TOPICS IN HIGH-SPEED CHANNEL MODELING - Advanced Signal Integrity for High-Speed Digital Design - page 417

Quasistatic techniques, which are used by most commercial transmission-line
calculators, provide reasonable values of the external inductance. Furthermore,
many tools also provide realistic values of the skin effect and dc resistance as
well as the internal inductance for ideal smooth conductors. Problems arise when
creating models of realistic conductors, which are usually purposely roughened
during the PCB manufacturing process to ensure adhesion of the metal traces
to the dielectric substrate. Fortunately, the shortcomings of frequency-invariant
quasistatic techniques can easily be overcome by using a tabular transmission-line
model.
Similar to the tabular model for the dielectric outlined in Section 10.1.2, the
easiest methodology is to use a conventional 2D transmission-line calculator that
assumes perfectly smooth conductors, and modify the output to establish the
proper frequency-dependent relationships between values R ( ω ) and L ( ω ). The
output of the transmission-line calculator will provide the values of the reference
matrices R ( ω ref ) and L ( ω ref ). The reference values for each entry in the matrix
are then scaled according to

K SR (f )R s f
f ref

when δ<t

R(f ) = (10-5)

R dc when δ ≥ t

R(f )
2 πf

L external + when δ<t

L(f ) = (10-6)

R(f δ = t )
2 πf δ = t

+

when δ ≥ t L external

where the frequency-dependent surface roughness correction factor K SR (f ) is
calculated with one of the modeling methods presented in Chapter 5 [i.e., Ham-
merstad equation (5-48), the hemispherical model in equation (5-58), or the Huray

model in equation (5-66)], R s √ f is the skin effect resistance assuming perfectly
smooth conductors, t is the conductor thickness, δ is the skin depth, f δ = t is the
frequency where the skin depth equals the thickness of the conductor, and f ref is
the reference frequency where the seed values are calculated [Hall et al., 2007].
If the 2D transmission-line calculation does not include the internal inductance,
the value of the reference frequency ( f ref ) is immaterial because all the current
is assumed to reside on the surface of the conductor and the internal inductance
is zero. However, if the calculation includes the internal inductance, as described
in Section 5.2.3, the reference frequency should be high enough that most of the
current is confined to a small layer near the surface, so that the internal inductance
is minimized. This simplifies the problem so that the reference inductance can
be considered to be purely external and the internal portion can simply be added.
Note that even at relatively low frequencies, the skin depth in copper is small
compared to the thickness of the trace. For example, at 1 GHz, the skin depth
is 2 m, meaning that for a typical PCB trace thickness of about 1 to 2 mils

µ