Microstrip and Printed Antennas: New Trends, Techniques and Applications

This book focuses on new techniques, analysis, applications and future trends of microstrip and printed antenna technologies, with particular emphasis to recent advances from the last decade

Attention is given to fundamental concepts and techniques, their practical applications and the future scope of developments. Several topics, essayed as individual chapters include reconfigurable antenna, ultra-wideband (UWB) antenna, reflectarrays, antennas for RFID systems and also those for body area networks. Also included are antennas using metamaterials and defected ground structures (DGSs). Essential aspects including advanced design, analysis and optimization techniques based on the recent developments have also been addressed.

Key Features:

• Addresses emerging hot topics of research and applications in microstrip and printed antennas
• Considers the fundamental concepts, techniques, applications and future scope of such technologies
• Discusses modern applications such as wireless base station to mobile handset, satellite earth station to airborne communication systems, radio frequency identification (RFID) to body area networks, etc.
• Contributions from highly regarded experts and pioneers from the US, Europe and Asia

This book provides a reference for R&D researchers, professors, practicing engineers, and scientists working in these fields. Graduate students studying/working on related subjects will find this book as a comprehensive literature for understanding the present and future trends in microstrip and printed antennas.

• Language: English
• Publisher: Wiley
• Release date: Feb 2, 2011
• ISBN: 9781119972983

Book preview

Preface

Microstrip technology has been popular for microwave and millimeter wave applications since the 1970s and recently has taken off, with the tremendous growth in communications, wireless, as well as space-borne/airborne applications, although the concept dates back to 1952 [1]. The basic microstrip configuration is very similar to a printed circuit board (PCB) used for low frequency electronic circuits. It constitutes a low-loss thin substrate, both sides being coated with copper film. Printed transmission lines, patches, etc. are etched out on one side of the microstrip board and the other copper-clad surface is used as the ground plane. In between the ground plane and the microstrip structure, a quasi-TEM electromagnetic wave is launched and allowed to spread.

Such a structure offers some unique basic advantages such as low profile, low cost, light weight, ease of fabrication, suitability to conform on curved surface, etc. All these have made microstrip technology attractive since the early phase of its development.

Within a year of the pioneering article Microstrip – a new transmission technology for the kilomegacycle range appearing [1], Deschamps [2] had conceived of microstrip as microwave antenna. But its practical application started nearly two decades later. Howell [3] and Munson [4] may be regarded as the pioneer architects of microstrip antenna engineering.

These early developments immediately attracted some potential research groups and the following studies were mainly concerned with theoretical analysis of different patch geometries and experimental verifications [5–12]. A parallel trend also developed very quickly and some researchers tried to implement conventional antennas such as dipole, wire, aperture, etc. in planar form [13–16]. They are commonly referred to as printed circuit antennas or simply printed antennas. Their operations and characteristics are completely different from those due to microstrip patches, although microstrip patch antennas, in many papers, are casually called printed circuit antennas. The topic printed antenna had acquired tremendous importance by the late 1970s and a three-day workshop held at New Mexico State University in Las Crises in October, 1979 was dedicated to Printed Circuit Antenna Technology.

The developments in microstrip antennas that occurred up to the late1970s were documented by Bahl and Bhartia in their famous book [17], published in 1980. The analysis and design aspects were addressed in another book by James, Hall and Wood [18], published in 1981. A contemporary article by Carver and Mink [19] discussed the fundamental aspects of microstrip antennas and this is still regarded as a good review paper for a beginner.

More activities in the area grew gradually and many applications were realized. The suitability of deploying such lightweight low profile antennas in airborne and space-borne systems initiated major developments in microstrip array technology. With the development of mobile and wireless communications, microstrip and other printed antennas attained a new focus to serve in different technology from the mobile handset to base station antennas. General information, gathered from journals, symposia and conference articles, reveals that about 50% of the whole antenna community has been active in microstrip or printed antenna practice for the past two or three decades.

The first handbook [20] was published in 1989, nearly a decade after the first book by Bahl and Bhartia [17]. Within another five years, microstrip antenna research had attained a level of maturity as is reflected in the title and topics of the microstrip antenna books published around the middle of 1990s [21–23]. The edited volume by Pozar and Shaubert [21] contains some published articles bearing the results of contemporary interests, such as bandwidth enhancement approaches, analysis and design techniques, aperture coupling and other feeding methods, active integrated antennas, conformal and phased arrays, etc. Narrow impedance bandwidth appears an inherent limitation of the microstrip element. The research and consequent developments in bandwidth enhancement were documented in [22]. Lee and Chen [23] covered some key areas of advances reported up to 1997.

The growing need and interest in microstrip antenna designs are reflected in three design handbooks [24–26] published at close interval from 2001 to 2004. Compacting, along with bandwidth widening of printed antennas, has attracted worldwide interest to support new wireless technology since the beginning of this century and its importance was reflected in titles [27–32] which appeared between 2002 and 2007.

