On Wire-Grid Representation for Modeling Symmetrical Antenna Elements

Abstract

This paper focuses on the combination of the method of moments and the wire-grid approximation as an effective computational technique for modeling symmetrical antennas with low computational cost and accurate results. The criteria and conditions for the use of wire-grid surface approximation from various sources are presented together with new recommendations for modeling symmetrical antenna structures using the wire-grid approximation. These recommendations are used to calculate the characteristics of biconical and horn antennas at different frequencies. The results obtained using different grid and mesh settings are compared to those obtained analytically. Moreover, the results are compared to those obtained using the finite difference time domain numerical method, as well as the measured ones. All results are shown to be in a good agreement. The recommendations used for building a symmetrical wire-grid of those symmetrical antenna elements provided the most advantageous parameters of the grid and mesh settings and the wire radius, which are able to give quite accurate results with low computational cost. Additionally, the known equal area rule was modified for a rectangular grid form. The obtained radiation patterns of a conductive plate using both the original rule and the modified one are compared with the electrodynamic analysis results. It is shown that the use of the modified rule is more accurate when using a rectangle grid form.

1. Introduction

The importance of designing new antenna elements (AE) is increasing since they have a significant impact on the rapid development of radio electronic systems. Difficulty in meeting the demand for the digital active phased array antennas has arisen due to the challenge in designing and manufacturing such antennas to meet expectations, which include, but are not limited to, high quality, low cost, and reliable final product. Nowadays, various computer-aided design (CAD) systems are used to estimate the effectiveness of the proposed technical solutions and adjust them if necessary. CAD is based on the numerical solution of Maxwell’s equations or their derivatives since analytical solutions are known only for particular AEs. The use of CAD allows researchers to significantly reduce the time of AE development, optimize their characteristics, as well as reduce the financial costs of design. Meanwhile, each numerical method of computational electrodynamics has its own field of application where they are most effective. Nevertheless, despite the achieved successes in the development of specialized CAD systems and numerical methods, it is currently unknown whether a universal numerical method suitable for solving all problems of electrodynamics can be created [1]. Consequently, one of the main challenges in solving electrodynamic problems is to choose the “optimal” method (calculation method) for solving a specific problem. According to [2], “the method is called optimal for solving a given class of problems with the required accuracy on a specified computing system if it allows you to get a result with minimal resource costs”. Currently, there are many popular methods to solve electrodynamic problems numerically. The finite difference method in the time domain (FDTD) was first described by K. Yee in 1966 [3], and the abbreviation of its name was proposed by A. Taflove [4]. The finite difference in the frequency domain (FDFD) is better able to incorporate material dispersion than FDTD and is easier to implement; however, it is only faster than FDTD for small structures and it can be restricted to two dimensional simulations without using an iterative method [5]. The finite integration method (FIT) was proposed by T. Weiland in 1977 as a computational tool for solving Maxwell’s equations [6]. The method can be implemented in both time and frequency domains. FDTD can be considered to be a special case of the FIT [7]. The finite element method (FEM) is widely used in the mechanical analysis of structures. Despite the fact that the mathematical interpretation of the method was proposed in 1943 by Courant [8], it was not used to solve electromagnetic problems until 1968. In general, these methods can solve problems of a complex structure and give accurate results. However, this happens at the expense of computational cost, especially when solving electrodynamic problems in the frequency domain, which will dramatically increase the costs to achieve the same required accuracy. The method of moments (MoM) seems to be most promising and more efficient for solving AE problems at relatively low computing costs. In relation to electrodynamic problems, the use of the MoM provides the following stages of solution. First, the metal parts of the studied AE are replaced by equivalent surface electric currents. Appropriate boundary conditions are imposed on the resulting solution of the metal elements to calculate the equivalent currents. Then, the problem of exciting the environment with these currents is solved. An important aspect of such a process is the discretization of the metal surfaces into elementary areas and the approximation of the current within each such area.

To approximate the curved boundaries of surfaces of an arbitrary shape, it is common to use the discretization into triangles, and to represent the current in them, using the vector basic functions RWG (Rao, Wilton, Glisson) [9]. This approach may be computationally costly because of the time and memory required to build the numerical mesh and also the increase in the costs due to the need to solve a problem with the obtained big mesh. Moreover, the obtained mesh is not symmetrical, and it is sometimes hard to adjust it. Since the surface current in the symmetrical AE structures seems to distribute symmetrically through the structure in a narrow frequency range, it is more logical to get a symmetrical mesh for a symmetrical structure. Therefore, this approach may not be efficient.

As is known, the analysis of wire linear AEs is reduced to solving the integral Pocklington [10] and Hallen [11] equations. The features of the solution of these equations are based on the thin-wire approximation [12,13,14,15,16,17,18,19]. In this approach, the wire is assumed to be an ideal conductor in the form of a cylinder located along one of the coordinate axes (a one-dimensional problem), with a radius much smaller than the wavelength of the excitation signal and the cylinder physical length. This simplification enables using a scalar current density function instead of a vector one, which greatly simplifies the complexity of the task. This approach is also applicable to the representation the AE surfaces by a wire-grid [20,21,22]. Such an approach in combination with formulations of integral equations of the electric field and the MoM is widely used in well-known software products, such as NEC [23]. Although, in comparison with the surface triangular approximation and the use of RWG functions, this approach demonstrates its inherent weaknesses in near-field analysis, it has proven its efficiency on a large number of far-field scattering and radiation problems associated with conducting bodies of an arbitrary shape [24,25,26,27]. Therefore, the wire-grid approximation continues to be widely used in computational electrodynamics [28,29] since it is very easy to adjust the obtained grid and it is more suitable for building a symmetrical grid for symmetrical structures.

