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Advanced Numerical Techniques for Near-Field Antenna Measurements 13
methods. In particular, it takes advantages of the interferometric approach to significantly
reduce the number of unknowns in the phase retrieval algorithm. Although the functional
to be minimized is highly non-linear, the lower number of unknowns, given by the phase
shifts, allows an accurate and fast convergence to the solution. Furthermore, the absence of a
reference antenna gives a simpler and more compact measurement setup.
3.1 Theoretical formulation of hybrid phase-retrieval technique
Let us consider an observation curve C over an arbitrary scanning geometry (Fig. 14), with
a sampling step Δs
= λ/2 and a separation d = iλ/2 between two adjacent interference
points, i being an integer greater than one. Two identical probes simultaneously moving
along the measurement curve (Fig. 14) are used to obtain four amplitude information, namely
(Costanzo et al., 2001; Costanzo and Di Massa, 2002; Costanzo et al., 2005):
|V
1
|
2
, |V
2
|
2
, |V
1
+ V
2
|
2
, |V
1
+ jV
2
|
2
(27)
where:
V
1
= |V
1
|·e
jϕ
1
, V
2
= |V
2
|·e
jϕ
2
(28)
are the complex signals on a pair of interference points along C.
y
z
x
AUT
s
Ds
Probe 1
Probe 2
d
C
Fig. 14. Observation curve C with probes positions.
Intensity data (27) are processed to give the phase shift Δϕ
= ϕ
1
− ϕ
2
by means of the
following interferometric formula (Costanzo et al., 2001; Costanzo and Di Massa, 2002):
Δϕ
= tg
−1
|V
1
+ jV
2
|
2
−|V
1
|
2
−|V
2
|
2
|V
1
+ V
2
|
2
−|V
1
|
2
−|V
2
|
2
(29)
Let be:
E
(s)=|E(s)|·e
jϕ(s)
(30)
the field radiated by the AUT on the observation curve C, where parameter s denotes the
curvilinear abscissa along C (Fig. 14). If we suppose to scan 2N+1 measurement points (N
even), the application of equation (29) gives a number of sets of complex near-field data equal
to i, namely (Costanzo et al., 2001; 2005):
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Advanced Numerical Techniques for Near-Field Antenna Measurements
14 Will-be-set-by-IN-TECH
E
s
(1)
, E
s
(2)
= ε
s
(2)
·e
jΔφ
1
, E
s
(3)
= ε
s
(3)
·e
jΔφ
2
, ...
..., E
s
(i)
= ε
s
(i)
·e
jΔφ
i−1
(31)
wherein:
s
(1)
=
−N
λ
2
,
(−N + i)
λ
2
,
(−N + 2i)
λ
2
, ...
s
(2)
=
(−N + 1)
λ
2
,
(−N + i + 1)
λ
2
,
(−N + 2i + 1)
λ
2
, ...
.
.
. (32)
s
(i)
=
(−N + i −1)
λ
2
,
(−N + 2i −1)
λ
2
,
(−N + 3i −1)
λ
2
, ...
The terms ε
s
(2)
, ε
s
(3)
, ..., ε
s
(i)
into expression (31) are known quantities and the
phase shifts Δφ
1
, Δφ
2
, ..., Δφ
i−1
are the unknowns to be determined. If we change
Δφ
1
, Δφ
2
, ..., Δφ
i−1
∈ [−π, π[, expression (31) gives the set S
m
of all fields compatible with
the measured data. The field radiated by the AUT is so given by the intersection S
m
∩ S
A
,
where S
A
is the set of all fields that the AUT can radiate.
