Statement:

     The maximum power transfer theorem states that the maximum amount of power will be delivered to the load resistance when the load resistance is equal to the Thevenin/Norton’s resistance of the network supplying power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, then the power delivered to load is less than the maximum.

$R_{L}=R_{TH}$

Explanation:

To obtain the condition for maximum transfer, refer the circuit shown in figure:

RL is the load resistance and RTH is the Thevinin’s equivalent resistance of a complex network.

The Current flowing through the circuit,

$I_L=\dfrac{V_{Th}}{R_{TH}{\ +\ R}_L}Amp$

Power received by the load,

$P=I_L^2R_L{\left[\dfrac{V_{Th}}{{R_{TH}+\ R}_L}\right]}^2\times{}R_L$

  The power will be the maximum, if its derivative with respect to load resistance is zero.

$i.e\ \dfrac{dP}{{dR}_L}=0$

$\dfrac{d}{{dR}_L}\left[\dfrac{{V_{Th}}^2R_L}{{{(R}_{TH}+\ R_L)}^2}\right]=0$

$\dfrac{{{(R}_{th}+\ R_L)}^2\ x\ {V_{Th}}^2\ -{{\ V}_{Th}}^2\ x\ R_L\ x\ 2(R_{Th}+R_L)}{{(R_{Th}+\ R_L)}^4}=0$

$R_{Th}^2V_{th}^2+2R_{Th}R_LV_{th}^2+R_L^2V_{th}^2-2V_{th}^2R_LR_{Th}-2V_{th}^2R_L^2=0$

$R_{Th}^2V_{Th}^2-R_LV_{th}^2=0$

$R_{Th}=R_L $

This is the condition for Maximum Power Transfer.