Environmental Engineering Reference
In-Depth Information
Fig. 3.4
Typical TDR waveform (black curve) and corresponding first derivative (red curve)
for a 30 cm-long three-rod probe in air, obtained through a Tektronix TDR80E04. The am-
plitude of the incident step-like signal is 0.25 V. The first derivative of the TDR waveform
emphasizes the impedance variations. The section
k
=
1 indicates where the sensing portion
of the considered probe begins
these two components have the major influence on the performance of the instru-
ment. In particular, the frequency content of the signal,
f
B
, is approximately given
by the following equation:
0
.
35
t
r
f
B
=
(3.8)
where
t
r
is the rise-time of the generated step signal
1
.
A higher bandwidth of the signal makes the TDR measurements significant
in a wider frequency range; this concept will be detailed in Sect. 3.4, when the
TDR/FDR combined approach is described. The lower the rise-time of the gener-
ated signal, the more sophisticated (and costly) is the TDR instrument.
It is worth mentioning that the minimum spatial resolution,
Δ
L
min
, is related to
the rise-time of the TDR step-like signal,
t
r
[10]:
t
r
c
2
√
ε
app
.
Δ
L
min
=
(3.9)
Equation (3.8) is a simplification of
f
B
=
ln
(
0
.
9
/
0
.
1
)
2
1
t
r
, which is an expression typically used
in electrical engineering to describe the frequency characteristics of low-pass filters. This
expression remains accurate when the energy contained in the voltage pulse is equally
distributed across the frequency bandwidth and when there is no significant dispersion
[31].
π
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