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'unknown' Cole-Cole parameters of the liquid filling the probe and of the 'unknown'
liquid levels. Through the minimization (over the unknown variables) between the
measured scattering parameter (
S
11
,
MEAS
) and the scattering parameter of the TL
model (
S
11
,
MOD
), the proposed procedure provides the accurate evaluation of the
Cole-Cole parameters and of liquid levels. Furthermore, this procedure can be ex-
tended to stratified liquids [22].
As will be detailed later, an additional accuracy enhancement was obtained by
realizing an
ad-hoc
fixture that would allow performing short-open-load (SOL) cal-
ibration measurements, which are essential to reduce the effect of systematic errors
(as seen in subsection 3.6.2).
Experimental measurements were performed considering different levels of lay-
ered samples (fuel/tap water). Since both fuel and tap water exhibit a strong dis-
persive behavior (especially, the imaginary part of their complex permittivity), a
complete spectral picture of the frequency-dependent dielectric properties can be
obtained through the five-parameter Cole-Cole formula.
4.3.1
Transmission Line Modeling of the Measurement Cell
The approach proposed herein is based on the adoption of a TL model of the
measurement cell. As aforementioned, the complex relative permittivity of fuel and
water,
ε
r
, can be appropriately modeled through the Cole-Cole formula. Such a
function, both for fuel and water, is incorporated in the
S
11
(
multiple-reflection
model described by (3.26), through (3.27)-(3.29). The TL model is parameterized
also as a function of the unknown liquid levels.
On such bases, as reported in in subsection 3.6.4, the minimization (over the
unknown variables) of the squared difference between the measured,
S
11
,
MEAS
f
)
(
)
f
,
(
)
and the modeled scattering parameters,
S
11
,
MOD
, leads to the estimation of
the Cole-Cole parameters and of the corresponding liquid levels. For the adopted
f
Fig. 4.6
Schematic
representation of the mea-
surement cell and of the
corresponding main sec-
tions used for modeling the
S
11
(
f
)
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