Maximum Power Transfer

The power transfer from a source to it’s load is maximum when their respective impedance are complex conjugate of each other.

Let Z_s = R_s + jX_s and Z_l=R_l+jX_l

The power delivered to load is given by

(1)   \begin{eqnarray*} P_l &=& {1 \over 2}V_l I_l^*\\ & = & {1 \over 2}{V_s \frac{R_l}{Z_s+Z_l}}{\frac{V_s}{(Z_s+Z_l)^*}}\\ & = & {1 \over 2}{V_s^2 R_l \over (R_s+R_l)^2+(X_s+X_l)^2} \end{eqnarray*}

The condition for maximum power transfer is

\frac{\partial P_l}{\partial R_l}=0  \mbox{ and } \frac{\partial P_l}{\partial X_l}=0

Therefore, R_l=R_s and X_l=-X_s

Under this condition, power delivered to load is

(2)   \begin{equation*} P_{l,max} = \frac{1}{2} {V_s^2 \over 4R_l}\end{equation*}

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