The book edited by Lee and Chen [23] was a timely effort to incorporate major technological developments that had occurred up to1997, under the same cover. Since then, more than a decade has passed during which many new trends, techniques and applications in planar antenna technology have been developed. For example, RFID (Radio Frequency Identification) is an ideal example to showcase the need to this day. This application needs low cost antennas, printed on paper or very thin substrate. Another example is printed antenna using unconventional and new innovations, such as using metamaterials and defected ground structures (DGSs). Replacing a large parabolic dish with a flat microstrip array with a special feeding mechanism is also a new area of activity. The design of small ultrawideband (UWB) antennas with good performance is a challenging area. Antenna for the body area network is another interesting new topic.

From our long experience in teaching and mentoring doctoral and post-doctoral students and working with practicing engineers, we certainly feel there is a need for a book that is to address more recent topics of microstrip and printed antennas. We have chosen some topics that have recently been developed or have considerably advanced during the past decade and at the same time appear to be important to the new generation of researchers, developers and application engineers. We shared the ideas with some of our colleagues and friends who are the real technical experts and potential developers in those selected topics. They fully agreed with our views, gave valuable suggestions and delivered on their promise to contribute. Our collaborative efforts have finally culminated in the present title.

As indicated by the title, the focus is on the New Trends, Techniques and Applications of Microstrip and Printed Antennas. The chapters are organized as follows: Chapters 1–4 address advances in design, analysis, and optimization techniques, Chapters 5–10 focus on some important new techniques and applications, Chapters 11 and 12 deal with engineered materials applied to printed antenna designs, and finally Chapter 13 addresses advanced methods and designs of printed leaky wave antenna.

Chapter 1 deals with numerical techniques, which are essential in analyzing and designing planar antennas of any arbitrary geometry. A brief overview of the commonly used methods are discussed and the finite difference time domain (FDTD) technique is elaborated on, with special emphasis on the recent developments that occurred after 2003. Chapter 2 presents the advances in computer aided designs (CAD) of microstrip antennas reported during 2001 and onwards. The aim of this chapter is to provide accurate closed form expressions, which can be reliably used to compute essential design parameters such as operating frequency, input impedance and matched feed-location for a given antenna involving single or multiple dielectric layers. Chapter 3 embodies the Generalized Scattering Matrix (GSM) approach to analyzing the multilayer finite printed array structures. The methodology is demonstrated through examples. Chapter 4 deals with antenna optimization techniques. Optimization in terms of performance, size and cost is discussed and the basic concept of stochastic optimization techniques is demonstrated.

Chapter 5 describes microstrip reflectarray technology, its general principle, design, operation, and applications. Microstrip's inherent demerit of narrow bandwidth is dealt with in terms of spatial and frequency dispersions and some of the techniques to suppress these factors are presented. Chapter 6 deals with Reconfigurable Microstrip Antennas, which use switches, tunable materials, or control circuitry to give additional degrees of operational freedom or to make a single element operative in multiple frequencies. A wide variety of reconfigurability is discussed. The emerging trends and directions for future research have also been indicated.

Chapter 7 describes wearable antennas for body area networks. The properties of the human body in terms of electromagnetic radiations and the performance of multiple antenna systems in presence of the human body are described. Chapter 8 presents printed wireless antennas. These include three primary configurations: microstrip patch, slot, and monopole showing multiband, wideband, or ultra wideband performances. Significant developments reported since 2000 are addressed in this chapter. Chapter 9 deals with printed antennas for RFID tags. An RFID system may be one of the following types: active, passive, or in between of these two, based on the nature of the devices used and also any of LF, HF, or UHF type based on the frequency of operations. Passive tags operating at UHF place several specialized requirements on the associated antenna structures and these are described in this chapter. Chapter 10 deals with printed antennas for ultra-wideband (UWB) applications. This incorporates the innovative technologies to minimize ground plane effects on the performance of small printed antennas.

Chapter 11 presents applications of metamaterials to planar antenna and radiative system designs. Both leaky wave and resonant metamaterial antennas are discussed with special emphasis on their recent and somewhat exotic applications. Chapter 12 deals with defected ground structures (DGS) applied to microstrip antennas. This is a recently developed topic and all the major developments that have occurred after 2002 are discussed, indicating the future scope of development. This is probably addressed here as an exclusive book chapter for the first time. Chapter 13 concludes with printed leaky wave antennas. It includes both theory and some applications based on recent advances in technology.

Each chapter is designed to cover the range from fundamental concepts to the state-of-the-art developments. We have tried to satisfy a wide cross-section of readers. A student or a researcher may consider this a guide book to understanding the strength and weaknesses of the contemporary topics. To a practicing engineer, we hope that the book will be a ready reference to many new areas of applications. To an educator, the book appears as a comprehensive review and a source of up-to-date information.

Our sincere efforts and exercise will be successful if our readers appreciate and find it useful for their respective purposes.

Debatosh Guha

Yahia M. M. Antar

References

1. D. D. Greig and H. F. Engleman, Microstrip – a new transmission technology for the kilomegacycle range, Proc. IRE, vol. 40, pp. 1644–1650, 1952.

