NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 232

d peak s , RMS

r e ,RMS

Figure 5-20 Equivalent surface represented by hemispheres with the same RMS volume
as that of the measured surface profile.

2.8

Hemisphere model (5-58)

2.4

2

Hammerstad model (5-48)

1.6

1.2

0.8

0 10 20 30 40 50

Frequency, GHz

Figure 5-21 Hammerstad correction factor (5-48) compared to the hemisphere model
(5-58). RMS roughness: h RMS

= 5 . 8 m ,d peaks , RMS = 9 . 4 m.

µ µ

performed using this method in HSPICE and Nexxim with no apparent nonphys-
ical aberrations observed.
The frequency dependence of the skin effect resistance and total inductance
using the hemisphere correction for surface roughness is implemented with
(5-62a) and (5-62b). † When the skin depth δ is larger than the conductor
thickness (which includes the roughness profile), the dc value of the resistance
and low-frequency inductance where the skin depth is equal to the total

conductor thickness should be used:

† Just as with equation (5.49b), the method shown here for calculating the internal portion of the
inductance ( L internal R ac /ω ) for a rough conductor is an approximation based on the result for a

smooth conductor. The approximation will induce causality errors that tend to be small enough to
ignore, so this method is generally acceptable. Appendix E derives the internal inductance using a
more rigorous approach based on the discussion in Chapter 8.

=