# The Maximum power transfer theorem

Earlier it was demonstrated the existence of internal resistance in the

power supply such as in a battery, and the effect that this resistance

has on the voltage supplied to the load was discussed.

The load voltage is the actual voltage given out by the power supply

after it has dropped a percentage of its EMF voltage across its

internal resistance.

How much voltage is dropped across the internal resistance depends

on the value of the internal resistance in relation to the value of the

load.

The relationship between the values of load resistance and internal

resistance is also important for another reason. Maximum power can

be developed in a load resistance only when the values of the load

resistance and the internal resistance of the source are equal.

This statement is known as the maximum power transfer theorem.

Figure shows a 12V EMF source of internal resistance 3 ohms

connected to a load resistance of 1 ohm. The total resistance in the

circuit is 4 ohms and the circuit current is therefore 3 amperes. The

power developed in the load (I2R) is therefore 9 watts.

Figure B shows the same source connected to a load resistance of 3

ohms. The total resistance is now 6 ohms and the current 2

amperes. The power developed in the load is now 12 watts.

Figure C shows the effect of inserting a load of 9 ohms. The total

resistance is now 12 ohms and the current 1 ampere. The power

developed in the load is now 9 watts.

The above examples have used the power formula I2R, but any of the

other

2 formulae, V2/R and I x V could be used.

Example ‘B’ using V2/R would give the same answer by measuring

the volts drop across the load resistance and then dividing the square

of that by the actual load resistance. Try it, it works!

The graph shown below shows these and other results by plotting the

power developed in different values of load resistance. It shows that

maximum power is developed in the load only when the load

resistance is equal in value to the internal resistance of the source

and, thus, illustrates the maximum power transfer theorem.

In many circuits we are interested in transferring the maximum

possible amount of power to a load circuit. To do this we must

‘match’ the load resistance to the internal resistance of the source.

Matching is very important in electronic circuits that usually have a

fairly high source resistance. A typical example is the ‘matching’ of

an audio amplifier to a loudspeaker and we shall consider this and

many others later in the book.

Note however that batteries, generators and other power supply

systems cannot be operated under maximum power transfer

conditions. It can be seen from the previous Fig that to do so would

result in the same amount of power being dissipated in the source as

was supplied to the load. This is obviously extremely wasteful of

energy and power supply systems are always designed to have the

minimum possible internal resistance to minimize losses.