Environmental Engineering Reference
In-Depth Information
For the sake of example, let us consider the series
R-L
case [4]. Also, for simplic-
ity, let us assume that the time is zero when the reflected wave arrives back at the
monitoring point.
At
t
E
i
. This is because the inductor will not accept
a sudden change in current: it initially behaves like an infinite impedance; hence, at
t
=
0, the reflected voltage is
+
=
will be unit.
Successively, the current in
L
builds up exponentially and its impedance drops
toward zero. Therefore, at
t
0,
ρ
=
∞
,
E
r
(
t
)
is determined only by the value of
R
:
R
−
Z
0
ρ
=
(3.2)
R
+
Z
0
The exponential transition of
E
r
(
) determined by the effec-
tive resistance seen by the inductor. Since the output impedance of the transmission
line is
Z
0
, the inductor sees
Z
0
in series with
R
:
t
)
has a time constant (
τ
L
τ
=
(3.3)
R
+
Z
0
A similar analysis can be carried out for the other three cases reported in Fig. 3.3.
It must be pointed out that an in-depth analysis of the impedance characteristics
of the SUT should rely also on a FD approach (in particular, through the evaluation
of the reflection scattering parameter). However, as seen from Fig. 3.3, the TDR
waveforms can provide a concrete idea on the impedance-behavior of the SUT.
This approach is particularly useful for TDR-based individuation of faults along
cables. In fact, faults do show specific impedance behavior, whose meaning can be
inferred directly from the TDR waveforms.
3.2.1
Typical TDR Measurements
When the TDR signal propagates along a transmission line, impedance mismatches
cause the reflection of a portion of the signal. The TDR signal's propagation veloc-
ity,
v
, is related to the relative dielectric permittivity (
ε
r
) of the medium, which is
assumed to be lossless or at least with negligible conductivity, and to the relative
magnetic permeability (
μ
r
), which is equal to 1 for most materials [4]:
c
√
ε
r
μ
r
v
=
(3.4)
where
c
is the speed of light in free space (
c
=
10
8
ms
−
1
).
For the sake of example, let us consider a typical three-rod probe, whose elec-
trodes have length
L
. A typical TDR waveform for such a probe, immersed in a
generic homogeneous dielectric, is represented in Fig. 3.4. The analysis of TDR
waveform directly leads to the evaluation of
L
app
(also called electric distance),
which can be directly associated to the dielectric characteristic of the medium. In
3
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