2. G. A. Deschamps, Microstrip microwave antennas, presented at the 3rd USAF Symp. on Antennas, 1953.

3. J. Q. Howell, Microstrip antennas, Dig. IEEE Int. Symp. Antennas Propagat., pp. 177–180, Dec. 1972.

4. R. E. Munson, Conformal microstrip antennas and microstrip phased arrays, IEEE Trans. Antennas Propagat., vol. 22, pp. 74–78, 1974.

5. T. Itoh and R. Mittra, Analysis of microstrip disk resonator, Arch. Elek. Ubertagung, vol. 21, pp. 456–458, Nov. 1973.

6. T. Itoh, Analysis of microstrip resonator, IEEE Trans. Microwave Theory Tech., vol. 22, pp. 946–952, Nov. 1974.

7. A. Derneryd, Linearly polarized microstrip antennas, IEEE Trans. Antennas Propagat., vol. 24, no. 6, pp. 846–851, 1976.

8. G. Dubost, M. Nicolas and H. Havot, Theory and applications of broadband microstrip antennas, Proc. 6th European Microwave Conference, pp. 275– 279, 1976.

9. P. Agrawal and M. Bailey, An analysis technique for microstrip antennas, IEEE Trans. Antennas Propagat., vol. 25, no. 6, pp. 756–759, 1977.

10. W.F. Richards, Y.T. Lo and D. D. Harrison, Improved theory for microstrip antennas, Electronics Letters, vol. 15, no. 2, pp. 42–44, 1979.

11. Y.T. Lo, D. Solomon and W. Richards, Theory and experiment on microstrip antennas, IEEE Trans. Antennas Propagat., vol. 27, no. 2, pp. 137–145, 1979.

12. P. Hammer, D. Van Bouchaute, D. Verschraeven and A. Van de Capelle, A model for calculating the radiation field of microstrip antennas, IEEE Trans. Antennas Propagat., vol. 27, no. 2, pp. 267–270, 1979.

13. K. Keen, A planar log-periodic antenna, IEEE Trans. Antennas Propagat., vol. 22, no. 3, pp. 489–490, 1974.

14. D.T. Shahani and Bharathi Bhat, Network model for strip-fed cavity-backed printed slot antenna, Electronics Letters, vol. 14, no. 24, pp. 767–769, 1978.

15. Inam E. Rana and N. G. Alexopoulos, On the theory of printed wire antennas, 9th European Microwave Conference, 1979, pp. 687– 691, 1979.

16. A. Mulyanto, and R. Vernon, A V-shaped log-periodic printed-circuit antenna array for the 1 to 10 GHz frequency range, Antennas and Propagation Society Intl. Symp., 1979, vol. 17, pp. 392–395.

17. I. J. Bahl and P. Bhartia, Microstrip Antennas, Artech House, Dedham, MA, 1980.

18. J. R. James, P. S. Hall and C. Wood, Microstrip Antennas: Theory and Design, Peter Peregrinus, London, 1981.

19. K. Carver and J. Mink, Microstrip antenna technology, IEEE Trans. Antennas Propagat., vol. 29, pp. 2–24, Jan. 1981.

20. J. R. James and P. S. Hall, Handbook of Microstrip Antennas, Peter Peregrinus, London, 1989.

21. D. M. Pozar and D. H. Schaubert, Microstrip Antennas, IEEE Press, New York, 1995.

22. J. F. Zürcher and F. E. Gardiol, Broadband Patch Antennas, Artech House, Boston, 1995.

23. K. F. Lee and W. Chen, Advances in Microstrip and Printed Antennas, John Wiley & Sons, Inc., New York, 1997.

24. R. Garg et al., Microstrip Antenna Design Handbook, Artech House, Boston, 2001.

25. R. Waterhouse, Microstrip Patch Antennas: A Designer's Guide, Springer, Berlin, 2003.

26. R. Bancroft, Microstrip and Printed Antenna Design, Noble Publishing, 2004.

27. Kin-Lu Wong, Compact and Broadband Microstrip Antennas, John Wiley & Sons, Inc., New York, 2002.

28. G. Kumar and K. P. Ray, Broadband Microstrip Antennas, Artech House, Boston, 2002.

29. Kin-Lu Wong, Planar Antennas for Wireless Communications, John Wiley & Sons, Inc., New York, 2003.

30. Zhi Ning Chen and Michael Yan Wah Chia, Broadband Planar Antennas: Design and Applications, John Wiley & Sons, Inc., New York, 2006.

31. Peter S. Hall and Yang Hao, Antennas and Propagation for Body-Centric Wireless Communications, Artech House, Boston, 2006.

