Environmental Engineering Reference
In-Depth Information
measurement setup, the considered multi-section TL model comprises three sec-
tions: section 0 is the probe termination, section 1 corresponds to the probe portion
filled with water, and section 2 corresponds to the portion filled with fuel.
To estimate the input scattering parameter
S
k
=
2
11
of this multi-section TL
model, the value of the reflection coefficient at the termination has to be assumed.
The used probe has an open-ended configuration and, therefore,
S
k
=
0
11
(
f
)
=
1isas-
sumed. Starting from this value and using (3.26), firstly for section 1 and then for
section 2, the input scattering parameter can be obtained. It is worth noting that
Γ
k
=
1
=
1, due to the open-ended termination, while
Z
0
,
k
=
50
Ω
for all the remain-
ing sections (due to the specific probe design).
The proposed model was implemented within the commercial microwave circuit
simulator AWR Microwave Office (MWO), which also makes optimization routines
directly available. In particular, the chosen objective function to be minimized is the
sum of the squared magnitude of the differences between the measured and the
MWO-modeled scattering parameters.
This way, since the Cole-Cole formula is included in the modeled scattering pa-
rameter, along with the liquid level, the minimization of the objective function suc-
cessfully provides an accurate estimation of the five dielectric parameters (over the
considered frequency range) and of the levels of liquids.
It must be pointed out that the optimization is rather complex because of the
large number of parameters involved: the fuel level (the water level is constrained
to the difference between the known probe length and the fuel level), four Cole-
Cole parameters for the fuel (the static conductivity is constrained to 0 S m
−
1
), and
five Cole-Cole parameters for the water. The optimum Cole-Cole parameters and
levels for fuel and water are evaluated through the following three-step optimization
procedure:
1. the Cole-Cole parameters for water and fuel are fixed at reasonable
initial guess
values (the dielectric properties of water are well known once the temperature is
measured), thus the fuel level is optimized. This gives a reasonable initial guess
for the fuel level;
2. starting from the previous values, all parameters are optimized through a
random
optimization
procedure that should take the objective function close to the global
minimum and should guarantee that the subsequent optimization step does not
fall into a local minimum;
3. using the previous results as the new starting inputs, a simplex optimization
procedure is applied, which forces the objective function to the final global
minimum.
The robustness and efficiency of the proposed procedure will be shown in the next
subsections.
4.3.2
Measurement Setup and Enhanced Probe Design
As aforementioned, the probe configuration design plays a crucial role in terms of
measurement accuracy. In fact, any impedance mismatch in the TL may affect the
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