Environmental Engineering Reference
In-Depth Information
probe (which is assumed lossless
2
);
b
is the inner diameter of the outer conductor;
and
a
is the outer diameter of the inner conductor.
On the other hand, for soil monitoring and for granular materials monitoring
in general, the multi-rod configuration is a widely used solution. In fact, this con-
figuration allows an easier insertion of the probe into the granular material. For a
two-wire transmission line, the characteristic impedance can be derived from (2.13)
and it may be written as
120
ln
2
D
d
Z
(
f
)=
ε
r
(
f
)
where
D
is the distance between the center of the conductors, and
d
is the diameter
of each conductor.
However, three-rod probes have become a widespread solution since their electric
behavior resembles coaxial cells [42]. Unfortunately, for configurations different
from a coaxial line, models are not always available [25], and the evaluation of the
probe impedance is not straightforward, especially for unbalanced probes. However,
for multi-rod probes, an analytical formulation relating the impedance of the probe
in air,
Z
p
, to the geometric characteristics can be found in [12]:
μ
ln
H
1
H
2
1
0
ε
0
Z
p
=
+
−
(3.15)
2
π
(
n
−
1
)
where
H
is given by
a
2
(
n
−
1
)
i
s
2
a
0
)
n
−
1
(
−
−
H
=
(3.16)
a
n
−
1
i
{
(
s
+
a
0
)
n
−
1
−
(
s
−
a
0
)
n
−
1
}
In the previous equations,
n
is the number of rods of the probe,
a
i
is the radius of
the probe center conductor,
a
0
is the radius of the outer conductors,
s
is the distance
between the middle of the center conductor and the middle of the outer conductor,
ε
0
=
10
−
12
Fm
−
1
π
×
10
−
7
Hm
−
1
is the magnetic permeability of free space. The cross-section of a four-
wire probe is shown in Fig. 3.10.
A simplification of (3.16) for three-rod probes is reported in [27]:
8
.
854
×
is the dielectric permittivity of free space,and
μ
0
=
4
μ
0
ε
0
ln
1
4
1
4
−
(
g
/
s
)
Z
p
,
three
−
rod
=
(3.17)
3
π
2
(
g
/
s
)
where
s
is the center-to-center rod spacing, and
g
is the rod radius.
Recently, the adoption of a single-rod probe for monitoring soil moisture was
investigated. This probe exploits transverse magnetic (TM) propagation, rather than
transverse electromagnetic (TEM) propagation. This kind of probe is simple to
2
It is worth pointing out that when dielectric losses of the material cannot be neglected,
equation (3.14) can be extended by considering the complex dielectric permittivity.
Search WWH ::
Custom Search