Environmental Engineering Reference
In-Depth Information
The scattering parameter at the generic section
k
, due to the sections up to the
k
th
is given by the following equation [20, 23, 38]:
S
k
−
1
11
s
e
−
2γ
L
k
)=
Γ
(
f
)+
(
f
)
S
11
(
f
(3.26)
S
k
−
1
11
s
e
−
2γ
L
k
1
+
Γ
(
f
)
(
f
)
where
S
k
−
1
11
th
(
f
)
is the scattering parameter at the
(
k
−
1
)
section due to the first
s
(
is the complex frequency-
dependent propagation parameter [23]; and
L
k
is the length of the
k
th
section. The
losses due to relaxation and conductivity for a TEM mode propagation,
k
−
2
)
sections;
Γ
(
f
)
is the reflection coefficient;
γ
γ
(
f
)
L
are
given by [25]
fL
ε
r
(
f
)
γ
(
f
)
L
=
i2
π
(3.27)
c
However, an assumption must be made about the reflection at the end of the probe
(i.e., section
k
0): for an open-ended probe,
S
11
(
=
f
)=
1; for a short-circuited probe
S
11
(
1; and for a matched probe
S
11
(
0.
The reflection coefficient between the different sections
k
of the transmission line
f
)=
−
f
)=
is
Z
k
−
1
(
f
)
−
Z
k
(
f
)
k
Γ
(
f
)=
(3.28)
s
Z
k
−
1
(
f
)+
Z
k
(
f
)
where
Z
k
(
f
)
is the impedance of the transmission line at section
k
and is given by
Z
0
,
k
ε
∗
(
Z
k
(
f
)=
(3.29)
f
)
Z
0
,
k
being the characteristic impedance of the transmission line at section
k
.
Equation (3.26) is recursive; therefore, to evaluate
S
11
(
f
)
for the whole system (i.e.,
at the
M
th
section), it is necessary to subsequently compute
S
11
(
f
)
for
k
=
1, for
k
M
[38].
The practical usefulness of the TL modeling will be discussed thoroughly in the
following chapters. However, it is useful to anticipate one major application, which
is the extraction of the Cole-Cole parameters of materials. The approach proposed
for the accurate evaluation of the Cole-Cole parameters is the combination of mul-
tiple strategies, and it comprises three major steps:
•
=
2, ... ,
k
=
First, the calibrated
S
11
is evaluated (either through direct VNA measurements
or through the TD/FD combined approach);
(
f
)
•
Secondly, the used measurement cell is accurately modeled as a TL, and the
considered material is parameterized through the Cole-Cole model;
•
Finally, through the minimization of the deviations between the measured
S
11
(
)
and the reflection scattering parameter of the implemented TL model, the Cole-
Cole parameters of the material are obtained.
f
These steps are summarized in the diagram in Fig. 3.19 and will be described in
detail in the following subsections.
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