Digital Signal Processing Reference
In-Depth Information
The Poynting vector, which was described in Section 2.6, gives the power of an
electromagnetic wave in units of watts per square meter:
×
H
0
)
=
|
S
|=
2
Re
( E
0
1
1
2
2
η
|
H
0
|
(5-52)
Subsequently, the total power absorbed or scattered is calculated by multiplying
(5-50) and (5-52) and dividing by 2 so the results are for that of a hemisphere:
1
2
η
|
H
0
σ
tot
Re
1
k
2
(α(
1
)
+
β(
1
))
1
2
2
3
π
2
P
hemisphere
=
|
=−
4
η
|
H
0
|
(5-53)
where
η
=
√
µ
0
/ε
0
ε
and
H
0
is the magnitude of the applied magnetic field. Note
that reasonable accuracy (at least up to 30 GHz) can be obtained when only the
first term (
m
=
1) in (5-50) is considered when calculating (5-53).
Equation (5-53) calculates the power loss of a hemisphere. Now, the
losses of the flat plane surrounding the protrusion must be accounted for.
The time-averaged power absorbed by a flat conducting plane per unit area is
calculated from equation (5-43):
dP
plane
da
µ
0
ωδ
4
2
=
|
H
0
|
(5-54)
To approximate the losses of a single hemispherical boss sitting on a flat plane
of finite conductivity, the power loss of the hemisphere is added to the loss of
the plane less the base area of the hemisphere:
−
Re
1
2
k
2
(α(
1
)
+
β(
1
))
+
2
3
π
µ
0
ωδ
4
2
−
A
base
)
(5-55)
where
A
tile
is the tile area of the plane surrounding the protrusion (see Figure
5-17) and
A
base
is the base area of the hemisphere. Note that (5-55) is an approx-
imation because it assumes that the magnetic field (
H
0
) on the tile is not affected
by the presence of the hemisphere and the loss of the surrounding plane is simply
a function of the area.
To gain an intuitive understanding of how a propagating electromagnetic wave
behaves in the presence of a protrusion, it is useful to observe the fields and solve
for the surface currents on a PEC (perfect electrical conducting) sphere, which is
a good approximation of how the current will flow at very high frequencies when
the skin depth is small compared to the sphere. In Section 3.2.1 the boundary
conditions for a PEC were described; the electric field must emanate from and
terminate normal to a perfectly conducting surface, and the magnetic field must be
tangential to the conductor surface. First consider Figure 5-18a, which depicts the
front cross-sectional view of a hemispherical protrusion sitting on a conducting
plane where the current flow is out of the page. Note that the electric fields are
drawn perpendicular to the conductor surface and the magnetic fields are tangent
P
tot
=
4
η
|
H
0
|
|
H
0
|
(A
tile
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