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, R_{S}. 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, R_{L} across the output terminals of the power source, the impedance of the load will vary from an opencircuit state to a shortcircuit 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 Circuit 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 R_{L} = R_{S} 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, R_{L} that will give the maximum power transfer in the following circuit.
Maximum Power Transfer Example No1.
Where: 
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


Using the data from the table above, we can plot a graph of load resistance, R_{L} against power, P for different values of load resistance. Also notice that power is zero for an opencircuit (zero current condition) and also for a shortcircuit (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, R_{L} is equal in value to the source resistance, R_{S} that is: R_{S} = R_{L} = 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, Z_{OUT} of the amplifier may be given as between 4Ω and 8Ω, while the nominal input impedance, Z_{IN} 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, Z_{LOAD} to output impedance, Z_{OUT} 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, Z_{LOAD} is purely resistive and the source impedance is purely resistive, Z_{OUT} then the equation for finding the maximum power transfer is given as:
Where: N_{P} is the number of primary turns and N_{S} 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,
Maximum Power Transfer Example No2.
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 Z_{1}, the load impedance is Z_{2} 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.
anjit neupane
plz also include thr numerical based on resistive and inductive fo the vaification of maximum power theorem
Wayne Storr
Hello Anjit, good idea, I will think of something.
Ravi Dhindhwal
Learning from your website easier than the boaring classrooms
Tom
Is there a publication date for this? I would like to use this as a reference.
John Wiltrout
Perhaps there is a difference in the conventions and I shouldn’t argue with the teacher, but, If RL=0 indicates an open circuit what does RL= infinity indicate? While both zero and infinity are theory values and meaningless in the real world they are the limits of a series. I would agree that GL = 0 (conductance of the Load = 0) should indicate an open circuit but I am still do not understand why plugging 0 into your equation I = VS/ (RL + RS) would not give an answer of 4 amps. I do agree that the power should be 0.
John
Wayne Storr
If there is no load resistor connected, then there is no closed circuit, then there is no 4 amps. If it makes you feel better, we will assume the load resistor is a complete short circuit, ie, 0 Ohms across the source. The current supplied by the source through this short will indeed by 4.0 amps (maximum value). The I^2R power delivered by the source will therefore be 4.0 Amps x 4.0 Amps x 0 Ohms = 0. The same as before. Then, if the load is infinity, no power is delivered to the load as maximum voltage exists but no current. If the load is zero, no power is delivered to the load as maximum current exists but no voltage. For maximum power to be transferred between the source and the load the two must have the same impedance.
ERICK SIAME
The maximum power theorem states that the power transferred from the supply source is at its maximum when the resistance of the load is equal to the the internal resistance.
John Wiltrout
I am continuing to learn and greatly enjoy your tutorials. In your Table above shouldn’t I = 4 when RL = 0?
John
Wayne Storr
No, RL = 0 represents an open circuit condition, i.e. no load resistor connected, then i = 0.
lazer
Please include source transformation techniques
Wayne Storr
Hello Lazer. Thanks for the suggestion.
Asif
Twenty Eight