In order to successfully retrieve the unknown phase shifts Δφ
1
, Δφ
2
, ..., Δφ
i−1
, a non redundant
representation is adopted which substitutes the original field (30) with the reduced field
(Bucci et al., 1998) F
(ξ)=E(ξ) · e
jψ(ξ)
, obtained after extracting a proper phase function
ψ
(ξ( s)) and introducing a suitable parameterization ξ(s) along the observation curve. A
proper choice of these parameters leads to approximate the reduced field by a cardinal series
of the kind:
F
(ξ)=
N
∑
n=1
E
(
ξ
n
)
·
e
−jψ
(
ξ
n
)
Φ
[
w(ξ − ξ
n
)
]
(33)
where Φ
(x) is the
sin(x)
x
function or the Dirichlet function, ξ
n
=
nπ
χ·W
are the positions of
non-redundant sampling points, while N’ represents the number of non redundant samples
falling i n the measurement interval.
The above relation, discretized in the M measurement points, say ξ
m
, m = 1, ..., M,canbe
written in matrix form as (Costanzo et al., 2005):
r
= A · s (34)
where s
is the array of the reduced field values in the non redundant sampling positions and
r
is the corresponding array of the reduced field values at the measuring points . Due to the
representation error and the presence of noise usually corrupting measurements, data do not
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Numerical Simulations of Physical and Engineering Processes
Advanced Numerical Techniques for Near-Field Antenna Measurements 15
belong in general to the range of matrix A. Consequently, the following generalized solution
is adopted:
inf
Δφ
1
, Δφ
2
,...,Δφ
i−1
d
S
m
, S
N
A
(35)
The term d
(., .) into equation (35) represents the distance between the two sets, while S
N
A
is the
set of all reduced field (evaluated at the M measurement points) that the AUT can radiate.
The distance d between the two sets is numerically evaluated by introducing the projector
operator P
= A A
+
onto the range of matrix A, A
+
denoting the pseudoinverse of A.
Consequently, the near-field phase retrieval involves the finding of:
min
Δφ
1
, Δφ
2
,...,Δφ
i−1
r(Δφ
1
, Δφ
2
, ..., Δφ
i−1
)−
P r( Δφ
1
, Δφ
2
, ..., Δφ
i−1
)
2
(36)
which can be easily performed by a suitable least-square procedure.
3.2 Experimental validations of hybrid phase-retrieval technique
The hybrid phase-retrieval technique is experimentally validated by designing a
multifrequency prototype properly working within X-band. Two rectangular waveguides
used as probes are connected to the microstrip circuit in Fig. 15(a) for obtaining the required
amplitude information. Measurements are performed on a standard X-band pyramidal horn
(Fig. 15(b)) by assuming a cylindrical scanning geometry of 47x85 points along z and φ,
respectively, with s ampling steps Δz
= λ/2 = 1.5cm and Δφ = 4.23
o
at different frequencies.
(a) (b)
Pyramidal horn (AUT)
Waveguides (Probes)
Microstrip circuit
Fig. 15. (a) Microstrip circuit and (b) test setup for phaseless near-field measurements.
The near-field directly measured at one output of the integrated probe is reported under Figs.
16-17 for both amplitude and phase at two different frequencies, namely f
= 8GHz and
f
= 10GHz. The interferometric formula (29) is used in conjunction with the minimization
procedure (36) to obtain the retrieved near-field phase, whose agreement with the exact one is
illustrated under Figs. 16(b)-17(b) along the cylinder generatrix at φ
= 90
o
.
The standard NF-FF cylindrical transformation (Leach and Paris, 1973) is then applied to
obtain the far-field patterns of Figs. 18-19. In particular, a good agreement between results
obtained from direct and retrieved near-field phase can be observed under Figs. 18(b)-19(b)
for the H-plane.
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Advanced Numerical Techniques for Near-Field Antenna Measurements
16 Will-be-set-by-IN-TECH
(a) (b)
Pyramidal horn (AUT)
Waveguides (Probes)
Microstrip circuit
Fig. 16. (a) Measured near-field amplitude on the cylindrical surface and (b) near-field phase
(retrieved and measured) on the cylinder generatrix at φ
= 90
o
:frequencyf = 8GHz.
(a) (b)
Pyramidal horn (AUT)
Waveguides (Probes)
Microstrip circuit
Fig. 17. (a) Measured near-field amplitude on the cylindrical surface and (b) near-field phase
(retrieved and measured) on the cylinder generatrix at φ
= 90
o
:frequencyf = 10GHz.
