Maximum power theorem

In electrical engineering, the maximum power theorem states that for the transfer of maximum power from a source with a fixed internal resistance to a load, the resistance of the load must be the same as that of the source. This theorem is of use when driving a load such as an electric motor from a battery.

It is important to note the condition that the source resistance be fixed. If the source resistance were variable, maximum power would be transferred simply by setting the source resistance to zero.

In radio electronics there is often a requirement to match the source (e.g. transmitter) impedance to the load (e.g. antenna) impedance, but this is to avoid reflections in the transmission line. The maximum power theorem is only a part of the reason for this requirement.

The maxim is also known as Jacobi's Theorem after its discoverer, Professor Moritz von Jacobi of St. Petersburg in Russia, although this is also the name of an unrelated theorem in mathematics. The theorem was originally misunderstood (particularly 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 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 realised that the theorem could be sidestepped by making the resistance of the source (whether a battery or a dynamo) 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.

Proof

Jacobi obtained his theorem by common sense, but a mathematical proof is as follows. In the diagram opposite, power is being transferred from the source, with voltage V and resistance R_S, to a load with resistance R_L, resulting in a current I. I is simply the source voltage divided by the total circuit resistance:

The power P_L dissipated in the load is the square of the current multiplied by the resistance:

We could calculate the value of R_L for which this expression is a maximum, but it is easier to calculate the value of R_L for which the denominator R_S²/R_L+2R_S+R_L is a minimum. The result will be the same in either case. Differentiating with respect to R_L:

For a maximum or minimum, the first derivative is zero, so

or

In practical resistive circuits, R_S and R_L are both positive. To find out whether this solution is a minimum or a maximum, we must differentiate again:

This is positive for positive values of R_S and R_L, showing that the denominator is a minimum, and the power is therefore a maximum, when R_S = R_L.