We have solved the problem of maximizing the power delivered by a power-supplying resistive network to a load R_{L}. Now we will talk about what is maximum average power transfer.

Representing the circuit in Thevenin equivalent circuit, we proved that the maximum power would be delivered to the load if the load resistance is equal to the Thevenin resistance R_{L} = R_{Th}.

We now extend that result to ac circuit. Make sure to read what is ac circuit first.

There are several types of power in ac circuit:

- Maximum average power transfer
- Voltage and current RMS
- Power factor and apparent power
- Power triangle and power complex
- Power ac conservation

## Maximum Average Power Transfer

Consider the circuit in Figure.(1), where an ac circuit is connected to a load **Z _{L}** and is represented by its Thevenin equivalent.

Figure 1. Finding the maximum average power transfer : (a) circuit with a load, (b) the Thevenin equivalent |

The load is usually represented by an impedance, which may model an electric motor, an antenna, and so forth.

In rectangular form, the Thevenin impedance **Z _{Th}** and the load impedance

**Z**are

_{L}(1a) |

(1b) |

The current through the load is

(2) |

The average power delivered to the load is

(3) |

Our objective is to adjust the load parameters **R _{L}** and

**X**so that

_{L}*P*is maximum.

To do this we set ∂P/∂R_{L} and ∂P/∂X_{L} equal to zero. From Equation.(3), we obtain

(4a) |

(4b) |

Setting ∂P/∂X_{L} to zero gives

(5) |

and setting ∂P/∂R_{L} to zero results

(6) |

Combining Equations.(5) and (6) leads to the conclusion that for maximum average power transfer, **Z _{L}** must be selected so that X

_{L}= -X

_{Th}and R

_{L}= R

_{Th}, i.e,.

(7) |

For

maximum average power transfer, the load impedanceZmust be equal to the complex conjugate of the Thevenin impedance_{L}Z._{Th}

This result is known as the *maximum average power transfer theorem* for the sinusoidal steady state. Setting R_{L} = R_{Th} and X_{L} = -X_{Th} in Equation.(3) gives us the maximum average power as

(8) |

In a situation in which the load is purely real, the condition for maximum power transfer is obtained from Equation.(6) by setting X_{L} = 0; that is,

(9) |

This means that for maximum average power transfer to a purely resistive load, the load impedance (or resistance) is equal to the magnitude of the Thevenin impedance.

## Maximum Average Power Transfer Examples

For better understanding let us review examples below :

**1. Determine the load impedance ****Z _{L}**

**that maximizes the average power drawn from the circuit of Figure.(2). What is the maximum average power?**

Figure 2 |

*Solution :*

First, we obtain the Thevenin equivalent at the load terminals. To get **Z _{Th}**, consider the circuit is shown in Figure.(3a). We find

Figure 3. Finding the Thevenin equivalent of the circuit in Figure.(2) |

To find **V _{Th}**, consider the circuit in Figure.(3b). By voltage division,

The load impedance draws the maximum power from the circuit when

According to Equation.(8), the maximum average power is

**2. In the circuit in Figure.(4), find the value of R _{L} that will absorb the maximum average power. Calculate that power.**

Figure 4 |

*Solution :*

We first find the Thevenin equivalent at the terminals of R_{L}.

By voltage division,

The value of R_{L} that will absorb the maximum average power is

The current through the load is

The maximum average power absorbed by R_{L} is