Source: ANTENNA ENGINEERING HANDBOOK P ● A ● R ● T ● Introduction and Fundamentals Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Introduction and Fundamentals Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Source: ANTENNA ENGINEERING HANDBOOK Chapter Fundamentals of Antennas, Arrays, and Mobile Communications Thomas F Eibert Universität Stuttgart John L Volakis The Ohio State University CONTENTS 1.1 INTRODUCTION 1-4 1.2 HUYGENS’ AND EQUIVALENCE PRINCIPLES 1-5 1.3 HERTZIAN AND FITZGERALD ELEMENTARY RADIATORS 1-7 1.4 FAR-FIELD ANTENNA PROPERTIES, POWER TRANSFER, AND RECIPROCITY 1-8 1.5 ANTENNAS AS ELECTROMAGNETIC CIRCUITS 1-11 1.6 POLARIZATION 1-14 1.7 DIRECTIVITY PATTERNS FROM CONTINUOUS LINE SOURCES 1-17 1.8 DIRECTIVITY PATTERNS FROM AREA SOURCE DISTRIBUTIONS 1-21 1.9 FUNDAMENTALS OF ANTENNA ARRAYS 1-27 1.10 BASIC CONCEPTS IN MOBILE COMMUNICATIONS 1-32 1-3 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Fundamentals of Antennas, Arrays, and Mobile Communications 1-4 CHAPTER ONE 1.1 INTRODUCTION * Antennas are key components of any wireless communication system.1,2 They are the devices that allow for the transfer of a signal (in a wired system) to waves that, in turn, propagate through space and can be received by another antenna The receiving antenna is responsible for the reciprocal process, i.e., that of turning an electromagnetic wave into a signal or voltage at its terminals that can subsequently be processed by the receiver The receiving and transmitting functionalities of the antenna structure itself are fully characterized by Maxwell’s equations and are fairly well understood.3 The dipole antenna (a straight wire, fed at the center by a two-wire transmission line) was the first antenna ever used and is also one of the best understood.1,2 For effective reception and transmission, it must be approximately l/2 long (l = wavelength) at the frequency of operation (or multiples of this length) Thus, it must be fairly long (or high) when used at low frequencies (l = m at 300 MHz), and even at higher frequencies (UHF and greater), its protruding nature makes it quite undesirable Further, its low gain (2.15 dB), lack of directionality, and extremely narrow bandwidth make it even less attractive Not surprisingly, the Yagi-Uda antenna (typically seen on the roof of most houses for television reception) was considered a breakthrough in antenna technology when introduced in the early 1920s because of its much higher gain of 8–14 dB Log-periodic wire antennas introduced in the late 1950s and 1960s and wire spirals allowed for both gain and bandwidth increases On the other hand, even today high gain antennas rely on large reflectors (dish antennas) and waveguide arrays (used for many radar systems) Until the late 1970s, antenna design was based primarily on practical approaches using off-the-shelf antennas such as various wire geometries (dipoles, Yagi-Uda, log-periodics, spirals), horns, reflectors, and slots/apertures as well as arrays of some of these The antenna engineer could choose or modify one of them based on design requirements that characterize antennas, such as gain, input impedance, bandwidth, pattern beamwidth, and sidelobe levels (see References and for a description of these quantities) Antenna development required extensive testing and experimentation and was, therefore, funded primarily by governments However, in recent years, dramatic growth in computing speed and development of effective computational techniques for realistic antenna geometries has allowed for low-cost virtual antenna design Undoubtedly, the explosive growth of wireless communications and microwave sensors, microwave imaging needs, and radars has been the catalyst for introducing a multitude of new antenna designs over the past decade and an insatiable desire for using modern computational techniques for low-cost designs Requirements for ∗ Heinrich R Hertz was the first to demonstrate the generation of radio waves at UHF using a gap dipole in 1885– 1886 at Karlsruhe University (Germany) Hertz was able to detect radio waves 20 m away using a high-voltage electrical spark discharge to excite the dipole gap From recorded conversations, Hertz did not seem to understand the impact of his experiments, but instead did it as a validation of the newly developed Maxwell’s equations Within ten years, Tesla at the Franklin Institute in the U.S., Marconi in Bologna, Italy, Popov in Russia, and Bose in India, demonstrated wireless telegraphy In 1892, Tesla delivered a widely distributed presentation at the IRE of London about “transmitting intelligence without wires,” and in 1895, he transmitted signals detected 50 miles (80 km) away Concurrently, in 1894 Bose used wireless signals to ring a bell in Calcutta, and Popov presented his radio receiver to the Russian Physical & Chemical Society on May 7, 1895 Marconi is certainly considered the key individual for his contributions to the commercialization of radio waves, and he received the Nobel prize for his work in 1909 Nevertheless, Marconi’s widely advertised first radio wave transmission experiment was in 1895, and his British patent application in 1897 was preceded by that of Tesla A culmination of Marconi’s experiments was the December 12, 1901, trans-Atlantic radio wave transmission of the Morse code for the letter S The success of this experiment is often disputed, possibly due to strong atmospheric noise during the time of the experiment, but by the 1920s the U.S had hundreds of radio stations, and in 1922, the BBC began transmitting in England Subsequent development of radio detectors, vacuum tubes, and the tiny transistor in 1947 played a critical role in the practical everyday use of radio waves for communication and wireless transmission of information Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Fundamentals of Antennas, Arrays, and Mobile Communications FUNDAMENTALS OF ANTENNAS, ARRAYS, AND MOBILE COMMUNICATIONS 1-5 conformal antennas (non-protruding) for airborne systems, increased bandwidth requirements, and multifunctionality have led to heavy exploitation of printed (patch) or other slot-type antennas4 and the use of powerful computational tools (commercial and noncommercial) for designing such antennas Needless to say, the commercial mobile communications industry has been the catalyst for the recent explosive growth in antenna design needs Certainly, the past decade has seen an extensive use of antennas by the public for cellular, GPS, satellite, wireless LAN for computers (WiFi), Bluetooth technology, Radio Frequency ID (RFID) devices, WiMAX, and so on However, future needs will be even greater when a multitude of antennas are integrated into say automobiles for all sorts of communication needs and into a variety of portable devices and sensors for monitoring and information gathering Certainly, future RFID devices will most likely replace the bar codes on all products while concurrently allowing for instantaneous inventorying For military applications, there is an increasing need for small and conformal multifunctional antennas that can satisfy a plethora of communications needs using as little space as possible In this first chapter of the handbook, we provide a summary of antenna fundamentals and introduce antenna parameters typically used