Page 431 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 413

and so we see that the incident and transmitted power densities are related through
2

Re 1/η ∗ 2 η 1 η 2 + η ∗
2
2
S 2 = |τ| S 1i = 2 |τ| S 1i (16)
Re 1/η ∗ 1 η 2 η 1 + η ∗ 1
Equation (16) is a relatively complicated way to calculate the transmitted power,
unless the impedances are real. It is easier to take advantage of energy conservation
bynotingthatwhateverpowerisnotreflectedmustbetransmitted.Eq.(15)canbeused
to find
2 (17)
S 2 = (1 −| | ) S 1i
As would be expected (and which must be true), Eq. (17) can also be derived from
Eq. (16).

D12.1. A 1-MHz uniform plane wave is normally incident onto a freshwater
lake ( = 78, = 0, µ r = 1). Determine the fraction of the incident power

r
r
that is (a) reflected and (b) transmitted. (c) Determine the amplitude of the
electric field that is transmitted into the lake.
Ans. 0.63; 0.37; 0.20 V/m

12.2 STANDING WAVE RATIO
In cases where | | < 1, some energy is transmitted into the second region and some is
reflected. Region 1 therefore supports a field that is composed of both a traveling wave
and a standing wave. We encountered this situation previously in transmission lines, in
which partial reflection occurs at the load. Measurements of the voltage standing wave
ratio and the locations of voltage minima or maxima enabled the determination of an
unknown load impedance or established the extent to which the load impedance was
matched to that of the line (Section 10.10). Similar measurements can be performed
on the field amplitudes in plane wave reflection.
Using the same fields investigated in the previous section, we combine the in-
cident and reflected electric field intensities. Medium 1 is assumed to be a perfect
dielectric (α 1 = 0), but region 2 may be any material. The total electric field phasor
in region 1 will be
E x1T = E + + E − = E x10 e − jβ 1 z + E + e jβ 1 z (18)
+
x1
x1
x10
where the reflection coefficient is as expressed in (9):
η 2 − η 1 jφ
= =| |e
η 2 + η 1
We allow for the possibility of a complex reflection coefficient by including its phase,
φ. This is necessary because although η 1 is real and positive for a lossless medium,