Digital Signal Processing Reference
In-Depth Information
Step 5:
Calculate the ac resistance at 2 GHz using (5-17). Note that (5-18)
could be used as well.
πµ
0
f
σ
1
w
+
1
6
h
R
ac, micro
=
0
.
0117
1
=
(
3 mils
)(
25
.
4
×
10
−
6
m/mil
)
1
+
6
(
2 mils
)(
25
.
4
×
10
−
6
m/mil
)
=
191
.
3
/
m
=
4
.
86
/
in.
Step 6:
Calculate the internal inductance using (5-30).
R
ac
ω
=
4
.
86
/
in.
2
π(
2
L
internal
=
10
9
Hz
)
=
0
.
387 nH/in.
×
Step 7:
Calculate the total inductance using (5-20).
L
total
=
L
internal
+
L
external
=
8
.
03
+
0
.
387
=
8
.
42 nH
/
in.
5.2.4 Power Loss in a Smooth Conductor
In high-speed digital design, surface treatment of the copper foils used to con-
struct printed circuit boards (PCBs) significantly affects the power losses experi-
enced by a signal propagating on a transmission line. In this section, the power
losses of an electromagnetic wave impinging on a flat, smooth plane are exam-
ined. In later sections we explore the consequences of rough conductor surfaces.
First, we assume that the fields in the vicinity of a good but not perfect con-
ductor will behave approximately the same as for a perfect conductor. In Section
3.2.1 it was shown that the electric fields terminate normal to a perfect conduct-
ing surface and the magnetic fields are tangential to the surface. Furthermore, in
Section 5.1.2 it was shown that fields inside a conductor will attenuate exponen-
tially and are measured in terms of the skin depth
δ
. At high frequencies, the
boundary condition shown in (3-3) is true for a good conductor, except for a thin
transitional layer.
To derive an equation to predict the loss for a smooth plane, we first assume
that just outside the conductor there exists only a normal component of the
electric field (
E
⊥
) and a tangential component of the magnetic field (
H
), which
are the identical boundary conditions used for perfect conductors. Following
the approach outlined by Jackson [1999], Maxwell's equations are then used to
calculate the fields within the transition layer.
If a tangential
H
exists just outside the surface, the same
H
must exist just
inside the conductor surface. With the neglect of the displacement current, (2-33)
and (2-34) become
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