Environmental Engineering Reference
In-Depth Information
fact,
L
app
can be considered as the distance that would be traveled by the EM signal,
in the same interval of time, if the signal were propagating at a speed
c
.
Typically, in TDR analysis, the apparent distance is evaluated through the indi-
viduation of the beginning and end of the probe, thus it can be associated to the
physical length,
L
, through the following equation:
L
app
=
ε
app
L
ct
t
2
=
(3.5)
where
t
t
is the travel time (round-trip time taken by the signal to travel between the
beginning and end points), and
ε
app
is referred to as apparent dielectric permittivity
of the material in which the probe is inserted [4].
A more rigorous expression for
ε
app
is given by the following equation [37]:
⎛
⎝
1
2
⎞
⎠
ε
0
⎧
⎨
⎫
⎬
)+
σ
0
2
r
ε
(
f
r
)=
ε
(
f
)
π
f
ε
app
(
f
1
+
+
(3.6)
r
⎩
⎭
2
ε
(
f
)
ε
0
=
10
−
12
Fm
−
1
where
8
.
854
×
is the dielectric permittivity of free space,
f
is the
ε
r
(
ε
frequency,
f
)
describes energy storage,
(
f
)
accounts for the dielectric losses,
r
and
0
is the static electrical conductivity.
However, when dealing with low-loss materials, the imaginary part of the com-
plex permittivity can be neglected. Additionally, considering low dispersive materi-
als, the dependence of
σ
ε
r
(
f
)
on frequency can be considered negligible. Under these
circumstances,
ε
(
f
)
can be considered approximately constant:
app
)
=
ε
r
(
)
=
ε
(
f
f
const
.
(3.7)
app
On the basis of the above described theory, TDR-based dielectric measurements
simply rely on the individuation of the points corresponding to the beginning and
to the end of the probe. To individuate these two points in the TDR waveform,
different approaches based on the so-called tangent method can be used [24, 41].
However, as can be intuitively deduced from Fig. 3.4,
L
app
can be easily evaluated
through the derivative of the TDR waveform, which typically exhibits prominent
peaks in correspondence of the probe-beginning and probe-end sections. Indeed,
not only is this 'derivative method' simple and quick, but it also proves particularly
advantageous when significant impedance changes are masked by other impedance
variations (for example, by the transition coaxial cable/rods in Fig. 3.4), as will be
clarified later in this topic.
3.2.2
Typical TDR Instrumentation
As aforementioned, the pivotal components of a TDR instrument are a high-
bandwidth oscilloscope and a high-speed pulse generator: the characteristics of
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