32. Zhi Ning Chen (eds.), Antennas for Portable Devices, John Wiley & Sons, Inc., New York, 2007.

List of Contributors

Yahia M. M. Antar, Royal Military College, Canada

Jennifer T. Bernhard, University of Illinois at Urbana-Champaign, USA

Arun K. Bhattacharyya, Northrop Grumman Corporation, USA

Sujoy Biswas, Institute of Technology and Marine Engineering, India

Christophe Caloz, École Polytechnique, Montreal, Canada

Reza Chaharmir, Communication Research Centre Canada, Ottawa, Canada

Zhi Ning Chen, Institute for Infocomm Research, Singapore

Daniel Deavours, University of Kansas, USA

Daniel Dobkin, Enigmatics, USA

Ramesh Garg, Indian Institute of Technology, Kharagpur, India

Debatosh Guha, Institute of Radio Physics and Electronics, University of Calcutta, India

Peter S. Hall, University of Birmingham, UK

Yang Hao, Queen Mary College, University of London, UK

Samir F. Mahmoud, University of Kuwait, Kuwait

Rabindra K. Mishra, Electronic Science Department, Berhampur University, India

Xianming Qing, Institute for Infocomm Research, Singapore

Shie Ping Terence See, Institute for Infocomm Research, Singapore

Lotfollah Shafai, University of Manitoba, Canada

Jafar Shaker, Communication Research Centre Canada, Ottawa, Canada

Satish K. Sharma, San Diego State University, USA

Jawad Y. Siddiqui, Institute of Radio Physics and Electronics, University of Calcutta, India

Acknowledgments

Editing a book, like this, is a rare experience involving both liberty and responsibility. We came up with this idea in 2008 and started consulting with the experts who could be potential authors for different chapters of this book. The idea has turned into reality only due to the unstinted cooperation of the authors, who could dedicate time from their extremely busy schedules and contribute to different topics. We are grateful to all of them for their spontaneous help and support.

We would also like to express our thanks to a number of our colleagues, researchers and students who helped with many tasks throughout the process. Mr. Sujoy Biswas of Institute of Technology and Marine Engineering, India, Dr. Jawad Y. Siddiqui of the University of Calcutta, India (currently associated with the Royal Military College, Canada), Mr. Chandrakanta Kumar of the Indian Space Research Organization, and Mr. Anjan Kundu of University of Calcutta have extended their constant help and technical support throughout the whole process. We have also received help from some of our students: Sudipto Chattopadhyay of Siliguri Institute of Technology, India, Symon Podilchak of Queen's University and the Royal Military College, Canada, and Mr. David Lee of CRC in Ottawa. Dr. Somnath Mukherjee of RB Technology, USA, helped us tremendously in resolving the organization of the book. We have received constant help and support from Sarah Tilley, Anna Smart, and Genna Manaog of Wiley, which made our job easy. We are extremely grateful to all of them.

We cannot but acknowledge the ungrudging support and cooperation received from our families and from our respective Institutions: the University of Calcutta and the Royal Military College of Canada. It is always challenging to bring so many people from different parts of the world to work together on one task at the same time. We express our indebtedness to all members of this team for contributing to this volume in their different capacities.

Chapter 1

Numerical Analysis Techniques

Ramesh Garg

Indian Institute of Technology, Kharagpur, India

1.1 Introduction

Microstrip and other printed antennas are constituted of, in general, patches, strips, slots, packaged semiconductor devices, radome, feed, etc. in a nonhomogeneous dielectric medium. Finite substrate and ground plane size are the norm. The dielectric used is very thin compared to the other dimensions of the antenna. The design of these antennas based on models such as transmission line model or cavity model is approximate. Besides, these designs fit regular-shaped geometries (rectangular, circular, etc.) only, whereas most of the useful antenna geometries are complex and do not conform to these restrictions [1]. The effect of surface waves, mutual coupling, finite ground plane size, anisotropic substrate, etc. is difficult to include in these types of design. The numerical techniques, on the other hand, can be used to analyze any complex antenna geometry including irregular shape, finite dielectric and ground plane size, anisotropic dielectric, radome, etc. The popular numerical techniques for antenna analysis include method of moments (MoM), finite element method (FEM), and finite difference time domain method (FDTD). MoM analysis technique, though efficient, is not versatile because of its dependence on Green's function. FEM and FDTD are the most suitable numerical analysis techniques for printed antennas. FDTD is found to be versatile because any embedded semiconductor device in the antenna can be included in the analysis at the device-field interaction level. This leads to an accurate analysis of active antennas. Maxwell's equations are solved as such in FDTD, without analytical pre-processing, unlike the other numerical techniques. Therefore, almost any antenna geometry can be analyzed. However, this technique is numerically intensive, and therefore require careful programming to reduce computation cost. We shall describe the advances in FDTD. Our reference in this respect is the classic book on FDTD by Taflove and Hagness [2].

A large number of FDTD algorithms have been developed. These can be classified as conditionally stable and unconditionally stable. The conditionally stable schemes include the original or Yee's FDTD also called FDTD (2,2), FDTD (2,4), sampling bi-orthogonal time-domain (SBTD) and their variants; and the unconditionally stable schemes include ADI (Alternate Direction Implicit), CN (Crank Nicolson), CNSS (Crank Nicolson Split Step), LOD (Local One-Dimensional) and their variants. The updating of fields in conditionally stable schemes does not require a solution of matrix equation as an intermediate step, and are therefore fully explicit. However, these schemes have a limit on the maximum value of the time step, which is governed by the minimum value of the space step through the Courant-Friedrich-Levy (CFL) condition.