for characterizing antenna properties often employed to evaluate the entire radio system We start with the radiation of an ideal (Hertzian) or infinitesimal dipole and proceed to the resonant l/2 dipole, antenna arrays, and mobile communication concepts 1.2 HUYGENS’ AND EQUIVALENCE PRINCIPLES The electromagnetic behavior and thus the functioning of antennas is governed by Maxwell’s equations,3 which must be solved for a particular antenna and a given excitation Typically, exact solutions of Maxwell’s equations are not available and thus numerical modeling is often used to compute approximate solutions for practical configurations A formal simplification of electromagnetic antenna problems can be achieved by employing the equivalence principle.3 If interest is restricted to the field solution in a limited region of space, the antenna configuration can be replaced by the equivalent electromagnetic sources located on the surface of a volume enclosing the antenna configuration (see Figure 1-1) Because the antenna materials are no longer there, these sources are usually radiating in a homogeneous solution space (such as free-space), and the corresponding fields can thus be calculated by evaluating the radiation integrals The equivalent sources are not uniquely defined, and there are many different ways of constructing them In general, the equivalent sources are a composition of electric and magnetic surface current densities representing the excitation terms in Maxwell’s equations A straightforward way of constructing equivalent sources is provided by Huygens’ principle.3 Huygens’ principle states that the field solution in a region V is completely determined by the tangential fields over the surface S enclosing V The corresponding electric and magnetic equivalent surface current densities are given by Electric current density: J = nˆ × H (1-1) M = − nˆ × E (1-2) Magnetic current density: Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Fundamentals of Antennas, Arrays, and Mobile Communications 1-6 CHAPTER ONE S n R= r−r M J r r z y x FIGURE 1-1 Replacement of an antenna by equivalent electric and magnetic surface current densities where both J and H are expressed in amperes per meter (A/m) and M and E are expressed in volts per meter (V/m) For the problem of a radiating antenna, as illustrated in Figure 1-1, the outer boundary of V is assumed to be located at infinity, where the fields radiated by the corresponding equivalent sources can be neglected As shown in the figure, the antenna can be replaced by equivalent sources on an arbitrary surface S enclosing it As already mentioned, these equivalent sources reproduce the radiated fields of the antenna, and they can be assumed as radiating in homogeneous space For a particular antenna configuration, the exact determination of J and M requires knowledge of the true field distribution on S However, for many practical antennas, an approximate determination of J and M is possible For instance, placing S to coincide with a metallic section of the antenna structure causes M to vanish on these portions of S The radiated fields from any antenna can be obtained by integrating the field contributions of the equivalent electric and magnetic current densities using the well-known radiation integral:3 e − jk0 |r-r '| e − jk0 |r-r '| E = − jωµ0 ∫∫ J(r ') 4π | r - r ' | + k02 (∇ ' ⋅ J(r '))∇ 4π | r - r ' | ds ' S + ∫∫ M(r ') × ∇ e − jk0 |r-r '| ds ' 4π | r - r ' | which for the far-field ( r → ∞) reduces to (see Figure 1-1) E = − jωµ0 where e − jk0r 4π r ∫∫ ( I − rrˆˆ) ⋅ J(r ') − ε0 rˆ × M( r ') e jkrˆ⋅r ' ds ' µ0 I = unit dyad r = defines location of observation point (see Figure 1-1) r = distance (in m) to observation point r' = defines location of the integrated surface current densities rˆ = unit vector in radial direction e0 = free-space permittivity m0 = free-space permeability Z0 = µ0 = free-space impedance ε0 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Fundamentals of Antennas, Arrays, and Mobile Communications FUNDAMENTALS OF ANTENNAS, ARRAYS, AND MOBILE COMMUNICATIONS 1-7 k0 Z = ωµ0 k0 = b = 2p /l l = wavelength (in meters, m) j = −1 E is given in volts per meter (V/m) H is given in amperes per meter (A/m) For the ideal (delta) or infinitesimal electric (Hertzian) or magnetic (Fitzgerald) dipole sources, the radiation integrals are eliminated and the fields can be given in closed forms The resulting field expressions can then be used to extract and study the usual antenna parameters 1.3 HERTZIAN AND FITZGERALD ELEMENTARY RADIATORS Considering the infinitesimal electric dipole J = zˆ Idz δ (z), as illustrated in Figure 1-2, the resulting rms (root mean square) electric and magnetic field components are given by Er = k02 Eθ = jk02 Hφ = jk02 µ0 Idz j − jk r − cosθ e , ε 2π ( k0 r ) ( k0 r )3 µ0 Idz 1 j − jk r − − sinθ e , ε 4π k0 r ( k0 r ) ( k0 r )3 (1-3) Idz j jk r − sinθ e , 4π k0 r ( k0 r )2 Eφ = H r = Hθ = where Idz = moment of the differential current element ( I is given in rms amperes, and dz is given in meters) FIGURE 1-2 Coordinate system for an electric dipole Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Fundamentals of Antennas, Arrays, and Mobile Communications 1-8 CHAPTER ONE A time factor of e jw t has been suppressed since a sinusoidally time-varying current excitation of constant frequency is assumed These are the exact fields, but antenna parameter evaluation is usually carried out using simplified far-fields, i.e., when r is much greater than the wavelength l Under these conditions, terms of order 1/r2 and greater are neglected to obtain Eθ = jk0 µ0 Idz e − jk0r sinθ = ε 4π r Hφ = jk0 Idz e − jk0r sinθ = 4π r µ0 H , ε0 φ (1-4) ε0 E µ0 θ The complex Poynting vector S3 in the far-field is given by S = E × H* = k02 µ0 Idz sin 2θ rˆ ε 16π r (1-5) showing that it is a purely real quantity and indicating that power transport is in the r direction (away from the elementary current) without any reactive energy Also, it is seen that the radiated power density (power flow per unit area) for any r = const depends on sin2q (independent of f ) This is referred to as the radiation pattern plotted in dB For an elementary (ideal or infinitesimal) magnetic dipole M = zˆ I m dz δ ( z ), the radiated fields can be found by duality3 (replacing E by H, H by –E, and I by Im) 1.4 FAR-FIELD ANTENNA PROPERTIES, POWER TRANSFER, AND RECIPROCITY Because antenna radiation can be represented by radiation integrals over equivalent current sources, as considered in the previous paragraph, no reactive field components will be found in the far-field of any antenna Further, far-field antenna characterization can be performed by considering power flow under the constraint of energy conservation However, the distance from an antenna to its far-field depends on the antenna, and it is commonly accepted that the far-field region starts after the distance R=r= 2D2 λ (1-6) where D is the largest dimension of the antenna This is due to the varying propagation distances of field contributions from different parts of the antenna to an observation point P, as illustrated in Figure 1-3 In the far-field, every antenna is considered a point source, and the far-field criterion in Eq 1-6 is derived under the assumption that the phase errors due to the varying propagation