(a) (b)
Pyramidal horn (AUT)
Waveguides (Probes)
Microstrip circuit
Fig. 18. (a) 3-d view of radiation pattern and (b) H-plane obtained from exact and retrieved
near-field phase: frequency f
= 8GHz.
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Numerical Simulations of Physical and Engineering Processes
Advanced Numerical Techniques for Near-Field Antenna Measurements 17
(a) (b)
Pyramidal horn (AUT)
Waveguides (Probes)
Microstrip circuit
Fig. 19. (a) 3-d view of radiation pattern and (b) H-plane obtained from exact and retrieved
near-field phase: frequency f
= 10GHz.
4. Conclusion
Innovative techniques for near-field antenna testing have been presented in this chapter.
Two primary aspects, namely the reduction of both measurement time and cost setup on
one hand, and the accurate near-field phase characterization on the other hand, have been
accurately faced. For what concerns the first focus point, accurate and fast near-field to
far-field transformations on new strategic geometries of helicoidal, plane-polar, bi-polar
and planar spiral type have been presented. On the other hand, the problem of accurate
phase retrieval at high operating frequencies has been faced by presenting a hybrid
interferometric/functional-kind approach to obtain the antenna far-field pattern from a
reduced set of amplitude-only near-field data acquired on a single scanning surface. All
discussed procedures have been successfully validated by numerical and experimental tests.
5. References
Abramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions,Dover, New York.
Bennet, J. C., Anderson, A. P., Mcinnes, P. A. & Whitaker, J. T. (1976). Microwave holographic
metrology of large reflector antennas. IEEE Trans. Antennas Propag., Vol. 24, (1976)
page numbers (295-303).
Bracewell, R. N. (2000). The Fourier Transform and Its Applications, McGraw-Hill, ISBN
0-07-303938-1, Singapore.
Bucci, O. M., Gennarelli, C. & Savarese, C. (1998). Representation of electromagnetic fields
over arbitrary surfaces by a finite and nonredundant number of samples. IEEE Trans.
Antennas Propag., Vol. 46, (1998) page numbers (351-359).
Costanzo, S., Di Massa, G. & Migliore, M. D. (2001). Integrated microstrip probe for phaseless
near-field measurements on plane-polar geometry. Electronics Letters, Vol. 37, (2001)
page numbers (1018-1020).
Costanzo, S. & Di Massa, G. (2002). An integrated probe for phaseless near-field
measurements. Measurement, Vol. 31, (2002) page numbers (123-129).
Costanzo, S., Di Massa, G. & Migliore, M. D. (2005). Integrated microstrip probe for phaseless
near-field measurements on plane-polar geometry. IEEE Trans. Antennas Propag.,Vol.
53, (2005) page numbers (1866-1874).
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Advanced Numerical Techniques for Near-Field Antenna Measurements
18 Will-be-set-by-IN-TECH
Costanzo, S. & Di Massa G. (2004). Far-field reconstruction from phaseless near-field data on a
cylindrical helix. Journal of Electromagn. Waves Applicat., Vol. 18, (2004) page numbers
(1057-1071).
Costanzo, S. & Di Massa G. (2006). Efficient near-field to far-field transformation from
plane-polar samples. Microwave Opt. Tech. Letters, Vol. 48, (2006) page numbers
(2433-2436).
Costanzo, S. & Di Massa G. (2006). Direct far-field computation from bi-polar near-field
samples. Journal of Electromagn. Waves Applicat., Vol. 20, (2006) page numbers
(1137-1148).
Costanzo, S. & Di Massa G. (2007). Near-field to far-field transformation with planar spiral
scanning. Progress In Electromagnetics Research, PIER, Vol. 73, (2007) page numbers
(49-59).
Costanzo, S. & Di Massa, G. (2008). Wideband phase retrieval technique from amplitude-only
near-field data. Radioengineering, Vol. 17, (2008) page numbers (8-12).