(1.1) equation

Due to the heterogeneous nature of the dielectric in the printed antennas, the wave velocity is less than c and may vary from cell to cell and from one frequency to another. We therefore introduce a safety margin and choose uniformly to simplify coding and avoid instability. Defining the Courant number q as

(1.2) equation

implies that q = 1/2 and the wave takes time to travel to the next node.

The value of puts a severe computational constraint on the structures as they have fine geometrical features such as narrow strips or slots or thin dielectric sheets. Since the simulation time of an antenna is independent of space and time steps, the number of updates of fields increases linearly with the decrease in the time step. This results in an increase in processor time. The limitation on is removed in some of the FDTD algorithms and these are therefore called unconditionally stable schemes. In these schemes one can use the same value of the time step over the whole geometry even if fine geometrical features exist without significantly affecting the accuracy of simulation results. Updating fields in unconditionally stable schemes is carried out in stages called time splitting and involves solving a set of simultaneous equations before going on to the next stage. These schemes therefore are more computationally intensive. However, their accuracy is similar to that of conditionally stable FDTD schemes.

The FDTD analysis of open region problems such as antennas necessitates the truncation of the domain to conserve computer resources. The truncation of the physical domain of the antenna is achieved through absorbing boundary conditions, either analytical ABC or material ABC. Material ABC in the form of PML can achieve a substantial truncation of domain with very low reflection. The design of PML should be compatible with the FDTD scheme employed for the rest of the antenna. A number of PML formulations are available. These are split-field and non split-field PML. Non split-field types are convenient for coding and are therefore preferred. Of the various PML formulations available now, uniaxial PML looks promising.

All the FDTD algorithms suffer from computational error, and the amount of error is related to the space and time step sizes employed. The error is quantified in the form of numerical dispersion. The goal of various FDTD schemes is to analyze multi-wavelength long complex geometries, efficiently and accurately. The complexity of the geometry may be in the form of fine geometrical dimensions, anisotropic dispersive medium, embedded packged semiconductor device, feed, mounting structure, etc. The efficient FDTD algorithms try to achieve this aim by increasing the permissible space step size without increasing dispersion, by an increase in the time step size compatible with fine geometrical features, the applicability of the algorithm to anisotropic and dispersive medium and reduced reflection from the PML medium. The presence of thin strips/slots makes uniform discretization an inefficient approach. New and efficient solutions are being tested in the form of a sub-cell approach, quasi-static approximation, etc. The treatment of PEC and PMC boundary conditions presented by irregular geometries is receiving due attention, while the interface conditions interior to the device are somewhat difficult to implement accurately. Modeling of fast variation of fields in metal, and analysis of curved geometries is being attempted. We shall now discuss the advances in FDTD analysis since 2003.

Yee's algorithm is outlined first in order to define the grid structure and the placement of electric and magnetic field components on the Yee cell. This grid will be used as a reference for other FDTD algorithms.

1.2 Standard (Yee's) FDTD Method

The FDTD method was first proposed by Yee in 1966 [3] and has been used by many investigators because of its host of advantages. However, computer memory and processing time for FDTD have to be huge to deal with the problems which can be analyzed using techniques based on the analytical pre-processing of Maxwell's equations such as MoM, mode matching, method of lines, FEM, etc. Therefore, the emphasis in the development of FDTD technique is to reduce the requirement for computer resources so that this technique can be used to analyze electrically large complex electromagnetic problems.

To determine time-varying electromagnetic fields in any linear, isotropic media with constants ε, μ, σ Maxwell's curl equations are sufficient; the curl equations are

(1.3a) equation

(1.3b) equation

The partial differential equations (1.3) are solved subject to the conditions that: (i) the fields are zero at all nodes in the device at t = 0 except at the plane of excitation; (ii) the tangential components of E and H on the boundary of the domain of the antenna must be given for all t > 0. For computer implementation of Equation (1.3), the partial derivatives are implemented as finite difference approximations, and are partly responsible for the inaccuracy of the solution. For better accuracy, the central difference approximation is used in FDTD and is defined as,

(1.4)

equation

where O(·) stands for the order of. Use of Equation (1.4) converts Equation (1.3) into the following form:

(1.5a)

equation

(1.5b)

equation

(1.5c)

equation

(1.5d)

equation

(1.5e)

equation

(1.5f)

equation

The indices i, j, and k define the position of the field nodes, such that . The time instant is defined by . To implement the finite difference scheme in three dimensions, the antenna is divided into a number of cells, called Yee cells, of dimension . One such cell is shown in Figure 1.1. Remarkably the positions of different components of E and H on the cell satisfy the differential and integral forms of Maxwell's equations. One may note from Figure 1.1 that the placements of the E and H nodes are offset in space by half a space step; it is called staggered grid. We note from Equation (1.5) that the time instants when the E and H field components are calculated are offset by half a time step, that is, components of E are calculated at nΔt and components of H are calculated at (n + 1/2)Δt. The alternate update of E and H fields is called leap frog and saves computer processing time.

