Maximum Power Transfer Theorem for DC circuits:-

• Maximum Power Transfer Theorem states that in a  active DC network, the maximum power will be transferred from source to the load when the external load resistance equals to the internal resistance of the source.

• Consider any circuit which is replaced by Thevenin’s voltage source in series with Thevenin’s equivalent resistance connected across the complex load RL.

• The current through the above circuit is given as,

$I = \dfrac{V_{TH}}{R_{TH}+R_L}$

• Power delivered to the load is given by

$P_L = I^2R_L$

• Subtituting current I in the above equation we have

$P_L = \dfrac{V^2_{TH}\times R_L}{(R_{TH}+R_L)^2}$

• To find the value of RL at which maximum power is delivered to the load then differentiate PL with respect to RL and equates it to zero.

$\dfrac{dP_L}{dR_L} = \dfrac{d}{dR_L}\left[ \dfrac{V^2_{TH}\times R_L}{(R_L+R_{TH})^2}\right]=0$

• Then we get 

$\ \ \ \ \ \ \ R_L+R_{TH}=2R_{L} \\ \Rightarrow R_L=R_{TH}$

• Therefore, in DC circuits, if RL = RTH, maximum power transfer takes place from source to load. This implies that maximum power transfer occurs when the resistance of the load is equal to the thevenin's resistance , i.e., RL = RTH
• When RL = RTH  then maximum power is given by

$P_{MAX} = \dfrac{V^2_{TH}}{4R_{TH}} = \dfrac{V^2_{TH}}{4R_{L}}$