Elliott, R. S. (2003). Antenna theory and design, IEEE Press, ISBN 0-471-44996-2, New York.
Isernia, T., Pierri, R. & Leone, G. (1991). New technique for estimation of farfield from
near-zone phaseless data. Electronics Letters, Vol. 27, (1991) page numbers (652-654).
Isernia, T., Leone, G. & Pierri, R. (1996). Radiation pattern evaluation from near-field
intensities on planes. IEEE Trans. Antennas Propag., Vol. 44, (1996) page numbers
(701-710).
Johnson, R. C., Ecker, H. A. & Hollis, J. S. (1973). Determination of far-field antenna
patterns from near-field measurements . Proc. of IEEE, Vol. 61, (1973) page numbers
(1668-1694).
Leach, M. W. & Paris, D. T. (1973). Probe compensated near-field measurements on a cylinder.
IEEE Trans. Antennas Propag., Vol. 21, (1973) page numbers (435-445).
Ludwig, A. C. (1971). Near-field far-field transformations using spherical-wave expansions.
IEEE Trans. Antennas Propag., Vol. 19, (1971) page numbers (214-220).
Pierri, R., D’Elia, G. & Soldovieri, F. (1999). A two probes scanning phaseless near-field
far-field transformation technique. IEEE Trans. Antennas Propag., Vol. 47, (1999) page
numbers (792-802).
Wang, J. J. H. (1988). An examination of theory and practices of planar near-field
measurements. IEEE Trans. Antennas Propag., Vol. 36, (1988) page numbers (746-753).
Yaghajian, A. D. (1986). An overview of near-field antenna measurements. IEEE Trans.
Antennas Propag., Vol. 34, (1986) page numbers (30-45).
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Numerical Simulations of Physical and Engineering Processes
16
Numerical Simulations of Seawater
Electro-Fishing Systems
Edo D’Agaro
Medical Veterinary Faculty
University of Udine
Italy
1. Introduction
In the last decades, several surveys and research works have reported a decrease in
pelagic fish resources in the Mediterranean sea, with the exception of the Adriatic sea. In
fact, in this area, an overall decrease of stocks of fish species was reported as opposed to
the simultaneous increase in others (Picinetti 2008). Fishing methods that use attractive
elements of fish such as light and the electric current are used in many parts of the world.
In this regard, the attraction of light, which exploits the phototropism of certain fish
species is widely used, for instance, by the famous Japanese method for squid catching or
electro-fishing techniques of bluefish in use throughout the Mediterranean. Also in
freshwater lakes and rivers is very common to use electro-fishers to attract and capture
fish. Regarding the electrical fishing in salt water, various experiments have been carried
out to develop this new technique (Kolz,1993; Kurk,1971,1972; Roth et al., 2006). These
studies were mainly carried out in the United States, France and Soviet Union
(Blabcheton,1971; Diner & Le Men, 1971; Kolz, 1993; Van Harreveld, 1938). The basic
elements that must be taken in consideration for the personnel who, for the first time, is
preparing to use a sea electric attraction system are, first and foremost, the safety of
operators and possible damage to fish. To understand these effects, it is necessary to know
some basic principles of electrical circuits and the chemical-physical characteristics of
water and fish subjected to different types of current. Regarding the former, it is
important the knowledge of circuit features such as the power and characteristics of an
electric generator, the current type, shape and use of electrodes (anode and cathode). The
application of electric fields in non homogeneous systems consisting of fish and salt water
is far more difficult than in freshwater conditions. This point is of fundamental
importance and its understatement, in fact, may impair or reduce the efficiency of
electrical fishing. Electric fishing is based on the principle of introducing an electric
potential gradient in the water body, between one or more cathodes and one anode. The
perception of this potential gradient by fish is function of their position towards
electrodes and of their conductivity in respect to water’s, as well as of temperature, size
and species. The potential gradient produces different effects on fishes depending on the
intensity and type of current used. Those effects are known and described since the end of
1800 (Van Harreveld, 1938). Currents used in electro-fishing can be continuous (DC),
alternate (AC) or pulsed (PDC), depending on environmental characteristics
Numerical Simulations of Physical and Engineering Processes
340
(conductivity, temperature) and fish to be sampled (species, size). The three current types
(DC, AC, PDC) produce different effects. Only DC and PDC cause a galvanotaxis reaction,
as an active swim towards the anode. With AC this phenomenon is not possible due to the
continuous changing in polarity of the electrodes. Fundamental limit to the application of
electric fishing in sea water is given by the high conductivity of salted water, that being
much greater than animal tissues causes the current to flow around the fish instead of
passing through it. In high conductive water, PDC is the mainly used current form,
because of the lower power demand, at parity of result, compared to DC (Le Men,
1980;
Beaumont
et al., 2002), and also causes galvanotaxis in fish (Kurc et al., 1971). Fish in fact
swims towards the anode under the effect of the muscle contraction given by each electric
impulse (electrotaxis) until narcosis occurs (tetanus) (Beaumont
et al., 2002).
