Preamble

Vol. 43, No. 3

September 2025

PROGRESS IN ASTRONOMY Vol. 43, No. 3 Sept., 2025
doi: 10.3969/j.issn.1000-8349.2025.03.10

Optimized Design of Surface Antenna Panel Fitting Based on Particle Swarm Algorithm

ZHANG Wenyang¹,², WU Yang¹,², LIU Shengwen¹,²
(1. The 54th Research Institute of China Electronics Technology Group Corporation, Shijiazhuang 050081, China;
2. Hebei Key Laboratory of Radio Astronomy Technology, Shijiazhuang 050081, China)

Abstract

Taking a 40 m offset Cassegrain antenna as an example, this paper analyzes different panel fitting schemes to investigate the relationship between panel design and surface accuracy. We verify that the surface accuracy of the reflector is proportional to the square of both the circumferential and axial lengths of the panels. Simultaneously, simulation software is used to validate different schemes and explore the impact of panel size variations on antenna gain. The results demonstrate that the annular spacing of sub-panels affects only the horizontal energy distribution, while the segmentation width within the same ring determines the vertical energy distribution. The particle swarm optimization algorithm is employed to optimize the panel segmentation scheme, thereby improving surface accuracy to reduce far-field sidelobes and mitigate radio frequency interference. Simulation experiments on a 3 m panel scheme confirm the reliability of this approach, compensating for 3.6% of the energy loss in the central beam caused by surface deformation while effectively reducing newly generated sidelobe power by 1.15 dB, 0.77 dB, 0.52 dB, 1.02 dB, and 2.1 dB, respectively.

Keywords: surface antenna; panel fitting; surface accuracy; large aperture antenna; optimization model
Classification: P111.44
Document code: A

1 Introduction

Reflector antennas represent a major branch of antenna design, offering advantages such as strong directivity and low manufacturing costs, which have led to their widespread application in satellite communications and radio telescopes. With technological advancements driving new explorations into deep space, increasingly stringent requirements have been placed on antenna sensitivity and resolution. For reflector antennas, increasing the aperture size is one of the most effective ways to enhance sensitivity and resolution.

In practical engineering applications, large-aperture reflector antennas must be fabricated in segments, where the number and variety of panels directly influence manufacturing costs. To reduce expenses, small flat panels are often substituted for the ideal continuous reflector surface [1], introducing deviations from the ideal parabolic surface from the very beginning of the design process [2]. These deviations, commonly expressed as root-mean-square (RMS) values, are referred to as surface accuracy. Higher surface accuracy means the designed reflector more closely approximates the ideal parabolic surface, but this imposes stricter constraints on the dimensions of the flat panels. Consequently, the trade-offs between panel variety and quantity, their impact on surface accuracy, and the ultimate effect on the electrical performance of the reflector antenna have become critical issues.

To better observe the 21 cm hyperfine transition emission from neutral hydrogen at different redshifts, Brazil has led an international collaboration to design and construct the BINGO (Baryon acoustic oscillations for Integrated Neutral Gas Observations) radio telescope. BINGO employs neutral hydrogen intensity mapping technology to precisely measure baryon acoustic oscillations in the key redshift range (z = 0.13–0.45). Its primary objectives are to investigate the nature of dark energy, test the standard cosmological model, and accurately measure key cosmological parameters. This radio telescope features a 40 m offset Cassegrain antenna [3] operating in the 0.98–1.26 GHz frequency band. Using small panels to approximate the reflector surface can effectively reduce construction costs. This paper models the reflector surface of the BINGO telescope and calculates variations in surface accuracy and antenna radiation patterns under different segmentation schemes.

2 Theoretical Foundation

Variations in surface accuracy directly affect the optical path of electromagnetic waves from the far field to the feed, thereby influencing the focal plane field distribution. Consequently, surface accuracy serves as a key criterion in antenna design, making accurate and rapid antenna modeling and surface accuracy calculation essential first steps [4]. Current research on antenna modeling primarily focuses on measuring and modeling fabricated antennas using Coons surface fitting [5] or least-squares-based surface fitting [6].

This paper addresses surface error modeling during the antenna design phase and conducts electrical performance simulations. This section introduces the computational methods employed and the approach to antenna electrical performance simulation.

2.1 Calculation of Surface Accuracy for Arbitrary Polygons

Any polygon can be decomposed into multiple triangles, transforming the problem into one of selecting three points on the parabolic surface, connecting them to form a plane, and calculating the deviation between this plane and the original parabolic surface. To simplify calculations, we introduce global and local coordinate systems [7]. The global coordinate system is established with the parabolic vertex as the origin, while the local coordinate system uses the fitted plane as its reference frame, requiring only the projection of the parabolic surface onto the local coordinate system.

The local coordinate system can be considered as derived from the global coordinate system through translation and rotation. For any point P₁ in space, the coordinate transformation relationship is:
𝑿𝑩 = 𝑩 𝑨 𝑹 · ( 𝑿𝑨 − 𝑩 𝑹 = 𝑩

where the coordinates in the global and local systems are composed, and 𝑩 𝑨 𝑹 = ( ˆ𝑥′, ˆ𝑦′, ˆ𝑧′). 𝑨 𝑹⁻¹ is the rotation matrix from global to local coordinates, formed by the unit vectors of the local coordinate system in the global frame.

With the transformation relationship between the two coordinate systems established, substituting the parabolic surface expression in the global coordinate system yields the functional relationship of the original surface in the local coordinate system, denoted as 𝑓 (𝑥′, 𝑦′, 𝑧′) = 0. Similarly, the deformed surface expression is 𝑔(𝑥′, 𝑦′, 𝑧′) = 0. The domain of (𝑥′, 𝑦′) is then determined by the triangle's position in the local coordinates, and the difference in distances to the 𝑥′𝑂′𝑦′ plane at any point represents the error at that location.

Since all three vertices of the triangle lie on the paraboloid of revolution, the relationship between their coordinates and the focal length 𝑓 can also be obtained:

Substituting equation (2) into equation (1) further simplifies the calculation to yield:
𝑧 − 𝑧₁ = [2𝑥₁(𝑥 − 𝑥₁) + 2𝑦₁(𝑦 − 𝑦₁) + (𝑥 − 𝑥₁)² + (𝑦 − 𝑦₁)²]/4𝑓

Using equations (2) and (3), the error value at any point can be expressed as:
△𝑧′ = 𝑧′ 𝑓 − 𝑧′

Through integration, the mean square error over the triangular region can be calculated:
△𝑧′d𝑥′d𝑦′ d𝑥′d𝑦′

2.2 Panel Segmentation Scheme and Electrical Simulation

This experiment uses the BINGO telescope as a case study, focusing analysis solely on the main reflector. To reduce panel variety, an annular segmentation scheme is adopted where panels within each ring are identical. The length and width of the panel projections in the 𝑋𝑂𝑌 plane are adjusted to vary the number of panel types and total count.

[FIGURE:1] shows the vertex projections in the 𝑋𝑂𝑌 plane for ring spacings of 2 m and 5 m with a panel width of 2 m. (a) illustrates the segmentation scheme with 2 m ring spacing and 2 m width; (b) shows the scheme with 5 m ring spacing and 2 m width.

The reflector surface point cloud data is imported into the GRASP simulation software to construct the reflector model using triangular planar facets, as shown in the schematic diagram in [FIGURE:2]. During simulation, the feed and sub-reflector remain unchanged while only the main reflector shape is varied to investigate changes in surface accuracy and far-field radiation patterns under different schemes.

2.3 Particle Swarm Optimization Algorithm

For reflector antennas approximated by small panels, the overall surface accuracy is positively correlated with electrical performance—higher surface accuracy yields performance closer to that of an ideal surface and thus higher gain. Surface accuracy is directly influenced by the panel segmentation scheme. This paper first compares the electrical performance of different panel sizes, then performs fine adjustments after determining the basic dimensions to compensate the segmented reflector and improve antenna gain.

In segmentation point selection, the shape of each panel is interdependent, and the relationship between surface accuracy and point positions cannot be expressed by simple functional forms. Therefore, this optimization problem can be abstracted as a multivariate unknown function optimization problem. To address this, the particle swarm optimization algorithm [8] is introduced to search for a relatively optimal solution through a combination of global and local search. Finally, electrical simulation of the reflector antenna validates the feasibility of using the particle swarm optimization algorithm to improve antenna gain.

3 Experimental Verification

The previous chapter introduced the antenna design methodology. This chapter adjusts panel dimensions to calculate surface accuracy under various conditions and compares antenna electrical performance across different schemes.

3.1 Relationship Between Panel Size and Surface Accuracy

In engineering practice, engineers typically calculate the surface accuracy of fitted planes using a discretization method—selecting points on the panel according to certain rules and computing their distances to the ideal surface. This approach is limited by point selection methods and density; only with uniform and sufficiently dense sampling can the results be considered accurate. In contrast, this paper employs a coordinate transformation method that preserves the functional relationship of the original surface coordinates, enabling more accurate and efficient calculations.

Initially, segmentation uses approximately square rectangular panels with equal ring spacing and width, gradually increasing these dimensions to study surface accuracy variations. The results are shown in [FIGURE:3]. As panel side length increases, the rate of RMS error growth accelerates. To understand this trend, we calculated the slope variation rate, shown in [FIGURE:3b), revealing a quadratic relationship between panel surface accuracy and side length.

The first experiment used near-square panels; the square of the side length equals panel area, making surface accuracy linearly proportional to area for square panels. For rectangles with unequal sides, the relationship between area and surface accuracy requires further experimental verification. We separately varied the annular spacing and width of the panels during segmentation to observe surface accuracy changes, as shown in [FIGURE:4]. Both annular spacing and width exhibit quadratic relationships with surface accuracy. Using 𝑥 and 𝑦 to represent annular spacing and width respectively, surface accuracy 𝛿 follows:
𝛿 = 𝑎𝑥² + 𝑏𝑦²

where 𝑎 and 𝑏 are constants related to the ideal reflector shape.

