For a short transmission line of impedance ( R + j X ) ( R + j

For a short transmission line of impedance $(R+jX)$ ohms per phase, show that the maximum power that can be transmitted over the line is

$$P_{max}=\frac{V_R^2}{Z^2}(\frac{Z{V}_S}{V_R}-R)\text{where}Z=\sqrt{R^2+X^2}$$

when the sending-end and receiving-end voltages are fixed, and for the condition

\[
\mathrm{Q}=\frac{-\mathrm{V}_{\mathrm{R}}^{2} X}{\mathrm{R}^{2}+X^{2}} \quad \text { when } d \mathrm{P} / d \mathrm{Q}=0
\]