Transmission line relative amplitude question

[timhunderwood]

Homework Statement

An annular sheet (i.e. a flattened ring) , of thickness t = 1mm and made of a material of resistivity 0.5 Ohm m 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.

Homework Equations

The Attempt at a Solution

I firstly calculated the resistance/impedance of the annular sheet which i found to be:

Z(sheet)= ln(r2/r1)*p/(2*Pi*t) where r2 and r1 are radii of coaxial cable, t is thickness of sheet and p is conductivity.

I then used the formula
Reflection coefficient = (Z(sheet)-Z(characteristic))/(Z(sheet)+Z(characteristic))
from which the unknowns nicely cancel.

But this gives the wrong answer by a factor of about 1/2.

Am I missing something? I assumed that the termination by characteristic impedance completely absorbed all waves reaching end of transmission line... Is my derivation of the impedance of the annular sheet correct?

Thanks

 

[marcusl]

Should the impedance at that point be the parallel combination of Zs and Zc?

 

[timhunderwood]

Thank you very much, this gives the right answer...

However, could you explain why?

I don't think I really understand why the impedance at that point should be considered as the parallel combination of Zc and Zs, although I recognise that the circuit looks like resistors in parallel...

 

[marcusl]

Resistors in parallel is the correct mental picture for this problem.