Numerical Methods for Ports in Closed Waveguides Christer Johansson Stockholm 2003 Licentiate Thesis Stockholm University Department of Numerical Analysis and Computer Science Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framlägges till offentlig granskning för avläggande av licentiatexamen tisdagen den 28 oktober 2003 kl 16.00 i sal D31, Kungl Tekniska Högskolan, Lindstedtsvägen 5, Stockholm. ISBN 91-7283-598-2 TRITA-NA-0319 ISSN 0348-2952 ISRN KTH/NA/R-03/19-SE c Christer Johansson, October 2003 Högskoletryckeriet, Stockholm 2003 Abstract Waveguides are used to transmit electromagnetic signals. Their geometry is typi- cally long and slender their particular shape can be used in the design of computa- tional methods. Only special modes are transmitted and eigenvalue and eigenvector analysis becomes important. We develop a finite-element code for solving the electromagnetic field problem in closed waveguides filled with various materials. By discretizing the cross-section of the waveguide into a number of triangles, an eigenvalue problem is derived. A general program based on Arnoldi’s method and ARPACK has been written using node and edge elements to approximate the field. A serious problem in the FEM was the occurrence of spurious solution, that was due to improper modeling of the null space of the curl operator. Therefore edge elements has been chosen to remove non physical spurious solutions that arises. Numerical examples are given for homogeneous and inhomogeneous waveguides, in the homogeneous case the results are compared to analytical solutions and the right order of convergence is achieved. For the more complicated inhomogeneous waveguides with and without striplines, comparison has been done with results found in literature together with grid convergence studies. The code has been implemented to be used in an industrial environment, to- gether with full 3-D time and frequency domain solvers. The 2-D simulations has been used as input for full 3-D time domain simulations, and the results have been compared to what an analytical input would give. ISBN 91-7283-598-2 • TRITA-NA-0319 • ISSN 0348-2952 • ISRN KTH/NA/R-03/19-SE iii iv Acknowledgments I would like to thank all the people involved in the GEMS project. Particularly I want to thank Erik Abenius and Anders Ålund for their help in debugging the code. I would also like to thank the industrial partners that give a broader perspective to my work. I would also want to thank my advisor Björn Enquist for finding time for sci- entific discussions. He also arranged visits for me to University of California, Los Angeles and Princeton University. This work was supported with computing re- sources at Paralelldatorcentrum (PDC), Royal Institute of Technology. Special thanks to Anders Höök for the geometrical setup of Clavin antenna, and Katarina Gustavsson for proofreading my thesis. Financial support has been provided by NADA, KTH, NUTEK (PSCI). v vi Contents 1 Introduction 1 2 General Framework 5 2.1Waveguidemodeling.......................... 6 2.2Staticfield................................ 8 2.3Full-Waveanalysis........................... 10 2.3.1Homogeneousmedia...................... 10 2.3.2 Inhomogeneousmedia..................... 12 2.4Analyticalsolutions.......................... 12 2.4.1Rectangular........................... 13 2.4.2 Circular............................. 13 2.4.3 Coaxial............................. 14 2.4.4 Transverseresonancetechnique................ 15 3 Variational Formulation 17 3.1Potential................................. 17 3.2Vector.................................. 18 4 Domain Decomposition and Hybrid Methods 19 5 Finite Dimensional Approximation 21 5.1Potential................................. 23 5.2Vector.................................. 23 5.3 Spurious modes . 25 6 Generalized Algebraic Eigenvalue Problem 27 6.1Arnoldi’smethod............................ 27 6.2Reordering............................... 31 vii viii Contents 7 Numerical Results for 3-D Waveguides 33 7.1Homogeneouswaveguide........................ 33 7.1.1 Rectangularwaveguide..................... 33 7.1.2 Circularwaveguide....................... 41 7.1.3 Coaxialwaveguide....................... 41 7.2Inhomogeneouswaveguide....................... 42 7.3EnclosedStripline........................... 46 7.3.1Homogeneous.......................... 46 7.3.2 Inhomogeneous......................... 47 8 Direct Numerical Simulation of 3-D Waveguides 49 8.1Coaxialwaveguide........................... 52 8.2Rectangularwaveguide......................... 55 8.3Clavinantenna............................. 58 9Conclusion 59 A Element Matrices 61 A.1Laplace................................. 61 A.2Helmholtz................................ 62 B Algorithm Description 63 B.1Laplace................................. 