# Maximum power transfer theorem

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:[2][3] 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. Contents 1 Maximizing power transfer versus power efficiency 2 Impedance matching 3 Calculus-based proof for purely resistive circuits 4 In reactive circuits 4.1 Proof 5 Notes 6 References 7 External links Maximizing power transfer versus power efficiency Simplified model for powering a load with resistance RL by a source with voltage VS and resistance RS.

The theorem was originally misunderstood (notably by Joule[4]) 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 red curve shows the power in the load, normalized relative to its maximum possible. The black curve shows the efficiency η.

The efficiency η is the ratio of power dissipated by the load resistance RL to total power dissipated by circuit (which includes the voltage source's resistance of RS as well as RL): {displaystyle eta ={frac {P_{mathrm {L} }}{P_{mathrm {Total} }}}={frac {I^{2}cdot R_{mathrm {L} }}{I^{2}cdot (R_{mathrm {L} }+R_{mathrm {S} })}}={frac {R_{mathrm {L} }}{R_{mathrm {L} }+R_{mathrm {S} }}}={frac {1}{1+R_{mathrm {S} }/R_{mathrm {L} }}},.} Consider three particular cases: If {displaystyle R_{mathrm {L} }/R_{mathrm {S} }to 0} , then {displaystyle eta to 0.} 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. Note, voltage sources must have some resistance. If {displaystyle R_{mathrm {L} }/R_{mathrm {S} }=1} , then {displaystyle eta ={tfrac {1}{2}}.} Efficiency is only 50% if the load resistance equals the source resistance (which is the condition of maximum power transfer). If {displaystyle R_{mathrm {L} }/R_{mathrm {S} }to infty } , then {displaystyle eta to 1.} 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 Main article: 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 simplified model of powering a load with resistance RL by a source with voltage VS and source resistance RS, then by Ohm's law the resulting current I is simply the source voltage divided by the total circuit resistance: {displaystyle I={frac {V_{S}}{R_{mathrm {S} }+R_{mathrm {L} }}}.} The power PL dissipated in the load is the square of the current multiplied by the resistance: {displaystyle P_{mathrm {L} }=I^{2}R_{mathrm {L} }=left({frac {V_{S}}{R_{mathrm {S} }+R_{mathrm {L} }}}right)^{2}R_{mathrm {L} }={frac {V_{S}^{2}}{R_{mathrm {S} }^{2}/R_{mathrm {L} }+2R_{mathrm {S} }+R_{mathrm {L} }}}.} The value of RL for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of RL for which the denominator {displaystyle R_{mathrm {S} }^{2}/R_{mathrm {L} }+2R_{mathrm {S} }+R_{mathrm {L} }} is a minimum. The result will be the same in either case. Differentiating the denominator with respect to RL: {displaystyle {frac {d}{dR_{mathrm {L} }}}left(R_{mathrm {S} }^{2}/R_{mathrm {L} }+2R_{mathrm {S} }+R_{mathrm {L} }right)=-R_{mathrm {S} }^{2}/R_{mathrm {L} }^{2}+1.} For a maximum or minimum, the first derivative is zero, so {displaystyle R_{mathrm {S} }^{2}/R_{mathrm {L} }^{2}=1} or {displaystyle R_{mathrm {L} }=pm R_{mathrm {S} }.} In practical resistive circuits, RS and RL 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: {displaystyle {frac {d^{2}}{dR_{mathrm {L} }^{2}}}left({R_{mathrm {S} }^{2}/R_{mathrm {L} }+2R_{mathrm {S} }+R_{mathrm {L} }}right)={2R_{mathrm {S} }^{2}}/{R_{mathrm {L} }^{3}}.} This is always positive for positive values of {displaystyle R_{mathrm {S} }} and {displaystyle R_{mathrm {L} }} , showing that the denominator is a minimum, and the power is therefore a maximum, when {displaystyle R_{mathrm {S} }=R_{mathrm {L} }.} The above proof assumes fixed source resistance {displaystyle R_{mathrm {S} }} . When the source resistance can be varied, power transferred to the load can be increased by reducing {displaystyle R_{textrm {S}}} . For example, a 100 Volt source with an {displaystyle R_{textrm {S}}} of {displaystyle 10,Omega } will deliver 250 watts of power to a {displaystyle 10,Omega } load; reducing {displaystyle R_{textrm {S}}} to {displaystyle 0,Omega } 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.[5] 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.) This means that the source and load impedances should be complex conjugates of each other. In the case of purely resistive circuits, the two concepts are identical.

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.