ELECTRICAL PROPERTIES OF DIELECTRICS - Advanced Signal Integrity for High-Speed Digital Design - page 272

Passivity A physical system is denoted as passive when it is unable to generate
energy. For example, an n -port network is said to be passive if
t

v T (τ ) · i (τ ) dτ ≥ 0 (6-38)

−∞

where v T (τ ) is the transpose of a matrix containing the port voltages and i (τ )
a matrix containing the currents. Integral (6-38) represents the cumulative net
energy (power) absorbed by the system up to time t . In a passive system, this
energy must be positive for all t . This requirement is satisfied if (1) the system
absorbs more energy than it generates, and (2) any generation of energy happens
after the absorption. A noncausal system that generates energy before it absorbs
it would be considered to be nonpassive.

Summary

It has been shown that a realistic dielectric model must be:

1. Causal , meaning that the model cannot respond to a stimulus until some
time after it has been applied.
2. Analytic , which says that the function must satisfy the Cauchy-Riemann
equations, meaning that the real and imaginary parts of the permittivity are
related.
3. Real in the time domain, which means that its Fourier transform must
satisfy the complex-conjugate rule (6-37).
4. Passive , which means that the model cannot generate energy.

The dielectric models derived in Section 6.3 satisfy these mathematical con-
straints.

6.5 FIBER-WEAVE EFFECT

As described briefly in Section 3.1, printed circuit boards (PCBs) are most com-
monly constructed from an FR4-type material. Many dielectric materials used
in the electronics industry, including FR4, have been treated historically as a
homogeneous dielectric medium for the propagation of digital signals in PCBs.
As data rates for system buses push into the multi-Gb/s range, the composi-
tion of FR4, in which woven fiberglass bundles are embedded in epoxy resin,
causes this assumption to break down. At high frequencies, local variations in
dielectric constants create spatially dependent values of the time delay τ d and the
characteristic impedance Z 0 . If not controlled properly, the spatially dependent
transmission-line parameters can severely degrade voltage and timing margins,
especially in differential signal-based buses, described in detail in Chapter 7. In
this section we demonstrate how PCB physical structures create local, system-
atic dielectric constant variation, and describe material and design options for
mitigating the effect.