Radio Science

E-plane radiation pattern analysis of rectangular horn antennas with V-shaped corrugations by UTD-PO formulation

Authors

  • José-Víctor Rodríguez,

    Corresponding author
    1. Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Cartagena, Spain
      Corresponding author: J.-V. Rodríguez, Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Antiguo Cuartel de Antiguones, Plaza del Hospital, 1, ES-30202 Cartagena, Spain. (jvictor.rodriguez@upct.es)
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  • Fernando Quesada-Pereira,

    1. Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Cartagena, Spain
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  • Leandro Juan-Llácer,

    1. Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Cartagena, Spain
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  • Elisa Verdú-Rodríguez,

    1. Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Cartagena, Spain
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  • José-María Molina-García-Pardo,

    1. Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Cartagena, Spain
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  • Alejandro Álvarez-Melcón

    1. Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Cartagena, Spain
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Corresponding author: J.-V. Rodríguez, Departamento de Tecnologías de la Información y las Comunicaciones, Universidad Politécnica de Cartagena, Antiguo Cuartel de Antiguones, Plaza del Hospital, 1, ES-30202 Cartagena, Spain. (jvictor.rodriguez@upct.es)

Abstract

[1] A new method for the calculation of the radiation pattern of corrugated E-plane rectangular horn antennas based on a hybrid uniform theory of diffraction-physical optics (UTD-PO) formulation is presented. The method, which allows for the analysis of horns in which V-shaped corrugations have been considered, has been validated with numerical data obtained through the application of both an electric field integral equation (EFIE) solved by the method of moments (MoM) and the finite element method (FEM)-based full-wave electromagnetic analysis software, HFSS. In this sense, the proposed solution is mathematically less complex and computationally more efficient than existing techniques.

1. Introduction

[2] At present, corrugated horn antennas play a very important role in the development of current and future communication systems, due to their symmetric radiation patterns compared to conventional horns, low cross-polarization, low sidelobes and back-lobes, and broadband performance [Mentzer and Peters, 1976; Borgioli et al., 1997; Bird, 1991; Baldwin and Mcinnes, 1975; Clarricoats and Olver, 1984; Mentzer et al., 1975]. However, when it comes to corrugated horns placed in the mm-band and below (with several corrugations per wavelength), the analysis of their radiation patterns is usually based on complex numerical techniques which can result in high computational cost [Peterson et al., 1998; Salazar-Palma et al., 1991].

[3] This work presents an easier, faster method—based on a hybrid uniform theory of diffraction-physical optics (UTD-PO) formulation—for the calculation of the E-plane radiation pattern of E-plane rectangular horn antennas with V-shaped corrugations equally spaced and distributed along the faces of the horn.

2. Theoretical Model

[4] The calculation of the E-plane radiation pattern of a conventional E-plane rectangular horn (without corrugations), whose E-plane scheme (2-D geometry) is depicted inFigure 1, can be obtained, in a ray-based UTD approach, by means of the evaluation of the total fieldET(θ)—normalized to the value at θ = 0—at the point of Figure 1 where the emitting source (a magnetic current line) is located, provided that we assume via reciprocity that a plane wave is impinging on the horn.

Figure 1.

Scheme of the E-plane of an E-plane conventional rectangular horn antenna (without corrugations).

[5] It should be noted that a 2-D approach can be used for the calculation of the E-plane radiation pattern in such a rectangular horn, since the illumination in this plane is completely constant [Stutzman and Thiele, 1998]. In this way, three basic possible field contributions at this point can be considered, namely, the direct ray (r), the ray diffracted at Q1 (r1), and the ray diffracted at Q2 (r2). Therefore, as can be seen in Figure 1, depending on which contributions reach the receiving point at a given θ (which ranges from 0 to π), four different zones can be distinguished (Z1–4). In view of this, the radiation pattern can be calculated as

display math

where ET(θ) takes different expressions depending on the zone considered:

display math
display math
display math

being

display math
display math
display math

where E0 is the relative amplitude of the incident plane wave, k is the wave number, D(ϕ, ϕ′, L) is the diffraction coefficient for an edge given in Kouyoumjian and Pathak [1974], and

display math

[6] It should be noted that, in equations (6) and (7), a factor of 1/2 has been included since, although reciprocity is being considered, the rays are actually being emitted from the real source and they are incident at a grazing angle on the arms of the horn [Luebbers, 1989].

