Impedance of Transmission Line

In a satellite receiver operating at 12.0 GHz there is need for a reactive impedance having reactance of X = – 67 Ohm. Using an open circuit stub formed from 50 Ohm lossless transmission line having a phase velocity of 1 x 10^8 m/s determine the shortest length of this line required to provide the reactance.


$Z = -j*Z0*tan(\beta*L)$

if C is the speed of wave in air ($c =3*10^8 m/s$) we have

$\beta = \omega/C =(2*\pi*F)*1/C =(2*\pi*12*10^9) / 3*10^8 =251.32$

$Z = -j*67 = -j*50*tan(251.32*L)$

$tan(251.32*L) =67/50$

$251.32*L =0.9297 rad$

$L =0.0036992 m =3.6992 mm$