Synthesis of the Sparse Uniform-Amplitude Concentric Ring Transmitting Array for Optimal Microwave Power Transmission
International Journal of Antennas and Propagation
Synthesis of the Sparse Uniform-Amplitude Concentric Ring Transmitting Array for Optimal Microwave Power Transmission
Hua-Wei Zhou 0
Xue-Xia Yang 0
Sajjad Rahim 0
0 Shanghai Institute of Advanced Communication and Data Science, Key laboratory of Specialty Fiber Optics and Optical Access Networks, Joint International Research Laboratory of Specialty Fiber Optics and Advanced Communication, Shanghai University , Shanghai 200444 , China
Beam capture efficiency (BCE) is one key factor of the overall efficiency for a microwave power transmission (MPT) system, while sparsification of a large-scale transmitting array has a practical significance. If all elements of the transmitting array are excited uniformly, the fabrication, maintenance, and feed network design would be greatly simplified. This paper describes the synthesis method of the sparse uniform-amplitude transmitting array with concentric ring layout using particle swarm optimization (PSO) algorithm while keeping a higher BCE. Based on this method, uniform exciting strategy, reduced number of elements, and a higher BCE are achieved simultaneously for optimal MPT. The numerical results of the sparse uniform-amplitude concentric ring arrays (SUACRAs) optimized by the proposed method are compared with those of the random-located uniform-amplitude array (RLUAA) and the stepped-amplitude array (SAA), both being reported in the literatures for the maximum BCE. Compared to the RLUAA, the SUACRA saves 32% elements with a 1.1% higher BCE. While compared to the SAA, the SUACRA saves 29.1% elements with a bit higher BCE. The proposed SUACRAs have higher BCEs, simple array arrangement and feed network, and could be used as the transmitting array for a large-scale MPT system.
Microwave power transmission (MPT) technology transfers
power from one location to another by the microwave beam,
which could be applied in supplying power to the space
power satellites, unmanned aerial vehicles, the far-reached
areas, and so on . For a large-scale MPT system, the most
important parameter is the beam capture efficiency (BCE),
which is the ratio of the captured microwave power by the
receiving antenna array to the transmitted power by the
transmitting antenna array .
In 1974, Dr. Brown performed an MPT experiment with
a distance of 1.7 m in the laboratory. The overall efficiency up
to 54% and the BCE is 95% . However, the MPT
experiment carried out next year only obtained an overall efficiency
of 7% and BCE of 11.3% when the range was 1.54 km .
Until now, the overall efficiency of a MPT system is not
higher than 10% because of a low BCE .
The transmitting aperture illuminated by the Gaussian
amplitude distribution can obtain a maximum beam capture
efficiency BCEmax higher than 99% because of the broad
beam width and low side lobe level in the far field .
Discrete transmitting aperture, namely, antenna array, is
more practical for expanding the MPT system to a large
scale. The optimized excitation amplitudes of a planar
array for the BCEmax can be achieved by solving generalized
eigenvalue problem . Nevertheless, owing to the
continuous amplitude distribution, many different amplifiers would
be required for every distinct element, which results in a
complex transmitting array. To reduce the kinds of
amplifiers, Baki et al. and Li et al. [7, 8] proposed the isosceles
trapezoidal distribution (ITD) and stepped-amplitude arrays
(SAAs), respectively. The design and implementation of
transmitting array could be greatly simplified if all elements
are uniformly excited . The random-located
uniformamplitude array (RLUAA) comprising of 100 elements was
optimized by particle swarm optimization (PSO) algorithm
with a BCE of 89.96% being obtained . However, the
computation amount would grow up rapidly as the element
number increases, which could not be applied in a
largescale transmitting array design.
Besides the exciting strategy, the sparsification of a
largescale transmitting array has a practical significance. Sparse
arrays can not only reduce the complexity of the feed
networks but also can decrease the weight. Most studies on
sparse antenna arrays [
] are focused on reducing the
number of elements, the peak side lobe level, the
computational effort, and so on but not considering the power
transmission efficiency. In the MPT scenario, the element
numbers of antenna arrays were reduced to 65% and
64% of the original one through compressive sensing
(CS) and convex programming (CP) methods, respectively,
]. By combining these two methods, the element
number was reduced to 54% of the original one and the
BCE was improved about 3.16% . Unfortunately, arrays
] were not uniformly illuminated. Moreover, CS
and CP would not be efficient for the large-scale array design
due to strong nonlinear relationship between the array factor
and the element positions [
]. The Bessel-approximation
array factor of a concentric ring array (CRA) is only related
to the radius and excitation of each ring, which would reduce
the computation amount and could be used in optimizing a
PSO algorithm was firstly introduced by Kennedy and
Eberhart in 1995 [
]. Due to its high search efficiency,
PSO has been widely used in enhancing antenna gain [
and beam pattern synthesis [
] and improving BCE of a
MPT system . In this work, the synthesis of the sparse
uniform-amplitude transmitting array is discussed for the
optimal MPT. The exciting strategy, element number, and
BCE are considered simultaneously for the MPT system.
The outline of this paper is organized as follows. Section 2
describes the calculation equations of BCE of the sparse
uniform-amplitude CRA (SUACRA). Section 3 introduces
the optimization model for the SUACRA, and Section 4
presents the numerical results of SUACRAs, which have been
compared with those of RLUAA discussed in  and SAA
proposed in .
2. Theoretical Foundation
As shown in Figure 1, the transmitting array is a CRA located
in the XOY plane with an element in the center, and the
radial space between the (m ? 1)th and the mth rings is
denoted by ??m m = 1, ? , M . All elements are excited by
the identical phase and amplitude.
The receiving array is in the far region of the transmitting
array, namely, Dtr ? 2Dt2/?, where Dtr is the distance between
transmitting and receiving array, Dt is the radius of
transmitting array, and ? is the wavelength. As a result, the CRA array
factor can be written as [
M M Nm
AF = ? Fm = ? ? Im exp jum cos ? ? ?mn ,
m=0 m=0 n=1
where Fm is the array factor of mth ring, ?m, Nm, and Im
represent the radius, the element number and the excitation
amplitude of the mth ring, respectively, k denotes the
wavenumber, and ?mn is the azimuth angle of the nth
element located on the mth ring.
Nm elements are distributed with the same space on the
mth ring. When Nm is large enough, array factor of the mth
ring can be approximated as [
Fm ? Tm J0 um
In (4), Tm = ImNm and J0 is the zero-order Bessel function
of the first kind. To evaluate the precision, the power error
index ?m is defined as
?m = 10 log
? Fm 2 ? Tm J0 um 2 d?
? Fm 2d?
where ? is the visible region (0 ? ? ? ?, 0 ? ? ? 2?) of the
transmitting array. Under the constraint of keeping a higher
precision of (4), the minimum element number Nmmin can
be found by increasing Nm from 2 till to reach ?m ? ?30 dB.
The line labeled by numerical results in Figure 2 lays out
the Nmmin for radiuses ranging from 0.5? to 19.5? with an
interval of 0.5?.
By applying the fitting method, Nmmin can be
approximated to the following formula:
Nmmin = 3 + 6 7?m , m ? 0,
where ? ? stands for mapping number to the least integer
that is greater than or equal to the original number. The
The raduis of ring array (?)
fitting line is plotted in Figure 2. It can be seen that the fitting
results are consistent well with the numerical results.
Equation (6) could be used to estimate Nmmin on a ring of a
With the element numbers on each ring not less than the
minimum ones, namely, Nm ? Nmmin m = 1, ? , M , the CRA
array factor of (1) can be rewritten as
where T = T0, T1, ? , TM , J = J0 u0 , J0 u1 , ? , J0 uM ,
and superscript ?H ? stands for transpose and complex
conjugate. The CRA power pattern is
AF = TJH ,
S = F 2 = TJH JTH
Therefore, the BCE can be calculated by the
where ? shows the receiving region, P? is the received power,
and P? is the total transmitting power. A and B are all
M + 1 ? M + 1 matrixes. The elements of A and B are
J0 up J0 uq d?,
J0 up J0 uq d?
In the above equations, p = 0, 1, ? , M and q = 0, 1, ? , M.
Considering the uniform exciting strategy, namely, Im = 1
for m = 0, 1, ? , M, T can be simplified as T = N = 1,
N1, ? , NM . N is the element number matrix. The BCE of
When the ? is defined, the matrices of A and B can be
calculated from u = u0, u1, ? , uM . According to (2), u is
directly determined by applying the radial space matrix
?? = ??1, ??2, ? , ??M . Therefore, from the given ??
and N, the BCEU can be calculated by using (11).
3. Optimization Model of SUACRA
Sparsification ratio ? of a transmitting array is defined as the
ratio of the saved element number of the sparse array to that
of the original one. In this paper, the radial space matrix
?? and the element number matrix N are simultaneously
optimized to improve the ? at the most extent while the
BCEU is kept as high as possible. The optimization model
can be established as follows:
Find ??1, ? , ??M, N1, ? , NM ,
Max f itness = wBBCEU + w?? + pBhB,
S T ??m ? dmin, ?m = 1, 2, ? , M,
where wB and w? are weights of the BCEU and ?, respectively,
and wB + w? = 1. In order to investigate the impact of BCEU
on the ?, the penalty factor pB and penalty function hB have
been introduced in (13). The hB is defined as
hB = min BCEU ? BCE0, 0
BCE0 in (15) is the threshold value of BCEU, which is set
to be 1% lower than the maximum BCEU of the SUACRA, as
denoted by BCEUmax.
From (15), hB < 0 when BCEU < BCE0. In order to
maintain BCEU ? BCE0, pB should be large enough, such
as 105, to magnify the impact of the hB on the fitness
function. In this way, hB could change toward zero in the PSO
procedure. Otherwise, pB should be set to zero to invalidate
The radial space ??m should not be less than dmin and
can be guaranteed by (14), where dmin is the minimum space
between the adjacent elements on the planar array. In order
to ensure the space along ring path between the adjacent
4. Numerical Results
elements larger than dmin, the element number of mth ring
Nm should satisfy this condition:
Nm ? Nmmax =
2d?m?inm , m ? 0,
where ? ? stands for mapping number to the greatest
integer that is less than or equal to the original number. Nm
should not be less than N min m to keep the accuracy of
(4), namely, power error index ?m ? ?30 dB. Moreover, the
size of SUACRA is confined by ?M ? Dt/2, in which Dt is
the expected maximum diameter.
The variable set is ??1, ? , ??M, N1, ? , NM , while the
fitness function is stated in (13). Each particle of the swarm
characterizes a candidate solution, which can be evaluated
by the fitness function. After each iteration, the optimal
particle is obtained, and each particle is updated. When the
termination condition is satisfied, the optimal variable values
can be obtained as the optimal particle over the iteration
history. For details of the PSO, readers could refer to [
and the references therein.
The SUACRAs are optimized by the proposed procedure and
method. The numerical results will be compared with those
of the RLUAA in  and the SAA in  on the condition
of the same transmitting aperture size and the same
4.1. Synthesis of the SUACRA Compared with the RLUAA.
The RLUAA, the first model optimized in , consists of
100 elements distributed arbitrarily on an aperture of
4.5? ? 4.5?, in which the minimum element distance dmin
was 0.4? and the inception angle was ?0 = 0.201. The
optimization of the SUACRA is carried out for the same
transmitting aperture and the same inception angle, which
is denoted by SUACRA 1. In order to further improve the
sparsification ratio of ?, the SUACRA 2 is investigated
with a 1% decrease of BCEU compared to that of SUACRA
1. The numerical results are listed in Table 1, and the layouts
of SUACRA 1 and 2 are given in Figure 3.
In order to improve the sparsification ratio of ? and keep
BCEU as high as possible, in the optimization of SUACRA 1,
wB and w? are set to 0.99 and 0.01, respectively. Penalty
factor pB is set as 0 to invalidate hB. As shown in Table 1,
SUACRA 1 has a BCEU of 91.06%. Compared to the RLUAA,
SUACRA 1 saves 32% elements and has a 1.1% higher BCEU.
With 1% decrease of BCEU of SUACRA 1, the threshold
of BCE0 is set to 90.06%. In order to improve ? at the most
extent, wB and w? are set to 0 and 1, respectively. The penalty
factor pB is set as 105 to guarantee BCEU ? BCE0. The
numerical results show that the ? of the SUACRA 2 is improved by
12% with 0.98% decrease of BCEU, which means that the
element number is reduced to 56% of the original RLUAA.
As shown by the SUACRAs? power patterns given in
Figure 4, SUACRAs can concentrate microwave power on
the receiving region. When ? is close to ?/2, the pattern of
SUACRA 2 is not symmetrical with respect to the center
(? = 0, ? = 0), because the element numbers of rings is close
to the minimum ones. Nevertheless, the difference is just
0.15% between the accurate BCE (90.08% obtained by
(1)) and the approximate one (90.23% obtained by (7)).
Moreover, the power patterns (? = 0) comparison of two
SUACRAs and the original RLUAA are given in Figure 5.
SUACRA 3 SUACRA 4
Compared to the RLUAA, the two SUACRAs have lower
side lobes and higher main lobe levels. Therefore, the
two SUACRAs have higher BCEU of 91.06% and 90.08%,
respectively. The side lobe of SUACRA 1 is a little bit
lower than that of the SUACRA 2, which results in the
difference of 0.98% BCEU.
The initial and optimized parameters of SUACRA 1 and
2 are given in Table 2. The initial ??m and Nm are set to dmin
and Nmmin, respectively. Because the diameter of the 6th ring
of the initialized array will be larger than the maximum
aperture size 4.5?, parameter M is set as 5. The symbol
?del? in Table 2 means that the according ring is deleted
because the diameter of the ring is larger than 4.5?. In the
optimization, the population size is set as 60. As shown in
Figure 6, the fitness values of the two SUACRAs rapidly reach
the convergence points within 50 iterations.
4.2. Synthesis of the SUACRA Compared with the SAA. The
first discrete aperture example, namely, the SAA, in 
consists of 316 elements distributed on a circular aperture
of diameter Dt = 9 5?, in which the inception angle ?0 was
0.107. The optimization of the SUACRA is carried out for
the same transmitting aperture and the same inception angle,
which is denoted by SUACRA 3. In order to further improve
the sparsification ratio of ?, the SUACRA 4 is investigated
with 1% decrease of BCEU compared to that of SUACRA 3.
The numerical results are listed in Table 3, and the layouts
of SUACRA 3 and 4 are given in Figure 7.
Compared to the SAA, SUACRA 3 saves 29.1% elements
and has a bit higher BCEU, while the ? of the SUACRA 4 is
improved by 9.8% with 1% decrease of BCEU.
The power patterns of SUACRA 3 and 4 are given
in Figure 8, and the special power patterns (? = 0)
comparison of SUACRA 3, 4 and the original SAA are
given in Figure 9. It could be seen that the main beams
of the three arrays are almost the same although their side
lobes are different. SUACRA 3 and the SAA have the
same BCEU of 92.5%. The side lobe of SUACRA 4 is a
little bit higher than that of SUACRA 3, which results
in 1% decrease of BCEU.
The initial and optimized parameters of SUACRA 3 and
SUACRA 4 are given in Table 4. The parameter M is set as
11, and the population size is set as 120. As shown in
In this paper, the synthesis of the sparse uniform-amplitude
transmitting array is discussed. As a result, uniform exciting
strategy, reduced element number, and a higher BCE are
achieved simultaneously for the optimal MPT. Accordingly,
the fabrication, maintenance and feed network design of
the transmitting array are greatly simplified without loss of
BCE. The numerical results show that the SUACRAs
optimized by the proposed method have fewer elements than
the random array and the stepped one on the same BCEU.
Compared to the RLUAA, the SUACRA saves 32% elements
with a 1.1% higher BCEU, while the sparsification ratio ? is
improved by 12% with 0.98% decrease of BCEU. Compared
to the SAA, the SUACRA can save 29.1% with a bit higher
BCEU, and the sparsification ratio ? is improved by 9.8%
with 1% decrease of BCEU. The proposed SUACRAs have
higher BCEs, simple array arrangement and feed network,
and could be used as the transmitting array for a
largescale MPT system.
The data used to support the findings of this study are
available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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