Environmental Engineering Reference
In-Depth Information
i
S
11
sc1
,
B
−
S
11
sc2
,
B
+
S
11
sc0
,
B
−
S
11
sc1
,
B
cot
S
22E
=
−
(
β
l
1
)
+
A
S
11
sc0
,
B
−
S
11
sc2
,
B
cot
(
β
)
l
2
−
(5.18)
A
where
A
is given by
i
S
11
sc1
,
B
−
S
11
sc2
,
B
−
S
11
sc0
,
B
−
S
11
sc1
,
B
cot
A
=
−
(
β
l
1
)+
+
S
11
sc0
,
B
−
S
11
sc2
,
B
cot
(
β
l
2
)
S
11
sc0
,
B
,
S
11
sc1
,
B
,and
S
11
sc2
,
B
are the reflection coefficients measured at section B for
the short circuits located at
l
=0,
l
l
2
, respectively.
Once the error network is characterized by means of the TSC procedure, the
measurements performed at section B (
S
11
,
B
) can be corrected so as to obtain the
corresponding reflection coefficient at the beginning of the rods (
S
11
,
A
):
=
l
1
,and
l
=
S
11
,
B
−
S
11
,
E
S
11
,
A
=
S
11
,
E
S
22
,
E
.
(5.19)
S
12
,
E
−
S
11
,
B
S
22
,
E
+
A crucial point of the TSC procedure is the optimal choice of the distances
l
1
and
l
2
.
The impedance along a line presents a periodic behavior, with a period equal to
half wavelength. This means that at the frequency for which
l
2
is equal to half wave-
length, the distal short appears again as a short at section A, causing a singularity in
the error network parameters (as a matter of fact, only two of the three loads remain
independent). On the other hand, choosing very short distances for the three shorts
would compromise measurement accuracy at low frequencies, because the phase
differences between the shorts tend to vanish.
Since the TDR unit used for these measurements has an operating bandwidth of
approximately 1.5 GHz, a distance
l
2
equal to 10 cm was chosen, yielding a singu-
larity exactly at the upper bound of the bandwidth. For symmetry,
l
1
was set equal
to 5 cm.
5.4.2.1
Evaluation of Moisture Content
Soil moisture content was evaluated through the minimization of the squared mag-
nitude of the difference between the
S
11
,
MOD
(
)
f
modeled through (5.12) and (5.13),
(
)
and the
S
11
,
MEAS
that was derived from TDR measurements and referred to the
corresponding calibration section (i.e., section A in Fig. 5.5). The measurement
cell is modeled as the portion of a transmission line, and the soil is characterized
by means of an appropriate dielectric model. Subsequently, the reflection scatter-
ing parameter of the implemented model,
S
11
,
MOD
(
f
f
)
, is evaluated as a function
of the unknown value
θ
. Finally, through the aforementioned minimization routine
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