ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 259

to drop and the imaginary portion begins to increase, implying a relationship
between the real and imaginary parts of the complex permittivity. This relation-
ship is discussed in detail in Section 6.4. Finally, the value of the permittivity
settles out at a new level after the operating frequency ω has surpassed the natural
frequency ω 0 .

6.3.3 Anomalous Dispersion

Perhaps the most interesting part of the curve shown in Figure 6-6 is the area
where the relative dielectric permittivity drops below 1. This area is often referred
to as anomalous dispersion . When this is first encountered, it often causes great
confusion, as demonstrated by observing the definition of the speed of light in a
vacuum, which was introduced in Section 2.3.4 in units of meters per second:

1
√ µ 0 ε r ε 0

c ≡ when ε r = 1

This implies that if ε r < 1, the velocity will exceed the speed of light in a

10 8 m/s). One of the consequences of Einstein's theory of
special relativity is that speeds greater than c are not attainable, yet the phase
velocity of a wave traveling at frequencies where ε r < 1 in Figure 6-6 appears
to break this fundamental law of physics.
The apparent conflict with the laws of special relativity comes from a widely
mistaken assumption that all quantities with units of velocity must obey this rule.
In fact, special relativity only places an upper value on the speed of material
bodies that include signals, or information. Since a single-frequency harmonic
plane wave is not a material body and is not a signal, it cannot be used to transmit
information by itself. To understand this, the definition of phase velocity must
be examined from Chapter 2. 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 ).
However, the only way to measure the velocity of a signal is to turn on the
transmitter, time how long it takes for a response to arrive at the receiver, and
divide by the distance. When velocity is measured in this way, it cannot exceed
c . To understand why, consider the propagation of information, such as a digital
pulse. If the pulse is decomposed into its Fourier components, each will propagate
with its own velocity, some slower than c and some faster than c ; however, when
all components are combined, the total velocity of the pulse cannot exceed the
speed of light . A question that often arises when this subject is discussed is the
possibility of modulating a narrowband signal with a frequency that coincides
with an area of anomalous dispersion to sidestep the laws of special relativity and
transmit information faster than c . However, a single-frequency plane wave still
cannot carry information unless another signal is combined with it. For example,

vacuum ( c ≈ 3 ×