There are many studies concerning summarizing and forming the criteria and conditions for the use of wire-grid surface approximation for AE modeling [20,23,26,30,31,32,33]. In these sources, the proposed recommendations are not suitable for any AE structure and depend on the used weight and basics functions.

This work aims to propose new recommendations for modeling symmetrical AE structures using the wire-grid surface approximation approach in combination with MoM.

In addition to the introduction and the appendices, this article contains another four sections. In the next section we attempt to justify the choice of the used method and provide more details about it and the applied boundary conditions. Moreover, a summary of the recommendations for the criteria and conditions, extracted from various sources, will be formulated for the use of wire-grid surface approximation in AE modeling. The third section is about the equal area rule modification for a rectangular grid form, the use of the combination of MoM and the wire-grid approximation with the proposed recommendations for modeling and calculating the characteristics of biconical and horn antennas, and the verification of the obtained results. The fourth section discusses the obtained result, while the outcome from this work is summarized in the conclusion section.

2. Materials and Methods

None of the existing numerical methods are suitable for all electrodynamic modeling problems. Thus, MoM program codes are practically unsuitable for describing inhomogeneous nonlinear dielectrics. FEM codes cannot effectively solve large scattering problems. Unfortunately, there are tasks where it is necessary to take into account all these features, for example, when evaluating radiation from a printed circuit board and, therefore, the analysis cannot be performed by any of these methods. One of the solutions to this problem is to combine two or more methods in one program code. In this case, one of the methods is often MoM [34,35,36,37,38,39,40,41]. Each method is applied to the task field for which it is best suited. Appropriate boundary conditions are applied at the interfaces between these fields. Separately, it should be noted that the FDTD, FEM, and MoM are the most universal methods, and the largest number of books are devoted to their consideration where these methods are considered together, for example [1,42,43,44,45]. The choice of a particular numerical method is based on the use of some criteria, such as the complexity of the geometry of the analyzed object, its electrical dimensions, available computing power, etc.

Using MoM has an advantage since the modeled object can have a complex shape. In addition, MoM is most effective for open geometries with linear and homogeneous media. The method is excellent for hybridization with other methods and asymptotic procedures, for example, FMM and MLFMM (fast multipole method and multilevel fast multipole method), as well as UTD and GTD (homogeneous and geometric diffraction theory) [46]. Unlike FEM and FDTD methods, MoM, when constructing the grid, does not require discretization of the volume, in which the analyzed object is enclosed. It only requires discretization of the surface of this object, which gives a relatively small cost for this procedure. Similarly, as with any other method, MOM has disadvantages; in particular, it is difficult to model internal problems and heterogeneous media with this method. Furthermore, the calculation speed is sometimes low for objects of a relatively simple shape. However, this is easily eliminated by increasing the speed of workstations and the development of numerical procedures, which makes it possible to speed up the overall solution process using MoM [47].

Based on the above, to solve antenna problems, it seems most effective to use the MoM, which is “surface” and not “volumetric” like the others since its use does not require the boundary conditions that emulate remote boundaries to be artificially set. This reduces the dimension of the problem, reducing, for example, a three-dimensional problem to a two-dimensional one. In addition, the method allows for hybridization with other numerical methods. Therefore, if necessary, the functionality of the software module can be significantly expanded in the future. The computational aspects of MoM are considered in detail in [48,49,50,51].

When using a particular numerical method, the important issue is related to its convergence rate and the accuracy of the results obtained with its help. Convergence when using MoM directly depends on the operator, the basis and test functions, as well as their number. At the same time, the effectiveness of using the method to obtain a result with a given accuracy is determined by the computational costs (the time and the memory of the workstation used). As a result, when calculating by the method of moments, an unknown quantity (for example, a field or current density) depending on spatial coordinates is approximated by a finite number of known basis functions. These functions are multiplied by the coefficients to be found. This gives a linear operator equation. Both sides of the resulting equation are multiplied by a suitable function (called a test or weight function) and integrated over the domain in which the test function is defined. Then, the linear operator equation reduces to a system of a linear algebraic equation (SLAE). Repeating this procedure for a set of independent test functions, the number of which should be equal to the number of basic functions, SLAEs are obtained. The SLAE solution gives unknown coefficients and allows finding an approximate solution to the operator equation. Further, the characteristics of interest are determined from the SLAE solution.

The MoM is widely applicable in modeling antennas of various classes. In addition, the method is most effective if the AE contains only perfectly conductive elements. In MoM, the choice of basic and test functions is determined by the shape of the entire conductive surface (structure). The most commonly used basic functions are called the piecewise functions. They have a constant value in each meshing cell and are equal to zero outside of it [52,53]. One of the frequently used test functions that are not used as basic ones are Dirac delta functions (single impulse functions). When both piecewise and Dirac delta functions are employed, the MoM is known as the collocation method [54]. In addition, when similar basic and testing functions are employed, the MoM is known as the Galerkin method [49,50].

As in [53], the MoM solution used here assumes the radiation condition as the boundary one and ε = ε₀, μ = μ₀ for the environment. Considering a conducting body S in a known incident field Eⁱ, the integral equation for the charge density and the current on it can be obtained. The charge density and current on S produce a scattered field E^s which can be expressed as retarded potential integrals. Then, the boundary condition on S n × (Eⁱ+ E^s) = 0 is applied only to the axial component of E at the wire surface. Each wire in the grid is considered as N short segments connected together with N pairs of terminals to form an N port network. By short-circuiting all the network ports and calculating the open circuit voltages at them, the wire object is obtained, and then, by applying a current source to each port in turn, the impedance matrix can be determined. This procedure involves only current elements in empty space. The thin-wire approximation applied here assumes that the current is approximated by filaments on the wire axis, and it flows only in its direction. Assuming that the current in each segment is constant, the integrals can be approximated by the sum of integrals over N segments. An extra 1/2 interval is added at each wire end to insure zero current if the wire terminates.

As mentioned above, an important aspect of using the MoM is the discretization of metal surfaces into elementary areas and the approximation of the current within each such area. Therefore, a summary of the recommendations for criteria and conditions, extracted from various sources, can be formulated for the use of wire-grid surface approximation in AE modeling [20,23,26,30,31,32,33]:

• The assumption of a thin-wire must be fulfilled, i.e., the length of the wire L and the wavelength λ must be much larger than its radius a;
• The wires of the grid should not intersect with each other along the length without forming a node (Figure 1). Hence, if two wires are electrically connected at their ends, then for current interpolation it is necessary to use the same coordinates to specify these ends;
• Parallel wires should not be too close to each other. Hence, the distance between the axis of two (or more) wires connected in parallel should exceed their largest radius by more than 4 times;
• The minimum number of basic functions for electrically shorted wires should be 3, especially if they are part of a wire-grid. For electrically short AEs, this number should be between 8 and 10;
• For an electrically short dipole, the number of segments (basic functions) must be between at least 12 and 16;
• For each individual wire, it is necessary to separately set the number of segments (basic functions) based on the frequency of the excitation signal (single calculation) or the highest frequency if the wire is modeled in the frequency range (multiple calculations). At the same time, it is not recommended to use less than between 8 and 10 segments;
• When setting the excitation at the intersection of the wires, an additional wire consisting of 3 segments is used. The source is set in the central part of this wire (Figure 2). In this case, all segments in contact with the segment where the excitation source is located and near it must have the same length;
• In most cases, the length of the segments should be less than λ/10, and when describing complex geometric transitions it should be less than λ/20. For large extended wire sections, the length of the segments can be increased, but it should not be less than λ/5. At the same time, using very short segments with a length of less than 0.0001 λ should be avoided;
• In most cases, the length of the segments should be more than between 8 and 10 times their radius;
• In most cases, the appropriate wire-grid mesh is λ/10 for the middle of the frequency range;
• The radius of the wire when using a wire-grid is determined through the surface area of the wires (the equivalent area rule). Thus, Figure 3 shows a grid element with a size of ∆ × ∆ formed by four wires, and the surface area of a wire with a diameter of 2a. As a result, with the known grid step ∆, the required wire radius is defined as [55]:

  a = ∆/2π.

  (1)
• If the wire-grid covers a surface with an irregular shape, then the equivalent area rule can be generalized as in [33]. Figure 4 shows an example of an approximation of a curved surface consisting of two regions A₁ and A₂. The required wire radius is calculated as:

  a = (A₁ + A₂)/4∆π

  (2)

Using the above recommendations is not enough, and there are some problems that occur during the modeling process. In what follows, some cases will be presented in which considering the traditional recommendations for modeling symmetrical AE structures using wire-grid approximation does not give correct results, and it is necessary to adjust these recommendations to get appropriate results. Moreover, some recommendations will be presented for modeling such structures, and the results will be validated by comparing them with those obtained analytically and experimentally, as well as numerically with another method. To validate the results obtained by MoM, according to the standard [56], one may use the results obtained by closed-form equations, measurements, and any other alternative modeling method. These three validation types are used later in this work. Moreover, the above-mentioned standard in terms of modeling methods recommends comparing the integral technique results (in this case, MoM) with the results of another modeling method based on differential equations, which is preferable because the nature of these approaches to the solution is different. Therefore, we used the FDTD method, implemented into the commercial software EMPro.

To do this, and based on the wire-grid approximation and the MoM, a program module was developed using C++ programming language with OpenMP, VTK, and Eigen libraries and the QT cross-platform software (Figure 5). This module (further on referred to as the wire-grid) is used to calculate all the obtained results in this work. Moreover, the Octave mathematical package was used to estimate the software implementation of the analytical models used later in this work to validate the results obtained using the wire-grid.

3. Problems, Solutions, and Results

3.1. Modeling of a Monopole on Plate Using Wire-Grid

In the MoM, when wires are used to split the surfaces of the conductive parts in the structure into elementary platforms and approximate the current within them, a scalar current density function is used instead of a vector function, which greatly simplifies the solution. In this case, a preliminary choice of the number of the wires and their radius is required. The equal area rule is typically employed for constructing a square grid of wires when solving scattering problems [26,28,33,57,58]. However, the construction of such a grid is not always feasible, so it is necessary to use a different grid. Thus, we present a simple modification of the equal area rule for modeling structures using the wire-grid when solving the radiation problem.

The known equal area rule is based on replacing a square polygon with a grid of wires, the radii of which are based on the size of the polygon [28]. The influence of the wire-grid size constructed considering this rule on the accuracy of the obtained results is investigated in [57]. The rule is generalized to the case of approximation of a surface consisting of square and triangular polygons [33]. The physical interpretation of the equal area rule is given in [58]. According to this rule, to approximate a square polygon with a separate wire, its radius a is determined so that the surface areas of the polygon and a wire length ∆ coincide as in (1) (see Figure 6a) [28].

However, this rule cannot be applied in the case of a rectangular grid. In this case, it is proposed to use the length of the smaller side of the rectangle ∆₂ (Figure 6b), i.e.,:

a = ∆₂/2π

(3)

Testing was performed on the example of a dipole oriented along the x axis on a conducting plate 25 × 50 mm² located on the yz plane. The dipole with a length of 12.5 mm (the length of its shoulders was 5, and the gap was 2.5 mm) and a radius of 0.015 mm was located at a distance of λ/4 from the end of the plate. The excitation frequency was 7.56 GHz (λ ≈ 39.655 mm). The wires with the radii calculated by (1) and (3) were modeled using the wire-grid approximation. The EMPro (FDTD) system [59] was used for verification. The excitation port had the following characteristics: an amplitude of 1 V; an internal resistance of 0 ohms; the signal form was “broadband”. The segment length in both systems was assumed to be λ/10. Using the wire-grid approximation, the plate was approximated by a grid L_y × L_z where L_y and L_z are the step numbers of the grid along the y and z axes. When a square grid was used, these values were assumed to be 8 and 4, respectively (the lengths of the wire segments on both axes were 6.25 mm). According to (1), the radius of the wires a = 0.9947 mm was obtained. At λ/10, the length of the segments was 3.125 mm, which satisfies the conditions in the recommendation 1 and 10 (further on, will be referred to as conditions (1, 10)). As a result, each piece of wire was divided into 2 segments. The total number of segments N was 157. When a rectangular grid was used, L_y = L_z = 8 was assumed (the lengths of the wire segments along the y and z axes were 6.25 and 3.125 mm). The radius of the wires according to (3) was 0.5 mm, and the length of the segments was 3.125 mm, which also satisfies the conditions (1, 10). The wire segments along the y axis were divided into 2 segments and z were not divided; the value of N was 221. The obtained radiation patterns (RP) are shown in Figure 7. At θ = 0, 1, …, 180°, φ = 0°, 45° and 90° (Figure 8).

From Figure 7 and Figure 8 it can be seen that the results of EMPro with the wire-grid by (3) agree very well, and the maximum of the electric field strength magnitudes |E| are very close: 0.479 and 0.475 V/m (the difference is less than 1%). When using (1), the maximum |E| was 0.548 V/m (13% difference from EMPro). Thus, one can say that the proposed rule (3) gives more accurate results compared to (1).

3.2. Modeling Symmetrical Biconical AEs

When modeling a complex structure, such as the symmetrical biconical AE presented in Figure 9, it is inappropriate to use the traditional recommendations for modeling. The number of wires and their radii are difficult to determine, as well as the radius of the wire where the excitation source is located. Moreover, to answer the questions about wire location in the grid (horizontal and vertical grid elements) and the effect of the segment step on the results is not a trivial issue here. Therefore, the following model-testing based on the wire-grid approximation model was performed in order to reveal an optimal setting for modeling such an AE structure.

When the wire-grid approximation was employed based on the convergence of the results obtained by increasing the number of wires (4, 8, 16, 32, 64, and 128) approximating the AE surface, it was found that their optimal number is 64. The horizontal grid elements were not used here because the amplitudes of the distributed currents among them are very small and their contribution to the radiated electric field is negligible. The values of the length of the segment S_l, the radius of the structure wires a_s and the radius of the wire in the gap (excitation source) a_gap, were determined as S_l = λ/n, a_s = S_l/10, a_gap = S_l/5, respectively. The wire in the gap was approximated by one segment. A general view of the biconical AE at specified frequencies is shown in Figure 10.

Calculations of the radiotechnical characteristics (RTC) of the biconical AE were performed using the wire-grid, as well as the EMPro at frequencies of 0.1, 0.5, and 1 GHz with a gap length of 20 mm. The obtained results are compared to each other and to those obtained using well-known analytical expressions [60,61,62] (see Appendix A) to verify them and to prove that the proposed recommendations can give accurate results with low computational costs.

In general, the resulting inaccurate solution is not necessarily related to the used computational algorithm of the numerical method that was employed; it may be related to the solved mathematical problem itself. Even with an accurate calculation, the solution of the problem can be very sensitive to the disturbances in the input data. The qualitative concept of this sensitivity and its quantitative measure, which is called conditionality, are related to the influence of the disturbances in the input data on the resulting solution. Therefore, a problem is called insensitive or well-conditioned if this relative change in the input data causes a reasonably commensurate relative change in the solution. On the contrary, the task is considered sensitive or poorly conditioned if the relative change in the solution can be much greater than it is in the input data. For quantitative evaluation, the condition number of the problem is used in the form of the ratio of the relative change in the solution to the relative change in the input data. Then, the task is poorly conditioned or sensitive if its condition number is much greater than one. Otherwise, the task is considered well-conditioned.

