Maximum Power Transfer Theorem

Maximum power transfer theorem deals with the power transferred to the load on a circuit with a network of various sources or components on it. The maximum power transfer theorem defines the condition under which the maximum power is transferred to the load in a circuit.

Note: Here we are talking about maximum power transferred to the load only, not about the maximum power transferred to the load and internal components or resistance of the source combined, Under the condition of Maximum power transfer we only deal with the power transferred to the load and does not consider the power dissipated in internal circuits or resistance of the source so we are not talking about the maximum efficiency of power transfer but instead maximum possible power transfer from a source to a load.

The Maximum Power Transfer Theorem states that:

The power transferred from a source or circuit to a load is maximum when the resistance of the load is made equal or matched to the internal resistance of the source or circuit providing the power to the load.

Let,
 V= EMF supplied to the load.
RL = Load resistance.
Ri = Internal resistance of the source.

I = Current flowing through the load, internal resistance and the source of the circuit.
PL = Power transferred to the load.
Pi = Power dissipated at internal resistances.

Then,                                                    

Power transferred to the load = PL = I²RL 

or,
 =  = 

Now using the theorems of Differential calculus , If we keep the RL variable and want to calculate the maximum value of PL then we need to differentiate the PL with respect to RL and equate it with zero.
Thus,
Under Maximum power transfer to load condition:

or,

or,

Learning's:

The Maximum Power Transfer Theorem has a wide range of usage on real life situation. The theorem is used to maximize the power output to a load from any circuit. So they can be used to design circuits where the maximum output performance is desired. 

Maximum power transfer theorem is applied in radio electronics; for example: In Antenna Signal amplifier for radio and TV receivers; and various other fields where maximum performance is required but the maximum efficiency is not desired for example to match an Amplifier with a Loudspeaker to yield maximum power to the speaker and thus produce maximum sound.