Figure 1.1 Geometry of Yee's cell used in FDTD analysis

1.3 Numerical Dispersion of FDTD and Hybrid Schemes

The finite difference form of derivative (1.4) has an error term O . As a result, Equations (1.5 a–f) are second-order accurate, resulting in an approximate solution of the problem. The first sign of this approximation appears in the phase velocity for the numerical wave being different from that in the continuous case. This phenomenon is called numerical dispersion. The amount of dispersion depends on the wavelength, the direction of propagation in the grid, time step and the discretization size . The above algorithm is second-order accurate in space and time, and is therefore called FDTD(2,2). The numerical dispersion for plane wave propagation may be determined from the following expression

(1.6)

equation

where is the wave number for the numerical wave. The phase velocity is determined by solving Equation (1.6) as a function of discretizations and propagation angle . The phase velocity is found to be maximum and close to the velocity of light for propagation along the diagonals and minimum for waves propagating along the axis.

1.3.1 Effect of Non-Cubic Cells on Numerical Dispersion

Devices with high aspect ratio may be analyzed by using uniform or non-uniform cell size. An alternative is to employ non-square or non-cubic cells. The influence of the aspect ratio of the unit cell on the numerical dispersion of FDTD(2,2) has been reported by Zhao [4]. It is found that the dispersion error increases with the increase in aspect ratio of the cell but reaches an upper limit for aspect ratios greater than 10. For N (number of cells per wavelength, ) = 10, the maximum dispersion error for non-cubic cells is 1.6% which decreases to 0.4% for N = 20, showing second-order accuracy. In general, the maximum error for non-cubic cells is about 1.5 times that of the corresponding error for cubic cells. For the non-square cells, this ratio is twice that of square cells [4]. For guidance, the minimum mesh resolution required to achieve a desired phase velocity error is plotted in Figure 1.2 for the cubic and non-cubic cells. It may be noted from Figure 1.2 that 0.5% accuracy in phase velocity is achieved for N = 18.5, and N = 13 is needed for 1% accuracy when non-cubic cells are employed. This study shows that unit cells with very high aspect ratio may be used by sacrificing a small amount of accuracy in phase velocity. FDTD(2,2) is also employed for benchmarking other schemes.

Figure 1.2 Comparison of minimum mesh resolution required for a given accuracy of phase velocity when non-cubic (with high aspect ratio) or cubic unit cells are employed.

Reproduced by permission of ©2004 IEEE, Figure 8 of [4]

1.3.2 Numerical Dispersion Control

The numerical dispersion can be reduced to any degree that is desired if one uses a fine enough FDTD mesh. This, however, increases the number of nodes and therefore also increases the computer memory and processor time required. An alternative way to decrease numerical dispersion is to improve upon the finite difference approximation of Equation (1.4). Higher-order finite difference schemes, also called multi-point schemes, are available to reduce the error in approximating the derivatives. The fourth-order-accurate schemes called FDTD(2,4) employ four nodal values located at and on either side of the observation point , and the space derivative is defined as [5]

(1.7)

equation

Another algorithm with lower dispersion called SBTD (sampling bi-orthogonal time domain) has been proposed. It is an explicit scheme with leap-frog update. It is conditionally stable wavelet-based scheme in which spatial discretization of FDTD is replaced with sampling bi-orthogonal discretization [6]. The field is expanded in wavelets or scale functions as basis functions in space domain, while the time domain expansion is in pulse functions. The coefficients of expansion of wavelets are determined by testing Maxwell's equations with the scaling functions. For the two-dimensional TM case, the expression for the fields for SBTD is of the form [6]

(1.8a)

equation

(1.8b)

equation

(1.8c)

equation

where

(1.9)

equation

The field expressions (1.8) and (1.5) are very similar. The number of terms on the RHS of (1.8) are six compared to four for the fourth-order accurate finite difference scheme (1.7), and might be responsible for lower dispersion property of SBTD. The SBTD scheme belongs to the family of multiresolution time-domain (MRTD) schemes using Cohen-Daubechies-Feauveau (CDF) wavelets [7]. The MRTD schemes simultaneously address issues of higher-order approximation of fields, multigrid structure, and accurate treatment of the interface between different media, unlike the piecemeal approach of FDTD schemes [7].

The phase velocity for the two-dimensional TM case for SBTD and FDTD(2,2) schemes are compared in Figure 1.3 [6]. The number of nodes per wavelength or spatial resolution N is 20 and q = 0.5. It is observed from the graph that the phase velocity for SBTD scheme is 1.001c independent of the direction of travel of wave. The error is also less compared to FDTD(2,2).

Figure 1.3 Comparison of dispersion curves for SBTD and FDTD(2,2), (q = 0.5).