Since the MoM reduces the surface integral equation of the electric field to an SLAE, which integrally contains all the information about the object under study, the properties of the obtained SLAE have to be analyzed in order to assess the stability of the problem solution; in the case under consideration—RTC AE. Therefore, when using MoM, many elements of matrix Z and of vectors V are obtained by numerical integration. Consequently, small errors in integration can be “increased” by a poorly conditioned Z matrix. In general, it is customary to evaluate the measure of conditionality of a matrix using its condition number [63]:

cond(Z) = ||Z||_F·||Z⁻¹||_F

(4)

where || · ||_F is the matrix Frobenius norm. Conditionality characterizes the sensitivity of the SLAE solution to the changes in the values of the matrix elements. The higher the condition number of the matrix, the worse it is conditioned (for a single element matrix it is equal to 1). As the condition number increases, the error of the solution increases because floating-point numbers are represented by a finite number of digits [64]. One of the important consequences of this is that a correct solution of SLAE cannot be obtained by the Gauss method with a small bit representation of numbers because of the large condition number of matrix Z [65]. As a result, the required numerical accuracy of the calculated result must exceed the probable number of the lost digits since the errors are rounded during calculations. This number of digits can be estimated by the conditionality number of the matrix. Therefore, basically, if cond(Z) = O(10^p) and the source data have an error in the kth decimal point, then, regardless of the used method for solving the SLAE equation ZI = V, it is guaranteed to get its solution with no more than k—p decimal points [66].

For example, suppose that when the convergence of a solution is investigated by increasing the number of basic functions, the condition number of matrix Z is of the order of 10⁶. Then, if it is necessary that the results for the input impedance converge to the fifth decimal point, the elements of matrix Z must be calculated with an accuracy of at least 11 digits, and this number of digits must be preserved in all calculations [66]. Remember that the condition number does not give an exact value of the maximum error of the method used, but only allows estimating its order. Therefore, you can specify an upper bound for the condition number, if it is exceeded, the solution of the SLAE at a particular workstation will lead to false results. Thus, if:

cond(Z) ≥ 1/eps

(5)

where eps is the machine epsilon, then the solution is considered unreliable [63,65,66].

According to IEEE 754–2008, for calculations with single and double precision, the eps value is O(10⁻⁷) and O(10⁻¹⁶), respectively. Often, when solving large-scale problems, even the conditionality number of matrix Z is large, but it is less than 1/eps, so the MoM is workable [67]. However, the conditionality of the SLAE matrix worsens when the excitation frequency approaches the frequencies of natural oscillations (internal resonances) of the structure under study, especially at low frequencies [68]. Since the excitation frequency is selected discretely, there is a possibility of such a coincidence. The solution at this frequency may not be obtained at all, and at close frequencies it may be inaccurate or completely incorrect. For example, refs. [69,70,71,72,73] are devoted to demonstrating the presence of imaginary resonances that do not exist in reality. To avoid this, it is necessary to increase the bit depth of the number representation [74] or to use the iterative methods [65,75].

The influence of the mesh step (segments), set as λ/n, on the accuracy of the calculated RTC of AEs at n = 10, 20, and 40 was studied. The chosen value of n = 10 is according to recommendation 8 and the other ones were used to control the convergence of the solution by increasing the mesh density.

The required orders of the formed matrices Z of the SLAE and their condition number (cond(Z)) calculated by (4) when using the wire-grid are summarized in

Table 1. The matrix Z orders N and their condition numbers, for the biconical AE with a = 508 mm and Θ₁ = Θ₂ = 53.1° at frequencies of 0.1, 0.2, and 1 GHz and mesh step (segment size) λ/n.

Frequency (GHz)	n	N	cond(Z)
0.1	10	257	1.4 × 105
	20	513	2.2 × 106
	40	897	3.1 × 105
0.5	10	1153	2.7 × 105
	20	2177	5.7 × 105
	40	4353	9.6 × 106
1	10	2177	8.2 × 104
	20	4353	1.6 × 105
	40	8705	6.4 × 105

.

The values of the input impedance of the biconical AE obtained using the wire-grid and the EMPro (FDTD) system are summarized in

Table 2. The input impedance values for the biconical AE with a = 508 mm and Θ₁ = Θ₂ = 53.1° at frequencies of 0.1, 0.2, and 1 GHz and a mesh step (segment size) λ/n.

Frequency (GHz)	n	(A1), (A11)	EMPro	Wire-Grid
0.1	10	40.40 + j28.32	29.73 + j15.78	0.46 − j25.48
	20		39.51 + j12.95	192.31 − j58.89
	40		–	28.01 + j25.59
0.5	10	101.80 − j9.38	117.48 + j64.27	114.01 − j6.29
	20		112.89 + j74.69	102.02 + j33.36
	40		–	90.44 + j48.18
1	10	88.64 + j16.68	116.40 + j89.92	103.11 + j41.34
	20		117.84 + j112.36	88.29 + j56.82
	40		–	84.62 + j71.55

. The table shows that the results obtained in the developed wire-grid software, as well as the EMPro (FDTD) system, converge when the grid step decreases.

Table 3. The deviation (%) of the input impedance magnitudes from Table 2 calculated using the wire-grid and EMPro compared to those obtained analytically.

Frequency (GHz)	n	EMPro	Wire-Grid
0.1	10	31.78	48.35
	20	15.73	307.65
	40	–	23.10
0.5	10	30.99	11.69
	20	32.41	4.99
	40	–	0.23
1	10	63.08	23.16
	20	80.52	16.41
	40	–	22.86

shows the deviations of the input impedance magnitude values obtained in EMPro and using the wire-grid, relative to those obtained by analytical expressions. It can be seen that reducing the grid step gives values closer to the analytical ones, and using the wire-grid approximation gives closer results to the analytical ones from EMPro with less computational costs. In the EMPro system at n = 40, no results were obtained due to machine memory limitations so they are not listed in the tables (marked as «–»). The elapsed simulation time (T) and the used memory (M) for simulating the RTC of the biconical AE using the wire-grid and EMPro, are presented in

Table 4. The elapsed time and used memory for simulating the biconical AE using the wire-grid, with a = 508 mm and Θ₁ = Θ₂ = 53.1° at frequencies of 0.1, 0.2, and 1 GHz and a mesh step (segment size) λ/n.

