Q. Show Maximum Power Transfer?

In order to investigate the power transfer between a practical source and a load connected to it, let us consider Figure , in which a constant voltage source v with a known internal resistance RS is connected to a variable load resistance RL. Note that when RL is equal to zero, it is called a short circuit, in which case vL becomes zero and iL is equal to v/RS. When RL approaches infinity, it is called an open circuit, in which case iL becomes zero and vL is equal to v. One is generally interested to find the value of the load resistance that will absorb maximum power from the source.

The power PL absorbed by the load is given by

P_L = i_L²R_L

where the load current iL is given by

For given fixed values of v and RS, in order to find the value of RL that maximizes the power absorbed by the load, one sets the first derivative dP /dR equal to zero,