FEM Analysis of Horn Antennas

Transcription

1 FEM Analysis of Horn Antennas 1 Bhavana Varshney, 2 Manish, 3 A. K. Arora 1 M.Tech, ABES Engineering College, Ghaziabad MTU, Noida Bhavanavarshney.ece@gmail.com 2,3 (Department of Electronics and Communication Engineering, ABES Engineering College, Ghaziabad UP) ABSTARCT The field of antennas is vigorous and dynamic and over the past fifty years, antenna technology has been an indispensable partner of the communication revolution. There is a plethora of antennas and antenna elements, many of which exhibit intricate configurations. With the advent of super fast computers, numerical solution techniques are adapted extensively to analyze the problems with different boundary conditions and complex geometries. Over the past few years, there has been an increased reliance on Finite Difference Time Domain (FDTD) and Finite Element Method (FEM) for the characterization of electromagnetic problems. I. INTRODUCTION TO ANTENNAS The IEEE Standard Definitions of Terms for Antennas (IEE Std ) defines an antenna as a means for radiating or receiving radio waves. In other words antenna is the transitional structure between free space and a guiding device. The guiding device may take the form of a co-axial line or a waveguide, and is used to transport electromagnetic energy from transmitting source to antenna, or from the antenna to the receiver. magnetic flux density;? A,? V,? l are small changes in area,volume and length and is free charge density. Field problems are broadly classified as: Static Field problems Quasi-static Field problems Transient Field problems Work presented in the thesis belongs to Transient field problems. There are two domains in electromagnetism. First, includes the study of electromagnetic waves and propagation, where the displacement current cannot be neglected. This is the high frequency domain. Second, includes the major part of electromagnetic devices like transformers, Motors etc. This domain refers to the low frequency domain. The work presented in the thesis belongs to high frequency domain. II. ELECTROMAGNETIC FIELD PROBLEMS In the arena of Electromagnetics, nothing can move ahead without the involvement of Maxwell s equations. The four Maxwell s equations in are listed below: Ñ.D = Ñ.B = 0 Ñ E=- B/ t Ñ H = J + B/ t where E and H are electric and magnetic fields respectively; D and B are electric displacement and Figure 1.1 Types of field problems

2 Electromagnetic Field problems can be formulated using Maxwell s equations and need to satisfy certain constraints. These problems along with their constraints are expressed in the form of partial differential equations. Method of Solution 1. Numerical Method 2. Analogue Methods 3. Graphical Methods 4. Analytical Methods 1. Numerical Methods Today, computers and numerical methods provide an alternative for complicated calculations. To overcome the drawbacks faced by above listed methods, numerical methods are extensively been used. Different numerical techniques have been developed in recent past. Numerical methods are extremely powerful problem-solving tools. They are capable of handling large systems of equations, nonlinearities and complicated geometries that are not uncommon in engineering practice and they are impossible to solve analytically. As such they greatly enhance our problem solving skills. Numerical methods can be broadly classified as: Finite Difference Method Boundary Element Method Finite Element Method III. FINITE ELEMENT METHOD (FEM) The term finite element was first coined by Clough in 1960 and in the early 1960s; though till early 60 s, engineers used the method for approximate solutions of problems in stress analysis, fluid flow, heat transfer, and other areas. But for the first time, Zienkiweicz and Cheung [35] applied FEM to field problems in 1965.First order elements were used by Chari and Sylvester [36] in 1971 to analyze turbo alternators. Anderson [37] solved non-linear magnetic field problems by finite element method using iterative techniques. The formulations for periodicity conditions, flux line boundaries and Neumann s boundary conditions using triangular elements were considered. The concept of adaptivity and automatic mesh generation became important because it helped in reducing time involved in data preparation and eliminates human errors. Adaptive FE analysis is based on coupling of two different aspects of FE methodsmesh generation and error analysis. Denlaunay derived a simple procedure for triangulating and arbitrary set of points on a plane in such a way that the sum of minimum angle in each triangle is maximized. In 1990 s, several mesh generation algorithms using Denlaunay property were proposed by Joe [38] and S.Rebay[39]. There is plethora of antenna elements, many of which exhibit intricate configurations. To analyze each as a boundary value problem andobtain solutions in closed form, the antenna structure must be described by orthogonal curvilinear coordinate system. This places severe restrictions on the type and number of antenna systems that can be analyzed using such a procedure. Therefore, other exact or approximate methods are often pursued. Finite element has gained momentum in its applications to antenna problems. The development of quasi-optical, non-reciprocal devices in beam waveguide environment using FEM was proposed by W Epp, D.J.Hoppe,G.C. Chinn and Jin-Fa Lee[40] in W Epp, D.J.Hoppe and Jin-Fa Lee[41] proposed a new symmetric formulation of the hybrid finite element method (HFEM) which combines elemen ts of the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) for the exterior region along with the finite element solution for the interior region. The formulation was applied to scattering by inhomogeneous bodies of revolution. To avoid spurious modes in the interior region a combination of vector and nodal based finite elements were used. Integral equations in the exterior region were used to enforce the Sommerfeld radiation condition by matching both the tangential electric and magnetic fields between interior and exterior regions.the finite element method (FEM) was used in conjunction with the method of moments (MoM) and the mode matching technique (MM) to calculate reflection coefficients and radiation patterns for axisymmetric waveguide fed horns by G.C Chinn, D.J Hoppe, L.W[42] in One advantage of this approach was