[7] Therefore, in the same way, by considering the final UTD-PO solution for the multiple diffraction of plane waves by an array of wedges presented inJuan-Llácer and Rodríguez [2002], the radiation pattern of an E-plane rectangular horn withnwedge-shaped (triangular) corrugations, equally spaced and distributed along the entire length of the arms of the horn, with interior angleγ, height d, and separated by a constant distance w—whose E- plane scheme is depicted inFigure 2—can be calculated, assuming that the emitting source is located at a distance w from the preceding wedge, through the equation (1) with the following values of ET(θ), which, again, depend on the zone considered (in this case, five zones can be distinguished):

display math

with

display math

being the term that represents the field of both the contributions r and r1, where D(ϕ, ϕ′, L) is the diffraction coefficient for a conducting wedge given in Luebbers [1989], E(0) = E0, and Q1 and Q′1 form the first wedge over which the formulation proposed in Juan-Llácer and Rodríguez [2002] for the multiple diffraction of a plane wave by an array of wedges assuming a positive incidence angle—which implies the calculation of both Er and Er1—is applied. Furthermore,

display math

where n′ = n − 1, D(ϕ, ϕ′, L) is the diffraction coefficient for a conducting wedge given in Luebbers [1989], and dw is assumed, so that Q2 and Q′2 can be considered to be the same point, with

display math

where D(ϕ, ϕ′, L) is the diffraction coefficient for an edge given in Kouyoumjian and Pathak [1974], and

display math
Figure 2.

Scheme of the E-plane of a corrugated rectangular horn antenna.

[8] As can be noted, in this case, Er2 is calculated by considering one first diffraction over the edge of the end of the lower arm of the horn, since Q2 and Q′2 are approximated by the same point (as in the case of a conventional horn), so that the field existing over the subsequent wedge (E2) is achieved. Then, the multiple diffraction over the remaining wedges is calculated by assuming grazing incidence over them, since the direct ray was already considered in (10), where the effect of the multiple diffraction of plane waves is much more relevant due to the lower angle of incidence (θE − θ) over the wedges compared to that over the array of corrugations of the lower arm of the horn (θE + θ).

[9] Z2 (θE < θ ≤ π/2 if γ < 2θE, or θE < θ ≤ θE + π/2 − γ/2 if γ > 2θE): consider (3), with

display math

where D(ϕ, ϕ′, L) is the diffraction coefficient for a conducting wedge given in Luebbers [1989], and Er2 is calculated with expressions (11), (12), and (13).

[10] Z3 (π/2 < θ ≤ π/2 + θE − γ/2 if γ < E, or π/2 + θE − γ/2 < θ ≤ π/2 if γ > 2θE): a) if γ < 2θE (π/2 < θπ/2 + θE − γ/2): consider (4), with Er1 calculated with expression (14). b) if γ > 2θE (π/2 + θE − γ/2 < θ ≤ π/2): consider (3), where

display math

where n′ = n − 1, D(ϕ, ϕ′, L) is the diffraction coefficient for a conducting wedge given in Luebbers [1989], and dw is assumed, so that Q1 and Q′1 are considered to be the same point, with

display math

where D(ϕ, ϕ′, L) is the diffraction coefficient for an edge given in Kouyoumjian and Pathak [1974], and

display math

[11] In this case, Er1 is calculated in the same way that Er2 was obtained in Z1 in order to avoid incidence of the plane wave from inside the first wedge, which would have no meaning. Er2 is again calculated with expressions (11), (12), and (13).

