Maximum power transfer:

Every source of EMF has internal resistance.  If it is required to develop the maximum possible amount of power in an external load, then the load resistance must be equal in value to the internal resistance of the source.

This may be shown by calculating the power developed in R_Load for different values of R_Load.

This illustrates that maximum power is developed in the load when R_Load equals R_Internal.

Matching is very important in electronic circuits which usually have a fairly high source resistance.  A typical example is the 'matching' of a loudspeaker to an audio amplifier.  Note however that:

• For a power source with variable internal resistance and given load (R_L), the smaller the internal resistance, the higher the power transfer to the load. The highest power transfer is achieved here when the internal resistance is zero.

Batteries, generators and other power supply systems are not operated under maximum power transfer conditions, since to do so would result in the same amount of power being dissipated in the source as was supplied to the load, which is wasteful of energy.  Thus power systems are designed to have the minimum internal resistance to minimise loses in the power supply