3 that it allowed for the presence of inhomogeneous materials to be included in the modelling using FEM. C.A Balanis[43] analyzed the scattering and radiation from a ferrite loaded cavity backed slot antennas mounted on infinite ground plane using a hybrid FEM/MoM approach in1997. Analysis of curved boundaries in FDTD and FEM was done by A.Marwaha and S.Marwaha[44] in 2001.The field pattern of Dielectric loaded pyramidal horns was done in 2001 by A.Marwaha, I.S.Hudiara and S.Marwaha[45] in A major breakthrough was the analysis of rigid conical horn in FEM by Zhang Zhijan, Liu Bo, and Chen Hongjie [47] in 2002 which is a strong basis for the present work. With Taflove s, (1980, 1988) continued use of the FDTD as an important analytical tool for solving a multitude of different electromagnetic problems, many investigators began to use it (Lau et al.,1986), (Sullivan et al.,1987,1988) for modelling biological systems exposed to RF fields. The FDTD method has recently become the tool of choice for characterization of the SAR patterns in the human head exposed to cellular telephones. Gandhi (1995) perfected an FDTD model for calculating the SAR from exposure to RF fields from cellular telephones. His model is based on Magnetic Resonance Imager (MRI) scans on a human volunteer, obtained every 3 mm from the top of the head to the feet, which providing a resolution of about 2 mm per pixel for each body cross sectional slice. IV. ANALYSIS OF CONICAL HORN IN FEM Now suppose that the longitudinal field is F, which fulfills the Hehmholtz equations (2.1) Where Ñt 2 is a cross laplacian operator and k 2 = k 2 - _ 2 c, k 2 =w 2me, ß is transmission constant of waveguide. To TE wave F =Hz and for TM wave, F =Ez. Boundary condition for _ is / n c = 0 for TE wave (2.2) and _ c = 0 for TM wave, (2.3) where C is the closed boundary of the waveguide and n is the direction of outer normal line. Corresponding variational problems are (2.4) where s is the cross-section of the waveguide. In the process of dealing with the above variational problem, unknown function F can be expressed by primary function of proper interpolation. If we take triangular split and liner interpolation, the problem can be consequently changed into an extreme value problem of multivariate function: (2.5) According to the variational principal, the above formula is equal to the equation set: (2.6) where n is the sum of joints. Thereby, we can derive a FEM equation of waveguide. (2.7) As to variational problem (2.4), this equation should be processed according to special boundary condition. According to the transmission line theory, the reflection coefficient of the tapered portion on the input

