Page 427 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 409

or
η 2 − η 1
E x10 = E x10
−
+
η 2 + η 1
The ratio of the amplitudes of the reflected and incident electric fields defines the
reflection coefficient, designated by ,
E − η 2 − η 1
= x10 = =| |e jφ (9)
E x10 η 2 + η 1
+
It is evident that as η 1 or η 2 may be complex, will also be complex, and so we
include a reflective phase shift, φ. The interpretation of Eq. (9) is identical to that
used with transmission lines [Eq. (73), Chapter 10].
The relative amplitude of the transmitted electric field intensity is found by
combining (9) and (7) to yield the transmission coefficient, τ,

E + 2η 2
τ = x20 = = 1 + =|τ|e jφ i (10)
E x10 η 1 + η 2
+
whose form and interpretation are consistent with the usage in transmission lines
[Eq. (75), Chapter 10].
Let us see how these results may be applied to several special cases. We first let
region 1 be a perfect dielectric and region 2 be a perfect conductor. Then we apply
Eq. (48), Chapter 11, with = σ 2 /ω, obtaining

2

jωµ 2
η 2 = = 0
σ 2 + jω 2
in which zero is obtained since σ 2 →∞. Therefore, from (10),
E + = 0
x20
No time-varying fields can exist in the perfect conductor. An alternate way of looking
at this is to note that the skin depth is zero.
Because η 2 = 0, Eq. (9) shows that
=−1
and
E + =−E −
x10
x10
The incident and reflected fields are of equal amplitude, and so all the incident
energy is reflected by the perfect conductor. The fact that the two fields are of opposite
sign indicates that at the boundary (or at the moment of reflection), the reflected field
is shifted in phase by 180 relative to the incident field. The total E field in region 1 is
◦
E xs1 = E xs1 + E −
+
xs1
= E x10 e − jβ 1 z − E + e jβ 1 z
+
x10