ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 270

induced by the frequency-invariant approximation are amplified, especially for
longer line lengths. When models are used that do not obey these rules, incorrect
solutions are determined, lab correlation becomes difficult or impossible, and
the design time is increased significantly. In this section we introduce specific
limitations that dielectric models must obey to remain physically consistent with
nature. Specifically, the conditions of causality, analytic functions, reality , and
passivity are explored.

Causality An in-depth analysis of the relationship between the real and imag-
inary parts of complex permittivity was done by Ralph Kronig and Hendrik
Anthony Kramers in the early twentieth century [Balanis, 1989]. They devel-
oped the Kramers-Kronig relations, which describe the relationships between
the real and imaginary parts of any complex function that is analytic in the upper
half-plane:

∞

ω ε r (ω )
(ω ) 2

2
π

ε (ω) =

− ω 2 dω

1 + (6-34a)

0

∞

− ε r (ω )
(ω ) 2

2 ω
π

1

ε (ω) =

− ω 2 dω

(6-34b)

0

The Kramers-Kronig formalism is applied to response functions. In physics, a
response function χ(t − t ) describes how a property P(t) of a physical system
responds to an applied force F(t ) . For example, P(t) could be the angle of a
pendulum and F(t ) the applied force of a motor driving the pendulum motion.
The response χ(t − t ) must be zero for t<t since a system cannot respond
to a force before it is applied. Such a function is said to be causal . From a
commonsense point of view, it makes sense that an effect cannot precede its
cause, which is the fundamental principle of causality that every physical model
must respect, as expressed mathematically by

h(t) = 0 when t< 0 (6-35)

The causality requirement is described mathematically in Chapter 8.

Analytic Functions The response of a dielectric to an applied electric field is
quantified in terms of ε , and therefore a causal dielectric model cannot respond
prior to the electric field being applied, which makes sense for a physical system.
It can be shown that this causality condition implies that the Fourier transform of
the complex dielectric permittivity ε(ω) is analytic [Jackson, 1998], which in turn
implies a specific relationship between the real ( ε ) and imaginary ( ε ) parts .
For a complex function f(x + jy) = u(x, y) + jv(x,y) ,if u and v have con-
tinuous first partial derivatives and satisfy the Cauchy-Riemann equations, f
is said to be analytic [LePage, 1980]. The Cauchy-Riemann equations dictate