Instrumentation & Measurement Magazine 25-9 - 12

It illustrates an electric dipole with a sensor line (acts as one-dimensional
" electroreceptive skin " ) in between the two emitter
poles (circles marked with + for the positive (red) and − for the
negative (blue) emitter) as an analytical simulation model. The
sensor line is shown in green and consists of n sensors with a
spacing of 5 mm in the following simulations. The distance between
sensor line and emitters is visually enlarged, so that the
individual vectors to the sensors and difference locations are visible.
An object to be detected (gray) is placed above the sensor
line and distorts the electric field. In Fig. 2 there are two different
coordinate origins (Oo
and Os

), and the second one Os
), Oo
is in the middle of the object to
calculate the object-based field distortion (vectors denoted with
r
is placed in the middle between the two
emitter poles (i.e., center of the fish body) as a sensor-centric coordinate
system and can be moved arbitrarily.
The sensor positions on the sensor line can be described according
to Fig. 2 using:

S S id
i  

2
(1)
with i being the index of the respective sensor and d the distance
between two adjacent sensors.
Each sensor detects the potential of the electric field. In this
work, the potential is measured differentially, as this measurement
method does not require a reference electrode and has
the advantage of common mode rejection in real world measurements.
The potential differences are measured between
the direct neighbors of the sensor line
s ss s s s 1.
2 13 2
D

, ,, nn 
This results in n-1 differential pairs if n electrodes are placed
along the sensor line. The location i
of the potential difference
V can be assigned to a position between two adjacent
sensors and is described analytically in:

i
DS0 21i  d
  

D

(2)
Therefore, the potential difference V results from the potential
of two neighboring sensors at the location i
, as described
by:
V D VS VSii 1



   
 
i

(3)
If an object is placed in an electric field, it distorts the electric
field and thus the measured potential, also at the sensor
line. Analytically, the field distortion r
at any position r

relative to the object can be calculated with the help of (4), as
published by Rasnow [13]:
 rE r

Here, dipole
E
is the initial electric dipole field at object position,
a is the radius of the object, and χ is the contrast factor (χ=1
for perfect conductor). A detailed derivation for the field distortion
of round objects in an electric dipole field comparable
to that of a weakly electric fish can be found in [14]. Thus, the
field distortion of objects ( r

ence at a location ( , i
12
VD

, (4)) and the potential differ(3))
results in (5):
   

dipole


r


a

3
(4)
VD r  rSi


i

Si1
dipole Si1
Si1
   

 
 Er    


 

 

Si1
   
dipole
Ea3  
Si1
rr
rr
dipole Si

Si



rrSi


33
Since the field distortion according to (4) is calculated from
the object for any location in space, the vectors are transformed
into the sensor-centric coordinate system Os
. In this work, the
,
field distortion is needed at the certain sensor positions i
thus (6) results from the vector transformations r SoSi 11i
and Si
S

 


r So 

i
:
 S o So
VD a E ii

i
VD a Edipole

i   
S00i do S ido
 S i21 do S id o

3 0


 



   

21 2


VD

If the analytical description of the electric dipole field
 on the sensor line:
i
E

dipole
from [14] is taken into account, (8) is obtained. This equation
can be used to calculate the field distortion at an arbitrary difference
location
  21 2

VD   


i 3 
 
Qa op o n
33


S i21 do S id o

0  
4     
 
o p on S00i do S ido


3

(8)
The resulting voltage profile (a function of  plotted
over the positions i ) on the sensor line is therefore the sum of
VD

D

the individual voltages. It is characteristic for a field distortion
by a round object in an electric dipole field:
n
Voltage Profile   V D

i
1
i
(9)
With this simple simulation model of a weakly electric fish,
shown in Fig. 2, the field distortion can be calculated analytically
and in a time-efficient manner, since no calculation of
the distortion at each location in space has to be performed, as
this would be the case in general numerical electrical field simulations.
Therefore, no mesh grids are needed, and only the
distorted potentials at the sensor positions are used. The feature
extraction for object localization based on the voltage profile on
the sensor line is still done numerically in simulation.
Object Localization with Static Sensor Line
The electric dipole field, whether undistorted or distorted by
an object, results in a one-dimensional measurement profile on
IEEE Instrumentation & Measurement Magazine
December 2022
i
0 
2
3  

33 (7)
0 
2

  


dipole




1

ii
33

S o So
1
Equation (7) results from (6) and (1):
(6)
3 

aa



Si
33
  (5)
E r  
Instrumentation & Measurement Magazine 25-9

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