Optimal pattern synthesis of circular antenna arrays with improved effective aperture and beam area

Abstract

Circular antenna array (CAA) is one of the most widely used antenna array designs. This paper addresses the design challenges of the CAA with the non-uniform single ring, which is placed in an X-Y plane with the best sidelobe level (SLL) and improved first null beamwidth (FNBW). It has been solved using differential evolution, craziness-based particle swarm optimization (CRPSO), and novel particle swarm optimization (NPSO) techniques. An optimal combination of feeding current and inter-element spacing provides an array pattern with the best SLL and improved FNBW, as well as some other parameter calculations of the antenna array like maximum directivity, maximum effective aperture, total effective aperture, maximum beam area, total beam area, circumference, and radius of the CAAs using these techniques. There are six designs of CAAs with different antenna elements (i.e., 10-, 12-, 16-, 20-, 36-, and 64-elements) which have been taken into account. Simulations are done in MATLAB. Based on various simulation results, we can analyze the performance of SLL and FNBW with other parameters using NPSO and compare them with different techniques of CAAs, as shown in the numerical analysis and simulation result section.

Introduction

It is possible to design having a greater degree of freedom when asymmetric feeding current and non-uniform inter-element spacing are used [1]. The growing electromagnetic environment pollution has spurred the study of array pattern strategies, i.e., to reduce the sidelobe level (SLL) while maintaining the beamwidth. Circular antenna arrays (CAAs) have gained significant attention in wireless communication due to their ability to provide directional radiation patterns with high gain and low SLLs. The circular geometry of the array ensures uniform distribution of power in all directions [2], making it ideal for applications requiring omnidirectional coverage, such as satellite communication, mobile networks, and radar systems for minimizing performance loss in the signal-to-noise ratio caused by unwanted interference [3]. Equally spaced, uniformly stimulated antenna arrays [4, 5] have good directivity but frequently have high SLL. To reduce the SLL, optimize the inter-element spacing and excitation amplitude of the arrays’ elements.

In recent years, CAAs have been extensively studied to improve their performance regarding radiation patterns, beamforming, polarization diversity, and bandwidth of different antenna arrays. To address the demand for long-distance communication in various applications, constructing antennas with very directional properties is required. An antenna array combines radiating elements in an electrical and geometrical configuration. The total field of the antenna array is calculated by adding the fields emitted by each component vectorially [6]. Each element is positioned along the circle’s perimeter in a circular array. As opposed to other array shapes, circular arrays have gained popularity recently since they can execute scans in all directions without significantly changing the beam pattern and offer 360° azimuth coverage. Additionally, because they lack edge elements, compared circular arrays to linear and rectangular arrays, they are less susceptible to mutual coupling [7].

The first metaheuristic technique has been deployed towards the CAA design problem in the work of Panduro et al. [8], who applied genetic algorithm (GA) to optimize the feeding current and between-element spacing of the CAAs’ elements. Mohammad et al. [9] used the particle swarm optimization (PSO) technique and got better results than GA. Munish et al. [10] have applied simulated annealing (SA) for the same design problem and got better results as compared to the PSO technique. CAAs have also been used in radar systems, where they can provide high-resolution imaging and accurate target detection. In Tabesh et al.’s paper [11], a circular array was designed for radar applications, and the results showed that the array had improved resolution and sensitivity compared to traditional linear arrays. Ram et al. in paper [12] have used the seeker optimization algorithm (SOA) to minimize the SLL of CAAs’ design problem. In addition, CAAs have been used in satellite communication systems, where they can provide reliable and high-speed data transmission. In paper [13], a circular array was used for satellite communication, and the results showed that the array had higher gain and lower noise levels than traditional linear arrays.

The primary goal of this article is to design CAAs with a small 3-dB beamwidth and low SLL. These two primary CAA design requirements are attained by selecting the optimum current amplitude excitation and inter-element spacing while maintaining a zero-phase difference between elements. Alberto et al. in paper [14] have discussed the approach to reducing SLL and the isoflux radiation requirement for geostationary Earth orbit (GEO) satellites using the PSO technique. The paper also discusses the challenges and future directions of research in concentric ring arrays (CRAs). Alberto et al. [15] also examined the concentric ring 61 disk patch antenna array for a reconfigurable isoflux pattern, including mutual coupling, using the cavity model at a frequency of 2.8 GHz. In this paper, harmony search algorithm (HSA) and PSO techniques are implemented in the optimization problem. In paper [16], Ibarra et al. designed a sparse CRA for isoflux coverage to the earth’s surface for the low Earth orbit satellite using optimization of the angular position of antenna elements and excitation amplitude. In paper [17], Ibarra et al. have also designed concentric ring antenna arrays (CRAAs) for GEO and medium Earth orbit satellites by getting isoflux radiation. In this paper, trade-off curves have been generated between SLL and isoflux mask error using the differential evolution (DE) technique. In paper [18], HSA and PSO techniques are implemented to get a wide coverage pattern of CRAA.

In this proposal, the variable used to alter the radiation pattern of both SLL and first null beamwidth (FNBW) with a maximum reduction is taken from the feeding current values and inter-element separations of the antenna elements using DE, CRPSO, and novel particle swarm optimization (NPSO) techniques. The NPSO technique is more efficient in improving directivity, effective aperture, and beam area problems for implementation in an application with a wide range.

The rest of the paper is arranged as follows: Mathematical formulation of CAA section presents the mathematical formulation of CAA. Formulation of fitness function section briefly discusses the formulation of the fitness function. In Optimization techniques employed section, the employed optimization technique has been discussed for the design of CAA. Numerical analysis and simulation results are explained in Section Numerical analysis and simulation results. Finally, concludes the paper.

Mathematical formulation of CAA

A CAA is a group of single antenna elements with non-uniform or uniform spacing arranged circularly. The current feeding distribution and inter-element spacing are identical for all the radiating elements in uniform CAA. A non-uniform CAA, however, exhibits tapering distributions. The standard form of a circular array with the “$N$” number of elements in the X-Y plane with radius “$a$” and finding at a point P in the far field is shown in Fig. 1. The array radiation pattern may be mathematically represented by its array factor as [19] if the components of the CAA are conceived of as isotropic radiating sources.

(1)\begin{equation}AF(\theta ,\varphi ) = \sum\limits_{n = 1}^N {{I_n}.{e^{j[\beta .a.\sin \theta .\cos (\varphi - {\varphi _n}) + {\alpha _n}]}}} \end{equation}

Figure 1. A non-uniform N-element CAA placed in X-Y plane with radius “a”.

where

${I_n}$ is the excitation amplitudes of the array, β is the propagation constant $\left( {\beta {\text{ }} = {\text{ }}2\pi /\lambda } \right)$, λ is the wavelength, θ indicates the incident angle of electromagnetic wave, ϕ is the azimuthal angle, ${\varphi _n}$ is the angular position of the nth elements, ${\alpha _n}$ is the excitation phase of the nth elements.

(2)\begin{equation}{\varphi _n} = \left( {2\pi /\beta a} \right)\sum\limits_{k = 1}^n {{d_k}} \end{equation}

(3)\begin{equation}{\alpha _n} = - \beta .a.\cos ({\varphi _0} - {\varphi _n})\end{equation}

(4)\begin{equation}\beta .a = \sum\limits_{k = 1}^n {{d_k}} = \frac{{2\pi a}}{\lambda }\end{equation}

After putting the value in Equation 1 from Equation 3, the total array factor can be as in Equation (5).

(5)\begin{equation}AF\left( {\theta ,\varphi } \right) = \sum\limits_{n = 1}^N {{I_n}.{e^{j.\beta .a\left[ {\sin \theta .\cos \left( {\varphi - {\varphi _n}} \right) - \cos \left( {{\varphi _0} - {\varphi _n}} \right)} \right]}}} \end{equation}

where

$d = \left[ {{d_1},{\text{ }}{d_2},{\text{ }}{d_3}, \ldots \ldots ,{\text{ }}{d_N}} \right]$, and $I = {\text{ }}\left[ {{I_1},{\text{ }}{I_2},{\text{ }}{I_3}, \ldots \ldots ,{\text{ }}{I_N}} \right]$. The excitation amplitude of nth elements and the distance from nth element to (n + 1)th element are represented by I_n and d_n, respectively. In our design, at the most top radiation angle ϕ ₀ = 0⁰ in $\phi = {\text{ }}[ - \pi ,{\text{ }}\pi ]$, the global maximum is getting. To get the radiation pattern with the maximum reduction of SLL and improved FNBW in the preferred direction ϕ, the study’s goal is to choose the best combination of I_n and d_n values to modify the antenna design.

The directivity of CAAs is described by [15]

(6)\begin{equation}DIR = \frac{{P\left( {\theta ,\varphi } \right)}}{{\frac{1}{{4\pi }}\int\limits_{4\pi }^\infty {P\left( {\theta ,\varphi } \right)d\Omega } }}\end{equation}

For a half-wave dipole antenna array, the effective aperture is given by [20]

(7)\begin{equation}{A_{eff}} = \frac{{DIR.{\lambda ^2}}}{{4\pi }}\end{equation}

The beam area is given by [21]

(8)\begin{equation}\Omega (sr) = \frac{{4\pi }}{{DIR}}\end{equation}

Formulation of fitness function

The motivating factor of any optimization technique is the fitness or objective function. It is denoted as CF, given in Equation (9). The fitness or cost values of each solution string generated throughout the search process are computed using it. In this design problem, each solution string represents different element excitation and inter-element spacing for different test cases. The objective is to design 8-, 10-, 12-, 16-, 20-, and 36-element CAAs with the lowest SLL, improved FNBW, and enhanced directivity. The fitness is designed as follows to achieve the ultimate goal:

(9)\begin{align}CF =& {W_1} \times \frac{{\prod\limits_{\varphi = - {{180}^o}}^{LFN} {\left| {AF\left( {{\varphi _{sll}},{I_n}} \right)} \right|} }}{{{{\left| {AF\left( {{\varphi _0},{I_n}} \right)} \right|}_{\max }}}} + {W_2} \nonumber\\ &\times \frac{{\prod\limits_{\varphi = RFN}^{{{180}^o}} {\left| {AF\left( {{\varphi _{sll}},{I_n}} \right)} \right|} }}{{{{\left| {AF\left( {{\varphi _0},{I_n}} \right)} \right|}_{\max }}}} + {W_3} \times \left| {FNB{W_0} - FNB{W_R}} \right| \nonumber\\ &+ {W_4} \times \frac{1}{{DIR}} + {W_5} \times \left| {{C_0} - {C_R}} \right|\end{align}

where $C = \sum\limits_{k = 1}^N {{d_k}} $; d_k is the inter-element spacing [10].

W ₁, W ₂, W ₃, W ₄, and W ₅ are the weighted coefficients with equal values chosen, such as the optimal value of SLL remains more imposing than the optimal value of FNBW and CF never being negative. LFN is the left side of the first null, and RFN is the right side of the first null. ${\varphi _{sll}}$ is the maximum SLL angle of the main beam side lobe on either side (i.e., LFN and RFN). FNBW₀ is the computed first null beamwidth in the non-uniform case, and FNBW_R is the required first null beamwidth in the uniform case. ${\left. {FNB{W_0} \lt FNB{W_R}} \right|_{{I_n} = 1}}$ condition must be satisfied to get an optimal set of feeding current and inter-element spacing otherwise discarded. DIR_max is the maximum directivity. C₀ is the computed circumference, and C_R is the required circumference.

Optimization techniques employed

The DE and craziness-based particle swarm optimization (CRPSO) techniques are briefly discussed in papers [22] and [23], respectively. Therefore, the steps of DE and CRPSO are not discussed due to the space limitation. In the context of PSO, the velocity and position of the particle vector can be mathematically adjusted using the following equations, as described in papers [24–26].

(11)\begin{equation}V_i^{k + 1} = w \times V_i^k + {c_1} \times {r_1} \times \left( {pb_i^k - X_i^k} \right) + {c_2} \times {r_2} \times \left( {gb_i^k - X_i^k} \right)\end{equation}

(12)\begin{equation}X_i^{k + 1} = X_i^k + V_i^{k + 1}\end{equation}

where ‘r ₁’ and ‘r ₂’ denote random values within the range of zero to one; The variables ‘c ₁’ and ‘c ₂’ represent the local acceleration and global acceleration constants of the particle, respectively. The symbol “w” stands for a weighting factor; “V_i^k” refers to the velocity of the ith particle in the kth iteration; “pb_i^k” corresponds to the local best of the ith particle in the kth iteration; “gb_i^k” signifies the global best among all particles in the kth iteration; lastly, “X_i^k” pertains to the current position of the ith particle in the kth iteration.

When considering Equation (11), it is important to note that the two random values, r ₁ and r ₂, are interrelated. If both of these random parameters have large values, it can lead to excessive reliance on individual and collective experiences, potentially causing the particle’s local optimum value to shift too far. On the other hand, if both r ₁ and r ₂ have small values, there is an incomplete utilization of individual and collective experiences, which, in turn, slows down the convergence speed of the optimization process.

Utilizing the modification described below leads to a significant enhancement in the global search capacity of the traditional PSO. This enhanced version of PSO is referred to as NPSO. NPSO technique is more efficient in global search, has few parameters to adjust, and has a high convergence speed compared to DE and CRPSO techniques.

To address this, instead of using independent r ₁ and r ₂, a single random number, r ₁, is selected. By doing so, when r ₁ is large, (1 − r ₁) is small and vice versa. However, to maintain control, another random value, r ₂, is introduced. In certain exceptional situations, akin to fish schooling for food, after the particle’s location has changed in accordance with Equation (12), a fish may not swim in the region it believes offers the best chance of finding food due to inertia. Alternatively, it may head in a direction opposing the expected prosperous regions. Therefore, in the subsequent step, the fish must adjust its travel direction to return to the promising area, and the presence of “sign(r₃)” serves this purpose. Consequently, both the analytical and social components undergo modification,

where

(13)\begin{equation}\begin{matrix} {sign\left( {{r_3}} \right)} & { = - 1} & {when} & {{r_3} \le 0.05} \\ {} & { = 1} & {when} & {{r_3} \gt 0.05} \\ \end{matrix} \end{equation}

Ultimately, the enhanced velocity of the “ith” particle is formulated as [14].

(14)\begin{align}V_i^{k + 1} =& r{d_2} \times sign\left( {{r_3}} \right) \times V_i^k + \left( {1 - {r_2}} \right) \times {c_1} \times {r_1} \times \left( {pb_i^k - X_i^k} \right) \nonumber\\ &+ \left( {1 - {r_2}} \right) \times {c_2} \times \left( {1 - {r_1}} \right) \times \left( {gb_i^k - X_i^k} \right)\end{align}

The NPSO technique is implemented for the optimization of feeding current and inter-element spacing of CAA.

Numerical analysis and simulation results

In this section, the numerical outcomes of several CAA designs are described. The radiation pattern of CAA with the primary lobe pointed to ϕ₀ = 0° is considered. It is assumed that there are six non-uniform CAAs with 10-, 12-, 16-, 20-, 36-, and 64-elements. The NPSO algorithm has been implemented with a population size of 120 and an iteration size of 400. The best-proven control parameter values of the NPSO technique have been discussed in Table 13. Optimized feed current and inter-element spacing with reduced SLL and improved FNBW are discussed below in test case 1, test case 2, test case 3, test case 4, test case 5, and test case 6 for 10-, 12-, 16-, 20-, 36-, and 64-element non-uniform CAA, respectively. Based on the simulation result, it is clear that the NPSO technique is more efficient in getting improved directivity, effective aperture, and beam area problems for implementation in an application with a wide range concerning DE, CRPSO, and previous work.

Test case 1: N = 10 elements

In test case 1, N = 10 elements of non-uniform CAAs have been considered. The DE, CRPSO, and NPSO techniques have been deployed to optimize excitation amplitude and inter-element spacing with low SLL and improved FNBW. From Table 1, it is clear that NPSO is outperforming concerning SLL of −14.140 dB as compared to uniform, DE, CRPSO, GA [8], SA [10], BBO [2], SOA [12], PSO [9], OBA [27], and FA [28] with SLL of −7.93 dB, −11.204 dB, −11.810 dB, −9.811 dB, −13.00 dB, −13.95 dB, −12.83 dB, −12.307 dB, −14.06 dB, and −13.30 dB, respectively. NPSO is also outperforming concerning FNBW of 48.24° compared to uniform FNBW of 56.12°. After comparing these values with previously published results, it can be analyzed that the NPSO technique is more efficient in getting low SLL, improved directivity, and FNBW. Based on these optimized values, other parameters of CAAs can be calculated, as shown in Table 2. Figure 2 shows the radiation pattern with maximum SLL reduction and improved FNBW using DE, CRPSO, and NPSO. From Fig. 2, it is clear that maximum SLL was reduced using the NPSO technique as compared to DE and CRPSO techniques. Figure 3 shows the convergence curve using DE, CRPSO, and NPSO. From Fig. 3, it is clear that the NPSO technique is more efficient converges than the DE and CRPSO techniques. Figure 4 shows the optimal values of feeding current and inter-element spacing of 10-element non-uniform CAAs using the NPSO technique.

Table 1. Comparison of results obtained using the NPSO technique with other techniques for N = 10 elements

NR* = not reported.

Table 2. Other parameters’ comparison using NPSO technique with different algorithms for N = 10 elements

Figure 2. Radiation pattern of 10-element non-uniform CAAs.

Figure 3. Convergence curve of 10-element non-uniform CAAs.

Figure 4. Optimal values of 10-element non-uniform CAAs using NPSO technique.

Test case 2: N = 12 elements

In test case 2, N = 12 elements of non-uniform CAAs have been considered. The DE, CRPSO, and NPSO techniques have been deployed to optimize excitation amplitude and inter-element spacing with low SLL and improved FNBW. From Table 3, it is clear that NPSO is outperforming concerning SLL of −14.401 dB as compared to uniform, DE, CRPSO, GA [8], SA [10], BBO [2], SOA [12], PSO [9], OBA [27], and FA [28] with SLL of −7.90 dB, −11.610 dB, −13.130 dB, −11.830 dB, −13.910 dB, −13.95 dB, −13.77 dB, −13.670 dB, −14.30 dB, and −14.21 dB, respectively. NPSO is also outperforming concerning FNBW of 36.72° compared to uniform FNBW of 45.36°. After comparing these values with previously published results, it can be analyzed that the NPSO technique is more efficient in getting low SLL, improved directivity, and FNBW. Based on these optimized values, other parameters of CAAs can be calculated, as shown in Table 4. Figure 5 shows the radiation pattern with maximum SLL reduction and improved FNBW using DE, CRPSO, and NPSO. From Fig. 5, it is clear that maximum SLL was reduced using the NPSO technique as compared to DE and CRPSO techniques. Figure 6 shows the convergence curve using DE, CRPSO, and NPSO. From Fig. 6, it is clear that the NPSO technique is more efficient and converges than the DE and CRPSO techniques. Figure 7 shows the optimal values of feeding current and inter-element spacing of 12-element non-uniform CAAs using the NPSO technique.

Table 3. Comparison of results obtained using the NPSO technique with other techniques for N = 12 elements

NR* = not reported.

Table 4. Other parameters’ comparison using NPSO technique with different algorithms for N = 12 elements

Figure 5. Radiation pattern of 12-element non-uniform CAAs.

Figure 6. Convergence curve of 12-element non-uniform CAAs.

Figure 7. Optimal values of 12-element non-uniform CAAs using NPSO technique.

Test case 3: N = 16 elements

In test case 3, N = 16 elements of non-uniform CAAs have been considered. The DE, CRPSO, and NPSO techniques have been deployed to optimize excitation amplitude and inter-element spacing with low SLL and improved FNBW. From Table 5, it is clear that NPSO is outperforming concerning SLL of −14.524 dB as compared to uniform, DE, and CRPSO with SLL of −7.94 dB, −11.926 dB, and −13.051 dB, respectively. NPSO is also outperforming concerning FNBW of 31.68° compared to uniform FNBW of 34.56°. After reaching the NPSO’s value with other values, it can be analyzed that the NPSO technique is more efficient in getting low SLL, improved directivity, and FNBW. Based on these optimized values, other parameters of CAAs can be calculated, as shown in Table 6. Figure 8 shows the radiation pattern with maximum SLL reduction and improved FNBW using DE, CRPSO, and NPSO. From Fig. 8, it is clear that maximum SLL was reduced using the NPSO technique as compared to DE and CRPSO techniques. Figure 9 shows the convergence curve using DE, CRPSO, and NPSO. From Fig. 9, it is clear that the NPSO technique more efficiently converges than the DE and CRPSO techniques. Figure 10 shows the optimal values of feeding current and inter-element spacing of 16-element non-uniform CAAs using the NPSO technique.

Table 5. Comparison of results obtained using the NPSO technique with other techniques for N = 16 elements

Table 6. Other parameters’ comparison using NPSO technique with different algorithms for N = 16 elements

Figure 8. Radiation pattern of 16-element non-uniform CAAs.

Figure 9. Convergence curve of 16-element non-uniform CAAs.

Figure 10. Optimal values of 16-element non-uniform CAAs using NPSO technique.

Test case 4: N = 20 elements

In test case 4, N = 20 elements of non-uniform CAAs have been considered. The DE, CRPSO, and NPSO techniques have been deployed to optimize excitation amplitude and inter-element spacing with low SLL and improved FNBW. From Table 7, it is clear that NPSO is outperforming concerning SLL of −14.770 dB as compared to uniform, DE, and CRPSO with SLL of −7.91 dB, −12.624 dB, and −13.695 dB, respectively. NPSO is also outperforming concerning FNBW of 26.28° compared to uniform FNBW of 27.36°. After comparing the NPSO’s value with other values, it can be analyzed that the NPSO technique is more efficient in getting low SLL, improved directivity, and FNBW. Based on these optimized values, other parameters of CAAs can be calculated, as shown in Table 8. Figure 11 shows the radiation pattern with maximum SLL reduction and improved FNBW using DE, CRPSO, and NPSO. From Fig. 11, it is clear that maximum SLL was reduced using the NPSO technique as compared to DE and CRPSO techniques. Figure 12 shows the convergence curve using DE, CRPSO, and NPSO. From Fig. 12, it is clear that the NPSO technique is more efficiently converged than the DE and CRPSO techniques. Figure 13 shows the optimal values of feeding current and inter-element spacing of 20-element non-uniform CAAs using the NPSO technique.

Table 7. Comparison of results obtained using the NPSO technique with other techniques for N = 20 elements

Table 8. Other parameters’ comparison using NPSO technique with different algorithms for N = 20 elements

Figure 11. Radiation pattern of 20-element non-uniform CAAs.

Figure 12. Convergence curve of 20-element non-uniform CAAs.

Figure 13. Optimal values of 20-element non-uniform CAAs using NPSO technique.

Test case 5: N = 36 elements

In test case 5, N = 36 elements of non-uniform CAAs have been considered. The DE, CRPSO, and NPSO techniques have been deployed to optimize excitation amplitude and inter-element spacing with low SLL and improved FNBW. From Table 9, it is clear that NPSO is outperforming concerning SLL of −15.145 dB as compared to uniform, DE, and CRPSO with SLL of −12.96 dB, −13.463 dB, and −14.381 dB, respectively. NPSO is also outperforming concerning FNBW of 12.96° compared to uniform, DE, and CRPSO with FNBW of 15.12°, 14.76°, and 13.68°, respectively. After comparing the NPSO’s value with other values, it can be analyzed that the NPSO technique is more efficient in getting low SLL, improved directivity, and FNBW. Based on these optimized values, other parameters of CAAs can be calculated, as shown in Table 10. Figure 14 shows the radiation pattern with maximum SLL reduction and improved FNBW using DE, CRPSO, and NPSO. From Fig. 14, it is clear that maximum SLL was reduced using the NPSO technique as compared to DE and CRPSO techniques. Figure 15 shows the convergence curve using DE, CRPSO, and NPSO. From Fig. 15, it is clear that the NPSO technique is more efficient converges than the DE and CRPSO techniques. Figure 16 shows the optimal values of feeding current and inter-element spacing of 36-element non-uniform CAAs using the NPSO technique.

Table 9. Comparison of results obtained using the NPSO technique with other techniques for N = 36 elements

Table 10. Other parameters’ comparison using NPSO technique with different algorithms for N = 36 elements

Figure 14. Radiation pattern of 36-element non-uniform CAAs.

Figure 15. Convergence curve of 36-element non-uniform CAAs.

Figure 16. Optimal values of 36-element non-uniform CAAs using NPSO technique.

Test case 5: N = 64 elements

In test case 6, N = 64 elements of non-uniform CAAs have been considered. The DE, CRPSO, and NPSO techniques have been deployed to optimize excitation amplitude and inter-element spacing with low SLL and improved FNBW. From Table 11, it is clear that NPSO is outperforming concerning SLL of −16.105 dB as compared to uniform, DE, and CRPSO with SLL of −7.89 dB, −14.296 dB, and −14.743 dB, respectively. NPSO is also outperforming concerning FNBW of 8.64° compared to uniform, DE, and CRPSO with FNBW of 9.28°, 9.14°, and 9.00°, respectively. After comparing the NPSO’s value with other values, it can be analyzed that the NPSO technique is more efficient in getting low SLL, improved directivity, and FNBW. Based on these optimized values, other parameters of CAAs can be calculated, as shown in Table 12. Figure 17 shows the radiation pattern with maximum SLL reduction and improved FNBW using DE, CRPSO, and NPSO. From Fig. 17, it is clear that maximum SLL was reduced using the NPSO technique as compared to DE and CRPSO techniques. Figure 18 shows the convergence curve using DE, CRPSO, and NPSO. From Fig. 18, it is clear that the NPSO technique is more efficiently converged than the DE and CRPSO techniques. Figure 19 shows the optimal values of feeding current and inter-element spacing of 64-element non-uniform CAAs using the NPSO technique.

Table 13 shows all optimizing parameter values for different antenna design problems using the NPSO technique.

Table 11. Comparison of results obtained using the NPSO technique with other techniques for N = 64 elements

Table 12. Other parameters’ comparison using NPSO technique with different algorithms for N = 64 elements

Table 13. Optimizing parameter of NPSO technique for N = 10-, 12-, 16-, 20-, 36, and 64-elements

Figure 17. Radiation pattern of 64-element non-uniform CAAs.

Figure 18. Convergence curve of 64-element non-uniform CAAs.

Figure 19. Optimal values of 64-element non-uniform CAAs using NPSO technique.

Conclusion

Many applications may find circular arrays’ relatively high SLLs unacceptable. Increased interference and a diminished capacity to distinguish between desired and unwanted signals can be caused by sidelobes. Because of their circular form, CAAs can be more challenging to design and optimize than linear arrays. Achieving the required radiation patterns may be difficult due to this intricacy, particularly when exact control over the SLLs and beam shape is needed. This article uses a meta-heuristic approach to discuss an unusual method of building antenna arrays. DE, CRPSO, and NPSO techniques have been deployed to get optimal feeding current and inter-element spacing with ultra-low SLL and improved FNBW. After analyzing the result, the NPSO technique is more efficient for different design problems of CAAs. For 10-, 12-, 16-, 20-, 36-, and 64-elements, a maximum reduction of SLL and improved FNBW can be obtained using the NPSO technique. Maximum directivity of the antenna array has been achieved. The circular design may limit the array’s capacity to gather and transmit energy efficiently because it may result in a lower aperture efficiency when compared to alternative array layouts. The maximum effective aperture of the half-wave dipole antenna array is recorded as shown in test cases. The maximum beam area covered by the antenna array is shown in the table also. All these values are the best using the NPSO technique compared to other methods.

Further research will be planned to handle various geometry and constraints.