distances are less than p/8 Consider an antenna located at the origin of a spherical coordinate system, as illustrated for the electric current element in Figure 1-2 Assume that the antenna is transmitting and let ● Pt = power accepted by the antenna (in Watts) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Fundamentals of Antennas, Arrays, and Mobile Communications FUNDAMENTALS OF ANTENNAS, ARRAYS, AND MOBILE COMMUNICATIONS 1-9 FIGURE 1-3 Schematic representation of an antenna aperture showing the observation point P and the distances to the observation point from two points on the antenna ● ● Prad = power radiated by the antenna (in Watts) h = radiation efficiency (unitless) These quantities are related by η= Prad Pt (1-7) Let ● St(q,f) = power density (in Watts/square meter, W/m2) and remark that St (q,f) is independent of the distance from the antenna r, a characteristic of the far-field region The total radiated power can be obtained by integrating the power density over a surface enclosing the antenna Such a surface can be of any shape and can be as close to the antenna as desired However, for simplicity, it is convenient to choose the surface to be a sphere, giving Prad = 2π π ∫0 ∫0 St (θ ,φ )r sinθ dθ dφ (1-8) with the average power density being Pavg = Now, let ● Prad (1-9) 4π r Dt(q,f ) = directivity (unitless) Directivity is a measure of the antenna to concentrate radiated power in a particular direction, and it is related to power density as Dt (θ , φ ) = St (θ , φ ) Pavg = St (θ , φ ) Prad / ( 4π r ) (1-10) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas 59-16 CHAPTER FIFTY-NINE FIGURE 59-8 Performance of a single-patch antenna simulated using Sonnet (left: Smith chart, right: resistance and reactance spectrum, with inset figure illustrating resonant currents) For example, the following integral equation32 can be used to represent dielectric materials with metallization E i = E + [ jω ( A D + A M ) + ∇( Φ D + Φ M )] E itan = [ jω ( A D + A M ) + ∇( Φ D + Φ M )]tan r ∈V r ∈ SM (59-16) In Eq 59-16, the volume of dielectric (V) and metal surface (SM) are assumed to be in free space The vector potentials (A) and scalar potentials (Φ) are associated with the dielectric or metal and these are denoted by the subscript D or M, respectively With this, the substrate need not be infinite in two dimensions and very thin dielectric layers can be simulated without unknown current surfaces that are too close In addition, the substrate can be inhomogeneous In contrast to most MoM implementations that utilize loworder expansion functions, higher-order functions are presented in the reference section.33 FIGURE 59-9 S-parameters for a two-element array as computed using SonnetLite (the geometry is shown as an inset figure) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas COMPUTATIONAL ELECTROMAGNETICS FOR ANTENNAS 59-17 Higher-order functions can be used to more accurately represent a boundary and to reduce the meshing requirements They not always lead to fewer unknowns since there are more degrees-of-freedom per element associated with a higher-order function as compared to a low-order function Finite Element (FE) Method The finite element (FE) method provides one of the most powerful formulations for antenna modeling provided that the antenna has a dielectric load, particularly if the material load is inhomogeneous or anisotropic The FE method can be used for antennas comprised of solely conducing material; however, in general the MoM is more efficient for such radiators The finite element method is an approach for solving Maxwell’s equation by solving the vector wave equation For time-harmonic (also known as continuous wave or frequency domain) problems, Eq 59-2 can be recast as ì E = j àr à0 H × H = jω ε r ε E + J ρ ∇ ⋅ (ε r E ) = e ε0 ∇ ⋅ ( µr H ) = (59-17) where the radial frequency w = 2pf is related to the frequency ( f ) and the surrounding space may contain an anisotropic, inhomogeneous relative permittivity (er) and/or an anisotropic, inhomogeneous relative permeability (mr) However, for simplicity, isotropic materials are assumed In Eq 59-17, the current density (J) represents sources within the computational domain This set of first-order partial differential equations (PDE) may be re-cast as a second-order PDE, known as the vector wave equation, by taking the curl of one or the other curl expressions in (59-17) and making use of the remaining expression for simplification So, the vector wave equation is given by 1 ∇ × ∇ × E − k02 ε r E = − jk0 Z J µr J 1 ì ì H k02 àr H = ∇ × ε r ε r (59-18) where the upper equation is in terms of the total electric field while the lower equation is the case when the total magnetic field is needed In Eq 59-18, one of the equations or the other is used depending on which of the two fields, electric or magnetic, is being sought and which of the two is most amenable for enforcement of boundary conditions Note that when solving Eq 59-18 for the electric field, should the magnetic field be needed in post-processing, the appropriate equation from Eq 59-17 may be used It is impractical to enforce these equations at every point in space This means that a so-called weak form of the vector wave equation is typically specified, either through minimization of a functional or by application of the Rayleigh-Ritz method For the latter, one of the wave equations (in terms of the total electric field from now on, for convenience) is tested with a vector function In this, the dot-product of the vector function is taken and the resulting scalar expression is integrated throughout the domain of the test function As a result, Eq 59-18 in the weak form is written after some manipulation (such as use of the First Vector Green’s Theorem34 to transfer a curl operator off of the unknown field onto the testing function, Wi) as shown in ∇ × Wi ⋅ ∇ × E − k02 ε r Wi ⋅ E dV − jk0 Z ∫ S Wi ⋅ (nˆ × H) dS = − jk0 Z0 ∫V Wi ⋅ J dV µr (59-19) ∫V Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas 59-18 CHAPTER FIFTY-NINE where the subscript i refers to the testing function Several observations can be made regarding this weak-form equation In Eq 59-19, the surface magnetic field will be provided either by a local condition or by an integral relation In either case, the surface magnetic field is proportional to the surface electric field It’s a given for this expression that isotropic materials have been assumed, and the resulting matrix developed from Eq 59-19 will be symmetric (since the electric field and testing functions may be interchanged without changing the form of the expression) In addition, the surface term in Eq 59-19 allows the introduction of boundary conditions on S = ∂V Through these boundary conditions, an external source may be included such as an impinging plane wave A series of relatively easy-to-read papers for FEM beginners have been featured in the EM Programmer’s Column of the IEEE Antennas and Propagation Magazine in recent years.35–36 The reader is encouraged to read such papers and their references for a greater understanding of the FE method Most implementations of the finite element method for electromagnetics primarily differ in the elements used to expand the unknown field (and usually the testing functions as well since these are similar) and the