NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 200

preserve causality , which is discussed in Chapter 8 and Appendix E. In prior days
it was possible to utilize simpler conductor models for digital designs because
bandwidth demands were much lower. However, as digital data rates increase, the
assumptions and approximations of traditional conductor models begin to break
down. Consequently, the signal integrity engineer is now required to learn new
techniques that compensate for variables that were insignificant in past designs.
So far in this topic we have covered the topics of electromagnetic theory
for signal integrity engineers, transmission-line fundamentals, and crosstalk. Up
to this point, the conductors were assumed to be infinitely conductive and the
dielectric was assumed to be a perfect insulator. In this chapter we develop
modeling techniques to predict properly the electrical behavior of conductors
used to design transmission lines on printed circuit boards, multichip modules,
and chip packages. First, classic electromagnetic theory will be used to derive the
frequency dependence of resistance and inductance for smooth conductors with
finite conductivity. Next, three different methodologies for modeling the effects
of rough copper on the electrical parameters of the transmission lines will be
introduced. Detailed analysis of how currents flow on a rough surface will give
physical insights into the mechanisms of surface roughness losses. Finally, a new
circuit model for a transmission line that accounts for realistic conductors will
be introduced along with a modified version of the telegrapher's equations that
account for realistic conductor losses.

5.1 SIGNALS PROPAGATING IN UNBOUNDED CONDUCTIVE MEDIA

The topic of uniform plane waves propagating in a lossless media was discussed
in Section 2.3, where the influences of the material properties µ and ε were
observed. In Chapter 3 we described how the waves propagated when confined
to the physical dimensions of a transmission line, yet the problem was still
idealized because it was assumed that the dielectric was a perfect insulator and
the conductor was infinitely conductive. To derive the equations that govern
the propagation of waves on realistic transmission lines, it is first necessary to
comprehend how an electromagnetic wave propagates in unbounded, conductive
media.

5.1.1 Propagation Constant for Conductive Media

To derive the equations that govern electromagnetic waves propagating in conduc-
tive or lossy media, we begin with the loss-free forms of Maxwell's differential
equations presented in Chapter 2 and modify them appropriately to obtain a
wave equation that accounts for loss. To begin, the time-harmonic forms of
Maxwell's equations are repeated here:

∇× E + jω B =

0 (2-33)

∇× H = J + jω D

(2-34)