Question

A load with impedance $Z_L=(25-j50)\Omega$ is ti be connected to a loss-less transmission line with characteristic impedance $Z_0$, with $Z_0$ chosen such that the standing wave ratio is the smallest possible. Then what should be the value of $Z_0$?

Answer 1

Minimum possible $VSWR\left(\rho \right)$ also gives the minimum possible ${\Gamma }_L$ and for that matter the minimum possible $|{\Gamma }_L{|}^2$

$0=\frac{\partial }{\partial Z_0}|{\Gamma }_L{|}^2=\frac{\partial }{\partial Z_0}{\left|\frac{R_L+j{X}_L-Z_0}{R_L+j{X}_L+Z_0}\right|}^2$


$=\frac{\partial }{\partial Z_0}\left[\frac{(R_L-Z_0{)}^2+X_L^2}{(R_L+Z_0{)}^2+X_L^2}\right]$

$=\frac{4R_L(Z_0^2-(R_L^2+X_L^2\left)\right)}{\left(\right(R_L+Z_0{)}^2+X_L^2{)}^2}$

Therfore $Z_0^2=R_L^2+X_L^2$

or

$Z_0=|Z_L|=\sqrt{{25}^2+(-50{)}^2}$

$=55.9\Omega$