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

In this book, the authors address topics such as reconfigurable antennas, ultra-wideband (UWB) antennas, reflectarrays, antennas for RFID systems and wearable antennas for body area networks. Antennas using metamaterials and defected ground structures (DGSs) are also explained. The authors discuss essential aspects including advanced design, analysis and optimization, and cover fundamental concepts and techniques, their practical applications and the future scope of developments.

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

Microstrip and Printed Antennas: New Trends, Techniques and Applications 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 insightful.

Debatosh Guha is Professor in Radio Physics and Electronics of the University of Calcutta, India. He has been active in research and teaching at post-graduate level in the areas of microwaves and antennas for last 18 years. He has published more than 100 Journal and Conference articles and has developed CAD for various microstrip antennas, which were reported in IEEE and IEE Journals. He has also developed some high gain microstrip antennas for a North American Wireless Industry, which have been implemented as commercial products.

Yahia M. M. Antar is Professor of ECE Department of the Royal Military College of Canada in Kingston, Ontario.  He has authored or co-authored over 150 journal papers, holds several patents, chaired conferences and sessions in many conferences, and supervised or co-supervised over 60 Ph.D. and M.Sc. theses at the Royal Military College and at Queen’s University, of which several have received the Governor General of Canada Gold Medal as well as best paper awards in major symposia.  He served as the Chairman of the Canadian National Commission for Radio Science (CNC, URSI,1999-2008), holds adjunct appointment at the University of Manitoba, and, has a cross appointment at Queen’s University in Kingston.

Microstrip and Printed Antennas

New Trends, Techniques and Applications

John Wiley & Sons

Copyright © 2011 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-68192-3

Chapter One

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. 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.

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 timedomain (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-FriedrichLevy (CFL) condition.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

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 Δt = (1/2)Δt_CFL uniformly to simplify coding and avoid instability. Defining the Courant number q as

q = Δt/Δt_CFL (1.2)

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

The value of Δt_CFL 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 Δt_CFL 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 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3b)

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,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5d)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5e)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5f)

The indices i,j, and k define the position of the field nodes, such that x = iΔx; y = jΔy; z = kΔz. The time instant is defined by t = nΔt. To implement the finite difference scheme in three dimensions, the antenna is divided into a number of cells, called Yee cells, of dimension ΔxΔyΔz. 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 Eand 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 Eand H field components are calculated are offset by half a time step, that is, components of E are calculated at nΔt and components of Hare calculated at (n + 1/2)Δt. The alternate update of E and H fields is called leap frog and saves computer processing time.

1.3 Numerical Dispersion of FDTD and Hybrid Schemes

The finite difference form of derivative (1.4) has an error term O(Δu)². 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 v_ph 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 Δt and the discretization size Du. 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where [bar.k] is the wave number for the numerical wave. The phase velocityv v = ω/[bar.k] is determined by solving Equation (1.6) as a function of discretizations Δx; Δy; Δz;Δt; and propagation angle θ φ. 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. It is found that the dispersion error (c–[bar.v])/c 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, λ/Δ) = 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. 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.

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). Higherorder 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 Δu=2 and 3Δu=2 on either side of the observation point u₀, and the space derivative is defined as [5]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

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. 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8c)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

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. 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.

The phase velocity for the two-dimensional TM case for SBTD and FDTD(2,2) schemes are compared in Figure 1.3. 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).

The normalized phase velocity for FDTD(2,2) and SBTD schemes for a cubic mesh with N = 20 are compared in 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. The phase velocity is plotted as a function of spatial sampling rate N in Figure 1.5. 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.

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) andSBTD is called HSBTD1. 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. 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. 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.

(Continues…)

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