method to enforce the boundary conditions The most popular elements for solution of the finite element equations are the brick, right prism, and tetrahedron These are shown in Figure 59-10 with the brick on the left, the prism in the center, and the tetrahedron on the right By far the most popular, due to its flexibility, is the tetrahedron Nevertheless, both the brick and right prism have their uses in that they are particularly easy to implement and, most importantly, rather easy to generate appropriate meshes for Usually, the expansion functions are lowest mixed-order edge elements (for example, the field representation is constant for component of the field parallel to the edge and linear for the component of the field normal to the edge) However, higher-order expansion functions37 and hierarchical functions38 have been used by a number of researchers The user will find that the major discriminator between finite element codes is not the element shape necessarily, nor even the order of the expansion; rather, it is the inclusion of a powerful mesh generator It is not an understatement to say that a large portion, and sometimes the majority, of effort expended by antenna engineers in using a finite element code is in mesh generation rather than data collection The second major discriminator is the mesh termination condition The following three methods are most popular: ● ● ● Absorbing boundary conditions39 Perfectly matched layers40 A boundary integral41 The first method, absorbing boundary conditions (ABCs), is widely used since it is particularly easy to implement without having to use user-defined parameters that FIGURE 59-10 Typical finite element shapes: brick (left), right prism (center), and tetrahedron (right) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas COMPUTATIONAL ELECTROMAGNETICS FOR ANTENNAS 59-19 impact performance The cost of using these ABCs is usually a larger computational domain since the mesh boundary must be maintained some distance from the structure being implemented The distance required is dependent on the particular ABC and on the problem In general, antenna simulations require a larger stand-off distance than scattering problems due to the need for accurate near-field quantities such as driving-point impedance Perfectly matched layers (PMLs) are another local (or in other words, dependent only on field behavior in the vicinity of the PML) mesh truncation technique that generally requires less stand-off distance than ABCs However, there are user-defined (or at least programmerdefined) parameters and typically, the condition number of the resulting matrix is poorer as compared to the ABC The final method, a boundary integral (BI), is a global rather than local condition It requires no stand-off distance; however, the resulting matrix is partially full due to the BI, and as a result, the demand for memory and solution time can be quite high In addition, other aspects of the method formulation and implementation can impact solution performance For example, the use of adaptive meshing42 is a common feature in commercial programs these days that largely reduces the need for specialized meshing expertise so that the essential field behavior is captured As an example of how the finite element method can be used for antenna analysis and design, consider a 40mm by 30mm cavity-backed antenna placed in a cavity that is 80 mm by 60 mm The cavity is filled with 30 mil thick Rogers Duriod/5870 This particular antenna will be analyzed using the computer program provided on the author’s web page (to obtain a copy of the source code, see the URL cited in the next section) This program utilizes right prism elements43 and is written in MATLAB with a functional graphical user interface (GUI) The program is initiated in MATLAB by running the msGUI.m script The cavity is divided into 32 × 24 cells that are then subdivided into triangles The program then extrudes these triangles into right prism elements Since the substrate is electrically thin, one layer of elements is sufficient to resolve the field behavior along the normal direction The surface mesh, Smith Chart (2.2-2.4 GHz), and normalized electric field within the cavity are shown in Figure 59-11 The fields shown are the normal component of the electric field The patch antenna is excited using a probe-feed from the base of the cavity to the patch slightly left of the antenna center along the top-to-bottom centerline This method of feeding excites the fundamental mode for this antenna Notice that the normal fields are zero along the centerline of the antenna while those fields are a peak near the edges of the patch This field distribution can be represented using a cosine function The phase of the field distribution left of the centerline as compared to right of the centerline is 180 degrees out-of-phase resulting in radiation This particular example does not utilize the most significant feature of a finite element formulation: the ability to model inhomogeneous antenna loading materials Simulation of grounded, homogeneous dielectrically-loaded antennas is most efficiently analyzed using an integral equation method (see the previous example using Sonnet) However, if the antenna has inhomogeneous (or anisotropic) materials in its construction, the finite element method is often the most efficient and accurate method to use for simulating resonant structures FIGURE 59-11 Surface mesh (left), Smith chart (center), and normal field distribution (right) for a cavitybacked patch antenna simulated using the FE-BI method Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas 59-20 CHAPTER FIFTY-NINE (for example, the class of antennas that is best simulated using a frequency-domain solver) This is also true for the case of cavity-backed antennas, when the cavity is odd-shaped regardless of the cavity-fill material One comment is in order concerning simulation of realistic antennas in the MATLAB environment Unless considerable effort is undertaken to vectorize (express the mathematical manipulations in terms of data vectors and matrices), parallelize, or compile as executable code, MATLAB is not an efficient platform for computing large problems The definition of large is in the “eye of the beholder,” in the sense that for a given algorithm and the time a user is willing to wait for an answer, the computational size that constitutes a large problem varies Nevertheless, for understanding the underlying concepts of various computational methods, it is an effective platform for discovery It is particularly useful for students due to their general familiarity with MATLAB Other Frequency-Domain Methods The method of moments and finite element methods are arguably the most prevalent frequency-domain CEM solvers in use today Nevertheless, there are excellent alternative methods that have very specific advantages in certain circumstances and other methods that are growing in popularity and impact One of the most effective means for computing the radiation by antennas in the presence of very large structures involves the use of high frequency (asymptotic) approximations such as the Uniform Theory of Diffraction (UTD) This method was extensively investigated by various research groups around the world during the 1970s and beyond, most notably by