When it comes to analyzing electrical circuits, a few theorems can be used. Among the most popular are **Norton’s theorem**, Thevenin’s theorem, and **maximum power transfer **(MPT). However, some people consider that the last mentioned theorem is not an analytical method. Instead, it aids the system design.

The MPT is a basic law that ensures the circuit to transfer maximum power. And yet, maximum power does not mean maximum efficiency. The application of the theorem will not result in high or maximum efficiency, which is more essential for AC.

## Maximum Power Transfer (MPT) Description

**Maximum power transfer theorem** mentions that in a linear **DC circuit**, load resistance can dissipate maximum power only if it is equal to Thevenin resistance. When the load resistance is higher or lower, the dissipated power will not reach maximum level.

Series resistance is equal to the load resistance when a circuit is an independent voltage source. Meanwhile, parallel resistance is equal to the load resistance when a network is an independent current source.

In electrical circuit, the electrical energy is transferred to the load. This is where the energy will be converted into a useful work. And yet, heating effect will not make the supplied power present at the load. For this reason, there will be difference among delivering and drawing power.

The amount of transferred power is affected by a number of factors, including the load size. Therefore, the theorem ensures the condition so that the load can receive maximum power.

*Never forget your history*. Historically, the theorem was published by Moritz van Jacobi in 1840. For this reason, the theorem is also known as Jacobi’s law. Until today, the theorem is applied in everyday practices especially in electronics.

## Maximum Power Transfer vs Power Efficiency

There is a misunderstanding that maximum power is the same as maximum efficiency. Joule, who notably misunderstood the theorem, implied that a system powered by a battery could not deliver more than 50% of efficiency. It was because the power lost as heat was always equal to power transferred to the motor.

And yet, around 1880 Edison or Francis Robbins Upton shown that it was a wrong assumption. He found that **maximum power transfer** was not the same as maximum efficiency. To obtain maximum efficiency, the source’s resistance should be made close to zero.

With this understanding, they achieved around 90% efficiency. At the same time, they also found a practical alternative to heat engine, which was electric motor. Additionally, it proved the condition of MPT does not lead to maximum efficiency,

## Impedance Matching Applications

Among the most useful impedance matching applications is the output of amplifier circuits. The circuit is made to provide **maximum power transfer** among the source and load. To achieve maximum sound output, the signal transformers are employed to match the loudspeakers to the amplifiers.

The following signal transformers are known as matching transformers. The maximum power transfer can be achieved even when the load and output impedance are not the same. Turns ratio can be used on the transformer by corresponding load impedance to output impedance.

Besides, the MPT is also used in radio frequency transmission and other electronics. Oftentimes, the source impedance found at the transmitter is required to be matched to the load impedance such as antenna. This is conducted to avoid reflections that potentially damage the transmitter.

## Maximum Power Transfer Proof

The theorem ensures that the load resistance value is delivered to the load. Proving the theorem can be conducted by using a DC two terminal circuit, in which the condition is determined for maximum power. Thevenin equivalent then replaces the original circuit. Define the current that passes the load.

The power transferred depends on the value of Thevenin Resistance (R_{TH}) and load resistance (R_{L)}. But as Thevenin equivalent is constant, the power transferred to the load depends on the load resistance. From here, you can define the value of load resistance.

When Th resistance is equal to the load resistance, this is when the **maximum power transfer** happens. It can be concluded that the theorem mentions that MPT only occurs when the resistance is equal to the load resistance.

Please note that the transferred power is zero when the load resistance is zero. During this condition, voltage drop does not occur. It also means current does not flow through the load.

## Applying MPT to DC and AC Circuit

The theorem can be applied to both DC and AC circuit and it is not as complex as **star delta transformation** theorem. The following steps can be a great example to understand how to apply MPT theorem to a DC circuit. The condition has been adjusted to support maximum power.

- Identify the load resistance, then disconnect the load resistance. Define the Th voltage and resistance.
- To calculate Thevenin resistance, replace source with internal resistances. You can assume that the circuit is shorted so the voltage source has 0 internal resistance.
- To achieve the theorem, R
_{L}must be equal to the R_{TH}.

Meanwhile, the following example of AC circuit has load impedance. The reactive and resistive part of the networks can be customized. For this case, calculate the value of load impedance to figure out the value of maximum power. The following steps should help you apply MPT to AC circuit.

- First, find equivalent circuit by using Thevenin’s theorem to obtain the load impedance value. Find the voltage by disconnecting the load impedance.
- After shorting the voltage source, the equivalent impedance can be calculated and equivalent circuit can be drawn.
- Calculate the load impedance to obtain the maximum power.

## MPT in Reactive Circuits

In case the source or load is not fully resistive, the **t**heorem is also applicable. The refined **maximum power transfer **theorem indicates that the load and source impedance have to be complex conjugates. The two of them are identical in case of pure resistive circuits. Some circuits with capacitive or inductive components are usually not purely resistive. Therefore, applying the theorem to these kinds of circuit by considering complex conjugate does exist. When a source is capacitive, then the load would receive 100 percent of the source’s energy.

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