ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 57

To obtain the time-averaged value of (2-120), remember that sin 2 θ + cos 2 θ = 1.

Therefore, over a complete cycle, the average of cos 2 θ

is equal to the average

of sin 2 θ and sin 2 θ = cos 2 θ =

1

2 . Subsequently, the time-averaged value of a
cosine-squared term over a complete cycle is

1
2 :

= a z (E + ) 2
2 η

S ave

(2-121)

Similarly, the Poynting vector can be expressed in terms of the magnetic field:

S = E × H

a x ηE + cos (ωt − βz) ] a y H + cos (ωt − βz) ] = [ × [

= a z η(H + ) 2 cos 2 (ωt − βz) (2-122)

a z η (H + ) 2
2

S ave

=

2.7 REFLECTIONS OF ELECTROMAGNETIC WAVES

So far, we have considered the propagation of electromagnetic waves in a sim-
ple, infinitely large medium. However, most practical problems involve waves
propagating in multiple dielectric media. Since each medium will have different
electric characteristics, it is essential to understand how a propagating electro-
magnetic wave will behave when it enters a region where the properties of the
medium change.
Generally, when an electromagnetic plane wave propagating in medium A
enters region B , where the properties of the dielectric change, two things happen:
(1) a portion of the wave is reflected away from region B , and (2) a portion
of the wave is transmitted into region B . When these plane waves encounter
planar interfaces, both the reflected and transmitted waves are also planar, so
their directions, amplitudes, and phase constants can easily be calculated. The
simultaneous existence of both the transmitted and reflected waves is a direct
result of the boundary conditions that must be satisfied when solving Maxwell's
equations at the interface between the two regions. We begin with a plane wave
incident on a perfect conductor.

2.7.1 Plane Wave Incident on a Perfect Conductor

Consider a plane wave propagating in medium A in the z -direction. Assume
that the medium in region A is a simple, loss-free medium and medium B is
a perfectly conducting metal plane, as shown in Figure 2-23. Assume that the
electric field is oriented in the x -direction, necessitating that the magnetic field