Digital Signal Processing Reference
In-Depth Information
n
A
Dielectric
(Region 1)
h
Conductor
(Region 2)
n
Figure 3-3
Surface used to calculate the boundary conditions at
the dielectric-
conductor interface.
First, we must choose a surface to integrate and compare the normal components
of
ε E
on both sides of the conductor-dielectric boundary. A convenient surface
is a cylinder, as depicted in Figure 3-3. Since we are observing the behavior of
fields at the surface, the height (
h
) of the cylinder can be made infinitely small,
reducing (2-59) to
ε E
·
ε
1
E
1
ε
2
E
2
=
(n
·
ε
1
E
1
)A
+
(
−
n
·
ε
2
E
2
)A
=
ρA
d
s
=
·
d
s
1
+
·
d
s
2
S
S
S
(3-2)
where
ε
1
is the dielectric permittivity of the dielectric in region 1 and
ε
2
describes
the dielectric permittivity of region 2, which in this example is a perfect con-
ductor. To interpret this equation, recall the discussion in Section 2.7.1, where it
was shown that the electric field (
E
2
in this case) must be zero inside a perfect
conductor. Setting
E
2
0 allows us to simplify (3-2) for the special case of a
boundary between a dielectric and a perfect conductor:
=
n
·
ε E
=
ρ
C/m
2
(3-3)
Equation (3-3) means that
the electric field must emanate normal from and ter-
minate normal to the conductor surface
. Since equations (2-29) and (2-30) say
that the magnetic field must be orthogonal to the electric field, we can conclude
that
the magnetic field must be tangential to the conductor surface
. These two
rules make it easy to visualize electric and magnetic fields on transmission-line
structures with perfect electrical conductors. Simply draw the electric field lines
so that they are always perpendicular to the conductor surface, emanating from
the high and terminating in the low potential conductor, and then draw the mag-
netic field lines so that they are always perpendicular to the electric field lines
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