the group at The Ohio State University, along with their colleagues from around the world who spent time at the ElectroScience Laboratory as students, post-doctoral researchers, and as visiting scientists.44–45 Another method of considerable note is the finite integration technique (FIT) mentioned earlier (in its time-domain incarnation) FIT is formulated using the integral form of Maxwell’s equations, either in the time- or frequency-domains FIT can also be used in conjunction with a high frequency method such as UTD.46 The method of lines is another method for simulating electromagnetic phenomena and it has been used to simulate conformal antennas.47 A good reference for various conformal antenna formulations is provided by a recent text.48 Another approximate technique for simulation of antennas installed in a large structure involves the use of reciprocity.49 In this situation, the antenna radiating currents are computed and then integrated with the currents due to the structure resulting in the radiation pattern of the antenna in the presence of the structure Although the paper is used for circular cylindrical structures, it can be used for more general shapes These hybrid formulations are very powerful, and at the dawn of the 21st century, the only practical method for solving installed antenna design problems when the structure is electrically very large It is probable that through advances in computer technology and in full-wave solution methods (for example, fast multipole methods among others) asymptotic methods will be needed less and less Having said that, if history teaches us anything, design problems will become more complex and larger over time and thus, asymptotic and hybrid methods may be needed long into the future 59.4 SUMMARY OF PUBLIC DOMAIN AND COMMERCIAL CODES: CAVEAT EMPTOR Computational electromagnetics has undergone a typical evolution during the past half century: What was merely a research topic in the 1960s through the 1980s is now a thriving industry Accordingly, many CEM tools are available for antenna design Some of these Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas COMPUTATIONAL ELECTROMAGNETICS FOR ANTENNAS 59-21 are in the public domain (some even provide the source code so that users can modify the program for their own needs), while others are commercial offerings As the section title indicates, the user (or buyer) must beware in assuming a particular program will solve their problem, or even be correct, either in formation, assumption, or coding In general, free programs have absolutely no guarantee as to their accuracy while commercial codes are generally easier to use and typically have technical professionals (such as application engineers) to help adopters generate solutions One prevailing dictum of CEM users should be: “Purgamentum init, exit purgamentum”! Inexperienced users of any CEM computer program are cautioned to validate both the program and their use of the program by comparing their simulation with reference data It is better to compare a user-generated case rather than running examples provided by the program developers since in that manner, the user will not only be testing the program, but their interpretation of how to give instructions to the program By all means, initial users should explore the examples that are now standard for CEM codes; but at some point, users must try their own hand at developing a model before they try to solve the design challenge that originally prompted their need (or at least desire) for the CEM code A brief listing of public domain and commercial codes is given next In no manner does this constitute a comprehensive list and over the lifetime of this handbook, it can be expected that Internet links will be broken and that companies will no longer offer these codes Hence, “Let the buyer beware!!!” Disclaimer: This section is not meant in any way to be an endorsement of any methods, computer programs, or company’s offerings It is merely meant to be an information source for antenna designers interested in using powerful CEM design tools This is not a comprehensive list by any means and any omissions in the interest of brevity are regrettable, but not intentional Public Domain Codes Many computer programs are offered free-of-charge by the developers, in the interest of promoting the adoption of the technology or as a means of technology transfer This was particularly true in the early days of CEM; it is less so in the 21st century as the intellectual property value of such codes is better understood Nevertheless, very powerful computer programs may be downloaded, and in most cases, modified by the user since source code is available Codes may be downloaded from ● ● www.egr.msu.edu/~kempel/HandbookCodes This is where you can find the MATLAB scripts used in this chapter The FD-TD code is provided by Mr Steven Cossmann and the FE-BI code by Mr Lanwu Zhao (developed with partial support from the National Science Foundation under grant DUE-0231312) http://www.emclab.umr.edu/codes.html The University of Missouri at Rolla (UMR) maintains an excellent listing of public domain codes available on the Internet The reader is encouraged to investigate this site when looking for a free program, especially the finite element code—EMAP—developed by the students and faculty at UMR Antenna Textbooks with Codes Many antenna textbooks now come with accompanying computer programs to aid the student and engineer Sometimes these codes come with a CD-ROM disk, while in some cases a related web site is provided for users to download code This latter method has the distinct advantage in that program updates may be provided in a convenient manner Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas 59-22 CHAPTER FIFTY-NINE Unfortunately, it has the disadvantage in that the site must be maintained and in practice, links to that web site may break over time Nevertheless, the software provided with modern antenna textbooks is very useful for users, both to learn how antennas function and to provide validation data in the context of accurate CEM codes Some texts have been discussed previously; however, for the convenience of the reader, they are repeated in the following list Note that there are undoubtedly many more antenna textbooks with complementary computer programs; brevity necessitates an unfortunately incomplete list: ● ● ● ● ● David Davidson from the University of Stellenbosch in South Africa has written a textbook25 describing various CEM techniques A major application of that program is antenna design and performance analysis Software for the text is provided via a resources web site maintained by Cambridge University Press (http://www.cambridge.org) Both MATLAB scripts and FEKO files are provided that support the material in the book Note that Appendix F lists additional resources found on the Internet Sergey Makarov of Worchester Polytechnic Institute wrote a textbook26 that describes the modeling of various antenna geometries (such as patches, slots, inverted-F, and so on) using the method of moments A new web site (http://ece.wpi.edu/mom) compared to the one cited earlier in the chapter provides MATLAB scripts and a user manual for using the scripts or for modifying them as needed for a different type of antenna A companion