Maximum power transfer theorem

In electrical engineering, the maximum power transfer theorem states that, to obtain "maximum" external power from a source with a finite 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 across the circuit, and not maximum "efficiency". If the resistance of the load is made larger than the resistance of the source then efficiency is higher, since a higher percentage of the source power is transferred to the load, but the "magnitude" of the load power is lower since the total circuit resistance increases.
If the load resistance is smaller than the source resistance, then most of the power ends up being dissipated in the source, and although the total power dissipated is higher, due to a lower total resistance, it turns out that the amount dissipated in the load is reduced.
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, 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.
Recent expository articles illustrate how the fundamental mathematics of the maximum power theorem also applies to other physical situations, such as:
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, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor. 
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 condition of maximum power transfer does not result in maximum efficiency. 
If we define the efficiency as the ratio of power dissipated by the load, "R", to power developed by the source, "V", then it is straightforward to calculate from the above circuit diagram that
Consider three particular cases:
The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistance approaches infinity, though the total power level tends towards zero. 
Efficiency also approaches 100% if the source resistance approaches zero, and 0% if the load resistance approaches zero. In the latter case, all the power is consumed inside the source (unless the source also has no resistance), so the power dissipated in a short circuit is zero.
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 that could overload or damage the transmitter.
Calculus-based proof for purely resistive circuits.
In the diagram opposite, power is being transferred from the source, with voltage and fixed source resistance , to a load with resistance , resulting in a current . By Ohm's law, 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 :
For a maximum or minimum, the first derivative is zero, so
or
In practical resistive circuits, and are both positive, so the positive sign in the above is the correct solution. 
To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again:
This is always positive for positive values of formula_16 and formula_17, showing that the denominator is a minimum, and the power is therefore a maximum, when
The above proof assumes fixed source resistance formula_16. When the source resistance can be varied, power transferred to the load can be increased by reducing formula_20. For example, a 100 Volt source with an formula_20 of formula_22 will deliver 250 watts of power to a formula_22 load; reducing formula_20 to formula_25 increases the power delivered to 1000 watts.
Note that this shows that maximum power transfer can also be interpreted as the load voltage being equal to one-half of the Thevenin voltage equivalent of the source.
In reactive circuits.
The power transfer theorem also applies when the source and/or load are not purely resistive. 
A refinement of the maximum power theorem says that any reactive components of source and load should be of equal magnitude but opposite sign. ("See below for a derivation.") 
Physically realizable sources and loads are not usually purely resistive, having some inductive or capacitive components, and so practical applications of this theorem, under the name of complex conjugate impedance matching, do, in fact, exist.
If the source is totally inductive (capacitive), then a totally capacitive (inductive) load, in the absence of resistive losses, would receive 100% of the energy from the source but send it back after a quarter cycle.
The resultant circuit is nothing other than a resonant LC circuit in which the energy continues to oscillate to and fro. This oscillation is called reactive power. 
Power factor correction (where an inductive reactance is used to "balance out" a capacitive one), is essentially the same idea as complex conjugate impedance matching although it is done for entirely different reasons.
For a fixed reactive "source", the maximum power theorem maximizes the real power (P) delivered to the load by complex conjugate matching the load to the source.
For a fixed reactive "load", power factor correction minimizes the apparent power (S) (and unnecessary current) conducted by the transmission lines, while maintaining the same amount of real power transfer. 
This is done by adding a reactance to the load to balance out the load's own reactance, changing the reactive load impedance into a resistive load impedance.
Proof.
In this diagram, AC power is being transferred from the source, with phasor magnitude of voltage formula_26 (positive peak voltage) and fixed source impedance formula_27 (S for source), to a load with impedance formula_28 (L for load), resulting in a (positive) magnitude formula_29 of the current phasor formula_30. This magnitude formula_29 results from dividing the magnitude of the source voltage by the magnitude of the total circuit impedance:
The average power formula_33 dissipated in the load is the square of the current multiplied by the resistive portion (the real part) formula_34 of the load impedance formula_28:
where formula_37 and formula_34 denote the resistances, that is the real parts, and formula_39 and formula_40 denote the reactances, that is the imaginary parts, of respectively the source and load impedances formula_27 and formula_28.
To determine, for a given source voltage formula_43 and impedance formula_44 the value of the load impedance formula_45 for which this expression for the power yields a maximum, one first finds, for each fixed positive value of formula_34, the value of the reactive term formula_40 for which the denominator
is a minimum. Since reactances can be negative, this is achieved by adapting the load reactance to
This reduces the above equation to:
and it remains to find the value of formula_34 which maximizes this expression. This problem has the same form as in the purely resistive case, and the maximizing condition therefore is formula_52
The two maximizing conditions
describe the complex conjugate of the source impedance, denoted by formula_55 and thus can be concisely combined to: