# Sunflower Array Antenna with Adjustable Density Taper

Authors: Maria Carolina Viganó; Giovanni Toso; Gerard Caille; Cyril Mangenot; Ioan E. Lager
Journal: International Journal of Antennas and Propagation
 (2009)

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## Abstract

A deterministic procedure to design a nonperiodic
planar array radiating a rotationally symmetric pencil beam pattern
with an adjustable sidelobe level is proposed. The elements
positions are derived by modifying the peculiar locations of the
sunflower seeds in such a way that the corresponding spatial
density fits a Taylor amplitude tapering law which guarantees
the pattern requirements in terms of beamwidth and sidelobe
level. Different configurations, based on a Voronoi cell spatial
tessellation of the radiative aperture, are presented, having as
a benchmark the requirements for a typical multibeam satellite
antenna.


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## Body

## 1. Introduction

Communication satellites use multiple beam antennas
providing downlink and uplink coverages over a field of view for high data
rate, multimedia, or mobile personal communication applications. High gain,
multiple overlapping spot beams, using both frequency and polarization reuse,
provide the needed coverage. In order to generate high gain spot beams,
electrically large antenna apertures are required. These apertures may be
generated by either reflectors or phased arrays. Phased arrays would be a
natural choice to generate multiple beams but up to now the poor efficiency,
the high cost, and the deployment complexity of active arrays have been their
main drawbacks, limiting their use onboard satellites. These drawbacks are
mainly due to the required distributed and tapered power amplification which is
inducing poor power efficiency.Aperiodic arrays with equiamplitude elements permit to
mitigate these limitations and represent a valid alternative to traditional
periodic phased arrays with amplitude tapering. Resorting to aperiodic arrays
with equiamplitude fed elements is particularly effective for the design of
large arrays working in transmission. This
type of antenna architecture is considered extremely promising for achieving a multibeam
coverage on the Earth from a geostationary satellite [1–4].Unequally spaced arrays have several interesting
characteristics and may offer some potential advantages with respect to periodic
arrays [2]. Firstly,
aperiodic arrays allow the reduction of the sidelobe level (SLL) without
resorting to an amplitude tapering. A second useful property of aperiodic
arrays is the possibility to reduce the number of elements in one assigned
aperture without major impact on the beamwidth. The reduction in the number of elements,
with respect to the corresponding periodic array, depends on the required
aperture efficiency and on the field of view where the assigned sidelobe level
is imposed (based on the desired scanning range and regulatory aspects). Thirdly, sparse arrays can
effectively be employed for spreading out the energy that would otherwise
accumulate in grating lobes (GL) due to the wide interelement spacing.In terms of limitations, nonperiodic arrays exhibit a
reduced aperture efficiency when identical, non-equispaced elements are used.
As a consequence, a reduced maximum equivalent isotropically radiated power
(EIRP) is obtained if not compensated by an increase of the power radiated by
each active chain. Furthermore, implementation constraints, as a nonregular
lattice, may jeopardize the use of generic building blocks, with consequences
on the costs. This particular drawback may be mitigated by implementing a set
of different types of subarrays to fill the whole aperture.Up to now, sparse and thinned arrays have been rarely
used, essentially because of the complexity of their analysis and synthesis
with a reduced knowledge, as a consequence of their radiative properties. The
main concern in the design of sparse arrays is to find an optimal set of
element spacing to meet the array specifications, while assuming a uniform
excitation for practical convenience.The synthesis of aperiodic arrays is a known problem
in the antenna community [5–22]. It is interesting to observe that while in the 1960s
and 1970s mainly deterministic solutions have been proposed, in the last years
procedures based on statistic global optimization techniques have been mainly
presented. Recently, in [22], the simple and elegant spatial tapering
deterministic solution introduced in the papers of Doyle [11] and Skolnik [17], and qualitatively in the work of Willey [7], has been revisited and
improved. Deterministic solutions present two important advantages with respect
to statistical algorithms: they allow obtaining results in real time and offer
a solution with a controllable accuracy. Moreover, the results obtained
applying deterministic solutions may be directly used or adopted as starting
point for a further optimization based on a numerical technique which can take
into account other constraints (like maximum and minimum spacing between the
radiators, etc.).The problem of aperiodic arrays has recently gained a
renewed interest especially for the design of multibeam satellite antennas
[1–4]. Most of the techniques
presented for the design of aperiodic arrays deal with the case of linear
arrays. When approaching the planar case, the designer has a higher number of
degrees of freedom but the problem increases also in complexity. In some
papers, an aperiodic planar lattice organized in rings has been proposed
[12–14]. Arrays organized in rings permit having a pattern
with good symmetry properties, allow the reduction and control of GL, and their
design is simplified because the elements positions can be expressed as the
product of two functions one controlling the angular position of the elements
and one controlling the distance of the rings from the center.In this paper, the equiamplitude elements constituting
the aperiodic array are placed on a lattice reproducing the positions of the
sunflower seeds, opportunely adjusted according to a desired amplitude tapering.
This type of lattice is selected essentially because it guarantees a really
good radial and azimuthal spreading in the element positions. As a consequence,
the pattern in the sidelobes and grating lobes region tends having a
plateau-like shape [9, 16], avoiding the presence of high narrow peaks.
Moreover, by adjusting the element positions using a simple parametric
equation, a beamwidth can be selected and the SLL kept under an assigned value.An aperiodic planar array with the elements organized
according to a sunflower lattice has been already proposed in [23]. However, the spatial
density of the elements in [23] is uniform. As a consequence, since the elements are
equiamplitude, the equivalent amplitude tapering is uniform as well so that
this type of array guarantees only a good suppression of the GL, without the
ability of controlling the SLL.The hereby proposed sunflower lattice is completely
adjustable in order to follow stringent requirements on the beamwidth and the
SLL without using any amplitude taper. This planar array can be considered in
the design of a transmitting direct radiating array for a satellite
communication antenna on a geostationary satellite.The paper now proceeds as follows. In Section2 the
radiation pattern is introduced for a generic array, in Section 3 the
definition of the element density function is given and discussed for the
uniform case. In Section 4 a procedure to adjust the spatial density according
to a Taylor amplitude tapering is presented. In Section 5 the requirements of a
typical telecommunication multibeam antenna are introduced. Finally, in Section 6 some numerical results, in order to test the functionality of the proposed
configuration, are presented. Additionally, the appendixes provide more general
information on spirals and discuss the normalized element density that is used
in the course of the element placement.


## 2. Array Radiation Pattern

The antenna radiation pattern of a planar array is
given by(1)E(θ,ϕ)=∑n=1NanFnexp(jk[xnsin(θ)cos(ϕ)+ynsin(θ)sin(ϕ)]),where N is the total number of elements, an represents the excitation coefficient for the nth element and Fn its radiation pattern, xn and yn are the nth element positions in the xOy plane, θ is the elevation angle measured from the Oz axis, and ϕ is the azimuth angle measured in the xOy plane with respect to the Ox axis. Because the variables to be derived (xn and yn) appear inside an exponential function, the
optimization problem is not linear. Moreover, Fn,
unlike in periodic arrays, can change from element to element if a requirement
on the minimum aperture efficiency (hence on the minimum gain) is enforced.In the following section, a particular spiral
configuration will be introduced as a starting point before the space taper is
applied.


## 3. Spiral Array with Uniform Spatial
Density

A well-known spiral is the Fermat one (see Figure12)
which has the property of enclosing equal areas within each turn. This spiral
is often found in nature, as indicated in the appendix. The elements are placed
along this spiral according to the following equations:(2)ρn=snπ,forn=1,…,N+1,(3)ϕn=2πnβ1,forn=1,…,N+1,where ρn is the distance from the spiral center to the nth element, the parameter β1 controls the angular displacement ϕ between two consecutive elements, and the
parameter s denotes the distance between the elements in
the xOy plane. Assume a sparse array deployed on a
circular aperture of radius Rap along the Fermat spiral, with element
locations given by (2) and (3). Note that ρN is taken to correspond to Rap,
whereas ρN+1 is a virtual element places outside the
aperture, its use is becoming obvious in what follows.Let us now introduce a normalized element density
function:(4)d˜(ρn)=(Rn2−Rn−12)min(Rn2−Rn−12),forn=1,…,N,where Rn−1 and Rn are the inner and outer radii of the annular
rings enclosing the nth element, respectively, with R0 being always taken as zero (see the appendix)
for a justification for this choice of defining the normalized density
function). Here, a choice is made to take the radii Rn as(5)Rn2=ρn+12+ρn22forn=1,…,N.As recognizable from (4), the
normalized density function corresponds to the current of a single element
divided by the area of the relevant annular ring.The lattice in [23] (see Figure 1) is characterized by a uniform density.
On account of (5) and (2), it can be easily shown that(6)d˜(ρn)=2(Rn2−Rn−12)min(ρn+12−ρn−12)=πRap2Ns2=1.This property is attractive when
the interest focuses on avoiding GL only, without a control of the SLL. As for
the SLL, it remains around 17dB,
irrespective of the number of elements in the array and the spacing factor s.
This is consistent with the element distribution replicating a uniform current
distribution on a circular aperture.Figure 1
Distribution of the 250 elements in the uniform
sunflower array antenna, as reported in [23].It is now clear that the only possibility to control
the SLL as well is by introducing a density taper. In the following section, it
will be demonstrated how, by translating a Taylor amplitude tapering law
[24] into a
corresponding spatial density law, the SLL can be drastically reduced.


## 4. Spiral Array with Density Tapering

The spiral aperiodic lattice with a uniform element
density introduced in the previous section is an excellent starting point to
apply a space tapering process. The spreading of the elements in the spiral
arms guarantees an optimal behavior in terms of GL even when the interelement
spacing is larger thanλ.
In order to be able to control the SLL, it is possible to vary the elements
positions with respect to the array center, thus obtaining an effect similar to
an amplitude taper.The space taper technique presented here consists of
choosing a reference amplitude distribution whose pattern satisfies the
assigned requirements and emulates it by varying the radiator distance from the
center. Concretely, a Taylor amplitude taper law with a certain SLL andn¯ [24] is selected as a reference. The locations of the
elements in the sparse array are determined by means of a simple, 2 step
algorithm: firstly N circles of increasing radii ρn, n=1,2,…,N are selected by sequentially applying the
relations(7)2π∫Rn−1ρnA(r)rdr=2π2N∫0RapA(r)rdr,(8)2π∫Rn−1RnA(r)rdr=2πN∫0RapA(r)rdr,starting from R0 that is taken to be 0. Here, A(r) denotes the Taylor amplitude taper and Rap is the radius of the complete circular
aperture. Note that (7) emulates the desired taper by equating the surface
integral over the annular ring delimited by Rn−1 and ρn to half of the Nth part of the total aperture excitation.
Subsequently, the element positions are determined by choosing their pertaining
azimuth angle ϕn according to (3). The result of this placement
strategy is illustrated in Figure 2 where a 56 λ aperture is filled with 250 elements
distributed in a manner such to obtain a pattern similar to the one achievable
with a Taylor amplitude law characterized by SLL = 32 dB and n¯  = 4. A total number N=250 is selected as a good compromise between the
performance in scanning and the cost. The choice for these values will be
clearer in the following sections.Figure 2
Distribution of
the 250 elements in the tapered sunflower array antenna.


## 5. Typical Requirements for a Multibeam
Satellite Application

The transmitting antenna considered in this study is
operating in Ka-band (19.7–20.2 GHz) and may have a maximum diameter of 1.3 m.
The starting point considers the circular direct radiating array with
dimensions deemed as sufficient to provide the required maximum gain and
beamwidth. The array must generate 64 spot beams. The total frequency band is
divided into 4 subbands, and each of them being assigned to a set of beams so
that there are no adjacent pencil beams using the same resource. Figure3 shows
the footprint on the Earth of the 64 pencil beams.Figure 3
European multibeam coverage in a 1:4 frequency re-use
scheme from a geostationary satellite.In the last 3 rows of Table1, the maximum sidelobe
level in three different regions has been specified. The value in dBi has been
preferred to the dB one as the configurations analyzed in Section 5 have
different maximum directivity values.Table 1
Mission requirements.
Number of spots64Spot
diameter0.65°Inter-spot
distance0.56°Rx
band29.5–30.0GHzTx
band19.7–20.2GHzFrequency
reuse1:4EOC
gain43.8dBiSLL
in the first∓4°20dBiSLL
in the first∓8°25dBiSLL
in the first∓90°30dBiIn Figure4, the array factor of the configurations
presented in Figures 1 and 2, respectively, is plotted for two different ϕ cuts. In both uniform and tapered sunflower
configurations, the array factor is remarkably stable in ϕ,
resulting in the area of interest, in practically rotationally symmetric
radiation patterns. The array factor in Figure 4(b) is following the expected
behavior until a certain θ angle at which the effects of the first
pseudograting lobe (the energy of which being spread over a wider θ interval due to the nonperiodic placement)
become visible. Considering the previous example in which the maximum
interelement distance Dmax=8.43λ,
the first contribution of the grating lobe is expected at(9)θGL=sin−1(1Dmax)=6.81°.In fact, in Figure 4(b), the
pattern starts exceeding the imposed SLL around this θGL.Array Factor, two differentϕ cuts (ϕ=0° in red dotted line and ϕ=90° in black), for the array configuration in
Figures 1 and 2, respectively.
(a)(b)


## 6. Validation of the Technique

The locations provided by the space taper process (see
Figure2) have been used as phase centers of the radiators in a planar array.
Two different techniques are used to select the radiators.


### 6.1. First Approach: Using the Entire Aperture

The circular aperture with a maximum radius of56λ is completely filled with patches disposed on
a regular lattice. The triangular grid is chosen because of its better
performances compared to the regular rectangular one. The analytical equations
in [25] are used to
express the field of the elementary patch antenna, radiating on a ground plane,
with side lengths equaling 0.42 λ and a rectangular cell surrounding it with
side lengths 0.8 λ  ×  0.85 λ.
In order to cover the complete aperture, with this patch choice, more than 15
thousands patches are required. These patches are then collected into
subarrays.The positions of the phase centers of the subarrays
forN = 250 are derived with the formulation
presented in the previous section following a Taylor taper with SLL = 32 dB and n¯  = 4. The subarray positions are then
superimposed on the uniform array, as indicated in Figure 6. Each patch center
is assigned to the closest subarray center that can be interpreted as assigning
the relevant patch to the Voronoi cell [26] corresponding to the computed phase center. Note that
the Voroni surface division (see Figure 5) provides an optimum tessellation of
the available real estate.Figure 5
The Voronoi tessellation consisting of the cells
enclosing the chosen phase centers.Figure 6
Subarray allocation and aperture subdivision
corresponding to the Voronoi tessellation in Figure5.To obtain the total radiation pattern, each radiation
pattern of the subarrayFn has been calculated and multiplied by the
exponential that takes into account the positions of the phase center of that
subarray given in (1).Since the Voronoi cell shapes are close to circular
ones, the subarray patterns result to be almost rotationally symmetric. This is
an important property when the beam is scanned.In Figure7, the pattern for the beam pointing at
boresight is plotted for 360 ϕ cuts (one at every degree). The red line in
this figure indicates the mask requirements given in Table 1. Figure 8
represents the radiation pattern when the beam is pointed at the Europe's
edges: the blue and red lines are iso-level curves at 20 dBi and 43.8 dBi,
respectively, while the black circles enclose the regions of the coverage in
which the same resource is used. As it can be seen from Figure 8, even when the
beam is pointed at the Europe's edges, the pattern remains complaint with the
requirements given in [1].Figure 7
Array pattern for the configuration depicted in Figure6. The beam is scanned to boresight. The red line corresponds to the
requirement mask.Figure 8
Array pattern for the configuration depicted in Figure6. The beam is pointing at Europe edges. The red line represents the iso-level
curves at 43.8 dBi, and the blue ones the iso-level curves at 20 dBi. The black
circles represent the interfering area.With this method, the entire surface available is used
maintaining at the same time a very small number of controls (one for each
subarray).


