NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 201

∇· D = ρ (2-35)

∇· B = 0 (2-36)

J

= σ E ,

D = ε E , and B = µ H , as derived from equations (2-6) through

where

(2-9).
Simplification of Ampere's law (2-34) by replacing the current density term
with σ E and

D = ε E yields

= σ E + jωε E = jω σ

jω + ε

E = jω ε − j σ
ω

∇× H

E

(5-1)

∇× H =
jωε E ) allows us to define the complex permittivity for a conductive or lossy
media by analogy:

Comparison to the solution of Ampere's law in a loss-free medium (

ε = ε − j σ

ω = ε − jε

(5-2)

where the real component is the dielectric permittivity discussed in Chapters 2
and 3 ( ε = ε 0 ε r ) and the imaginary component accounts for the losses in the
medium where the wave is propagating. The term σ can be thought of as the
conductivity of the material, which will be quite high for a metal and quite low
for a dielectric. If (5-2) is inserted into the time-harmonic solution of the electric
field derived in Section 2.3.4,

E x (z, t) = E x e − γz

+ E x e γz

(2-54)

then the complex propagation constant for a lossy media can be derived. The
complex propagation constant for a plane wave was derived in Section 2.3.4:

γ = α + jβ (2-42)

As discussed extensively in Chapter 2, if the wave is propagating in a loss-free
medium (where α = 0 ), (2-42) reduces to

jβ = ω √ µε

rad / m (5-3)

where ε µ r µ 0 . Substitution of the complex permittivity (5-2)

into (5-3) provides the form of the complex propagation constant of an electro-
magnetic wave traveling in a conductive medium:

= ε r ε 0 and µ =

= ω µ ε − j σ

ω

γ (5-4)

Setting (5-4) equal to (2-42) and separating into real and imaginary components
yields the general form of the attenuation constant α and the phase constant β for