Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem

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 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 the Maximum Power Transfer Theorem.

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.

Thevenin's 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.

Example No1.

Where: 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

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 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, R_L is equal to the source resistance, R_S so then: 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 occurs when the impedance of the external device matches that of the source. Improper impedance matching can lead to excessive power use and dissipation.

Transformer Impedance Matching

One very useful application of impedance matching to provide maximum power transfer is in the output stages of amplifier circuits, where the speakers impedance is matched to the amplifier output impedance to obtain maximum sound power output. This is achieved by using a Matching Transformer to couple the load to the amplifiers output as shown below.

Transformer Coupling

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,

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 Z1, the load impedance is Z2 and the turns ratio is given as N.

Generally, small transformers used in low power audio amplifiers are usually regarded as ideal so any losses can be ignored.