paper illustrates some of the capability of this design tool.32 This package was developed with partial support from the National Science Foundation under grant DUE-0231312 John D Kraus and Ron Marhefka from Ohio State University, in the third edition of their popular antenna textbook,27 provided a web site for several antenna design computer programs (http://www.antennas3.com) These accurate and efficient programs are useful for understanding antenna function as well as providing a method for validation Constantine Balanis from Arizona State University, in the third edition of his wellregarded antenna textbook,28 provides various antenna modeling codes in both MATLAB and Fortran (from the second edition) via a CD-ROM The scripts provided are especially useful for students and for antenna design engineers interested in validating a CEM tool Atef Elsherbeni and Matthew Inman from the University of Mississippi have developed a textbook50 that tightly integrates visualization of antenna properties (such as pattern impedance, and so on) Along with the book, the authors have developed software (www.adv-program.com) that implements standard formula for various wire-type antennas (such as dipoles, helices, and monopoles) in both single element and array configurations The program has a GUI and can provide reference data for other programs that implement CEM techniques Commercial Codes Commercial computer programs are not only powerful, but also designed to be easy-to-use That last term is something that is in the “eye-of-the-beholder.” What software designers consider easy-to-use and dummy-proof often does not correspond to the expectations of the user Nevertheless, commercial CEM codes are typically better designed from the user point-of-view than public domain codes They typically have a Graphical User Interface (GUI) and application engineers that can (for a fee) help users obtain the best possible calculations for a given problem During the 1990s and 2000s, the number and diversity of commercial CEM offerings have expanded greatly The following list describes some of the most popular commercial CEM codes No doubt during the lifetime of this edition, the offerings will change considerably: Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas COMPUTATIONAL ELECTROMAGNETICS FOR ANTENNAS ● ● ● ● ● ● ● ● 59-23 www.ansoft.com The Ansoft Corporation offers a variety of CEM solutions for antenna design, electromagnetic compatibility, and other applications The computer codes offered by Ansoft utilize the finite element method, the method of moments, and circuit simulators These computer programs are some of the most powerful in the industry and are widely used www.cst.com Computer Simulation Technology offers widely used CEM tools based on the finite integration technique (FIT) and the method of moments Both time-domain and frequency-domain solvers are available www.emssuusa.com Electromagnetic Software and Systems offers a hybrid method of moments, finite element, and high frequency (uniform theory of diffraction) computer code (also known as FEKO) This tool is widely used for antenna design and for EMC analysis in the automobile industry www.remcom.com Remcom offers a finite difference-time domain method computer program that can be used for antenna design, scattering analysis, and specific absorption rate (SAR) calculations, among other uses It comes with a well-developed GUI and support for distributed parallel computers (for example, clusters) www.sonnetusa.com Sonnet Software offers a suite of full-wave electromagnetic design tools based on a frequency-domain method of moments for planar geometries It provides a powerful GUI and allows for separation between the design workstation and the computational workstation A free reduced-capability version is available for download www.wipl-d.com WIPL-D is a powerful antenna design program capable of modeling antennas (and other structures) comprised of both metal and dielectric materials WIPL-D is becoming increasingly popular amongst antenna engineers due to its flexibility and accurate solutions www.comsol.com Comsol Software is based in Sweden with offices worldwide Their main product is FEMLAB, a finite element-based simulation tool Their claim to fame is the ability to include multiple physical and chemical models in the simulation Thus, simultaneous simulation of both thermal and electromagnetic properties is facilitated by FEMLAB It has a powerful mesh generator and various pre- and post-processing capabilities www.zealand.com Zealand offers the popular integral equation-based solver, IE3D This program is widely used for simulating planar, microstrip structures such as patch antennas with feed networks 59.5 SOURCES FOR PARALLEL PROGRAMMING INFORMATION Computer architectures have undergone considerable, and truly impressive, changes over the past 30 years In the 1970s, computers for engineering design were primarily housed in centralized computing centers and used in a time-share manner (often with punch cards as the input method of choice) In the 1980s, with the advent of the personal computer, it became possible to put a computer on the desk of every engineer This led to more detailed antenna designs, especially when the Numerical Electromagnetic Code (NEC) was released by Lawrence Livermore National Laboratory In the 1990s, the widespread use of powerful engineering workstations led to increasing use of powerful computational electromagnetics tools in antenna design During that decade, the workstation and personal computer merged (or more accurately, the personal computer became a workstation) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas 59-24 CHAPTER FIFTY-NINE leading to even more widespread use of CEM codes Simultaneously, during the latter 1980s and early 1990s, vector-based computing became available through CRAY Computers, leading to changes in computational algorithms that specifically made use of vector processors Once the practical limit of vector processors have been reached (in terms of computations per second versus cost), the next major change in computation involved the use of many processing units simultaneously These parallel processing computers offered the promise of nearly unlimited computational power as more and more “brains” were brought to bear on a problem In reality, the community quickly learned that parallel programming is not trivial and that very real limits exist on the scalability of many CEM approaches Nevertheless, for those fortunate souls with access to massively parallel computers, these new architectures offered unprecedented computing power Parallel computers traditionally are symmetric multiprocessing (SMP) or shared memory architectures That means that each processor has access to all of the memory in the system SMP machines are particularly easy to program since the processors and memory are tightly connected together However, large SMP machines are highly-engineered and therefore rather expensive Therefore, a popular parallel computer architecture available in the late 1990s and early 21st century is the so-called Beowulf cluster These computers are made from a number (sometimes a large number) of commodity compute nodes connected together with some network fabric The original network fabrics were 