Maximum power is transferred when

According to the maximum power transfer theorem, to obtain maximum power from external power from a source with finite impedance, the impedance of the load must be equal to source impedance.

Derivation:

From the above circuit, the current flowing through the load, ‘I’ is given as

\(I = \frac{{{V_{th}}}}{{{R_L} + {R_{TH}}}}\)

Power transferred to the load,

\({P_L} = {I^2}{R_L} = {\left( {\frac{{{V_{th}}}}{{{R_L} + {R_{TH}}}}} \right)^2}.{R_L}\)

In the above equation, RL is a variable, therefore the condition for maximum power delivered to the load is determined by differentiating load power with respect to the load resistance and equating it to zero.

\(\frac{{\partial {P_L}}}{{\partial {R_L}}} = V_{TH}^2\left[ {\frac{{{{\left( {{R_L} + {R_{TH}}} \right)}^2} - 2{R_l}\left( {{R_{TH}} + {R_L}} \right)}}{{{{\left( {{R_L} + {R_{TH}}} \right)}^4}}}} \right] = 0\)

→ (RL + RTH)2 = 2RL (RTH + RL)

→ RL + RTH = 2RL

→ RL = RTH