CROSSTALK - Advanced Signal Integrity for High-Speed Digital Design - page 152

boundary conditions. The inductance is then calculated from the capacitance
using the speed of light as a conversion mechanism. Most 2D quasistatic tools
will give the inductance and capacitance matrices (along with the resistance and
conductance matrices) per unit of conductor length, under the assumption that the
physical geometries and materials are uniform along the length of the transmis-
sion structures. They are typically easy to use and execute in a matter of seconds,
but because of the length uniformity assumption, they cannot handle complex 3D
transmission structures. Because they are static field solvers, they typically do
not calculate frequency-dependent effects such as internal inductance and skin
effect resistance. This is not a significant obstacle for printed circuit boards since
the transmission structures are often uniform, and we can use other methods
for including frequency dependence (refer to Chapters 5 and 6). The quasistatic
assumption also requires that the signals propagate in transverse electromagnetic
(TEM) mode, which means that the electric and magnetic fields are perpendic-
ular and there is no field component in the direction of wave propagation, as
described in Section 2.3.2. The TEM assumption is geometry dependent, but for
typical PCB traces used to design high-speed digital systems (50 , trace width
about 5 mils), it is valid for frequencies well past 20 GHz.
Full-wave 3D solvers, on the other hand, are capable of simulating complex
physical structures and will predict frequency-dependent losses, internal induc-
tance, dispersion, and most other electromagnetic phenomena, including radiation.
These tools essentially solve Maxwell's equations directly for an arbitrary geom-
etry. Complex structures such as edge connectors and packages may require 3D
tools to model their effects accurately at high data rates. The disadvantage of
full-wave solvers is that they require more expertise to use, and simulations typ-
ically take hours or days rather than seconds. Additionally, the output from a
full-wave simulator is often in the form of S -parameters, which typically require
additional processing in order to use them for interconnect simulations for digi-
tal applications. As a result, design engineers typically employ 2D field solvers
whenever possible, making use of 3D full-wave tools only where necessary.
4.2 COUPLED WAVE EQUATIONS
Before proceeding with the analysis of coupled systems, we first derive the wave
equations, to reinforce the notion of wave propagation and to allow us to study
the effects of mutual inductance and capacitance. The wave equation is central
to our analysis of transmission lines, and extension to coupled systems will lend
insight into the effects of crosstalk on the propagation of signals in a coupled
system.
4.2.1 Wave Equation Revisited
We start our derivation of the transmission-line equations by focusing on the
isolated line case shown in Figure 4-7. The circuit must satisfy Kirchhoff's laws.