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Medium of large loss We say the medium has large loss (for transmission of waves through that medium) when
σ À ω², i.e. conduction current À displacement current. Then the equations at the end
of Section 9.2 can be put in the different approximate form
γ ≈ (1 + j)
r
ωμσ
2
(9.10)
and
η ≈ (1 + j)
r
ωμ
2σ
. (9.11)
These approximations may be obtained from the formulae for the propagation constant
and wave impedance of a lossless medium by replacement of jω² by σ. Here we see that
displacement current has been replaced by conduction current. In terms of α and β
α =
q
ωμσ
2
β =
q
ωμσ
2
. (9.12)
We see that the rate of phase change (in radians per metre) and attenuation (in nepers
per metre) are equal. The distance for an attenuation by a factor e is defined as the skin
depth δ.
δ =
1
α
=
s
2
ωμσ
. (9.13)
It is common to define the surface resistivity per square Rs as
Rs =
r
ωμ
2σ
=
1
δσ
(9.14)
in terms of which the wave impedance is
η = (1+j)Rs . (9.15)
We note that Rs is equal to the resistance per square of a thin sheet of material of
thickness equal to δ in which the current is uniform. Of course the current here is decidedly
non-uniform in both amplitude and phase as the wave proceeds into the material; that is
why η 6= Rs . But this remark makes it easy to remember Rs when we know δ.
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