NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 224

that power loss is proportional to the surface area of the roughness structure ,
current flow transverse to the corrugated surface could increase the power losses
by 100%, and current flow parallel to the grooves increases losses by about 30%.
In 1975, a Norwegian scientist named Erik Hammerstad used Morgan's work to
fit the data to an arctan function, producing the Hammerstad equation shown in
(5-48), which became the standard equation in industry to account for the effects
of surface roughness [Pytel, 2007]. The model assumes that at high frequencies,
when the skin depth becomes small compared to the tooth height, the current
will begin to follow the contour of the corrugated surface, which will increase
the losses.
To understand why the Hammerstad equation breaks down for some rough-
ness profiles, the surfaces of relatively smooth and rough copper surfaces were
measured using an optical profilometer. The surface shown in Figure 5-16a [Hall
et al., 2007] can be described as corrugated with sparse protrusions on the sur-
face, suggesting that the Hammerstad equation (5-48) might be adequate for
approximating the surface roughness losses for a transmission line manufactured
with this copper foil. Figure 5-14a depicts a measurement of a transmission line
that was constructed with the copper foil depicted in Figure 5-16a compared to
a model created using (5-47) and (5-48). Note that the model and measurement
(a)
(b)
Figure 5-16 Surface profile measurement of (a) relatively smooth and (b) rough copper
foil used to construct PCBs.