63 B.2Helmholtz................................ 64 CTheGemsProject 65 Chapter 1 Introduction Waveguides are used for the transmission of electromagnetic signals. The electro- magnetic fields in these waveguides are given by the Maxwell equations, ∇·D = ρ ∇·B =0 ∂B = −∇ × E ∂t ∂D = ∇×H − J ∂t where, E(x, y, z, t) is the electric field, D(x, y, z, t) is the electric flux density, H(x, y, z, t) is the magnetic field, B(x, y, z, t) is the magnetic flux density, J(x, y, z, t) is the electric current density and ρ is the charge density. These equations describe general electromagnetic phenomena. The equations are linear but there are only in cases with special boundary interface conditions that analytical solutions can be obtained. Electromagnetic applications have become more complex and the need to predict their performance more important. Therefore numerical methods have become a useful tool based on faster computers and improved algorithms. One area of application is systems that rely on waveguides for the transmission of the signal, like antennas see figure 1.1. At high frequencies this is the only practical way of transmitting electromagnetic radiation. Waveguides can handle high power with low loss, but they are often bulky and expensive. In modern day an often used waveguide are planar transmission lines (striplines) used in integrated microwave circuits. These are very compact and with low cost. For many years, something called lumped element circuit [34], has been used for simulating transmission lines. This method makes the simplifying assumption that each element, such as resistors, inductors, capacitors, is an infinitesimal point in space, and that the wires connecting elements are perfect conductors. However, 1 2 Chapter 1. Introduction Figure 1.1. Cross-section of a horn antenna. The horn is feed by an waveguide which in turn is feed by a coaxial line. The dashed line shows there one may want to use ports this method is not valid at microwave frequencies where the dimension of the de- vice is of the same order as the micro wavelength. These high frequencies (short wavelength) are difficult to analyze but very important in applications like radar and communication systems, since • More bandwidth, more information can be transmitted • In radar systems the effective reflection area is proportional to the targets electrical size In resent years numerical methods has become a more useful tool when designing these waveguides. The simulation problem is a full 3-D problem, but by using separation of variables, the propagation direction is eliminated and the problem is reduced to a much smaller 2-D problem. This new problem is studied and used in the full 3-D simulation of for example antennas where the waveguide is the cable connecting to the antenna. With the introduction of more complex waveguides, it becomes important to be able to predict the wavenumber, or propagation constant for the medium [11]. One typically wants to evaluate propagation characteristics of waveguides with arbitrarily shaped cross section and the finite-element method (FEM) is often used. Other typical methods are methods of moments (MoM), spectral-domain and finite- difference methods (FD). One should note that FD may not work well on every type of waveguide because of geometric restrictions, but is generally faster and easier to implement. For complex geometries, with different materials one has to solve for all the field components to determine the electric and magnetic fields. Also, the normal components may suffer a discontinuity at the interfaces when we have domains with different materials. 3 We will focus on the FE approach in this thesis, since it allows complex geomet- ric structures containing inhomogeneous materials. By discretizing the cross-section of the waveguide into a number of elements, often triangles, and by using the vari- ational formulation, FEM can now be used to predict propagation characteristics of the waveguide. An often used basis in FEM is the nodal basis, but it is not straightforward since these elements cannot reproduce jump discontinuities easily. One way to solve this is to solve for the potentials, that are continuous. However the differentiation required to obtain the electric and magnetic fields suffers from loss of accuracy. Another a serious problem in the FEM is the occurrence of spurious solutions. This is due to improper modeling of the null space of the curl operator. An often used method to eliminate the spurious solutions is the use of a penalty function. The penalty function method does not remove the spurious modes but shift their cutoff frequency outside the frequency range of interest. Another approach to eliminate the spurious solutions, is to use tangential vector