Research Article | Open Access
Synthesis of the Sparse Uniform-Amplitude Concentric Ring Transmitting Array for Optimal Microwave Power Transmission
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 uniform-amplitude 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 large-scale transmitting array design.
Besides the exciting strategy, the sparsification of a large-scale 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 [11–13] 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, in [14, 15]. 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 in [14–16] 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 large-scale array.
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 . All elements are excited by the identical phase and amplitude.
The receiving array is in the far region of the transmitting array, namely, , where is the distance between transmitting and receiving array, is the radius of transmitting array, and is the wavelength. As a result, the CRA array factor can be written as  where is the array factor of mth ring, , , and represent the radius, the element number and the excitation amplitude of the mth ring, respectively, denotes the wavenumber, and is the azimuth angle of the nth element located on the mth ring.
elements are distributed with the same space on the mth ring. When is large enough, array factor of the mth ring can be approximated as 
In (4), Tm = ImNm and J0 is the zero-order Bessel function of the first kind. To evaluate the precision, the power error index is defined as where is the visible region (0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π) of the transmitting array. Under the constraint of keeping a higher precision of (4), the minimum element number can be found by increasing from 2 till to reach ςm ≤ −30 dB. The line labeled by numerical results in Figure 2 lays out the for radiuses ranging from 0.5λ to 19.5λ with an interval of 0.5λ.
By applying the fitting method, can be approximated to the following formula: where “” stands for mapping number to the least integer that is greater than or equal to the original number. The 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 on a ring of a large-scale array.
With the element numbers on each ring not less than the minimum ones, namely, , the CRA array factor of (1) can be rewritten as where , , and superscript “” stands for transpose and complex conjugate. The CRA power pattern is
Therefore, the BCE can be calculated by the following formula: where shows the receiving region, is the received power, and is the total transmitting power. and are all matrixes. The elements of and are calculated as
In the above equations, and . Considering the uniform exciting strategy, namely, for , can be simplified as . is the element number matrix. The BCE of the uniform-amplitude transmitting array, denoted by BCEU, can be achieved by
When the is defined, the matrices of and can be calculated from . According to (2), is directly determined by applying the radial space matrix . Therefore, from the given Δρ 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 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: where and are weights of the BCEU and , respectively, and . In order to investigate the impact of BCEU on the , the penalty factor and penalty function have been introduced in (13). The is defined as
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), when . In order to maintain , should be large enough, such as 105, to magnify the impact of the on the fitness function. In this way, could change toward zero in the PSO procedure. Otherwise, should be set to zero to invalidate the .
The radial space Δρm should not be less than and can be guaranteed by (14), where is the minimum space between the adjacent elements on the planar array. In order to ensure the space along ring path between the adjacent elements larger than , the element number of mth ring should satisfy this condition: where “” stands for mapping number to the greatest integer that is less than or equal to the original number. should not be less than to keep the accuracy of (4), namely, power error index ςm ≤ −30 dB. Moreover, the size of SUACRA is confined by , in which is the expected maximum diameter.
The variable set is , 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.
4. Numerical Results
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 inception angle.
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 4.5λ × 4.5λ, in which the minimum element distance was 0.4λ and the inception angle was θ0 = 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, and are set to 0.99 and 0.01, respectively. Penalty factor is set as 0 to invalidate . 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, and are set to 0 and 1, respectively. The penalty factor 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 θ is close to π/2, the pattern of SUACRA 2 is not symmetrical with respect to the center (θ = 0, φ = 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 (φ = 0) comparison of two SUACRAs and the original RLUAA are given in Figure 5. 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 and are set to and , respectively. Because the diameter of the 6th ring of the initialized array will be larger than the maximum aperture size 4.5λ, parameter 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 4.5λ. 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 , in which the inception angle 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 (φ = 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 is set as 11, and the population size is set as 120. As shown in Figure 10, the fitness values of the two SUACRAs reach the convergence points within 150 iterations.
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 large-scale 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.
This work was supported by Natural Science Foundation of China (Grant no. 61271062 and Grant no. 61771300).
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