Let me show that this maximum power occurs when *R _{L}* is equal to

*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 _{L}* and set the result equal to zero. We obtain

(3) |

showing that the maximum power transfer takes place when the load resistance *R _{L}* equals the Thevenin resistance

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

_{Th}*d*.

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

(4) |

Equation.(4) applies only when *R _{L }= R_{Th}*. When

*R*, we compute the power delivered to the load using Equation.(1).

_{L }≠ R_{Th}## Maximum Power Transfer Theorem 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)**

*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 _{1}*, we get

*i*= -2/3. Applying KVL around the outer loop to get

_{1}*V*across terminals

_{Th}*a-b*, we obtain

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