Reproduced by permission of ©2008 IEEE, Figure 1 of [6]

The normalized phase velocity for FDTD(2,2) and SBTD schemes for a cubic mesh with N = 20 are compared in [8] and plotted here as Figure 1.4. It is noted from Figure 1.4 that SBTD with q = 0.5 is isotropic and least dispersive. The combination of various spatial and temporal discretizations (q = 0.75) have been studied for their effect on numerical dispersion [9]. The phase velocity is plotted as a function of spatial sampling rate N in Figure 1.5 [9]. For each scheme, the phase velocity is bounded by two lines; the maximum (max) phase velocity occurs along the cell diagonal and the minimum (min) velocity occurs along the axis of the cell. It is noted from Figure 1.5 that except for FDTD(2,2), all other schemes generate fast (>c) waves.

Figure 1.4 Comparison of normalized phase velocity versus azimuth angle in a cubic mesh at a spatial sampling rate of 20 points per wavelength. q is the Courant number.

Reproduced by permission of ©2008 IEEE, Figure 1 of [8]

Figure 1.5 Comparison of phase velocities for SBTD and FDTD schemes as a function of spatial sampling rate N, q = 0.75.

Reproduced by permission of ©2009 IEEE, Figure 1 of [9]

The slow and fast wave behavior of various schemes, Figure 1.5, may be exploited to reduce numerical dispersion in FDTD. For this, hybrid FDTD schemes have been proposed. The hybrid scheme based on the combination of FDTD(2,2) and FDTD(2,4) is called HFDTD(2,4), and that based on FDTD(2,2) and SBTD is called HSBTD1 [9]. Numerical dispersion produced by the hybrid schemes has been compared with non-hybrid schemes and it is found that dispersion can be minimized by properly combining the schmes with slow and fast waves [9]. The lay-out of cells for such an experiment is shown in Figure 1.6 [9]. Most of the cells are updated using higher-order schemes. The cells marked black are updated using higher-order schemes whereas the cells marked white in each sixth row and column are updated with second-order schemes. For various schemes, the effect of spatial sampling rate or grid resolution on the error in resonant frequency of a two-dimensional cavity is compared in Figure 1.7 [9]. It is confirmed from Figure 1.7 that the hybrid schemes may be used to reduce the numerical error significantly. Further numerical experiments on a partially filled rectangular waveguide cavity confirm that the error in resonant frequency reduced by a factor of 3.1 when HFDTD(2,4) is employed; this factor increased to 22 when HSBTD1 is used. All these results are compared to standard FDTD(2,2). The spatial sampling rate used was 26.7. The processor times of the hybrid schemes are similar to those of higher-order schemes. The effects of numerical dispersion for layered, anisotropic media have been reported in [10].

Figure 1.6 Cell pattern for the field components normal to the view.

Reproduced by permission of ©2009 IEEE, Figure 2 of [9]

Figure 1.7 Comparison of relative error of various schemes for the resonant frequency of a two-dimensional rectangular cavity.

Reproduced by permission of ©2009 IEEE, Figure 3 of [9]

One area of challenge in applying the higher-order and hybrid schemes is in the treatment of boundary conditions which are inside the computational domain, e.g. antenna conductors, feed lines, pins, dielectric interface, etc. [5]. Some of these issues for standard FDTD method are discussed in [11]. Lossy curved surface in the form of surface impedance boundary condition is modeled in [12]. The metal-semiconductor interfaces may be defined by higher-order impedance boundary conditions [13].

Numerical dispersion exhibited by the various finite difference schemes have been reviewed and expressed in the form of a general expression [8]:

(1.10) equation

where is the complex eigenvalue (not equal to unity) of the amplification matrix M. The above expression is applicable to all known conditionally and unconditionally stable algorithms; within their stability limits for conditionally stable schemes. For specific FDTD schemes, expression for in terms of discretization parameters and numerical wave number is available in [8]. For FDTD(2,2), the eigenvalues of are obtained as , where d is given by [8]

(1.11a)

equation

(1.11b)

equation

(1.11c)

equation

1.4 Stability of Algorithms

The stability requirement of algorithm (1.5a) puts an upper limit on time step . This limit is necessary otherwise the computed field values might increase spuriously without limit as time marching continues. The reason for numerical instability is the violation of causality, that is, the minimum time required for the signal to propagate from one node to the other separated by is given by . Increasing the value of beyond this value to speed up the simulation will result in instability. The upper bound on Δt is called the CFL stability condition or sometimes Courant limit and is given by (1.1). For the special case of cubic cell , one obtains

(1.12) equation

The stability and numerical dispersion for ADI, CNSS and CN schemes are investigated in terms of their amplification matrix M [14]. The ADI and CNSS schemes are found to have the same dispersion,. and CN and CNSS are found to be unconditionally stable. However, the unconditional stablility of ADI is contingent upon space-time discretizations, and its amplification matrix M is given by

(1.13) equation

where the matrix M is a function of wavenumber(magnitude and propagation direction), and space-time discretizations . For a cubic cell, (1.13) reduces to

(1.14) equation

Another unconditionally stable FDTD scheme is LOD-FDTD. It is an efficient scheme compared to ADI, CN and CNSS schemes and is described in Section 1.5.

