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1
Antenna arrays for pattern control
J. A. R. Azevedo(1), A. M. Casimiro(2)
(1)University of Madeira(2)University of AlgarvePortugal
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
2
Contents
• Introduction to the proposed problem of antenna arrays synthesis
• Planar arrays
• Circular arrays
• Conclusions
• Results
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
• Basic Concepts
3
Introduction
Main objective of COST 296:Develop an increased knowledge of the effects imposed by the ionosphere on practical radio systems, and for the development and implementation of techniques to mitigate the deleterious effects of the ionosphere on such systems.
In Final Report of COST 284:Geo-stationary satellites were expected to require very large multi-feed reflector antennas and very versatile arrays and array feeds.
The mitigation of the interference can be performed by imposing appropriated nulls on the radiation pattern and by controlling the sidelobe levels.
Antenna arrays play an important role in the mitigation of the ionospheric effects.
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
4
Comparing with linear arrays, planar arrays provides a greater control of the radiation pattern.
Circular arrays permit a fast rotation of the main beam by 360º only by shifting the excitation of the elements.
Usually, the methods for the synthesis on planar arrays are iterative or based on the sampling of continuous apertures.
z
y x
The time consumed in the calculations is very high (several minutes or hours for the computation of the array excitation).
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
5
Basic Concepts
For the farfield, a spatial translation of any source only modifies the produced field in the phase:
x
z
y
θθ =z
φ
yθ
xθ
d
c0
c(x,y,z)V
),()(
),()(
1.β
1
11
φθ
φθ
Feaca
Fcj d
00dd
d →+
→
Fourier Relation:
[ ]),,()2(
),,(),,(13
)(
zyxc
dzdxdyezyxcF zyxjzyx
zyx
−
∞
∞−
∞
∞−
∞
∞−
++
=
= ∫ ∫ ∫π
βββ βββ
[ ]),,()2(
1
),,(),,(
3
)(
zyx
zyxzyxj
zyx
F
dddeFzyxc zyx
βββπ
ββββββ βββ
=
= ∫ ∫ ∫∞
∞−
∞
∞−
∞
∞−
++−
)(cos
)(sen)(sen)cos()(sen
θββφθββφθββ
=
==
z
y
x
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
6
Rectangular Arrays
z
y
x
θ
ydφ
)(sen)(sen)cos()(sen
φθββφθββ
==
y
x
For discrete arrays:
∫ ∫∞
∞−
∞
∞−
+= dxdyeyxcF yxjyx
yx )(),(),( ββββ
∫ ∫∞
∞−
∞
∞−
+−= yxyxj
yx ddeFyxc yx ββββπ
ββ )(2 ),(
)2(1),(
A planar array with elements in a rectangular grid provides two dimensions for the pattern control.
For a continuous array:
∫ ∫ ∑ ∑∞
∞−
∞
∞−
∞
− ∞=
∞
− ∞=
+−−=n m
yxjmnmnyx dxdyeyyxxyxcF yx )(),(),(),( ββδββ
)(),( mynx yxj
n mmn eyxc ββ +
∞
− ∞=
∞
− ∞=∑ ∑=
The fast Fourier transform can be applied.
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
7
Circular Arrays
[ ]
∫ ∫∫ ∫
∞ −
∞ +
=
=
0
2
0
)cos(
0
2
0
)sin().sin()cos().cos(
),(
),(),(π ϕψρ ξ
π ϕρψξϕρψξ
ϕρρϕρ
ϕρρϕρψξ
ddec
ddecF
j
j
Array factor in polar coordinates system
Source distribution in polar coordinates system
222222
)sin()cos(
)cos()cos(
yx
y
x
yxyx
ββξ
ψξβψξβ
ρϕρϕρ
+=
==
+===
For patterns with circular symmetry is more appropriated to use circular arrays.
The Fourier relation is applied in the same manner as presented previously. However, is more appropriated to deal with polar coordinate system,
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
8
For radial symmetry of the source distribution and taking into account that
the array factor is
∫=π α α
π2
0
)cos(0 2
1)( dexJ jx
ρρρ ξρπ
ρρϕρξπ ϕψρ ξ
dJc
ddecF j
∫
∫ ∫∞
∞ −
=
=
0 0
0
2
0
)cos(
)()(2
)()(
The well-known result for a continuous disc of uniform current.
For discrete arrays, the array factor can be also given by
)(),(),( mynx yxj
n mmnyx eyxcF ββββ +
∞
− ∞=
∞
− ∞=∑ ∑=
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
9Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
-2-1
01
2
-2-1
01
20
0.2
0.4
0.6
0.8
1
c(x,
y)
x(λ)y (λ)
-3-2
-10
12
3
-3-2
-10
12
30
0.2
0.4
0.6
0.8
1
c(x,
y)
x(λ)y (λ)
-25
-20
-15
-10
-5
0
|F(β
x,0)| dB
βx
-30
-25
-20
-15
-10
-5
0
-30/λ -20/λ -10/λ 0 10/λ 20/λ 30/λ-30
βx
-30/λ -20/λ -10/λ 0 10/λ 20/λ 30/λ
Comparing with equispaced grids, the periodicity is destroyed in circular arrays.
Visible window
|F(β
x,0)| dB
10
Using the previous changing of variables
mNπ2
z
y x
θ
φ
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
mn N
nπϕ 2= M – number of ringsNm – number of elements in ring m
This result could be also obtained sampling a continuous source distribution.
