Digital Signal Processing Reference
In-Depth Information
yielding
A
n
e
(nπ/d)h
sinh
nπ
d
h
=
B
n
which allows the equations for the potential to be written in terms of
A
n
alone:
A
n
cos
nπ
d
x
sinh
nπ
d
y
∞
when 0
≤
y<h
n
=
1
odd
(3-62a)
(x,y)
=
A
n
sinh
nπ
d
h
cos
nπ
d
x
e
−
(nπ/d)(y
−
h)
∞
when
h
≤
y<
∞
n
=
1
odd
(3-62b)
To get the electric field between the signal conductor and the ground plane, we
apply equation (2-65),
E
y
=−∇
φ
=−
∂/∂y
. Since
d(
sinh
ax)/dx
=
a
cosh
ax
and
d(e
ax
)/dx
=
ae
ax
, the electric fields become
∂y
A
n
cos
nπ
x
sinh
nπ
d
y
cos
nπ
d
x
cosh
nπ
d
y
∂
nπA
n
d
E
yn
=−
=−
d
for region 1 and
∂y
A
n
sinh
nπ
h
cos
nπ
d
x
e
−
(nπ/d)(y
−
h)
∂
E
yn
=−
d
sinh
nπ
d
h
cos
nπ
d
x
e
−
(nπ/d)(y
−
h)
nπA
n
d
=
for region 2, yielding
cos
nπ
d
x
cosh
nπ
d
y
∞
nπA
n
d
−
when 0
≤
y<h
n
=
1
odd
(3-63a)
sinh
nπ
d
h
∞
nπA
n
d
E
y
(x, y)
=
n
=
1
odd
cos
nπ
d
x
e
−
(nπ/d)(y
−
h)
×
when
h
≤
y<
∞
(3-63b)
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