4 port is 2D, a very fine mesh was used to achieve an excellent accuracy. (2.8) where ( z) K K 2 ( z) c _ = - ; K = 2p / l ;Kc(z) is a function of the cutoff wavelength along z. (2.9) V. RESULT Analysis of conical horn in FEM. FEMLAB is powerful simulation software which helps the user to build their own geometry, give the required boundary conditions and select the desired mesh elements. The user has just to define the problem either by specifying the geometry by construction or by importing images from real physical world to get a clear picture of the real world. The conical horn geometry is built in FEMLAB for analysis. The antenna geometry consists of a 0.2 m tall metallic cone with a top angle of 90 degrees on a finite ground plane of a m radius. The coaxial feed has a central conductor of 1.5 mm radius and an outer conductor (screen) of mm radius separated by a teflon dielectric of relative permittivity of The central conductor of the coaxial cable is connected to the cone, and the screen is connected to the ground plane. Domain Equations An electromagnetic wave propagating in a coaxial cable is characterized by transverse electromagnetic fields (TEM). Assuming time-harmonic fields with complex amplitude containing the phase information, the electric(e) and magnetic(h) fields are represented by: (3.1) (3.2) where, z is the direction of propagation and r, f, and z are cylindrical coordinates centered on axis of the coaxial cable. Z is the wave impedance in the dielectric of the cable, and C is an arbitrary constant. The angular frequency is denoted by w. The propagation constant, k, relates to the wavelength in the medium w as (3.3) In the air, the electric field also has a finite axial component whereas the magnetic field is purely azimuthal. Thus it is possible to model the antenna using an axisymmetric transverse magnetic (TM) formulation, and the wave equation becomes scalar in H: Boundary Conditions (3.4) Figure2.1: Geometry of the monoconical horn antenna. The rotational symmetry of the problem is taken advantage of, which allows modeling in 2D, using cylindrical coordinates. Since the modeling was done in The boundary condition for the metallic surfaces is n E = 0.At the feed point, a matched coaxial port boundary condition is used to make the boundary transparent to the wave. The antenna is radiating into free space, but only a finite

5 region can be discretized. Therefore, the geometry is truncated some distance from the antenna using a scattering boundary condition, allowing outgoing spherical waves to pass without being reflected. A symmetry boundary condition for boundaries at r = 0 is applied because the model is axisymmetric. Scattering boundary conditions are represented by the equation The generated mesh is shown below in figures 2.1 and 2.2. n (Ñ Hj)-jkHj=0. (3.5) The boundary at the antenna feed is used as a port because the excitation source is attached to it. For port, the boundary condition is S = ò(e - E 1 ) E 1 / ò E 1 E1 (3.6) Figure 2.3 Triangular Mesh formed inside the analysis plane Figure2.4 Enlarged view of the area with higher potential gradient in a denser mesh Figure 2.2 Illustration of various boundaries using different colours Mesh Generation: FEMLAB has an automatic meshgenerator. The initialized mesh consists of elements and the number of degrees of freedom are solved. There is an inbuilt facility for mesh refinement and selection of the type of finite elements. Figure2.5 2D Surface plot for z component of average power flow

6 Figure 2.6: Comparison of r component of electric field at three frequencies REFERENCES [1]. M. Hano, Finite-element analysis of dielectricloaded waveguides, IEEE Trans. Micro. Theo. Tech., vol. MTT-32, no. 10, Oct. 1984, pp [2]. S. Ahmed and P. Daly, Finite-element methods for inhomogeneous waveguides, Proc. IEEE, vol. 116, no. 10, Oct. 1969, pp [3]. Z.J. Csendes and P. Silvester, Numerical solution of dielectric loaded waveguides: I Finite-element analysis, IEEE Trans. Micro. Theo. Tech., vol.mtt- 18, no. 12, Dec. 1970, pp [4]. M.N.O. Sadiku, A simple introduction to finite element analysis of electromagnetic problems, IEEE Trans. Educ., vol. 32, no. 2, May 1989, pp [5]. O.W.Anderson, Iterative solution of Finite Element Equations in Magnetic Field Problems,IEEE Power Engg. Society Paper,C 72,July 1972,pp