NONIDEAL CONDUCTOR MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 235

10 8 c

f √ ε =

3 ×

λ =

10 9 √ 4 =

0 . 03 m

5 ×

2 π
λ =

209 . 3m − 1 k =

2 j

3 (kr e ) 3 1

− (δ/r e )( 1 + j)

α( 1 ) =−

1 + (δ/ 2 r e )( 1 + j)

10 − 6

0 . 935 ×

1 − ( 1 + j)

"

2 j
3 [209 . 3 ( 5

5 × 10 − 6

10 − 6 ) ] 3 =− ×

10 − 6 0 . 935 ×

1 +

10 − 6 ( 1

+ j)

2 · 5 ×

10 − 9 1

− 0 . 187 ( 1 + j)

=− j 0 . 764 ×

1 + 0 . 0935 ( 1 + j)

10 − 10 10 − 10 =− 1 . 78 × − j 5 . 53 ×

For this example, it can be shown that β (1) is negligible, so it is ignored.

Re η( 3 π/ 4 k 2 )α( 1 ) + (µ 0 ωδ/ 4 )(A tile

− A base )

K s =

(µ 0 ωδ/ 4 )A tile

10 − 12 10 − 13 1 . 8 × + 1 . 75 ×

= = 2 . 42

8 . 15 × 10 − 13

In this case, the series resistance of a transmission line manufactured with a cop-
per conductor with this roughness profile would be approximately 2.42 times
as high at 5 GHz as the same transmission line constructed with a smooth
conductor.

5.3.3 Huray Model

In 2006 at the University of South Carolina, Paul G. Huray was researching new
wideband modeling techniques for surface roughness that would provide better
accuracy than both the Hammerstad and hemisphere models. Upon observation
of scanning electron microscope (SEM) photographs of copper foil samples used
to manufacture printed circuit boards (PCBs), he observed that the structures
appeared to be constructed of conducting “snowballs,” as shown in Figure 5-23.
Subsequently, he formed a material and physical basis of a theoretical model that
is composed of a distribution of spherical shapes [Olufemi, 2007; Hurray, 2009].
Printed circuit boards are a “stackup” of layers of copper conductors and inter-
vening layers of an insulating propagating medium such as FR4 joined under heat
and pressure. To assure that the copper sheets do not delaminate from the dielec-
tric layers, manufacturers typically electrodeposit an additional surface layer of
copper on a relatively smooth copper foil that creates irregular features as large as
11 m, to promote good adhesion. The electroplated copper produces a surface

µ