Digital Signal Processing Reference
In-Depth Information
Finally, the charge can be calculated with (3-3):
n
·
ε E
=
ρ
r
(nπ/θ)
−
1
cos
nπ
θ
φ
∞
nπεα
n
θ
ρ(r,φ)
=
εE
φ
=
−
n
1
odd
=
φ
r
−
(nπ/θ
+
1
)
cos
nπ
θ
nπεα
n
θ
+
Since we are interested in the behavior of the charge in the immediate vicinity of
the edge,
r
is small and therefore only the first term, the summation, is important
(
n
=
1). Also, since for a flat plane the dependence on
r
should vanish and
the last term shows dependence on
r
for
θ
=
π
and
n
=
1, it is necessary that
α
n
=
0.
θ
φ
This yields the surface charge density as a function of the angle between two
conducting planes in the near vicinity of the corner:
r
0
cos
π
θ
φ
r
−
2
cos
π
πεα
1
θ
πεα
1
θ
ρ(r,φ)
=−
+
r
(π/θ)
−
1
cos
π
θ
φ
πεα
1
θ
ρ(r,φ)
=−
(3-85)
For a sharp corner,
θ
=
3
π/
2, as shown in Figure 3-20, 3.85 reduces to
2
εα
1
3
r
−
1
/
3
ρ(r,φ
=
3
π/
2
)
=
(3-86)
Since we are concerned with the charge density on the metal planes,
φ
will be
either
θ
or 0, meaning that the cosine term will equal 1 or -1.
Equation (3-86) means that for a transmission line built with a square con-
ductor, the charge density increases dramatically near the corner, as shown in
Figure 3-20. In fact, for a perfectly sharp edge, the charge density becomes
singular as
r
approaches zero. For the edge of a thin strip,
θ
=
2
π
and (3-85)
reduces to
εα
1
2
r
−
1
/
2
ρ(r,φ
=
2
π)
=
(3-87)
Equation (3-87) means that for an infinitely thin metal strip, as was assumed in
the derivation for the microstrip line in Section 3.4.3, the charge density also
approaches infinity near the edge. Since infinitely thin sheets or infinitely sharp
corners do not really exist in nature, the trends of (3-86) and (3-87) are what must
be understood.
The charge density increases dramatically near sharp corners on
metal conductors
. The coefficients
α
n
are dependent on the initial conditions
remote from the corner, which are not solved here.
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