[12] Z4 (π/2 + θE − γ/2 < θ ≤ π − θE if γ < 2θE, or π/2 < θ ≤ π − θE if γ > 2θE): consider (4), where Er1 is calculated with expressions (15), (16), and (17).

[13] Z5 (π − θE < θ ≤ π): consider (3), where Er1 is calculated with expressions (15), (16), and (17), and Er2 is calculated with expressions (11), (12), and (13), except for the fact that, in this case,

display math

3. Results

[14] In Figure 3, the E-plane radiation pattern of an E-plane rectangular conventional horn (without corrugations) withρE = 13.5λ, θE = 17.5°, and vertical/hard polarization, calculated with the UTD expressions given above, has been depicted.

Figure 3.

Comparison between the radiation patterns of a conventional horn (without corrugations) obtained through the UTD approach, the EFIE technique [Quesada-Pereira et al., 2007], the method based on Pauli's equations [Borgioli et al., 1997], the HFSS software, and measurements presented in Stutzman and Thiele [1998]. ρE = 13.5 λ and θE = 17.5°.

[15] Furthermore, for the sake of comparison, the same pattern obtained in this case according to three different methods is also shown: the formulation presented in Borgioli et al. [1997] which use Pauli's equations, the solution given in Quesada-Pereira et al. [2007]which is based on the application of a standard two-dimensional electric field integral equation (EFIE) solved by the method of moments (MoM), and the finite element method (FEM)-based full-wave electromagnetic analysis software, HFSS. Specifically, in the EFIE technique, the integral equation has been solved by expanding the unknown electric current density with subsectional triangular basis functions. The same set of functions has also been employed for testing the integral equation, following a Galerkin approach. In this way, for the simulation presented inFigure 3, a total number of 510 basis functions have been used, whereas a maximum fifth order Gauss-Legendre quadrature rule has been applied for the evaluation of the reaction integrals (which refer to the interaction between the basis and test functions through the Green's functions in the Method of Moments). Moreover, for the HFSS results, a 3-D model of the horn has been considered, in which the width has been taken electrically large for comparison purposes with the previously mentioned 2-D models. On the other hand, the fundamental mode TE10 of the feeding waveguide has been considered as the excitation of the horn and, in order to reduce the number of unknowns, two symmetry planes have been taken into account within the whole structure. Besides, measurements given by Stutzman and Thiele [1998], when considering the same horn, can be observed in the figure. As can be seen, solid agreement is found between the four methods and the measurements. It should be noted that the small step existing at θ = 90° in the curve obtained by the UTD approach is due to the fact that, at such an observation angle, the second-order diffracted field emanating from Q1 due to illumination from Q2has been neglected. However, the impact of such an approximation is very small, as can be seen from a comparison with the curves obtained by the other two methods, while at the same time it implies less mathematical complexity as well as higher computational efficiency. A multitude of second-order fields could also have been considered in the UTD approach, such as the second-order diffracted field from Q2 due to illumination from Q1 or, for example, the ray that is diffracted from Q1 and then reflected from the bottom face of the horn, but the effects of these fields on the total pattern would be practically insignificant [Stutzman and Thiele, 1998].

[16] Furthermore, the E-plane radiation pattern of a corrugated E-plane rectangular horn—obtained through the UTD-PO formulation proposed in this work—withρE = 13.5 λ, θE = 17.5°, 40 V-shaped (triangular) corrugations (equally spaced and distributed along the entire length of the faces of the horn),γ = 35° and d = 0.35 λ, is depicted in Figure 4.

Figure 4.

Comparison between radiation patterns of a corrugated horn antenna with 40 triangular corrugations—obtained with the proposed UTD-PO method, the EFIE technique [Quesada-Pereira et al., 2007], and the HFSS software—and that of a conventional horn (without corrugations). ρE = 13.5λ, θE = 17.5°, γ = 35°, and d = 0.35λ.

