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
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
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.

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