3.2 Relationship Between Segmentation Scheme and Radiation Pattern

The previous section analyzed the relationship between surface accuracy and panel segmentation schemes, but surface accuracy and antenna electrical performance are not linearly related. Therefore, this section validates different segmentation schemes through simulation.

Following the previous section's configuration, we compare the electrical performance of different square panel sizes by examining horizontal radiation patterns after reflection from various reflector surfaces. Multiple sidelobes with significantly increased gain are observed, which we term "bulges." The positions of these bulges and their impact on the main beam are recorded in [TABLE:1]. [FIGURE:5] shows the co-polarized radiation pattern, clearly illustrating that as panel side length increases, sidelobe bulges appear with varying severity, and their quantity is proportional to the rectangular side length.

The first bulge appears closer to the beam center as panel size increases. When panel side length reaches 5 m, this sidelobe appears within 3°, severely impacting target detection accuracy. At the beam center, gain degradation occurs at a faster rate than surface accuracy deterioration as panel side length increases.

Similarly, comparing vertical radiation patterns in [FIGURE:6] shows that significant bulges only appear when panel side length exceeds 3 m, with positions consistent with the horizontal direction. This further confirms that panel segmentation causes abrupt gain variations at different angles, related to square panel side length.

To further investigate segmentation shape effects on antenna gain, we separately varied panel length and width for electrical simulation comparison. Since pattern variations are minimal for panel side lengths under 1 m, we fixed one dimension at 1 m while varying the other. When only annular spacing was changed, the vertical pattern showed negligible variation, while the horizontal pattern exhibited bulges proportional to annular spacing, as shown in [FIGURE:7]. When only panel width was changed, horizontal gain remained essentially unchanged while only affecting vertical energy distribution, as shown in [FIGURE:8].

The combined results reveal that annular spacing primarily affects horizontal radiation, while width changes mainly influence vertical radiation. From an electromagnetic wave propagation perspective, increasing spacing enlarges phase variations in reflected waves, and after vector superposition, energy becomes more concentrated at corresponding angles, creating "bulges."

Based on these experiments, we find that surface accuracy affects overall electrical performance, but not through a simple linear relationship. Panel design schemes directly impact energy distribution in different directions. Among tested configurations, the 3 m × 3 m panel segmentation scheme exhibits prominent sidelobes with slight main beam degradation, providing a suitable baseline for optimization studies. Fixing panel width at 3 m for the 40 m aperture antenna yields 14 rings. Without changing the ring count, we adjust inter-ring distances to improve gain and reduce sidelobes.

With multiple ring spacings as variables exhibiting nonlinear effects on surface accuracy, traditional optimization methods cannot efficiently locate optimal solutions. Particle swarm optimization combines global and local search to find optimal points with minimal constraints, making it widely applicable. We therefore employ this method to improve the panel segmentation scheme, with optimized results shown in [FIGURE:9].

Experimental results demonstrate a 5 × 10⁻⁴ dB improvement in the central beam, compensating for 3.6% of the energy loss relative to the 0.0139 dB degradation. Simultaneously, the bulges on both sides show significant reduction, decreasing by 1.15 dB, 0.77 dB, 0.52 dB, 1.02 dB, and 2.1 dB respectively. Correction is more effective for sidelobes farther from the central beam. Different observation targets have varying sidelobe requirements [9]. Before optimization, the first bulge amplitude exceeded the first sidelobe, potentially causing false target detection. After optimization, this bulge drops below the first sidelobe and remains more than 30 dB below the central beam, reducing error rates in observations.

4 Summary and Outlook

Using the 40 m offset antenna of the BINGO project as an example, this paper analyzes different panel sizes, calculates antenna surface accuracy under various dimensions, and demonstrates that surface accuracy is proportional to the square of side lengths. Subsequent GRASP simulations of different reflector configurations show that increasing panel size introduces high-gain sidelobes of varying magnitude. Further investigation reveals that horizontal gain variations are primarily influenced by circumferential panel length—when annular spacing increases, generated sidelobes gradually approach the main beam with increasing gain. At 5 m spacing, sidelobes increase by 30 dB compared to the original. Similarly, with fixed annular spacing, panel width changes affect only vertical energy distribution, though less significantly than annular spacing changes, which is also related to the offset feed and feed polarization direction of the original antenna.

This paper provides an in-depth analysis of reflector surface design for large-aperture antennas, summarizing the effects of segmented surface design on surface accuracy and antenna energy distribution. The analysis shows that replacing curved surfaces with flat panels and varying circumferential and axial dimensions cause interference in reflected electromagnetic waves, generating sidelobes of varying degrees. Finally, global algorithm-based fine-tuning of the segmentation scheme can effectively reduce sidelobes and improve central beam gain.

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