ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 30

The terms in (2-42) have special meanings used throughout the topic to describe
the medium where the electromagnetic wave is propagating, whether it is in free
space, is an infinite dielectric, or is a transmission line. Specifically, α is the loss
term , which describes signal attenuation as it propagates through the medium. The
loss term accounts for the fact that real-world metals are not infinitely conductive
(except superconductors) and dielectrics are not perfect insulators (except free
space), both of which are discussed in detail in Chapters 5 and 6. The imaginary
portion of (2-42), β , called the phase constant , essentially dictates the speed at
which the electromagnetic wave will travel in the medium. To visualize these
waves propagating as described in (2-41), it is necessary first to recover the time
dependency removed in Section 2.3.3. Considering only the forward-propagating
component of a wave in a vacuum, replacing C 1 with the magnitude of the electric
field, restoring the time dependency as in (2-37), and applying the identity of
equation (2-31) yields

Re (E x e − γz e jωt ) = Re (E x e − αz e − jβz e jωt ) = e − αz E x E(z,t) = cos (ωt − βz)

(2-43)

0), Figure 2-12 depicts successive
snapshots of a wave propagating though space. To determine how fast the wave is
propagating, it is necessary to observe the cosine term for a small duration of time
t . Since the wave is propagating, a small change in time will be proportional to a
small change in distance z , which means that an observer moving with the wave
will experience no phase change because she is moving at the phase velocity ( ν p ).
Setting the term inside the cosine of (2-43) to a constant ( ωt − βz =

Assuming that the loss term is zero ( α =

constant)
and differentiating allows the definition of the phase velocity from the cosine
term in (2-43):

dz
dt =

ω
β

ν p = m / s (2-44)

E x cos ( w t

b z )

−

at t t 0

=

at t = t 1
at t

= t 2

∆ t

∆ z

= ν p

t

∆ z ∆

l

Figure 2-12 Snapshots in time of a plane wave propagating along the z -axis, showing
the definition of phase velocity.