I. Introduction
In recent years, the use of unfurlable reflector antennas in small satellites has gained immense popularity due to their capability to be stowed into small packages during launch [1]. Among the various unfurlable antennas, the deployable mesh reflector is especially preferred for its light weight, ability to expand into large apertures, and exceptional microwave electrical performance [2, 3]. Notably, deployable mesh reflector antennas are composed of a feed assembly and a reflector system, which primarily comprises a central hub, forming ribs, deployment structures, and a metallic mesh surface [4]. Usually, the mesh surface is woven into the periodic structures using gold-plated molybdenum wires.
To analyze the RF performance of the mesh surface, three models have been widely applied to calculate the reflection and transmission coefficients using the numerical analysis technique named the periodic method of moments: the wire-grid model, the strip-aperture model, and the surface-patch model.
The wire-grid model is based on Astrakhan’s reflection formulation, which employs three parameters of a grid cell—width a, length b, and wire diameter 2r0—to characterize the model [5]. This method has been extensively applied for mesh reflector analysis and has been utilized as a comparative reference in many studies [6]. However, since metallic mesh is woven in complex patterns and the wire-grid model usually analyzes single-wire knitted mesh in rectangular shapes, some researchers believe that the model cannot adequately describe interwoven wires with different geometries. Therefore, they proposed the strip-aperture model to account for more complicated structures [6, 7]. They used two groups, each with three parameters, to describe the model lattice and aperture configurations. However, defining one grid cell using six parameters greatly increases the amount of computation involved in the electromagnetic analysis of the entire reflector system. Later, the strip-aperture model was upgraded to the surface-patch model by simplifying the six parameters to three [4, 8]. Nevertheless, the surface-patch model is complicated and difficult to build. It uses equivalent flat strips to replace cylindrical metal wires. Notably, the equivalence theory is defined as follows: equivalent factor (β) = width of the strip (ω)/diameter of the cylindrical wire (d) [9]. In this context, the conclusion that the width of the equivalent strip should be about twice the wire diameter has been verified using waveguide experiments and simulations [9–11]. The feasibility of applying equivalence theory to a mesh with a square tricot knit pattern woven by a single wire has also been confirmed [12]. Moreover, the characteristics of the same mesh pattern in a square wire-grid model have been reported in [13, 14]. However, the shape of a real mesh is not always square, especially under different tensions in different directions. For instance, since the metallic mesh applied in spaceborne satellites is elastic and flexible, the shape of its grid cells is highly susceptible to tension. Moreover, there are a large number of metallic meshes with more complicated structures that are not limited to the square or rectangular shapes, such as the single Atlas and single Satin meshes. In addition, a complex mesh features a variety of weaving patterns. Furthermore, to the best of our knowledge, the characteristics of the rectangular wire-grid model have not been studied or verified using full-wave simulations. To address these issues, this paper proposes a method for building models in electromagnetic analysis software to analyze the electrical properties of complex meshes with irregular shapes.
This paper is structured as follows. In Section II, Astrakhan’s formulations for the wire-grid model are introduced and then approximated into succinct equations that can be conveniently utilized to calculate the reflection and transmission coefficients of the mesh while maintaining accuracy. Section III compares the accuracy of Computer Simulation Technology (CST) Studio Suite and High Frequency Structure Simulator (HFSS) with the results obtained from the original Astrakhan’s formulation for wire-grid models, programmed in MATLAB, as a reference, illustrating correlations between the electrical properties and influence factors of the rectangular wire-grid model. Section IV presents the measurement setup for a gold-coated molybdenum mesh with complex patterns, details the steps to simplify complex meshes with irregular pattern shapes for the wire-grid model, and verifies the applicability of the model for meshes with complex woven structures. Finally, Section V concludes this work.
II. Approximation of Astrakhan’s Formulation
Astrakhan’s formulations are used to find the reflection coefficients of plane wire grids with square or rectangular cells illuminated by plane electromagnetic waves of arbitrary polarization. The reflection and transmission matrix expressions are analyzed using the method of averaged boundary conditions in closed form, which is crucial for validating computer simulations. Notably, the original reflection coefficients for a wire-grid model have been presented in [6]. These rigorous exact formulations enable highly accurate numerical analysis while also accounting for lossy media. Furthermore, the wires in a wire-grid model can be assumed to be perfect electric conductor (PEC) since satellite mesh wires are usually plated with gold. In this section, we present approximated formulas that can help to easily obtain the reflectance and transmittance before measuring a real mesh or programming the original Astrakhan’s formulations in the software. The approximated formulas also represent a more efficient way to choose suitable mesh parameters for specific performance requirements.
Considering the normal incidence of a plane wave with θ = ϕ = 0° into a wire-grid model for PEC materials, the complicated formulas for reflection coefficients can be approximated through asymptotic expansion using the Taylor series and expressed in frequency and wire-grid parameters. The reflection coefficient formula is as follows:
(1)
The same procedure can be expressed as:
Furthermore, the transmission coefficients can be expressed as follows:
Meanwhile, the formulas for the short-hand notations are as follows:
where k is the wavenumber in free space, c is the speed of light in free space, n is the order of Taylor expansion, and a, b, and 2r0 are the length, width, and wire diameter of the wire-grid model, as shown in Fig. 1(a). Notably, α1, α2, δ1, δ2, γ1, γ2, and I0 are appeared in [6, 14]. With regard to the plane of incidence, TE and TM denote the perpendicular and parallel polarizations of the electric field, respectively.
To verify the accuracy of the approximated formulas, we calculated the reflectance of the wire-grid model considering 2r0 = 0.011512” and a = b = 1/12” at different frequencies and then compared the results with those obtained using the original formulations. Table 1 presents the comparison results, showing that the reflectivity values are almost the same, meaning that the difference caused by approximation is insignificant. Notably, reflectivity was calculated using the following equation:
III. Characteristics of the Wire-Grid Model
1. Accuracy Comparison of Different Software
It is crucial to select suitable software for antenna designs owing to their increasingly complex structures. In this context, the most significant criterion is simulation accuracy. In this section, the simulation results of the wire-grid model obtained using two well-known electromagnetic analysis software—HFSS and CST—are compared, with the original Astrakhan’s formulation results as reference. Notably, HFSS validation has already been reported in numerous papers [9, 10, 12]. In this study, the results obtained using CST are compared with those of HFSS to verify that it is more accurate to simulate the wire-grid model in CST.
Since mesh patterns are knitted periodically, a full-wave simulation with periodic boundary conditions could be performed [15]. The periodic cells of the wire-grid model in HFSS and CST are shown in Fig. 1(a) and 1(b), respectively. The finite element solver and Floquet ports were utilized in the HFSS along with the periodic boundary conditions. Meanwhile, in the CST, the unit cell was built to simulate the infinite planar with Floquet boundaries using the frequency domain solver. Furthermore, in CST, there were two options for the meshing type—hexahedral and tetrahedral. Both were used in the simulations and compared with the analytical solutions. The simulations in HFSS and CST were performed on the same computer with Intel Xeon Silver 4216 CPU @2.10 GHz and an NVIDIA GeForce GT1030 GPU.
Based on the findings from the Astrakhan’s formulations, it was determined that a variety of parameters could influence mesh performance, such as mesh opening size, wire diameter, angle of wave incidence, and the form of wire contact. In the case of square cells in PEC material, it has already been proved that the grid is isotropic in the azimuth plane, has no cross-polarization, the nature of contact has an insignificant influence on transmission coefficients, and the gain loss ΔG for the TE and TM polarization electrical fields is equal for plane wave incidence [13].
Notably, gain loss refers to power loss caused by leakage from mesh openings. Using Eq. (10), the simulation results for two specific cases were compared at a frequency of 9.65 GHz with PEC material. The first case pertained to investigating the effects of mesh opening size on gain loss, while the second case revealed the dependency on wire diameters. A commonly used parameter to define mesh opening size is the number of openings per inch (OPI), defined based on the distance between the centers of two adjacent wires. For square grids, as depicted in Fig. 1(a), a = b = 1/OPI. In the first case, the wire diameter was set to 0.003189” for various OPI, while considering a normal incident plane wave with θ = ϕ = 0°. Meanwhile, for the second case, the OPI was set to 12 for varying wire diameters, considering the same normal incident plane wave.
Fig. 2 shows that as the OPI increases from 10 to 40, the absolute value of the gain loss declines and approaches 0. This clarifies that the OPI is inversely proportional to the mesh opening size. As the OPI becomes larger, the width and length of the opening become smaller, and fewer waves are transmitted through the cell. However, more waves are reflected. Therefore, increasing the value of OPI can improve reflecting performance.
In addition, it is observed that the simulation results of CST with tetrahedral mesh are more consistent with Astrakhan’s formulation than CST with hexahedral mesh and HFSS. Furthermore, the root mean square error (RMSE) in Table 2 indicates a clear difference between Astrakhan’s formulation and the HFSS, CST tetrahedral, and CST hexahedral mesh types. Based on the results of Astrakhan’s formulation, the minimum RMSE of 0.0043 is attained by CST tetrahedral, followed by 0.0223 achieved by HFSS and 0.0533 by the CST hexahedral mesh type. The same conclusion was derived from the simulations conducted for the second case. Since the gain loss of both the CST tetrahedral mesh and HFSS matched Astrakhan’s curve, we calculated the RMSE as a reference. The RMSE for HFSS, CST tetrahedral, and CST hexahedral were 0.0059, 0.0049, and 0.0247, respectively, indicating that CST with the tetrahedral mesh type performs better in terms of accuracy.
2. Characteristics of the Rectangular Wire-Grid Model
While the square wire-grid model offers excellent performance on cross-polarization for a normal incident plane wave, it is crucial to know the characteristics of rectangular wire-grid models as well. Because of the elasticity of metallic mesh, the shape of the mesh structure is flexible to different tensions. Accordingly, the electromagnetic performance of the mesh varies. In this context, the wire-grid model combined with Astrakhan’s formulation offers an accurate and efficient method for studying the electromagnetic properties of square-shaped mesh. However, these formulations have limitations with regard to analyzing rectangular grids when the incident wave azimuth angle is 0° or 90°. The difference was observed when comparing the CST simulation results with those of Astrakhan’s formulations, as explained later in this study.
Based on the numerical comparison in previous sections, we built a wire-grid model in the CST and verified its characteristics with regard to the rectangular shape. Notably, since the model parameters for this analysis would be different, owing to the use of a different shape, we define a = 1/OPI and b = 1/OPI. Furthermore, the nature of contact for various metal meshes is also different. Wires that touch each other at a junction are referred to as “soft contact,” while physically soldered junctions indicate “hard contact,” as shown in Fig. 4. The impact of different contacts on the rectangular model was also examined through the following simulations.
Reflecting characteristics of the rectangular model were observed for four specific cases:
1. Different OPI1 for a fixed OPI2 and different OPI2 for a fixed OPI1 while maintaining the same wire diameter and angle of incidence in the form of hard contact.
2. Different wire diameters with fixed OPI1, OPI2, and angle of incidence in the form of hard contact.
3. Different values of θ with fixed OPI1, OPI2, and wire diameter for ϕ = 0° and ϕ = 0°, in the form of hard and soft contacts, respectively.
4. Different values of ϕ for a constant θ = 0° with fixed OPI1, OPI2, and wire diameter in the form of hard and soft contacts.
For the four cases mentioned above, the reflectivity estimated by CST using the tetrahedral mesh type was plotted and compared with the original Astrakhan’s formulations at 9.6 GHz for PEC wires. Figs. 5–8 reveal the correlations between reflectivity and different influence factors. Notably, Eqs. (1) and (2), before approximation, are the original Astrakhan’s formulations for calculating the reflectivity of TE- and TM-polarized waves, respectively.
The simulation results revealed some critical characteristics. First, Fig. 5 indicates that although OPI1 and OPI2 have different levels of influence on reflectivity, both parameters should definitely be considered influence factors. As OPI1 and OPI2 increase, the opening size decreases and the reflectivity increases. With regard to Astrakhan’s formulation, OPI1 and OPI2 successfully determined the reflection coefficients of the TE- and TM-polarized waves, respectively, for normal incident plane waves. The reflectivity of the TM mode remains stable when OPI1 varies, while that of the TE mode remains unchanged when OPI2 varies. Second, it is observed that reflectivity and wire diameter are positively correlated, as depicted in Fig. 6. However, compared to the wire diameter, the opening size seems to have a more significant effect on reflectivity. As OPI doubles from 8 to 16, the reflectance increases by about 0.25 dB. In contrast, when the wire diameter increases from 0.0032” to 0.0064”, the reflectivity difference is about 0.17 dB. Third, the reflection coefficients of the TE and TM modes are equal at ϕ = 45°, ϕ = 135° even when the grid is not square-shaped. Fig. 7 shows that the reflectivity for the TE mode increases with an increase in the elevation angle of incidence, while the opposite situation is prevalent for TM. Furthermore, reflectivity is relatively equal when θ is in the range of 0° to 30° for the TE and TM modes. Starting from 30°, the reflectivity diverges (ΔR > 0.01 dB). When observing the case of changing OPI1, it is apparent that the length of the grid perpendicular to the electric field direction of the TE mode plays a major role in determining reflectivity in the TE mode. The same rule applies to the TM mode as well. Moreover, Figs. 7 and 8 show that the performance difference between hard and soft contacts is negligible for both the elevation and azimuth angles, suggesting that the reflection coefficients are independent of the nature of wire contact. Finally, as shown in Fig. 8, it is found that the rectangular wire-grid model is not isotropic because its reflection coefficients are periodic functions of ϕ.
IV. Applicability Verification for Complex Mesh Patterns
Drawing on the numerical comparison between Astrakhan’s formulation and CST results for wire-grid models, the applicability of the model for mesh woven by multi-wires with complex knitting patterns is verified in the following experiments.
1. Mesh Performance Measurement
Photographs of the experimental setup are depicted in Fig. 9(a) and 9(b). Notably, the reflection coefficients were measured using a network analyzer, as presented in Fig. 9(a). The metallic mesh was placed between the X-band WR90 waveguide ports, as shown in Fig. 9(b). A warp-knitted gold-coated molybdenum mesh was purchased for measurement. According to the company, a gold-coated layer would not be easy to peel off even after a long time, as compared to gold-plated wires, since gold coating is a physical process that is more stable than the chemical electroplating process. Fig. 9(c) shows a magnified picture of the molybdenum wire covered by a gold coating of thickness 100–300 nm. The mesh was woven using multi-wires in a complex pattern and designed for X-band satellite deployable reflector antennas. Fig. 11(b) presents a comparison of the measured and simulated reflectance of the multi-wire mesh.
2. Mesh Model Establishment and Simulation
In contrast to the commonly used single atlas mesh and single satin mesh, the mesh used for this analysis featured a rare weaving pattern. Moreover, the metal thread used to knit the mesh pattern was composed of multi-thin wires, not a single wire. Thus, the thread diameter could not be directly utilized as the wire diameter for the mesh model. Fig. 10(a) presents a photograph of the real gold-coated molybdenum mesh with complex weaving patterns. Unlike the previously studied tricot knit mesh, in which strips can be used to easily replace the structure, the weaving pattern considered in this paper is more complex. Furthermore, the periodic cell is not a regular rectangle. If we simply used strips to cover the periodic cell, it would retain a one-piece structure. For this purpose, we needed to determine the strip width. The procedures followed to construct the wire-grid model are described below.