Frequency (GHz)	n	T (s)	M (MB)
0.1	10	0.34	1
	20	0.51	4
	40	0.60	12.28
0.5	10	1.02	20.29
	20	4.11	72.31
	40	19.29	289.13
1	10	4.03	72.31
	20	18.29	289.13
	40	109.06	1156.27

and

Table 5. The elapsed time to obtain the RTC at each frequency, the total simulation time (T_t), and the used memory for simulating the biconical AE using EMPro with a = 508 mm and Θ₁ = Θ₂ = 53.1° at frequencies of 0.1, 0.2, and 1 GHz and a mesh step (segment size) λ/n.

n	Frequency (GHz)	T (s)	Tt (s)	Physical/Virtual Memory (MB)	Required (GB)	Problem Size (Cells)
10	0.1	159	921	903/922	1.2	331 × 311 × 245
	0.5	181
	1	183
20	0.1	567	1892	4722/4753	6	584 × 581 × 448
	0.5	649
	1	676
40	0.1	–	–	–	36	1126 × 1120 × 855
	0.5	–		–
	1	–		–

, respectively.

Figure 11, Figure 12 and Figure 13 show the RP obtained using the wire-grid with decreasing mesh size (segments number) at frequencies of 0.1, 0.2, and 1 GHz, respectively. Additionally, for clarity, similar results obtained in EMPro are given.

In Figure 14, Figure 15 and Figure 16 the results obtained by using the wire-grid, the EMPro (FDTD) system (λ/10 mesh was used in the further comparisons since its results seem to be closer to the analytical ones), and the analytical expressions are summarized. In general, all the results are consistent with each other.

Table 6. The maximum differences of normalized RP values obtained at different mesh sizes λ/n and by (A8) at frequencies of 0.1, 0.2, and 1 GHz.

Frequency (GHz)	n	EMPro	Wire-Grid
0.1	10	0.0189	0.0259
	20	0.0189	0.0236
	40	–	0.0211
0.5	10	0.1080	0.1869
	20	0.1030	0.1395
	40	–	0.1395
1	10	0.1291	0.1569
	20	0.1268	0.1197
	40	–	0.1669

shows the largest differences in the calculated values compared to those obtained by (A8).

As can be seen from Table 2, Table 3, Table 4, Table 5 and Table 6 and Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, the estimated RTC AE results and their calculation error using the wire-grid were compared to those obtained analytically and using the EMPro system. It is shown how the decrease in the grid step affects the obtained values of the RP and the input impedance. The RPs obtained using the wire-grid are close to those obtained using the analytical expressions. At the same time, the computational costs when using the wire-grid are much less than those obtained using the EMPro system. The obtained results prove that using the proposed recommendations for modeling the symmetrical biconical AE can give accurate results and with less computational costs.

3.3. Modeling Horn AE

Another complex structure to be modeled is a horn AE presented in Figure 17. The cross-section of this AE is symmetrical in two planes. This AE is also simulated and tested using the wire-grid approximation, and new recommendations for modeling it are determined and provided further. The inner surface geometrical parameters of this AE are presented in

Table 7. The geometrical parameters (mm) of the inner surface of the horn AE.

Geometrical Parameter	ap	bp	a	b	ar	br	l	lr
Model	80	60	23	10	23	10	150	10
Prototype	79.9	59.8	23.0	10.0	23.0	10.0	149.9	10.0

. The wall thickness of the regular part of the AE is 1 mm, and the irregular part is 2 mm. For verifying the obtained results of modeling the horn RTCs, the results are compared with the measured ones for a manufactured prototype of this horn (its parameters after manufacturing are measured and also presented in Table 7). The AE was modeled using the wire-grid at an operating frequency of 8 GHz. The AE was excited by installing a wire at the junction of the regular and irregular parts of the horn between its wide walls. The length of the wire corresponded to the value b from Table 7. The wire was approximated by one segment. After performing model-testing based on the wire-grid approximation model, the following recommendations were obtained for modeling these types of AEs.

For common issues, a wide range of parameters was used. The RTCs of the AE were calculated with a decrease in the mesh step of the grid λ/n and an increase in the number of wires to approximate the surface of the regular part of the AE. When modeling, the wide walls of the regular part (the index r is used to denote the regular part of the AE) were divided into SZ_r parts along the z axis and SX_r parts along the x axis, the narrow sides were SZ_r and SY_r, and the shortened back wall was divided into SX_r and SY_r, respectively. The ratios SZ_r = SX_r = L_r, SY_r = L_r/2 were used. To approximate the extended part of the horn (the index e is used to denote the extended part of the AE), its wide walls were divided into SZ_e parts on the z axis and SX_e parts along the x axis, and narrow ones SY_e, parts along the y axis. It is assumed that SY_e = SX_e/2 and the ratios SZ_e = SX_e = L_e, SY_e = L_e/2 were used. The order N of the SLAE matrix Z with using the values of n = 10, 20, 40, L_e = 16 for the irregular part of the horn AE, and L_r = 2, 4, 8, 16 for the regular part, are given in

Table 8. Total wire number (TWN) and the order N of the SLAE matrix Z for the horn AE at different mesh step of the grid λ/n using the wire-grid and L_e = 16, and L_r = 2, 4, 8, 16; the condition number for matrix Z (cond(Z)) and the time consumed to calculate Z (T).

