Basic Electronics on the Go - Maximum Power Transfer

From http://www.electronics-tutorials.ws/dccircuits/dcp_9.html

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.

 When we connect a load resistance, R_L across the output terminals of the power source, the impedance of the load will vary 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:
  R_S = 25Ω
  R_L is variable between 0 – 100Ω
  V_S = 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) 0 4.0 0 5 3.3 55 10 2.8 78 15 2.5 93 20 2.2 97

RL (Ω) I (amps) P (watts) 25 2.0 100 30 1.8 97 40 1.5 94 60 1.2 83 100 0.8 64

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 in value to the source resistance, R_S that is: R_S = R_L = 25Ω. This is called a “matched condition”.

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

 

 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.