First, four periodic cells were picked from the mesh figure and nine nodes were linked using dotted lines. Next, using the dotted line as the midline, the concentrated wires were fully covered with strips while ignoring the diagonal wires. Subsequently, the average width of the six strips was calculated to determine the final strip width w, as shown in Fig. 10(b), where parameters a and b are the distances between the intersects of the equivalent strips. In Fig. 10(c), d is the equivalent wire diameter for the wire-grid model. The initial wire diameter was assumed to be half the strip width (β = 2) based on the strip-wire equivalence theory. Finally, the simulated and measured results of the reflectance of the wire-grid model were compared to see if they matched well. If not, the wire diameter was changed until the reflectance agreed well with the measured results to finally determine the wire diameter.
For the mesh considered in this study, the strip width was 0.02 inch, OPI1 and OPI2 were set to 12 and 13, respectively, and the wires in the model were assumed to be in hard contact in the CST simulation.
Research has already established that using half the strip width is a better choice than using the wire diameter used to make the mesh as the model diameter [12]. However, researchers have also concluded that this equivalent factor is not a fixed value [9]—it increases as the spacing between strips increases. Therefore, after the reflectivity was calculated using the assumed initial diameter, we changed the range of this factor from 1.85 to 1.9 and then compared the ensuing results with the measured results to improve the accuracy of the findings. In the X-band waveguide, the dominant TE10 mode was excited to identify the reflectivity of the metal mesh. Fig. 11(a) and 11(b) demonstrate comparisons between the measured reflectivity of the real mesh and the results for the TE-polarized wave obtained from Astrakhan’s formulation and CST using the tetrahedral mesh type, respectively. The trendline of the measurement results, calculated using the least squares method, is considered for comparison with the simulation results. Notably, the measured reflectivity presents a fluctuating and decreasing trend.
To further examine the comparison results, the simulation results with the equivalent factor equal to 1.86, 1.9, and 2 are depicted in Fig. 11. As shown in Fig. 11(a), the reflectivity curve for Astrakhan’s formulation, with an equivalent factor of 1.86, coincides well with the trendline of the measured reflectivity. Meanwhile, in the case of CST, the best matching equivalent factor is 1.9, as presented in Fig. 11(b), with the reflectivity difference between the best matching factor and the factor of 2 being about 0.01 dB. Furthermore, as analyzed in Section III, OPI2 does not affect the reflectivity of the TE mode in Astrakhan’s formulation. Therefore, even if the value of OPI2 is set to any random number, the reflectivity would remain the same as long as OPI1 is a fixed constant. This defect is not obvious when using a square-shaped wire-grid model. However, it limits the application of the model for rectangular grids. In such circumstances, using the wire-grid model in CST becomes indispensable.
V. Conclusion
This paper presents a novel method for constructing wire-grid models for metal mesh with irregular and complex structures. The factors that influence the electrical characteristics of metal mesh with complex woven structures were studied numerically and experimentally. First, Astrakhan’s formulations for the wire-grid model were approximated to conveniently acquire the reflectance and transmittance of the mesh while maintaining accuracy. Subsequently, an accuracy comparison between HFSS and CST, using Astrakhan’s formulation as a reference, revealed the feasibility of using the CST tetrahedral mesh type to simulate the wire-grid model for electromagnetic analysis. Next, the simulation results obtained using CST and Astrakhan’s formulation for four cases were plotted. The effects of mesh opening size, wire diameter, form of wire contact, and angle of incidence on the performance of a rectangular wire-grid model were investigated. Following this, the applicability of using the wire-grid model for complex mesh was verified. The steps to determine the strip width for an irregularly shaped intricate mesh and the final model diameter were described. Furthermore, the reflectivity of a purchased metallic mesh was measured and compared with the simulated results obtained using a corresponding wire-grid model. The procedures for determining model parameters for a complex mesh, building a wire-grid model in CST, and analyzing the characteristics of the mesh described in this paper are expected to be useful in the design of mesh reflector antennas with complex weaving patterns.
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