Lr	TWN	n
		10			20			40
		N	T (s)	cond(Z)	N	T (s)	cond(Z)	N	T (s)	cond(Z)
2	1609	3544	1.65	2.35 × 104	5931	4.40	8.68 × 104	11,448	16.80	3.76 × 105
4	1687	3636	1.71	2.41 × 104	6112	4.73	8.91 × 104	11,742	17.46	1.89 × 106
8	1999	3906	2.01	3.28 × 104	6480	5.25	1.02 × 105	12,464	19.25	4.48 × 105
16	3247	5178	3.44	3.69 × 105	7556	7.10	2.07 × 106	13,936	24.49	9.81 × 105

. Moreover, the condition numbers for matrix Z are presented along with the time consumed to calculate them, as well as the total numbers of wire used to model the regular and the extended parts of the AE. General views of the AE with the L_r values are shown in Figure 18. The values L_e = 16 and L_r = 8 seem to be minimal optimal values (see Appendix B) for modeling such AEs. The calculated values of the normalized magnitude of the electric field strength (|E|_N, dB) at φ = 0° и φ = 90° are shown in Figure 19, and demonstrate convergence with an increase in n. Therefore, the value n = 40 seems to be optimal. The values of the segment length S_l, the radius of the structure wires a_s, and the radius of the wire where the excitation was applied a_ex were determined as S_l = λ/n, a_s = S_l/10, a_ex = S_l/5, respectively. The wire where the excitation was applied was approximated by one segment.

The prototype of the horn AE with the parameters presented in Table 7 was manufactured, and its RTCs were measured in order to validate the results calculated using the wire-grid approximation. Figure 20 compares the results of the normalized RPs obtained using the wire-grid (L_r = 8, SX_e = SZ_e = 16) and experimentally at φ = 0° and φ = 90°.

Table 9. The calculated and measured RTCs of the horn AE.

RTC	Wire-Grid	Measured
BW (°), φ = 90°	25	31
BW (°), φ = 0°	34	31
SLmax (dB)	−12.61	−16.93

summarizes the calculated and the measured AE main beamwidths (BW) and the levels of the sidelobes (SL_max).

Then, we compared the results of simulation with the changing of the grid to the measured results. Figure 21, Figure 22 and Figure 23 show the RPs for the horn AE using L_r = 8, 16, 32 and L_e = 16, 32, and SZ_e = SX_e or equal to SX_e*2^±1. Figure 24 summarizes these results together for clarity and the numbers in the graphics legend refer to L_r, SX_e, SZ_e, respectively. As can be seen from Figure 21, Figure 22, Figure 23 and Figure 24, if the grid cells are refined to have a closer to square shape, the obtained results can be closer to the measured ones. Note that the results using the same value for SZ_e are very close except at the back lobe where the denser grid gives less radiation level. Therefore, one can say that the optimal grid form is symmetrical, denser, and squarer. This corresponds to SZ_e = SX_e = 32. Figure 25 compares the results considering this point. It can be seen that the results are almost identical and increasing the density of the grid will not produce more accurate results but will only increase the computational costs as can be seen from

Table 10. The calculated RTCs of the horn AE using the wire-grid with different mesh and grid settings.

	n
	10												20												40
Lr	8				16				32				8				16				32				8				16				32
SXe	16		32		16		32		16		32		16		32		16		32		16		32		16		32		16		32		16		32
SZe	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32	16	32
TWN	1584 + 415	3120 + 415	3168 + 415	6240 + 415	1584 + 1663	3120 + 1663	3168 + 1663	6240 + 1663	1584 + 6655	3120 + 6655	3168 + 6655	6240 + 6655	1584 + 415	3120 + 415	3168 + 415	6240 + 415	1584 + 1663	3120 + 1663	3168 + 1663	6240 + 1663	1584 + 6655	3120 + 6655	3168 + 6655	6240 + 6655	1584 + 415	3120 + 415	3168 + 415	6240 + 415	1584 + 1663	3120 + 1663	3168 + 1663	6240 + 1663	1584 + 6655	3120 + 6655	3168 + 6655	6240 + 6655
N	3906	5794	6674	9746	5178	7066	7946	11,018	10,218	12,106	12,986	16,058	6480	9120	10,688	14,464	7556	10,196	11,764	15,540	12,644	15,284	16,852	20,628	12,464	15,792	20,544	24,288	13,936	17,264	22,016	25,760	18,232	21,560	26,312	30,056
T (s)	6.42	15.12	21.60	52.72	12.43	24.05	31.47	64.34	54.28	83.22	97.87	167.24	19.10	41.30	61.06	131.29	27.56	55.39	77.57	157.34	92.08	154.39	186.79	311.35	90.19	161.69	311.84	473.22	121.86	202.88	372.68	542.52	219.29	352.75	570.93	851.60
cond (Z)	3.28 × 104	5.02 × 104	7.37 × 104	1.50 × 105	3.69 × 105	4.50 × 105	4.72 × 105	6.05 × 105	2.75 × 107	3.12 × 107	5.10 × 107	5.91 × 107	1.03 × 105	1.26 × 105	1.80 × 105	2.60 × 105	2.07 × 106	2.49 × 106	2.56 × 106	3.15 × 106	5.28 × 106	5.80 × 106	5.96 × 106	6.77 × 106	4.48 × 105	4.58 × 105	8.76 × 105	7.89 × 105	9.81 × 105	1.07 × 106	1.40 × 106	1.44 × 106	2.94 × 106	3.15 × 106	3.50 × 106	3.72 × 106
R (Ohm)	227.18	298.17	231.90	303.70	217.05	287.70	224.36	290.93	217.33	287.74	222.93	289.09	215.68	274.68	226.40	293.19	205.07	264.48	216.17	279.08	204.48	263.96	2113.53	275.19	210.86	242.17	221.74	275.12	199.63	234.12	212.97	266.42	193.60	228.65	207.58	259.63
X (Ohm)	267.54	274.63	267.14	288.83	275.49	290.37	275.55	304.27	277.11	292.70	276.17	305.65	330.36	306.99	326.06	321.58	341.40	328.61	339.68	346.79	345.87	335.80	342.88	352.96	400.66	366.40	389.78	368.76	415.03	388.28	407.21	399.02	419.44	395.82	413.16	409.90
|Z| (Ohm)	349.05	405.37	353.76	419.12	350.73	408.76	355.34	420.98	352.16	410.44	354.92	167.24	394.50	411.94	396.95	435.17	398.26	421.83	402.63	445.14	401.79	427.13	403.93	447.57	452.75	439.20	448.43	460.08	460.55	453.40	459.54	479.79	461.96	457.11	462.38	485.20

