# Maximum power transfer theorem

The maximum power transfer theorem refers to the condition in which the power delivered to a load is at its highest possible value. This condition is The theorem that posits that in a DC network with finite internal resistance, the optimal generation of external power can be achieved when the resistance of the load is equivalent to that of the source.

## Maximum power transfer theorem Statement

Maximum power transfer theorem states that,” Maximum power output is obtained from an A.C circuit when the load impedance is equal to the complex conjugate of the internal impedance of the circuit as seen from the terminals of the load.” Maximum power transfer theorem is particularly useful for analyzing communication circuits where the aim is to transfer maximum power from the source to the load with certain efficiency.

The maximum power transfer theorem aims at finding ZL such that the power dissipated in it is maximum. Any network can be converted into a single voltage source with series impedance (Thevenin’s equivalent circuit).

i.e the reactance of the load impedance is of opposite sign to the reactance of internal impedance of the circuit.

Therefore, maximum power will be transferred from source to load, if RL=R and XL=-X i.e. for maximum power transfer ,load impedance should be complex conjugate of the internal impedance of the circuit.

i.e. ZL=Z*

so, overall efficiency of a circuit supplying maximum power is 50%.

Maximum power transfer also states that,” Maximum power output is obtained from an A.C circuit when the value of the load resistance is equal to the magnitude of the internal impedance of the circuit as seen from the terminals of the load.”

i.e. maximum power will be transferred from sources to resistive load RL, if the value of RL is the magnitude of the internal impedance of the circuit.

## Various cases of maximum power theorem

Maximum power output is obtained from a circuit when,

(i) For D.C Circuit

(ii) For A.C circuit

(A) Load impedance=Complex conjugate of the internal impedance(Thevenin’s impedance)

(B) Load resistance=Magnitude of the internal impedance(Thevenin’s impedance)

(iii) For A.C circuit(where fixed reactance present in load)

If the load reactance XL is fixed and P is maximized by varying the load resistance RL, the condition for maximum power transfer is,

(iV) For A.C circuit(where internal impedance of the circuit is resistive but fixed reactance present in the load)

If the load reactance XL is fixed and P is maximized by varying the load resistance RL, the condition for maximum power transfer is,

(V) For A.C circuit like transformer:(where only the magnitude of the load impedance i.e. turn ratio can be varied not the angle)

Maximum power output is obtained from an A.C circuit when the magnitude of the load impedance is equal to the magnitude of the source impedance(or internal impedance of the circuit).

Let θ be the angle of the load impedance ZL.

On simplifying we get,

## Worked Examples of Maximum power transfer theorem

Determine the value of RL to be connected across AB

a) For maximum power transfer

b) Also calculate the maximum power absorbed by RL.

For Vth,

For Isc or IN

As Vx=o

current source is open circuited.