Digital Signal Processing Reference
In-Depth Information
5
4
3
2
1
0
x
−
0.5
−
0.3
−
0.1
0.1
0.3
0.5
w
w
2
w
2
−
Figure 3-21
Charge distribution across a microstrip signal conductor.
The problem arises that (3-88) is singular at
x
=
w/
2, so it cannot be integrated.
Fortunately, we can circumvent this problem by choosing a well-behaved function
to approximate (3-88) that can be integrated:
2
Q
πw
1
=
Q(ax
m
ρ(x)
=
+
c)
(3-89)
x
2
w
2
−
2
where
a
,
m
, and
c
should be chosen so that the polynomial approximates the real-
istic charge distribution. If the total charge is to remain the same as in (3-72), and
the width of the transmission line is normalized to
w
=
1, the charge distribution
must satisfy
w/
2
Q
=
w/
2
ρ(x)dx
=
1
(3-90)
−
Furthermore, the order of the polynomial must be high to approximate the sharp
increase in the charge density near the edges. If the order of (3-89) is chosen to
get a reasonable fit(
m
=
6), the polynomial will satisfy (3-90) when
a
=
100
and
c
=
0
.
77 for a strip width of 1,
100
x
6
ρ(x)
≈
+
0
.
77
(3-91)
where
0
.
5
0
.
5
(
100
x
6
+
0
.
77
)dx
≈
1
.
−
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