In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law".
The theorem results in maximum power transfer from the power source to the load, and not maximum efficiency of useful power out of total power consumed. If the load resistance is made larger than the source resistance, then efficiency increases (since a higher percentage of the source power is transferred to the load), but the magnitude of the load power decreases (since the total circuit resistance increases). If the load resistance is made smaller than the source resistance, then efficiency decreases (since most of the power ends up being dissipated in the source). Although the total power dissipated increases (due to a lower total resistance), the amount dissipated in the load decreases.
The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does not say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer from a voltage source is always zero (the hypothetical ideal voltage source), regardless of the value of the load resistance.
The theorem can be extended to alternating current circuits that include reactance, and states that maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance.
The mathematics of the theorem also applies to other physical interactions, such as:
mechanical collisions between two objects, the sharing of charge between two capacitors, liquid flow between two cylinders, the transmission and reflection of light at the boundary between two media.
Maximizing power transfer versus power efficiency
The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient, since the power dissipated as heat in the battery would always be equal to the power delivered to the motor when the impedances were matched.
In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer.
To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be (or should be) made as close to zero as possible. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine. The efficiency is the ratio of the power dissipated by the load resistance to the total power dissipated by the circuit (which includes the voltage source's resistance of as well as ):
Consider three particular cases (note that voltage sources must have some resistance): If , then Efficiency approaches 0% if the load resistance approaches zero (a short circuit), since all power is consumed in the source and no power is consumed in the short. If , then Efficiency is only 50% if the load resistance equals the source resistance (which is the condition of maximum power transfer). If , then Efficiency approaches 100% if the load resistance approaches infinity (though the total power level tends towards zero) or if the source resistance approaches zero. Using a large ratio is called impedance bridging.
Impedance matching
A related concept is reflectionless impedance matching.
In radio frequency transmission lines, and other electronics, there is often a requirement to match the source impedance (at the transmitter) to the load impedance (such as an antenna) to avoid reflections in the transmission line.
Calculus-based proof for purely resistive circuits
In the simplified model of powering a load with resistance by a source with voltage and source resistance , then by Ohm's law the resulting current is simply the source voltage divided by the total circuit resistance:
The power dissipated in the load is the square of the current multiplied by the resistance:
The value of for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of for which the denominator:
is a minimum. The result will be the same in either case. Differentiating the denominator with respect to :