10 or 100 MHz Ethernet connections (identical to those used to form a local area network) Currently, the most commonly used network fabric is a Gigabit Ethernet fabric It is becoming increasingly common to have higher speed (or lower latency) fabrics such as Myranet or Infiniband as the acquisition cost of these technologies is reduced Clusters are distributed memory architectures that often have a lower acquisition cost as compared to a comparable number of processors in an SMP system However, the programming requirements for a distributed cluster are often more significant than for an SMP system Recently, SMP machines are seeing a renaissance through the introduction of multiple core processors Traditionally, each processor had a single compute core Moore’s Law dictated a doubling of transistors every 18 months and as a result of this, it dictated an increase in computing power However, it became evident as the industry entered into the 21st century that such an expectation could not be sustained indefinitely One way to increase computing performance was to fabricate two (or dual) cores on each processor This way every single processor computer became an SMP while traditional thin (minimal capability) nodes for a cluster went from having two processors (a so-called 2-way node) to four cores (a so-called 4-way node) Most laptop computers offered for sale in 2006 now have dual-cores and so are properly SMP computers Realistically, with only two cores, efficient parallel programming is not practical due to overhead requirements Nevertheless, the trend is clear SMP computing is here to stay and will become ubiquitous Indeed, processors with four cores (e.g., quad-core) were released at the end of 2006 to original equipment manufacturers (OEMs) and to the public This means that computers will become 4-way SMP machines while thin cluster nodes will become 8-way SMP nodes It is therefore important for CEM code developers to become skilled at utilizing the dominant parallelization paradigms It is also important for consumers of either public domain or commercial CEM codes to understand the difference between the two types of parallel architectures (SMP vs distributed cluster) and the parallelization techniques used by those computer programs The two dominant parallel programming models in use today are OpenMP and MPI These two protocols are used to instruct the parallel computer on how to distribute the workload when more than one thread (akin to a core or processor) is being used simultaneously to solve a particular computational challenge Each of these application programming interfaces (API) will be discussed, briefly, in turn Finally, some brief comments on the use of the MATLAB distributed toolbox will be made along with the program environment, STAR-P Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas COMPUTATIONAL ELECTROMAGNETICS FOR ANTENNAS 59-25 OpenMP OpenMP is perhaps the simplest way to parallelize a program for use on an SMP computer Recall that an SMP computer has a number of threads within a single image (essentially, within a single node) The other major paradigm for parallelization of a program is Message Passing Interface (MPI) In MPI, the sending and receiving of data between nodes must be explicitly stated, as it is much more complex to implement than OpenMP However, MPI can be used effectively on a cluster system while OpenMP cannot With the emergence of quad-core processors, a four-processor computer has 16 cores and is in reality a 16-way system This is could be the optimal computer for CEM codes utilizing a matrix solver There are a large number of OpenMP tutorials on the Internet51–53 and textbooks including material on OpenMP.54-56 The examples in these texts are often in C (or C++); however, for those who still prefer Fortran, there are a large number of code examples in Fortran 77 and Fortran 90 These are the predominant languages used for scientific and engineering computing Parallelization of a program, at the most basic level, requires only a few commands describing how to partition the work amongst a number of threads for loop iteration The beginning of a parallel region must be declared as does the end of that parallel region Within the parallel region, loops that the programmer wishes to parallelize are identified individually along with some descriptions on how to parallelize the loop Data within a parallel region is either “shared” or “private” with the assumed state varying between implementations of OpenMP Shared data is available to all threads while private data is individual to a given thread It is important to understand how the data is identified, especially when reduction of data strings occurs (for example, addition of data vectors from various threads into a single data vector in the master thread) In general, users of OpenMP parallelized computer programs need to little to nothing if the program is written in a user-friendly manner One environment variable that may need to be set is the number of available threads In OpenMP, the command varies a bit based on the operating system and, for either Unix or Linux, the shell being used For example, in the Linux bash shell, the command export OMP_NUM_THREADS=# where “#” is the number of threads is used Programs then utilize that environment variable information during execution Typically this number should be set less than or equal to the number of available cores It is generally best to have a parallel region as large as possible to minimize overhead and to provide for the best synchronization of the program threads This is due to the fact that there is an implied barrier as the end of parallel region is reached What this means is that when an end of the parallel region is reached, all threads are synchronized This means that the slowest thread to be executed defines when the program execution outside the parallel region can commence Load balancing then becomes an important consideration for how well the program executes on a parallel machine In a similar light, use of the parallel region construct will generally be favored over loop-level parallelization; however, it comes with the cost of greater programming complexity (to ensure that all the code in the parallel region is compatible with parallel execution) MPI Message Passing Interface (MPI) is the second predominant parallelization paradigm in use at the dawn of the 21st century It has the distinct advantage (compared to OpenMP) of being useful on distributed memory computers, commonly referred to as clusters MPI requires considerably more investment by the programmer in terms of effort, since with MPI the programmer must tell the program how to share information between nodes Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas 59-26 CHAPTER FIFTY-NINE (these details are hidden from the user with OpenMP) For example, if a matrix-vector product is to be parallelized using MPI, the data for a given worker must be sent by the master (including the type of data and number of entries) where the master identifies which worker will receive the data Likewise, the worker must explicitly receive the data (knowing the format, data type, and so on), the work needed, and then send the resulting data back to the master In this case, the master must receive the data