NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 222

0

h RMS = 1.2 µ m

− 5 h RMS = 5.8 µ m

− 10

− 15

− 20

25

−

0 5 10 15 20 25 30

Frequency, GHz

Figure 5-13 Measured results of identical 7-in. transmission lines with relatively smooth
(h RMS = 1 . 2 m ) and rough (h RMS = 5 . 8 m ) copper showing how surface roughness µ µ

affects losses.

where R s √ f is the classic skin resistance for a smooth conductor as calculated
in (5-17) and (5-18) and K H is the Hammerstad coefficient:

arctan 1 . 4 h RMS
δ

(5-48)

2

2
π

K H = 1 +

where h RMS is the root-mean-square value of the surface roughness height and
δ is the skin depth [Hammerstad and Jensen, 1980; Brist et al., 2005]. The
Hammerstad coefficient is used to model the extra losses caused by the copper
surfaces on a transmission line that are often purposely roughened to promote
adhesion to the dielectric.
The frequency dependence of the skin effect resistance and total inductance
using the Hammerstad correction for surface roughness is implemented with †

K H R s √ f

when δ<t

R H (f ) = (5-49a)

R dc when δ ≥ t

R H (f )
2 πf

+

L external when δ<t

L H (f ) = (5-49b)

R H (f δ = t )
2 πf δ = t

L external + when δ ≥ t

† Note that 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. For the interested reader, Appendix E derives the internal inductance using a
more rigorous approach based on the discussion in Chapter 8.