### 6.2. Second Approach: 4 Different Types of Subarrays

In this case, a more technology-oriented approach is
considered: the aperture is filled as much as possible with predefined
hexagonal subarrays. A limited number of these subarrays is selected as a compromise
in order to keep the complexity and the cost limited while offering good
performances. Four subarrays with different sizes have been selected and used
to fill the array aperture. All the subarrays have a hexagonal shape and
consist of 2, 3, 4, or 5 rings of elements surrounding the central one on a
regular triangular lattice. The patches used in the subarrays are the same as
the ones described in the previous subsection.The procedure consists of computing for each cell the
radius of the maximum circle that can be inscribed in it. According to this
value, the best hexagonal sub-array among the four available is selected and
placed in the cell (see Figure9). After this first step, a postprocessing is
carried out in which every subarray is replaced by its larger version if no
overlapping with its neighbor occurs. In the following case, 117 of the 250
subarrays were substituted during the postprocessing and this is the reason why
the subarray placement in the array is not clearly divided into annular areas
around the array center enclosing the same type of subarrays. The
postprocessing allows increasing the aperture filling from 44.4% up to 60.9%.Figure 9
Hexagonal subarray positions and dimension after
postprocessing.In Figure9, the subarrays have been plotted in
different colors depending on their size. Since the element placement is more
dense in the center, smaller subarray dimensions are needed in the array middle
while at the periphery larger hexagonal subarray can be accommodated improving
the directivity but, depending on the maximum dimension, allowing for a limited
beam scan only.With the first approach, the results were exceeding
the requirements but the physical implementation of the array would be too
demanding since every subarray is different and has to be designed and tested
individually. With the second approach proposed here, only 4 subarrays need to
be generated and moreover, the feeding network will be easier to implement. The boresight radiation pattern for this configuration is depicted in Figure10. As it can be noticed, the SLL is still within the specifications; the maximum gain drops by approximately 2 dB but the EOC gain is enforced. As
a drawback, the array performance is optimal only when the beam is pointing at
boresight. When the beam is scanned to the Europe edges, it is possible to
maintain the SLL under the prescribed value in the area of interest but the
requirement on the EOC gain cannot be reached as shown in Figure 11.Figure 10
Array pattern for the configuration depicted in Figure9. The beam is scanned to boresight. The red line corresponds to the requirement
mask.Figure 11
Array pattern for the configuration depicted in Figure9. The beam is pointing at Europe edges. The red line represents the iso-level
curves at 43.8 dBi, and the blue ones the iso-level curves at 20 dBi. The black
circles represent the interfering area.Figure 12
The Fermat spiral and its associated coordinate
system.


## 6.1. First Approach: Using the Entire Aperture

The circular aperture with a maximum radius of56λ is completely filled with patches disposed on
a regular lattice. The triangular grid is chosen because of its better
performances compared to the regular rectangular one. The analytical equations
in [25] are used to
express the field of the elementary patch antenna, radiating on a ground plane,
with side lengths equaling 0.42 λ and a rectangular cell surrounding it with
side lengths 0.8 λ  ×  0.85 λ.
In order to cover the complete aperture, with this patch choice, more than 15
thousands patches are required. These patches are then collected into
subarrays.The positions of the phase centers of the subarrays
forN = 250 are derived with the formulation
presented in the previous section following a Taylor taper with SLL = 32 dB and n¯  = 4. The subarray positions are then
superimposed on the uniform array, as indicated in Figure 6. Each patch center
is assigned to the closest subarray center that can be interpreted as assigning
the relevant patch to the Voronoi cell [26] corresponding to the computed phase center. Note that
the Voroni surface division (see Figure 5) provides an optimum tessellation of
the available real estate.Figure 5
The Voronoi tessellation consisting of the cells
enclosing the chosen phase centers.Figure 6
Subarray allocation and aperture subdivision
corresponding to the Voronoi tessellation in Figure5.To obtain the total radiation pattern, each radiation
pattern of the subarrayFn has been calculated and multiplied by the
exponential that takes into account the positions of the phase center of that
subarray given in (1).Since the Voronoi cell shapes are close to circular
ones, the subarray patterns result to be almost rotationally symmetric. This is
an important property when the beam is scanned.In Figure7, the pattern for the beam pointing at
boresight is plotted for 360 ϕ cuts (one at every degree). The red line in
this figure indicates the mask requirements given in Table 1. Figure 8
represents the radiation pattern when the beam is pointed at the Europe's
edges: the blue and red lines are iso-level curves at 20 dBi and 43.8 dBi,
respectively, while the black circles enclose the regions of the coverage in
which the same resource is used. As it can be seen from Figure 8, even when the
beam is pointed at the Europe's edges, the pattern remains complaint with the
requirements given in [1].Figure 7
Array pattern for the configuration depicted in Figure6. The beam is scanned to boresight. The red line corresponds to the
requirement mask.Figure 8
Array pattern for the configuration depicted in Figure6. The beam is pointing at Europe edges. The red line represents the iso-level
curves at 43.8 dBi, and the blue ones the iso-level curves at 20 dBi. The black
circles represent the interfering area.With this method, the entire surface available is used
maintaining at the same time a very small number of controls (one for each
subarray).