The FDTD algorithm may have to be run for a large number of time steps, sometimes of the order of 100,000 steps until the time domain waveform converges. It is possible to accelerate the convergence of the algorithm by using matrix-pencil (or GPOF) approach [15]. In this approach, the late time waveform can be predicted from the early time sampling data. One may be able to save about 80% of the simulation time using GPOF [16].

1.5 Absorbing Boundary Conditions

The antennas have associated open space region. The FDTD simulation of the antenna in this form will require an unlimited amount of computer memory and processing time, which is impossible to arrange. Therefore, the domain must be truncated so that the associated reflection is minimal. For this, the solution domain is divided into two regions: the interior region and the exterior region as shown in Figure 1.8. The interior region must be large enough to enclose the antenna of interest. The exterior region simulates the infinite space. It is a limited free space enclosing the interior region on one side and terminated on the other side by a perfect electric conductor. When we apply the FDTD algorithm to the interior region, it simulates wave propagation in the forward and backward directions. However, only the outward propagation in the exterior region is desired so that infinite free space conditions are simulated. Reflections are generated at the interior-exterior region interface and from the perfect electric conductor terminating the exterior region. These reflections must be suppressed to an acceptable level so that the FDTD solution is valid for all time steps. The exterior region includes the domain of absorbing boundary condition or ABC for short.

Figure 1.8 A typical truncation of the physical domain by the exterior region in FDTD algorithms

The absorbing boundary condition can be simulated in a number of ways. These are classified as analytical (or differential) ABC and material ABC. The material ABC is realized from the physical absorption of the incident wave by means of a lossy medium, whereas analytical ABC is simulated by approximating the one-way wave equation on the boundary of interior region. Whereas the analytical ABC may be able to provide upto −60 dB of reflection, the material ABC can provide better absorption with the reflection reaching an ideal limit of −80 to −120 dB from PML at the boundaries. Various types of ABCs are summarized next.

1.5.1 Analytical Absorbing Boundary Conditions

Analytical ABC is a popular technique because of its simplicity of implementation. The reflection from analytical ABC could be as low as 0.1%, i.e. −60 dB. Analysis for this absorbing boundary condition is based on the works of Enquist and Majda [17], Mur [18], Higdon [19], and Liao [20]. For a plane wave incident on a planar boundary, the wave will propagate forward (without any reflection) if the field function F(x,y,z,t) satisfies one-way wave equation at the boundary. Such a boundary is therefore called the absorbing boundary. The PDE for the ABCs may be derived from the wave equation. Consider the following two-dimensional wave equation in the Cartesian coordinates

(1.15) equation

This expression may be factored as

(1.16)

equation

Applying each of the above pseudo-operators to F gives rise to the following PDEs for the analytical ABC as

(1.17a)

equation

(1.17b)

equation

where , etc. and . Direct implementation of the operators with square root is not possible. The operators are therefore approximated and result in various forms of ABC depending on the approximation employed.

The first-order ABCs are obtained by a very crude approximation in (1.17a); one obtains

(1.18) equation

for the left boundary at x = 0. Similar expressions can be written down by inspection for the other boundaries also. The expression (1.18) is called the Enquist-Majda (E-M) first-order ABC. The field function F represents the electric field tangential to the boundary. For a non-planar wave, Equation (1.18) should be applied to all the components tangential to the planar boundary. It is shown that (1.18) is unconditionally stable [18]. The absorbing boundary condition (1.18) is exact only for a plane wave at normal incidence. Hence the wave will be reflected for an oblique incidence. Higher-order ABCs are suitable for non-normal incidence, and are obtained by better approximation of the square root function . For this, let us approximate in a general form as

(1.19) equation

where the parameters a, b, d are optimized to define various forms of third-order ABCs such as Chebyshev, Pade and EM second-order. Use of Equation (1.19) in Equation (1.17a) gives

(1.20) equation

This expression reduces to:

a. first-order E-M ABC for d = 1, a = b = 0;

b. second-order E-M ABC for d = 1, b = ½, a = 0;

c. Pade approximation or Pade ABC for d = 1, b = ¾, a = ¼;

d. Chebyshev ABC for d = 0.99973, b = 0.80864, a = 0.31657.

Mur's ABCs are discretized versions of E-M ABCs, e.g. first-order Mur ABC for the boundary at is obtained by discretizing Equations (1.17a) or (1.20) at and is given by

(1.21) equation

Similar expressions may be obtained for other types of ABCs and for waves incident on different boundaries. One can realize about −40 dB reflection by using these ABCs. Liao's third-order ABC may be used to achieve about −50 to −60 dB reflection and is discussed next.

1.5.1.1 Liao's ABC

Liao's ABC is based on

Reviews for Microstrip and Printed Antennas

[Chonticha Yuangsuwan · Rating: 4 out of 5 stars]

good