ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 261

the resonance. The summation indicates that even low-frequency values of the
permittivity depend on the high-frequency resonances.
Although (6-26) is quite useful for developing an understanding of the physical
mechanisms that cause the dielectric to vary with frequency, it does not lend itself
well to simulating dielectrics for digital designs because minuscule details of the
molecular structure are required. A more pragmatic approach is to represent
(6-26) in terms of measured permittivity. If (6-26) is multiplied by ε 0 and the
right-hand term is simplified by dividing the top and bottom by 1 /ω 0 , the top
part, ( 1 /ω 0 ) [ N(q 2 /m) ], has units of farads per meter:

s 4 A 2 ·

1F =

m 2 · kg

1C = A · s

s 2 1
m 3

N q 2
m

A 2

s 2 1

ω 0

·

→ = F / m

kg

which are the same units as ε , allowing (6-26) to be rewritten as

n

ε i

σ d
ε 0 ω

ε = ε ∞ +

− ω 2 /ω 1 i + jω/ω 2 i − j

(6-27)

1

i = 1

where ω 1 i and ω 2 i [ ( 1 /ω i )(b/m)ω = ω/ω 2 i ] are the frequencies where the
dielectric variations are occurring, ω = 2 πf is the operating frequency, ε the

variation of dielectric permittivity over the frequency of interest, σ d the true
conductivity of the dielectric material, ε ∞ the dielectric permittivity value at

very high frequencies in the area of interest, and ε 0 the permittivity of free
space. The term j(ω/ω 2 i ) accounts for the damping of the molecular dipoles
(orientational polarization) in the mid-frequency ranges, the term ω 2 /ω 1 i
accounts for resonance of induced atomic and molecular dipoles (ionic and
electronic polarization), and the final term, j(σ d /ε 0 ω) (derived in Section 6.3.1),
accounts for the low-frequency loss of the dielectric, which is usually ignored.
Figure 6-8 shows a conceptualized plot of the real and imaginary dielectric
permittivity as a function of frequency. It should be noted that the figure does
not represent any particular dielectric material; rather, it is simply a guide to
help the reader conceptualize when the different forms of polarization begin to
become significant. For most dielectric materials used in digital design, labora-
tory measurements show that the permittivity is dominated by the damping factor
in (6-27)(the jω/ω 2 i term) and not resonances (the ω 2 /ω 1 i term). For orienta-
tional polarization (where polar molecules attempt to remain aligned with the
time varying electric field), the damping factor tends to be high. Consequently,
the classic model is derived from the concept of orientational polarization even
though other forms may affect the damping. When the high frequency resonance
term is ignored, the dielectric equation reduces to (6-28), which is applicable for