Digital Signal Processing Reference
In-Depth Information
w
eff
6
h
and an approximate formula can be derived for the ground-plane resis-
tance in a microstrip transmission line in units of ohms:
=
πµf
σ
l
σw
eff
δ
=
l
6
h
R
ac, ground
≈
(5-16)
The total resistance is the sum of (5-12) and (5-16) in ohms:
πµf
σ
πµf
σ
πµf
σ
1
w
+
l
w
l
6
h
1
6
h
R
ac, micro
=
+
=
(5-17)
Equation (5-17) should be considered a good “back of the envelope” estimation
of the ac resistance for a microstrip transmission line [Hall et al., 2000] A more
exact formula for the ac resistance of a microstrip was derived using conformal
mapping techniques by Collins [1992] and is shown in equation set (5-18). This
formula is significantly more cumbersome than (5-17), but should yield the most
accurate results.
LR
1
R
s
w
π
2
ln
4
πw
1
R
trace
=
π
+
t
where LR is given by
when
w
1
2
1
h
≤
LR
=
0
.
0062
w
h
0
.
132
w
when
1
2
<
w
2
0
.
94
+
h
−
h
≤
10
R
s
w
w/h
1
10
≤
w
h
≤
R
ground
=
when
10
(5-18)
(w/ h)
+
5
.
8
+
0
.
03
(h/w)
where
ωµ
2
σ
R
s
=
R
ac, micro
=
R
trace
+
R
ground
For practical micristrip lines, formulas based on smooth conductors should simply
be used as an approximation because realistic conductor surfaces are generally
rough, which will increase the conductor losses significantly at frequencies where
the skin depth begins to approach the magnitude of the roughness profile. The
extra losses caused by surface roughness are calculated in Section 5.3.
Example 5-1
Calculate the approximate frequency where ac resistance must be
used to calculate the ohmic losses of a microstrip transmission line constructed
with a copper conductor with a conductivity of
σ
10
7
(
·
m
)
−
1
and the
=
5
.
8
×
following cross-sectional dimensions:
w
=
5 mils,
h
=
3 mils,
t
=
2
.
1 mils.
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