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.

**Contents**show

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

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