Instrumentation & Measurement Magazine 25-9 - 13

Fig. 3. Illustration of different features for the object localization with a static (left column) and a moving (right column) sensor line.
the sensor line. The measurement profile (9), also called voltage
profile, is characteristic for the electrical field distorted by
a round object of a given conductivity.
The object localization based on the voltage profile can be
performed with a static-i.e., nonmoving-sensor line. Also,
an object localization is possible which takes advantage of
the sensor line's movement. That measurement strategy is explained
below. For both cases, different features of the voltage
profile are defined and used in this work to estimate the object's
distance to the sensor line and its size.
For the static case in this section, it is assumed that the
sensor line has moved through the aqueous medium until a
symmetrical voltage profile has been established on the sensor
line (Fig. 1c). This is the case when the object is located in
the middle between the two emitters at some distance to the
sensor line. To check for symmetry of the voltage profile, the
differences between the outermost, second outermost, etc.
sensor pairs on the sensor line are calculated and added up. In
the symmetrical case, the sum of these differences is less than
or equal to epsilon, where epsilon is a small number or even
zero in the ideal case. A formalization of the symmetry condition
is shown in:
n
2
 1
   
i0
ni i1

ss ,0
(10)
The static case is reached when (10) holds true, which
means that it has to be evaluated constantly. In Fig. 3a the
December 2022
object localization with a static sensor line is shown exemplary
for a round object in the electric dipole field. The measured
symmetric voltage profile in Fig. 3b was calculated based on
(8). As considered by von der Emde et al. [6] the amplitude
and the maximum slope of the voltage profile are characteristic
features for localization. In addition, Chen et al. proposed
the full-width at half-maximum (FWHM) as a characteristic
feature [15]. The FWHM is the width of the voltage profile at a
point where the voltage is half the maximum amplitude. With
these three features shown in Fig. 3b (amplitude, max. slope
and FWHM) and an additional one, the slope to amplitude
ratio [6], objects can be localized without any (scanning) movement
of the sensor line.
All features in Fig. 3 and the following simulations were
based on (8) and implemented in Matlab 2020b (The Mathworks
Inc., Natick, MA, USA).
The above features of the voltage profile were examined
to see whether they are suitable to distinguish between object
size and distance, either individually or in combinations.
For this purpose, two different simulations are shown in Fig. 4
(left column: different object sizes, right column: different object
distances). First, different object sizes were simulated with
the same vertical distance between static sensor line and object
center (Fig. 4a). The measured symmetrical voltage profile is
shown in Fig. 4b for three different object sizes. Fig. 4c shows
the numerical values of the features in an integrated table.
Fig. 4d shows the simulation of objects with the same size but
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Instrumentation & Measurement Magazine 25-9

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