knowing which worker sent a particular set of data MPI is very powerful in that it can be used on both SMP and cluster computers In general, the performance of a parallel program using MPI is greater than that of OpenMP, especially in terms of scalability Loosely speaking, scalability indicates the number of threads that can be efficiently used Many CEM algorithms by their nature not scale beyond 8-16 threads except for some hardware implementations (for example, the differences in memory access methods, interconnects, and so on, will impact scalability) One reason for this is the large amount of memory required for the method as compared to the relatively narrow data pipes between the compute cores and the main memory Note that FD-TD is the CEM method that is well-known to scale beyond 16 threads with even moderate levels of parallelization skill There are many textbooks that discuss MPI programming.55,57–58 For the user, the level of effort for running (rather than parallelizing) a program is not excessive; however, it is greater than what is needed for OpenMP Minimally, the program must be compiled and linked with a particular implementation of the MPI libraries These libraries are specific to the MPI implementation and to the interconnect fabric Hence, a program linked to the MPI library for Myranet will not operate properly on a cluster that uses Infiniband To run a program that has been parallelized with MPI, rather than running the executable code directly as one does with most Unix/Linux programs, the user must invoke the MPI execution wrapper For example, to run the program foo, many MPI implementations require the following command for execution: mpirun foo The user needs to consult the relevant documentation for both the cluster and the program in order to determine all the necessary environment variables and execution scripts MATLAB and STAR-P In addition to traditional high-level languages, such as C, C++, and Fortran 90, a new type of parallel environment is being distributed and continuously improved MATLAB, as stated previously in this chapter, is widely used in the engineering community MATLAB offers a Distributed Computing Toolbox (see http://www.mathworks.com) that simplifies the implementation of MATLAB scripts for parallel execution on cluster computers The user should be cautioned that placing this toolbox on a cluster system is not as straightforward as serial toolboxes since the distributed toolbox operation must be compatible with the scheduler and operating system implementation Note that if the program is not compiled, execution of large problems will not usually be very efficient due to the interpretive nature of the MATLAB programming environment STAR-P is another recent product that simplifies the development and utilization of parallel programs on distributed computers STAR-P is used in conjunction with interactive desktop tools, such as MATLAB It acts as a bridge between MATLAB scripts and a cluster (or SMP) computer So, its function is related to the Distributed Computing Toolbox offered by MathWorks More information concerning STAR-P can be found at: http://www interactivesupercomputing.com The particular solution for any given application and user is dependent on the cluster system implementation Nevertheless, it is prudent to test particular applications with the various options such as the Distributed Computing Toolbox, STAR-P, or other computing environments before promising results to customers! Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas COMPUTATIONAL ELECTROMAGNETICS FOR ANTENNAS 59-27 REFERENCES K S Yee, “Numerical Solution of Initial Boundary Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans Ant Propagat., 14, (May 1966): 302–307 G Burke and A Poggio, “The Numerical Electromagnetic Code (NEC),” LLNL Technical Report UCID-18834 (November 1980) S M Rao, D R Wilton, and A W Glisson, “Electromagnetic Scattering by Surfaces of Arbitrary Shape,” IEEE Trans Ant Propagat., vol 30 (May 1982): 409–418 J C Nédélec, “Mixed Finite Elements in R3,” Numer Math., vol 35 (1980): 315–341 J M Jin and J L Volakis, “A Finite Element Boundary-Integral Formulation for Scattering by Three-Dimensional Cavity-Backed Apertures,” IEEE Trans Ant Propagat., vol 39 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© 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas 59-28 CHAPTER FIFTY-NINE 23 P R Rousseau and P H Pathak, “Time-Domain Uniform Geometrical Theory of Diffraction for a Curved Wedge,” IEEE Trans Ant Propagat., vol 43 (December 1995): 1375–1382 24 R F Harrington, Field Computation by Moment Methods (New York: IEEE Press-Wiley, 1993) 25 D B Davidson, Computational Electromagnetics for RF and Microwave Engineering (Cambridge, Mass: Cambridge University Press, 2005) 26 S N Makarov, Antenna and EM Modeling with MATLAB (New York: Wiley & Sons, 2002) 27 J D Kraus and R J Marhefka, Antennas for All Applications, 3rd Ed (Boston: McGraw-Hill, 2002) 28 C A Balanis, Antenna Theory: Analysis and Design, 3rd Ed (New York: Wiley & Sons, 2005) 29 A F Peterson, S L Ray, and R Mittra, Computational Methods for Electromagnetics (Piscataway, NJ: Wiley-IEEE Press, 1997) 30 A F Peterson, Mapped 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given at the website Computational Electromagnetics for Antennas COMPUTATIONAL ELECTROMAGNETICS FOR ANTENNAS 59-29 48 L Josefsson and P Persson, Conformal Array Antenna Theory and Design, IEEE (New York: Wiley, 2006) 49 D H Werner, R J Allard, R A Martin, and R Mittra, “A Reciprocity Approach for Calculating Radiation Patterns of Arbitrarily-Shaped Microstrip Antennas Mounted on Circularly Cylindrical Platforms,” IEEE Trans Ant Propagat., vol 51 (April 2003): 730–738 50 A Elsherbeni and M Inman, Antenna Design and Visualization using MATLAB (Raleigh: NC: SciTech Publishing, 2006) 51 OpenMP web site: http://www.openmu.org 52 Boston University, http://scv.bu.edu/SCV/Tutorials/OpenMP/ 53 Texas A&M web site, http://sc.tamu.edu/help/ 54 H Jordan and G Alaghband, Fundamentals of Parallel Processing (Upper Saddle River, NJ: Prentice Hall, 2003) 55 M.J Quinn, Parallel Programming in C with MPI and OpenMP (New York: McGraw-Hill, 2004) 56 R Chandra, R Menon, L Daqum, D Kohr, D Maydan, and J McDonald, Parallel Programming in OpenMP (San Francisco: Morgan Kaufmann, 2000) 57 B P Lester, The Art of Parallel Programming, 2nd Ed (Fairfield, IA: World Publishing, 2006) 58 P Pacheco, Parallel Programming in MPI (San Francisco: Morgan Kaufmann, 1996) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Computational Electromagnetics for Antennas Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2007 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ... Mech Dish Ant Mech Dish Ant Helical Ant Mono-Dipole Mono-Dipole Mono-Dipole Mono-Dipole Mono-Dipole Mono-Dipole Mono-Dipole Mono-Dipole Yagi Yagi Mono- Dipole Electronic Scan Phased Array Electronic... performance is for point-to-point line-of-sight with a single antenna at the transmitter and receiver In order to achieve acceptable performance for non-line-of-sight use, MIMO and antenna gain are... C-band C-band Satcom Ka-band L-band Iridium Ka-band Satcom C-band W-band (4G) Broadway HIPERLAN/2 HIPERSPOT OFDM Ku-band S-band C-band (4G) 802.16 WiMax OFDM FDD/TDD Ku-band Satcom ISM band Industrial