Digital Signal Processing Reference
In-Depth Information
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Frequency, GHz
Figure 5-2
Skin depth
δ
in copper as a function of frequency.
provide the theoretical basis needed to derive frequency-dependent physically
consistent models of real-world conductors, which are often purposely rough-
ened to promote adhesion to dielectric layers during the manufacturing process.
The resistive loss induced by general transmission-line conductors can be broken
down into two components: low frequency, or dc, and high frequency, or ac.
First, the dc losses will be derived and the formulas will be modified to include
the frequency-dependent effects of ac resistance at high frequencies.
5.2.1 DC Losses in Conductors
Dc losses are of particular concern in small-geometry conductors, very long lines,
and
multiload
(also known as
multidrop
)
buses
. Long copper telecommunication
lines, for example, must have repeaters every few miles to receive and retransmit
the data because of signal degradation. Additionally, designs of multiprocessor
computer systems with long buses experience resistive drops that can encroach
on the logic threshold levels and reduce the noise margins.
The dc loss of a transmission line depends primarily on two factors: the con-
ductivity of the metal and the cross-sectional area of the conductor where the
current is flowing. Figure 5-3 shows the current distribution in a transmission
line at dc. Traditionally, dc resistance is defined to be the value at 0 Hz. How-
ever, for the purposes of this chapter, dc will be assumed to be valid for all
frequencies where the skin depth is larger than the conductor thickness
t
, which
At dc, current flows
through entire area
of the cross section where
Area
w
t
=
A
=
wt
Figure 5-3
Current distribution in a microstrip at dc.
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