Digital Signal Processing Reference
In-Depth Information
5
w
100,
m
=
6,
c
=
0.77
=
1,
a
=
4
3
2
1
0
−
0.5
−
0.3
−
0.1
0.1
0.3
0.5
w
Equation (3-88)
Equation (3-91)
Figure 3-22
Comparison of the charge distribution calculated analytically and the non-
singular polynomial approximation.
TABLE 3-1. Effect of Charge Distribution on the Impedance and Effective
Relative Permittivity
a
Z
0
()/ε
eff
1 when
Two-Dimensional
Numerical Field
Solver Results
|
x
|
<w/
2
ρ(x)
=
|
x
|
>w/
2
0 when
100
x
6
w/h
ρ(x)
=
+
0
.
77
4
37.6/3.26
31.6/3.37
32.5/3.38
2
59.7/3.02
52.6/3.08
51.4/3.13
1
84.8/2.86
77.0/2.90
74.6/2.96
0.667
99.9/2.81
92.1/2.83
89.3/2.89
0.5
110.8/2.78
99.1/2.76
99.9/2.84
a
d
=
100
h
, widths normalized to 1,
ε
r
=
4
.
0,
n
=
5000.
Figure 3-22 shows a comparison between the charge distribution approximated
with (3-91), which is not singular, and (3-88). Plugging (3-91) into (3-69) and
solving for
A
n
will provide a formula for the microstrip transmission line that
accounts for a realistic charge distribution on the signal conductor. Unfortunately,
the integration gets messy and the final form of
A
n
is ungainly and therefore is not
shown here. Mathematica was programmed to perform the integration and solve
(3-69) for
A
n
when the charge distribution was approximated with (3-91). Com-
parisons between the calculated impedance and the effective dielectric constant
assuming a uniform charge distribution using (3-64) and a realistic distribution
calculated with (3-88) and approximated with (3-91) is shown in Table 3-1. Note
that when a realistic charge distribution is used, the quasistatic approximations
are significantly more accurate when compared to a commercial two-dimensional
numerical field solver.
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