International Journal of Antennas and Propagation

Volume 2016, Article ID 4824873, 6 pages

http://dx.doi.org/10.1155/2016/4824873

## An Approximation Mathematical Formula of Pattern Analysis for Distorted Reflector Antennas considering Surface Normal Vector Variation

Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi’an 710071, China

Received 22 March 2016; Revised 10 June 2016; Accepted 19 June 2016

Academic Editor: Toni Björninen

Copyright © 2016 Shuxin Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An approximation mathematical formula to compute the distorted radiation pattern for reflector antennas is presented. In this approximation formula, besides the phase error caused by the structural deformation being added in the far field integral, the surface normal vector variation is also taken into consideration. The formula is derived by expanding the surface normal vector into a first-order Taylor series and the phase error into a second-order Taylor series. By assembling the integrals including the contributions of both surface normal vector variation and phase error, the far field electrical vector expressed as a function of structural nodal displacements is further obtained in a matrix form. Simulation results of a distorted reflector show the application of this formula.

#### 1. Introduction

Reflector antennas are widely used in radar, telecommunication, radio astronomy, and other applications. Reflector surface distortions are inevitable in outer working conditions owing to thermal, gravitational, and dynamic effects and manufacturing tolerance [1]. With the increasing frequency, the stringent requirements on reflector surface accuracy become demanding [2]. Towards the evaluation of surface accuracy on electromagnetic performance, several tolerance analysis techniques have been proposed in scientific literature for random error [3–6] and systematic error [7–11] in centered reflectors [7–10], offset reflectors [11], and membrane reflectors [11]. By performing tolerance analysis, the reflector antenna can be designed and manufactured with considering the surface tolerance.

The process of antenna design belongs to the area of multidisciplinary design, which contains two main steps of electromagnetic design and structural design. In the previous antenna design, the structural design procedure was implemented by simply reducing the surface root-mean-square (RMS) error to fulfill the requirements of antenna electromagnetic performance, which usually could not obtain the final satisfactory electromagnetic performance as observed in [10, 12]. Thus, the antenna design has been transformed from just reducing surface RMS distortion to an integrated structural-electromagnetic optimization design with a multidisciplinary optimization model [12–15]. The integrated structural-electromagnetic design concept is employed in various applications, which directly combines the electromagnetic performance gain or sidelobe levels with structural inputs other than simply reducing the surface RMS error. Some studies have been performed in the antenna design based on the integrated structural-electromagnetic concept, such as the shape control of antennas [12, 13], multidisciplinary optimization [14, 15], and electromechanical coupling analysis [9, 10]. In the integrated optimization design procedure, the antenna far field pattern will be calculated in the optimization iteration repeatedly, which requires an accurate and fast pattern analysis method for a given reflector with various deformations. An approximation radiation integral method for distorted reflector antennas was proposed in [16] using the decomposition orthogonal basis functions. The proposed method benefits the repeated calculations with allowable computation accuracy. Furthermore, referencing to the three-term Taylor series of phase error in [16], a matrix form formula between the surface nodal displacements obtained by structural finite element method (FEM) and far field electrical vector was derived in [17]. The far field pattern can be directly and easily calculated by substituting the surface nodal displacements into the matrix form, which benefits the iterative pattern analysis procedure of antenna design.

As for the distorted reflector antennas, phase error caused by the surface nodal displacement is usually added in the integral to evaluate its effect on far field radiation pattern [9–11]. The phase error is introduced based on the assumption that the reflector is located at the far field zone of the primary feed, and the amplitude variation of incident magnetic field is neglected. Strictly speaking, the surface distortion introduces not only phase error but also surface normal vector variation. In the previous study, the surface normal variation is neglected, which is based on the fact that, for small-amplitude, smoothly varying errors, the surface normal vector will not deviate substantially from the surface of an undistorted reflector [16]. The questions appear as what percentage of the surface normal variation contributes in the far field integral for distorted reflectors so as to be neglected, and what is the final expression for the matrix form in [17] when considering both the surface normal variation and phase error. The aim of this study, which is a continuation of the work in [17], is to further develop the matrix form formula considering both the phase error and the surface normal vector variation caused by surface nodal displacements.

In this study, the reflector surface is divided into many small elements as in structural FEM analysis, and the far field pattern is obtained by superposition of each element’s far field contribution. The surface normal vector variation caused by surface nodal displacement is expanded into a first-order Taylor series during the implementation, and the shape functions in [17] are also employed to express the integration point displacements as functions of nodal displacements. Both the phase error and the surface normal variation are included in these integrals, and the matrix form in [17] by assembling the integrals on elements is thus extended.

The main difference between this formula and the one in [17] lies in the normal vector variation. The approximated formula in [17] just takes the phase error caused by surface nodal displacements into account, while it fails to consider the surface normal vector variation. The derived formula in this study will take both the phase error and normal vector variation into account theoretically, which can be considered as an extended formula for the one in [17].

The study is organized as follows: Section 2 outlines the derivation of the approximation mathematical formula considering both the surface normal variation and phase error, a distorted reflector is calculated to show the proportion of the surface normal variation contribution in Section 3, and the conclusion is drawn in Section 4.

#### 2. The Approximation Mathematical Formula

The radiation integral for Physical Optics (PO) formula of reflector antennas is expressed as [16] where is the surface normal vector, is the incident magnetic field, the vector locates the integration point, is the imaginary unit, is the wavenumber, is the free-space wavelength at the working frequency, the unit vector is in the observation direction, and is the projected aperture region of reflector antennas.

Provided that a nodal displacement from an ideal undistorted reflector is defined as in the -direction shown in Figure 1, is a scalar function that defines the nodal deformation. Strictly speaking, in the distorted antenna analysis the surface deformation introduces not only the phase error but also the surface normal vector variation. Thus, the far field integral becomes where is the normal vector variation, is the undistorted spherical component in the feed coordinate system as shown in Figure 1, and defines the observation direction. Both and are in the ideal undistorted nominal state. As described in Figure 1, the solid line represents the ideal surface and the dashed line is for the distorted surface. A surface nodal displacement is shown in Figure 1. The global coordinate system is defined as the reflector unprimed coordinate system and is for the feed coordinate system. Shown in Figure 1, denotes the observation point and is the surface point vector in global coordinate system.