## 6.2. Second Approach: 4 Different Types of Subarrays

In this case, a more technology-oriented approach is
considered: the aperture is filled as much as possible with predefined
hexagonal subarrays. A limited number of these subarrays is selected as a compromise
in order to keep the complexity and the cost limited while offering good
performances. Four subarrays with different sizes have been selected and used
to fill the array aperture. All the subarrays have a hexagonal shape and
consist of 2, 3, 4, or 5 rings of elements surrounding the central one on a
regular triangular lattice. The patches used in the subarrays are the same as
the ones described in the previous subsection.The procedure consists of computing for each cell the
radius of the maximum circle that can be inscribed in it. According to this
value, the best hexagonal sub-array among the four available is selected and
placed in the cell (see Figure9). After this first step, a postprocessing is
carried out in which every subarray is replaced by its larger version if no
overlapping with its neighbor occurs. In the following case, 117 of the 250
subarrays were substituted during the postprocessing and this is the reason why
the subarray placement in the array is not clearly divided into annular areas
around the array center enclosing the same type of subarrays. The
postprocessing allows increasing the aperture filling from 44.4% up to 60.9%.Figure 9
Hexagonal subarray positions and dimension after
postprocessing.In Figure9, the subarrays have been plotted in
different colors depending on their size. Since the element placement is more
dense in the center, smaller subarray dimensions are needed in the array middle
while at the periphery larger hexagonal subarray can be accommodated improving
the directivity but, depending on the maximum dimension, allowing for a limited
beam scan only.With the first approach, the results were exceeding
the requirements but the physical implementation of the array would be too
demanding since every subarray is different and has to be designed and tested
individually. With the second approach proposed here, only 4 subarrays need to
be generated and moreover, the feeding network will be easier to implement. The boresight radiation pattern for this configuration is depicted in Figure10. As it can be noticed, the SLL is still within the specifications; the maximum gain drops by approximately 2 dB but the EOC gain is enforced. As
a drawback, the array performance is optimal only when the beam is pointing at
boresight. When the beam is scanned to the Europe edges, it is possible to
maintain the SLL under the prescribed value in the area of interest but the
requirement on the EOC gain cannot be reached as shown in Figure 11.Figure 10
Array pattern for the configuration depicted in Figure9. The beam is scanned to boresight. The red line corresponds to the requirement
mask.Figure 11
Array pattern for the configuration depicted in Figure9. The beam is pointing at Europe edges. The red line represents the iso-level
curves at 43.8 dBi, and the blue ones the iso-level curves at 20 dBi. The black
circles represent the interfering area.Figure 12
The Fermat spiral and its associated coordinate
system.


## 7. Conclusions

A deterministic procedure to design aperiodic planar
arrays which guarantees the control of SLL, GL, and beamwidth without
using any amplitude tapering has been introduced. Starting from an array
characterized by a uniform spatial density of the elements, the density
function has been modified in order to fit a reference amplitude tapering. The design
technique has been applied for the preliminary design of a Direct Radiating
Array for a multibeam satellite communication mission.



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*Source: 624035-2009-04-16.xml*