Editing Maximum power transfer theorem (section)

== Maximizing power transfer versus power efficiency ==
[[File:Source and load circuit.svg|thumb|Simplified model for powering a load with resistance ''R<sub>L</sub>'' by a source with voltage ''V<sub>S</sub>'' and resistance ''R<sub>S</sub>''.]]
The theorem was originally misunderstood (notably by [[James Prescott Joule|Joule]]<ref>{{Cite web |last=Magnetics |first=Triad |title=Understanding the Maximum Power Theorem |url=https://info.triadmagnetics.com/blog/maximum-power-theorem |access-date=2022-06-08 |website=info.triadmagnetics.com |language=en-us}}</ref>) to imply that a system consisting of an electric motor driven by a [[Electric battery|battery]] could not be more than 50% [[Electrical efficiency|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 [[Thomas Edison|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]].[[File:Maximum Power Transfer Graph.svg|thumb|The red curve shows the power in the load, normalized relative to its maximum possible. The dark blue curve shows the efficiency {{math|''η''}}.]]
The efficiency {{math|''η''}} is the ratio of the power dissipated by the load resistance {{math|''R''<sub>L</sub>}} to the total power dissipated by the circuit (which includes the voltage [[source resistance|source's resistance]] of {{math|''R''<sub>S</sub>}} as well as {{math|''R''<sub>L</sub>}}):

<math display="block">\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}} \, .</math>

Consider three particular cases (note that voltage sources must have some resistance):
* If <math>R_\mathrm{L}/R_\mathrm{S} \to 0</math>, then <math>\eta \to 0.</math> 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 <math>R_\mathrm{L}/R_\mathrm{S} = 1</math>, then <math>\eta = \tfrac{1}{2}.</math> Efficiency is only 50% if the load resistance equals the source resistance (which is the condition of maximum power transfer).
* If <math>R_\mathrm{L}/R_\mathrm{S} \to \infty</math>, then <math>\eta \to 1.</math> 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]].