[17] In order to validate the UTD-PO method, the pattern is compared with those provided –for the same structure and parameters– by both the above mentioned standard two-dimensional EFIE technique and the HFSS software. In this case, for the EFIE technique, a total number of 2900 basis functions have been used, whereas, again, a maximum fifth order Gauss-Legendre quadrature rule has been applied for the evaluation of the reaction integrals. Regarding the HFSS results, the same procedure as that employed for the conventional horn has been considered in this case for the corrugated horn.

[18] It can be noted how a solid agreement is obtained between the three methods. As can be seen from Figure 4, the agreement between the proposed UTD-PO method and both the EFIE numerical technique and the HFSS software is quite good especially in the main lobe, which is the most significant for antenna radiation pattern analysis. Since the back radiation level is much lower than the front level, when it comes to numerical techniques, a greater divergence is common in the region for back radiation as compared to the region for front radiation. In this sense, in the EFIE method, the back radiation results strongly depend on numerical parameters such as the mesh density, the kind of the basis functions and the integration quadrature rules, and in the HFSS method, it depends on the kind of the adaptive mesh or the number of unknowns. Therefore, the differences (ripples) observed inFigure 4 for angles far from the main lobe can be influenced by the above mentioned facts.

[19] It can also be pointed out, for the sake of comparison, that the UTD-PO calculations depicted inFigure 4required just 5.5 s to compute whereas the EFIE technique results took 101 s in the same computer, which means quite a longer time. On the other hand, the HFSS results took around one hour. Regarding the mentioned computational time comparison, it should be noted that most bench mark commercial electromagnetic simulation tools such as HFSS or FEKO require a 3-D model of the structures under analysis and therefore these software instruments do not provide a 2-D analysis module for fair comparison purposes with the new UTD-PO method presented in this paper. Furthermore, even when comparing to a 2-D FEM technique or a Finite Difference Time Domain Technique (FDTD), such numerical procedures require a full mesh of the structure under study as well as its surrounding empty space where appropriate termination boundary conditions are imposed. Therefore, the simulation time for FEM or FDTD would be much higher than the integral-equation based technique used for comparison in the paper. This is because integral equation techniques only need to mesh the horn conducting walls in order to solve the problem, therefore reducing the number of unknowns with respect to alternative methods. The only numerical technique possibly faster than the EFIE method that could be employed for simulation time comparison would be the Multilevel Fast Multipole Method (MLFMM) 2-D integral equation implementations. These MLFMM techniques are normally faster than a plain Method of Moments solution of the integral equation. However, the simulation time improvement of the MLFMM method is only significant when the number of unknowns is very high (in the order of 50000 unknowns). In our case, since the number of unknowns employed for the EFIE solution is about only 5000, the MLFMM time reduction with respect to the EFIE technique would be a 50% at best, which means that the UTD-PO solution would remain being the fastest method, since, as it was previously said, it is 18 times faster than the EFIE technique.

[20] Finally, as can be seen in the figure, the reduction in sidelobe level due to the presence of corrugations is considerable –which is one of the advantages of corrugated horns compared to conventional horns– and, besides, both the width of the main beam and its saddle are reduced compared to the conventional horn pattern, which is, again, shown in the plot.

4. Conclusions

[21] A method based on a hybrid UTD-PO formulation for the calculation of the radiation pattern of corrugated E-plane rectangular horn antennas with V-shaped corrugations has been presented. The method has been validated through comparison with the application of a standard two-dimensional electric field integral equation (EFIE) solved by the method of moments (MoM) and the finite element method (FEM)-based full-wave electromagnetic analysis software, HFSS. In this sense, it can be noted that the proposed UTD-PO solution shows better computational efficiency while at the same time being a straightforward approach.

Acknowledgments

[22] This work was supported by the Ministerio de Educación y Ciencia, Spain (TEC2010-20841-C04-03) and by the European FEDER funds.

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