Page 406 - Engineering Electromagnetics, 8th Edition

388 ENGINEERING ELECTROMAGNETICS

of travel along the surface. This is the mechanism for the resistive transmission line
loss that we studied in Chapter 10, and which is embodied in the line resistance
parameter, R.
As implied, a good conductor has a high conductivity and large conduction
currents. The energy represented by the wave traveling through the material therefore
decreases as the wave propagates because ohmic losses are continuously present.
When we discussed the loss tangent, we saw that the ratio of conduction current
density to the displacement current density in a conducting material is given by
σ/ω .Choosingapoormetallicconductorandaveryhighfrequencyasaconservative

.
8
6
5
example, this ratio for nichrome (σ = 10 )at 100 MHz is about 2 × 10 .We therefore
have a situation where σ/ω 1, and we should be able to make several very good

approximations to find α, β, and η for a good conductor.
The general expression for the propagation constant is, from (59),

σ

jk = jω µ 1 − j
ω
which we immediately simplify to obtain

σ
jk = jω µ − j
ω
or

jk = j − jωµσ
But
− j = 1 −90 ◦
and
√ 1
1 −90 = 1 −45 = √ (1 − j)
◦
◦
2
Therefore

ωµσ
jk = j(1 − j) = (1 + j) π f µσ = α + jβ (78)
2
Hence

α = β = π f µσ (79)
Regardless of the parameters µ and σ of the conductor or of the frequency of the
applied field, α and β are equal. If we again assume only an E x component traveling
in the +z direction, then
√
E x = E x0 e −z π f µσ cos ωt − z π f µσ (80)

5 It is customary to take = 0 for metallic conductors.