ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 255

relative dielectric permittivity ε r was introduced to account for the presence of
P in a dielectric. When the applied fields begin to alternate in polarity, the polar-
ization vector

P , and subsequently the dielectric permittivity, is affected. Since
the electric dipoles in a material will not align instantaneously with the applied
time-varying fields, the polarization and relative permittivity are a function of
the frequency of the alternating field. In this section we explore the frequency
dependence of dielectric materials and derive useful models that can be used to
simulate these effects.

6.3.1 DC Dielectric Losses
Dc losses imply a dielectric with conduction electrons that are free to move
according to Ohm's law:

J

= σ d E

(6-17)

J is the current density and σ d is the conductivity of the dielectric. Assum-
ing a time-harmonic field, substitution of (6-17) into Ampere's equation gives

where

∂ D(x,t)
∂t
= σ d E(x) + jω [ ε − jε ]

∇× H = J(x,t) +

E(x)

= jωε 0 ε r − jε r

σ d
ε 0 ω

E(x)

− j (6-18)

Rearranging (6-18) yields the dependence of Ampere's equation on permittivity
by adding the term σ d /ε 0 ω to (6-15) to account for the conduction electrons.
Do not get σ d in (6-18) confused with the term σ dielectric in (6-15), which is the
effective conductivity of the dielectric due to the energy it takes to polarize the
electric dipoles in the material. The term σ d used in (6-18) is true conductivity,
similar to that of a metal where electrons are not bound and are free to move. Note
that a pole is created at ω = 0 based on the rearrangement of (6-18), rendering

the equation invalid at dc. Since the dielectric conductivity σ d is very small

(σ d /ε 0 ω

in practical dielectrics 1 ) , the dc term is almost always neglected

[Huray, 2009].

6.3.2 Frequency-Dependent Dielectric Model: Single Pole
Following the derivation in Section 6.1.1, assume that an atom in the absence
of an applied electric field is represented by a positive nucleus and a negative
electric cloud with centers that coincide. Since protons and neutrons are much
heavier than electrons, we assume that when the external electric field is applied,
the nucleus remains stationary and the electron cloud moves. Therefore, when
the electric field is applied, the electron cloud is displaced, and when the field
is removed, the electron cloud returns to its original position, similar to the