Maximum Power Transfer Theorem
Maximum Power Transfer
We have seen in the previous tutorials that any complex circuit or network can be replaced by a single
energy source in series with a single internal source resistance, RS. Generally, this
source resistance or even impedance if inductors or capacitors are involved is of a fixed value in Ohm´s. However, when
we connect a load resistance, RL across the output terminals of the power source, the
impedance of the load will vary from an open-circuit state to a short-circuit state resulting in the power being absorbed
by the load becoming dependent on the impedance of the actual power source. Then for the load resistance to absorb the
maximum power possible it has to be "Matched" to the impedance of the power source and this forms the basis of
Maximum Power Transfer.
The Maximum Power Transfer Theorem is another useful analysis method to ensure that
the maximum amount of power will be dissipated in the load resistance when the value of the load resistance is exactly
equal to the resistance of the power source. The relationship between the load impedance and the internal impedance of
the energy source will give the power in the load. Consider the circuit below.
Thevenins Equivalent Circuit.
In our Thevenin equivalent circuit above, the maximum power transfer theorem states that "the
maximum amount of power will be dissipated in the load resistance if it is equal in value to the Thevenin or Norton
source resistance of the network supplying the power".
In other words, the load resistance resulting in greatest power dissipation must be equal in value to
the equivalent Thevenin source resistance, then RL = RS but if
the load resistance is lower or higher in value than the Thevenin source resistance of the network, its dissipated power
will be less than maximum. For example, find the value of the load resistance, RL
that will give the maximum power transfer in the following circuit.
RS = 25Ω
RL is variable between 0 - 100Ω
VS = 100v
Then by using the following Ohm's Law equations:
We can now complete the following table to determine the current and power in the circuit for different
values of load resistance.
Table of Current against Power
|RL (Ω)||I (amps)||P (watts)|
|RL (Ω)||I (amps)||P (watts)|
Using the data from the table above, we can plot a graph of load resistance, RL
against power, P for different values of load resistance. Also notice that power is zero for an open-circuit
(zero current condition) and also for a short-circuit (zero voltage condition).
Graph of Power against Load Resistance
From the above table and graph we can see that the Maximum Power Transfer occurs
in the load when the load resistance, RL is equal in value to the source resistance,
RS that is: RS = RL = 25Ω.
This is called a "matched condition" and as a general rule, maximum power is transferred from an active device such
as a power supply or battery to an external device when the impedance of the external device exactly matches the impedance
of the source.
One good example of impedance matching is between an audio amplifier and a loudspeaker. The output
impedance, ZOUT of the amplifier may be given as between 4Ω
and 8Ω, while the nominal input impedance, ZIN of the
loudspeaker may be given as 8Ω only. Then if the 8Ω speaker is
attached to the amplifiers output, the amplifier will see the speaker as an 8Ω load. Connecting
two 8Ω speakers in parallel is equivalent to the amplifier driving one 4Ω
speaker and both configurations are within the output specifications of the amplifier.
Improper impedance matching can lead to excessive power loss and heat dissipation. But how could you
impedance match an amplifier and loudspeaker which have very different impedances. Well, there are loudspeaker impedance
matching transformers available that can change impedances from 4Ω to 8Ω,
or to 16Ω's to allow impedance matching of many loudspeakers connected together in various
combinations such as in PA (public address) systems.
Transformer Impedance Matching
One very useful application of impedance matching in order to provide maximum power transfer between
the source and the load is in the output stages of amplifier circuits. Signal transformers are used to match the loudspeakers
higher or lower impedance value to the amplifiers output impedance to obtain maximum sound power output. These audio signal
transformers are called "matching transformers" and couple the load to the amplifiers output as shown below.
Transformer Impedance Matching
The maximum power transfer can be obtained even if the output impedance is not the same as the load
impedance. This can be done using a suitable "turns ratio" on the transformer with the corresponding ratio of load
impedance, ZLOAD to output impedance, ZOUT
matches that of the ratio of the transformers primary turns to secondary turns as a resistance on one side of the
transformer becomes a different value on the other. If the load impedance, ZLOAD
is purely resistive and the source impedance is purely resistive, ZOUT then the
equation for finding the maximum power transfer is given as:
Where: NP is the number of primary turns and
NS the number of secondary turns on the transformer. Then by varying the value of
the transformers turns ratio the output impedance can be "matched" to the source impedance to achieve maximum power
transfer. For example,
If an 8Ω loudspeaker is to be connected to an amplifier with an output
impedance of 1000Ω, calculate the turns ratio of the matching transformer required to
provide maximum power transfer of the audio signal. Assume the amplifier source impedance is Z1,
the load impedance is Z2 with the turns ratio given as N.
Generally, small transformers used in low power audio amplifiers are usually regarded as ideal so any
losses can be ignored.
In the next tutorial about DC Theory we will look at
Star Delta Transformation which allows
us to convert balanced star connected circuits into equivalent delta and vice versa.