Maximum Power Transfer Theorem - Maximizing Power Transfer Versus Power Efficiency

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 made close to zero. 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 to power developed by the source, then it is straightforward to calculate from the above circuit diagram that

\eta = {R_\mathrm{load} \over {R_\mathrm{load} + R_\mathrm{source} } } = { 1 \over { 1 + { R_\mathrm{source} \over R_\mathrm{load} } } }. \,\!

Consider three particular cases:

  • If, then
  • If or then
  • If, then

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 can be made close to zero. When the load resistance is zero, all the power is consumed inside the source (the power dissipated in a short circuit is zero) so the efficiency is zero.

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