2. Electro-fishing theory
2.1 Definition of an electric field
Materials consist of particles characterized by positive electric charges (protons) and
negative (electrons), while others have neutral charge (neutrons). In various materials, in
particular in metals, electrical charges have the ability to move. In reality, there is not a real
movement of electrons, but a transfer of energy through collisions between electrons. The
movement of charges, which occurs at a given time, is defined as movement of electric
current (I) and is measured in amperes (A). The relationship between the aforementioned
variables is as follows:
I=Q/t (1)
where:
Q=charge in coulombs
I = electrical current in amperes
t = time in seconds.
Table 1 shows the basic terms, definitions and units of measurement of variables used in
circuit theory and electric fields.
Term Symbol Unit
Electric charge Q coulomb
Voltage energy V volt
Current load/time I ampere
Electric resistance R ohm
Energy power/time P watts
Energy power*time W watt/hour
Resistivity fraction x
distance
p ohm/cm
Conductivity 1/P µS/cm
Voltage gradient variation e volts/
Current density J amp/cm
2
Power density D watt/cm
3
Table 1. Terms, symbols and unit used in the current field theory
Numerical Simulations of Seawater Electro-Fishing Systems
341
The electric current is made up of a flow of charges which tend to restore a state
of neutrality between two electrically charged bodies. If the two bodies become neutral,
the current ceases immediately to flow (because there is no more a force of attraction
between the two bodies). The circulation of electric current is higher in materials that have
a large amount of free electrons as conductors. In this way, the electric current flows from
a region with high negative charges to one with positive charges. The electric current (I)
is measured with the ammeter. The voltage (V) is defined as the potential difference
between two points of the electrical circuit and is measured with a voltmeter. With
a voltage V and a current density I, the power P can easily calculated as P = V x I. The
electric circuits can be classified into two main types: circuit in series or parallel. In the series
circuits, all components (generator, switch and the transformer) form a single path. Instead,
the circuits in parallel are divided into branches. If two different charged electrodes
are immersed in a liquid, several lines of force are created between the two poles. Along
these lines of force flows the electrical current. These lines of force coincide with the current
lines (Fig.1).
Fig. 1. Force lines are formed between the anode (positive) and cathode (negative) immersed
in a liquid
Now, let's take an example of a potential difference of 400 V between the two poles of the
field. This potential difference decreases gradually starting from the anode (+) going to the
cathode (-) to finally reach the value of 0 volts at the cathode. Consequently, we can see that
on the same force line, voltage values varie according to the position. We can also get lines
that have the same voltage values. These lines are called equipotential lines (Fig.2).
Numerical Simulations of Physical and Engineering Processes
342
Fig. 2. The equipotential lines are obtained by bringing together points of equal voltage
The figure obtained resembles a map in which lines mark the same altitude.
Fig. 3. Voltage curve between the anode (+) and cathode (-) located at a distance of 20 m
apart