Classical Electrodynamics · Problem 8.2

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · coaxial-transmission-line-tem-mode

Problem

A transmission line consisting of two concentric circular cylinders of metal with conductivity

\sigma

and skin depth

\delta

, as shown, is filled with a uniform lossless dielectric

(\mu, \epsilon)

. A TEM mode is propagated along this line. Section 8.1 applies. (a) Show that the time-averaged power flow along the line is

P = \sqrt{\frac{\mu}{\epsilon}}\, \pi a^2\, |H_0|^2 \ln\!\left(\frac{b}{a}\right)

where

H_0

is the peak value of the azimuthal magnetic field at the surface of the inner conductor. (b) Show that the transmitted power is attenuated along the line as

P(z) = P_0 e^{-2\gamma z}

where

\gamma = \frac{1}{2\sigma\delta}\sqrt{\frac{\epsilon}{\mu}}\frac{\left(\dfrac{1}{a}+\dfrac{1}{b}\right)}{\ln\!\left(\dfrac{b}{a}\right)}

Z_0

of the line is defined as the ratio of the voltage between the cylinders to the axial current flowing in one of them at any position

z

. Show that for this line

Z_0 = \frac{1}{2\pi}\sqrt{\frac{\mu}{\epsilon}}\ln\!\left(\frac{b}{a}\right)

(d) Show that the series resistance and inductance per unit length of the line are

R = \frac{1}{2\pi\sigma\delta}\left(\frac{1}{a}+\frac{1}{b}\right)

L = \left\{\frac{\mu}{2\pi}\ln\!\left(\frac{b}{a}\right) + \frac{\mu_c\delta}{4\pi}\left(\frac{1}{a}+\frac{1}{b}\right)\right\}

where

\mu_c

is the permeability of the conductor. The correction to the inductance comes from the penetration of the flux into the conductors by a distance of order

\delta

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