Digital Signal Processing Reference
In-Depth Information
r
r
=
0
V
Φ =
3
2
270
q =
° =
Figure 3-20
Charge distribution near a sharp corner.
3.4.5 Charge Distribution and Transmission-Line Parameters
Now that the behavior of the charge in the near vicinity of the edge is understood,
it is easy to imagine what the total charge distribution should look like across
the signal conductor of a transmission line. The charge density should be almost
uniform near the center of the strip but will increase dramatically near the edges,
as predicted by (3-85). A useful formula to predict the current distribution for a
microstrip, which works well for a variety of dimensions, was derived by Collins
[1992] using conformal mapping techniques:
2
Q
πw
1
ρ(x)
=
(3-88)
x
2
w
2
−
2
where
Q
is the total charge,
x
the distance from the conductor center, and
w
the width of the signal conductor. Figure 3-21 shows an example of a realistic
current distribution in the signal conductor of a microstrip transmission line, as
calculated with (3-88) and normalized so that the center of the conductor has a
charge density of 1.
To understand how the charge distribution will influence the transmission-line
parameters such as delay and impedance, the equations for the microstrip trans-
mission line derived in Section 3.4.3 can be improved by applying a more realistic
charge distribution to the signal conductor and solving (3-69) for
A
n
:
w/
2
d/
2
w/
2
ρ(x)
cos
nπ
x
dx
=
d/
2
A
n
k
cos
2
nπ
x
dx
(3-69)
d
d
−
−
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