, in which the calculated condition numbers and the input impedance (Z = R + jX) of the horn AE and the time consumed to calculate them at different mesh and grid settings using wire-grid approximation are presented.

4. Results and Discussion

However, the recommendations cannot be applied on arbitrary structures. They have only been tested on such AE as the ones used in this work. In the case of radical or complex changes in the structure, it is imperative to adjust them to meet all the special requirements which every unique structure demands and to eliminate any other unnecessary elements or dependences. For example, when modeling the biconical AE, we did not use the horizontal grid elements because the current distributed among them slightly affects the desired RTCs. On the contrary, in the case of the horn AE, the use of them was unavoidable since they have a remarkable impact on the horn AE RP back lobe. It is also worthwhile to consider the relationship between some elements and the wavelengths when building the grid of wires, which can lead to a number of restrictions on using the wire-grid approximation as, for instance, the segment length/radius ratio or the distance between the wires. This will lead to a number of limitations on the use of the wire-grid to deal with, added to those of the MoM.

The results obtained using the developed module in this work showed good agreement with the obtained analytical, numerical, and experimental ones. The slight differences between the calculated results are expected since different software implementation cannot give identical results even if they are based on the same numerical method. One of the reasons, for example, is the rounding and the bit representation of numbers. The inequality of the calculated and the measured results is a consequence of the modelling inability to consider some factors, such as the human mistake ones, which appeared here in the shown measured result where the asymmetry of the radiation pattern is clear due to the non-accurate angle of the horn position. Even more, at this stage of module development, it does not consider the losses in the wires, which we intend to deal with in the future.

Furthermore, building the grid by itself and testing it considering all these possibilities and limitations is a challenge and a time-consuming issue. Consequently, besides the fact that using this approach seems to be promising, it is relevant to make it easier and more efficient.

5. Conclusions

In this paper, we focused on the use of the MoM and wire-grid approximation as an effective computational approach for modeling symmetrical AEs with low computational cost and quite accurate results. For AE modeling, the MoM using RWG basic functions is widely used nowadays. The use of this method leads to reducing the integral equation to SLAE, the order of which determines the required machine memory. Therefore, it is advisable to search for other ways to save machine computational resources while maintaining the required modeling accuracy.

One such approach is the wire-grid, which replaces the conductive surfaces of the AE with a grid of wires. This will not only reduce the computational costs, but it may also give an equivalent structure of the AE with less mass if it is available to manufacture in its new form. However, this approach is complicated because it is necessary to construct the grid of wires correctly. To do this, there are several recommendations that are non-universal (they do not allow modeling arbitrary constructions) and difficult in software implementation.

Recommendations from various sources including criteria and conditions for the use of wire-grid surface approximation for AE modeling are summarized in this work. New recommendations are proposed for modeling symmetrical biconical and horn AE structures. The proposed recommendations are used to calculate the characteristics of biconical AE at different frequencies. The calculated results are compared to the ones obtained analytically using well-known Legendre and Hankel functions. These results showed good agreement, not only with analytical results, but also with those obtained using another numerical method, such as the FDTD method.

By the same token, we verified the proposed wire-grid approximation modeling recommendations on the commonly used horn AE. The received radiation patterns are compared to the measured ones and to the results obtained using the FDTD. The results are compared and analyzed using different grid and mesh settings. It was revealed that the closer symmetrical grid cells form into a square shape, the more accurate the results that can be obtained with low computational cost. Additionally, we show the modification of the known equal area rule on a rectangular grid form. The obtained radiation patterns of a conductive plate using the original rule and the modified one are compared with the FDTD electrodynamic analysis results. It is shown that the use of the modified rule is more suitable in this case.

In general, this approach along with the proposed recommendations showed its ability to give accurate results compared to the obtained analytical, numerical and experimental ones. Moreover, it surpasses other methods in the required computational time and memory to get the same results.

One of the perspective developments of the module used in this study is to add new capabilities which can enable it to import any AE geometrical model from a third-party computer aided design system and independently build and refine its equivalent wire-grid considering all the possible limitations and using the proposed modeling recommendations. Another future improvement is to create a smooth user interface for this module to enhance the user experience. Likewise, expansion of the module database of different AE structures and universalizing the recommendations proposed here can create a core for an advanced and expert AEs modeling software system.