Environmental Engineering Reference
In-Depth Information
5.5.2.1
TLM Method
In the TLM method, the TDR probe surrounded by the material is considered as a
portion of transmission line in which a TEM mode propagates [22, 64, 69]. Consid-
ering the proportionality between the static capacitance and the conductance for
such a line, and taking into account the relation between the static capacitance
per unit length and the characteristic impedance, the following expression can be
derived:
G
s
ε
0
cZ
p
L
σ
0
=
.
(5.25)
To apply (5.25),
Z
p
and
L
have to be accurately determined.
The probe length might be derived from a separate measurement or from the
probe datasheet. Nevertheless, the fringing effect at the open end of the probe would
be neglected. As for the evaluation of
Z
p
, for the adopted three-rod probe, an ana-
lytical formulation relates the impedance in air to the geometrical parameters of the
probe [5]. However, this formulation can only give an approximate value of
Z
p
,es-
sentially because the probe rods are not perfectly parallel, and their distance slightly
varies along the probe. Therefore, these first approximations would limit the ac-
curacy of
0
measurements. In order to overcome this limitation, a specific TLM
of the experimental setup was developed [21]. This way, by using a suitably imple-
mented optimization procedure it is possible to extrapolate the accurate values for
Z
p
and
L
.
More in detail, the experimental measurement system (TDR instrument, cable,
and three-rod probe) was modeled through a three-section transmission line, plus
some lumped elements in order to accurately take into account the parasitic effects
arising from the TDR instrument-to-cable connection and from the probe-head (see
Fig. 5.11). The analysis of measured TDR waveforms exhibited a purely induc-
tive effect at the TDR instrument-to-cable connection and a more complex behavior
(approximately resembled by a C-L cell) at the probe-head. The TLM was imple-
mented within the microwave circuit simulator AWR Microwave Office (MWO),
which allows computing the frequency dependent reflection coefficient at the input
section. Subsequently, an inversion processing, based on inverse fast Fourier trans-
form (IFFT), provides the corresponding TDR response.
σ
Fig. 5.11
TLM of the experimental setup [13]
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