)cos(1
0
1
0),( nm
mj
M
m
N
nnm ec ϕψξρϕρ −
−
=
−
=∑ ∑=
[ ])sin().sin()cos().cos(1
0
1
0),(),( nmnm
mj
M
m
N
nnm ecF ϕρψξϕρψξϕρψξ +
−
=
−
=∑ ∑=
11
The relation between polar and spherical variables is
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
For an array distribution with radial symmetry,
φφθβφθβ
ββ
ψ
θβφθβφθβββξ
=
=
=
=+=+=
)cos()(sin)(sin)(sinarctanarctan
)sin()(sin)(sin)(cos)(sin 22222222
x
y
yx
∑∞
− ∞=
−− =k
jkmk
kj nnm eJje )()cos( )( ϕψϕψξρ ξρ
∑ ∑
∑ ∑∑
∑∑ ∑
−
=
∞
− ∞=
+
−
=
−
=
−∞
− ∞=
∞
− ∞=
−−
=
−
=
=
=
=
1
0
2
1
0
1
0
2
21
0
1
0
)()(
)()(
)(),()(
M
m p
pjN
mpNmm
M
m
N
n
knN
j
k
jkmk
km
k
nN
jk
mkk
M
m
N
nnm
m
m
mm
mm
eJcN
eeJjc
eJjcF
πψ
πψ
πψ
ξρρ
ξρρ
ξρϕρξ
Usually, few terms of p are enough to obtain a good approximation of the arrays factor.
12
Fist terms of the summation:
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
0 2/λ 4/λ-0.5
0
0.5
1
6/λ 8/λ 10/λ 12/λ 14/λ 16/λ 18/λ 20/λ
0 2/λ 4/λ-0.5
0
0.5
1
6/λ 8/λ 10/λ 12/λ 14/λ 16/λ 18/λ 20/λ
0 2/λ 4/λ-0.5
0
0.5
1
6/λ 8/λ 10/λ 12/λ 14/λ 16/λ 18/λ 20/λ
J0(ρξ)
JN(ρξ)
J2N(ρξ)
J3N(ρξ)
λρ 5.0=
λρ =
λρ 5.1=
13
Source distribution for discrete circular array,
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
[ ]
∫ ∫
∫ ∫∞ −
∞ +
=
=
0
2
0
)cos(2
0
2
0
)sin().sin()cos().cos(2
),()2(
1
),()2(
1),(
π ϕψξρ
π ϕρψξϕρψξ
ψξξψξπ
ψξξψξπ
ϕρ
ddeF
ddeFc
nm
nmnm
j
jnm
For those cases where an independence of ψ exists,
∫
∫ ∫∞
∞ −
=
=
0 0
0
2
0
)cos(2
)()(21
)()2(
1)(
ξξξρξπ
ξξψξπ
ρπ ϕψξρ
dJF
ddeFc
m
jm
nm
resulting in the Hankel transform of zero order.
∫
∫∞
∞
=
=
0 0
0 0
)()(21)(
)()(2)(
qdqqrJqFrf
rdrqrJrfqF
π
π
14
Results
Rectangular array:
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
N=9d=0.5λ81 elements
βxβy
|F(β
x,βy)|
dB
/λ
/λ/λ
/λ/λ/λ
Imposing the peaks amplitude
-6/λ -4/λ -2/λ 0 2/λ 4/λ 6/λ-50
-40
-30
-20
-10
0dB
SLL=-13 dB
/λ /λ
βxβy
|F(β
x,βy)|
dB
/λ
/λ/λ
Uniform
-6/λ -4/λ -2/λ 0 2/λ 4/λ 6/λ-50
-40
-30
-20
-10
0
SLL=-30 dB
Visible window
15Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
M=6N1=6; N2=12; N3=18; N4=24; N5=20d=0.5λ between rings81 elements
βx
βy
|F(β
x,βy)|
dB
/λ
/λ /λ
/λ
- 1 0 -8 -6 - 4 - 2 0 2 4 6 8 1 0- 5 0
- 4 0
- 3 0
- 2 0
- 1 0
0
Uniform
SLL=-18.5 dB
/λ /λ /λ /λ /λ /λ /λ /λ /λ /λ
Visible window
Circular array:
16Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
Since the desired array factor does not depends of φ,
M=6N1=6; N2=12; N3=18; N4=24; N5=20d=0.5λ between rings81 elements ∫
∞=
0 0 )()(21)( ξξξρξπ
ρ dJFc mm
<
=otherwise0
1.31)(
ξξF
15.0:36.0:59.0:80.0:95.0:1:
5
4
3
2
1
0
ρρρρρρ
βx
βy
/λ
/λ /λ
/λ
|F(β
x,βy)|
dB
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 5 0
- 4 0
- 3 0
- 2 0
- 1 0
0
SLL=-31.5 dB
Current in each ring:
/λ /λ /λ /λ /λ /λ /λ /λ /λ /λ
Visible window
17Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
Source distributions:
-2-1.5
-1-0.5
00.5
11.5 2
-2
-1
0
1
20
0.5
1
|c(x
,y)|
x(λ)y(λ)
-3-2
-10
12
3
-3-2
-10
12
30
0.2
0.4
0.6
0.8
1
|c(x
,y)|
x(λ)y(λ)
Circular arrayRectangular array
The feed system of the circular array is simpler than the one of the rectangular grid since several elements have the same excitation.
18
Conclusions
Final meeting 20-21 – Warsaw, March 2009 General Review of COST296 activities and results
• The Fourier Relation method was applied to circular arrays in order to control the radiation pattern.
• Comparing with rectangular arrays, the uniform source distribution permits lower sidelobe levels.
• As it was demonstrated, lower sidelobes can be imposed only controlling the source distribution between rings. In each ring the distribution is uniform whilst for rectangular arrays usually it is necessary to control the value of all elements.
