Digital Signal Processing Reference
In-Depth Information
The power loss from a plane can also be put in more intuitive form. Equation
(2-7), which is simply Ohm's law, can be used to calculate the current density
from (5-40):
=
σ E
c
=
σ
µω
+
j)
n
×
H
||
e
−
z/δ
e
−
jz/δ
J
2
σ
(
1
Again, since
√
µω/
2
σ
=
1
/σ δ
, the equation can be rewritten as
+
j)
n
×
H
||
e
−
z/δ
e
−
jz/δ
1
δ
(
1
J
=
σ E
c
=
(5-44)
Equation (5-44) simply says that most of the current density will be confined to
a very small thickness, as described in Section 5.1.2. Note that the cross product
in (5-44) implies that the current will flow perpendicular to the magnetic field.
To derive a more intuitive form of the power dissipated by a flat plane, it
is first necessary to define an effective surface current. If the current density
(5-44) is integrated to get the total current, an equivalent surface current can be
calculated for use in the classic time-averaged power equation
P
1
2
RI
2
:
=
∞
J
eff
J dz
=
amperes
(5-45)
0
Equation (5-45) simply calculates the total current that is decaying exponentially
into the conductor surface. To calculate the power per unit area, we make the
approximation that all the current exists on the surface and use the real part of
the surface impedance in (5-28):
J
eff
1
2
RI
2
1
2
σδ
2
P
plane
=
=
W
/
square
(5-46)
5.3 SURFACE ROUGHNESS
To account properly for the frequency variation of
L
internal
and
R
ac
, the nonideal
effects of the copper surface must be considered. The problem is that most (if not
all) commercial 2D field solvers calculate the resistance and inductance assum-
ing smooth conductors. Real copper surfaces, however, are purposely roughened
to promote adhesion to the dielectric when manufacturing printed circuit boards.
The resulting copper surfaces have a “tooth structure” as depicted in Figure 5-12.
When the tooth height is comparable to the skin depth, the smooth copper assump-
tions break down. The root-mean-square (RMS) tooth height of common copper
foils used to manufacture printed circuit boards range from approximately 0.3
to 5.8
m, with peak heights exceeding 11
m [Brist et al., 2005]. The skin
µ
µ
depth in copper at 1 GHz is about 2
m, indicating that for many copper foils,
most of the current will be flowing in the tooth structure for multigigabit designs
µ
Search WWH ::
Custom Search