Basic Electrical Engineering Series
Maximum Power Transfer Theorem
This is perhaps the most important theorem in Electrical Engineering Science. The theorem has important applications in virtually every branch of Electrical Engineering. The theorem itself is quite simply stated and has an elegant proof, and for these reason the proof is included in this tutorial.
“For a source to transfer maximum power to a load connected to it, the load impedance must be the complex conjugate of the source impedance”.
Hence from the above definition, the load impedance in the diagram above is equal to the generator internal impedance except that the quadrature part of the impedance is negative, that is, the conjugate. So if the source impedance is inductive, then the load must be capacitive. Note the problem then reduces to one of matched resistances, since the reactive parts of the impedance are equal and opposite and therefore cancel – an example of series resonance. For a complex impedance load and source, the maximum power transfer therefore occurs at a singe frequency.
Discussion
Before prooving this theorem, it is perhaps worth while considering the hypothesis from an intuative perspective.
We know that the power in the load is the product of the current flowing through it and the potential difference across it – V x I. If we gradually reduce the size of the load towards zero, then the current through it will increase causing the power to increase. However this will coincide with a reduction in the potential across the load due to potential divider action with the source; which will cause the power to be reduced. Similarly if we gradually increased the size of the load then the load current will reduce but this will coincide with an increase in the potential across the load due to divider action. Intuitively then, it would appear that the 'best' situation would be obtained if the size of the load is somewhere in-between to give a 'medium' value for the current and potential difference. Indeed this is precisly the condition that occurs when the load is equal to the source.
Proof
1) The quadrature parts of the source and load cancel (by definition)
2) The power in the load PL is the product of the load potential and load current VL x IL
3) The load potential is
(Potential
divider action)
4) The load current is
(Ohms
Law)
5) Combining 3 and 4 the power in the load is therefore
6) Since E the generator emf is constant, then E2 is also constant, so replace with K
7) Expanding, the power in the load as a function of the load resistance is therefore
8) This function will be a maximum when its derivative is set equal to zero, hence
9) Recall the quotient rule
If
then
10) Hence from 7)
and
therefore
and
therefore
11) Hence substitute the results of (10) into (9)
12) Rearranging gives
13) The K constants cancel (I knew they would!) and multiply both sides by LHS denominator and cancel to give
14) Cancel like terms to give
15) Finally take square roots of both sides gives
- QED