An annular sheet, of thickness t=1mm and resistivity ρ=0.5Ωm, connects the inner and outer conductors of an air spaced coaxial transmission line at a point on the line. A low-frequency signal is fed into one end of the line and the other is terminated by its characteristic impedance. Calculate the relative amplitude of the wave reflected from the resistive sheet.
The Attempt at a Solution
The annular sheet results in a change in the impedance of the line at the point where the sheet is located. We know that, in general:
Z= sqrt((R+iωL)/(G+iωC)), R is resistance per unit length of cylinder, G is conductance per unit length of dielectric between cylinders
therefore, in this case: impedance at annulus=Za= sqrt((iωL)/((ρ/t)+iωC)), where it is assumed that the resistance in the cylinders is negligible.
I feel that the ρ/t term should be more complicated, but it is at least dimensionally consistent, and there doesn't seem to be enough information to do much else.
r=(Za-Z)/(Za+Z), where r is the reflection coefficient, Za is the impedance at the annulus and Z is the characteristic impedance of the line.
From these two equations it should be possible to calculate the reflection coefficient. However, in this case, absolutely no information is given about the line itself, the only values provided are t and ρ. Even after fiddling around with the equations, I haven't been able to get things to cancel out, leaving an expression for r in terms of t and ρ.
Am I going about this in completely the wrong way?
Any help is greatly appreciated.