Antenna Theory: A Review
`
`CONSTANTINE A. BALANIS, FELLOW, IEEE
`Invited Paper
`
`Thispaper is a tutorial on the theory of antennas, and it has been
`written as an introduction for the nonspecialist and as a review for
`the expert. The paper traces the history of antennas and some of
`the most basic radiating elements, demonstrates the fundamental
`principles of antenna radiation, reviews Maxwell’s equations and
`electromagnetic boundary conditions, and outlines basic proce-
`dures and equations of radiation. Modeling of antenna source
`excitation is illustrated, and antenna parameters and fisures-of-
`merit are reviewed. Finally, theorems, arraying principles, and
`advanced asymptotic methods for antenna analysis and design are
`summarized.
`
`I. INTRODUCTION
`For wireless communication systems, the antenna is
`one of the most critical components. A good design of
`the antenna can relax system requirements and improve
`overall system performance. A typical example is TV for
`which the overall broadcast reception can be improved by
`utilizing a high performance antenna. An antenna is the
`system component that is designed to radiate or receive
`electromagnetic waves. In other words, the antenna is the
`electromagnetic transducer which is used to convert, in the
`transmitting mode, guided waves within a transmission line
`to radiated free-space waves or to convert, in the receiving
`mode, free-space waves to guided waves. In a modern
`wireless system, the antenna must also act as a directional
`device to optimize or accentuate the transmitted or received
`energy in some directions while suppressing it in others [l].
`The antenna serves to a communication system the same
`purpose that eyes and eyeglasses serve to a human.
`The history of antennas [2] dates back to James Clerk
`Maxwell who unified the theories of electricity and mag-
`netism, and eloquently represented their relations through
`a set of profound equations best known as Maxwell’s
`Equations. His work was first published in 1873 [3]. He also
`showed that light was electromagnetic and that both light
`and electromagnetic waves travel by wave disturbances of
`the same speed. In 1886, Professor Heinrich Rudolph Hertz
`demonstrated the first wireless electromagnetic system. He
`
`Manuscript received November 26, 1990; revised March 1, 1991.
`The author is with the Department of Electrical Engineering, Telecom-
`munications Research Center, Arizona State University, Tempe, AZ 85287-
`7206.
`IEEE Log Number 9105093.
`
`was able to produce in his laboratory at a wavelength of 4
`m a spark in the gap of a transmitting X/2 dipole which
`was then detected as a spark in the gap of a nearby loop.
`It was not until 1901 that Guglielmo Marconi was able to
`send signals over large distances. He performed, in 1901,
`the first transatlantic transmission from Poldhu in Cornwall,
`England, to St. John’s, Newfoundland.
`His transmitting antenna consisted of 50 vertical wires
`in the form of a fan connected to ground through a
`spark transmitter. The wires were supported horizontally
`by a guyed wire between two 60-m wooden poles. The
`receiving antenna at St. John’s was a 200-m wire pulled
`and supported by a kite. This was the dawn of the antenna
`era.
`From Marconi’s inception through the 1940’s, antenna
`technology was primarily centered on wire related radiating
`elements and frequencies up to about UHF. It was not
`until World War I1 that modern antenna technology was
`launched and new elements (such as waveguide apertures,
`horns, reflectors, etc.) were primarily introduced. Much
`of this work is captured in the book by Silver [4]. A
`contributing factor to this new era was the invention of
`microwave sources (such as the klystron and magnetron)
`with frequencies of 1 GHz and above.
`While World War I1 launched a new era in antennas,
`advances made in computer architecture and technology
`during the 1960’~-1980’s have had a major impact on
`the advance of modern antenna technology, and they are
`expected to have an even greater influence on antenna
`engineering in the 1990’s and beyond. Beginning primarily
`in the early 1960’s, numerical methods were introduced
`that allowed previously intractable complex antenna system
`configurations to be analyzed and designed very accurately.
`In addition, asymptotic methods for both low frequencies
`(e.g., Moment Method (MM), Finite-Difference, Finite-
`Element) and high frequencies (e.g., Geometrical and Phys-
`ical Theories of Diffraction) were introduced, contributing
`significantly to the maturity of the antenna field. While
`in the past antenna design may have been considered a
`secondary issue in overall system design, today it plays
`a critical role. In fact, many system successes rely on
`the design and performance of the antenna. Also, while
`
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`APPLE 1021
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`in the first half of this century antenna technology may
`have been considered almost a “cut and try” operation,
`today it is truly an engineering art. Analysis and design
`methods are such that antenna system performance can be
`predicted with remarkable accuracy. In fact, many antenna
`designs proceed directly from the ’ initial design stage to
`the prototype without intermediate testing. The level of
`confidence has increased tremendously.
`The widespread interest in antennas is reflected by the
`large number of books written on the subject [5]. These
`have been classified under four categories: Fundamental,
`Handbooks, Measurements, and Specialized. In English,
`in all four categories, there were 4 books published in
`the 1940’s, 9 in the 1950’s, 17 in the 1960’s, 20 in the
`1970’s, 69 in the 1 9 8 0 ’ ~ ~ and 1 already in the 1990’s.
`This is an outstanding collection of books, and it reflects
`the popularity of the antenna subject, especially since the
`1950’s. Because of space limitations, only a partial list is
`included here [l], [4], [6]-[32]. Some of these books are
`now out of print.
`In this paper, the basic theory of antenna analysis, the pa-
`rameters and figures-of-merit used to characterize antenna
`performance, and significant advances made in the last three
`decades that have contributed to the maturity of the field
`will be outlined. It will be concluded with a discussion
`of challenging opportunities for the future. Some of the
`material in this paper has been borrowed from the author’s
`textbooks on antennas [ 11 and advanced electromagnetics
`[331.
`
`11. ANTENNA ELEMENTS
`Prior to World War I1 most antenna elements were of
`the wire type (long wires, dipoles, helices, rhombuses,
`fans, etc.), and they were used either as single elements
`or in arrays. During and after World War 11, many other
`radiators, some of which may have been known for some
`time and others of which were relatively new, were put
`into service. This created a need for better understanding
`and optimization of their radiation characteristics. Many
`of these antennas were of the aperture type (such as
`open-ended waveguides, slots, horns, reflectors, lenses,
`and others), and they have been used for communication,
`radar, remote sensing, and deep space applications both on
`airborne and earth based platforms. Many of these operate
`in the microwave region. In this issue, reflector antennas
`are discussed in “The current state of the reflector antenna
`art-Entering
`the 1990’s,” by W. V. T. Rusch.
`Prior to the 1 9 5 0 ’ ~ ~ antennas with broadband pattern
`and impedance characteristics had bandwidths not much
`greater than about 2:l. In the 1950’s, a breakthrough in
`antenna evolution was created which extended the max-
`imum bandwidth to as great as 40:l or more [34]-[36].
`Because the geometries of these antennas are specified
`by angles instead of linear dimensions, they have ideally
`an infinite bandwidth. Therefore, they are referred to as
`frequency independent. These antennas are primarily used
`in the 10-10 000 MHz region in a variety of applications
`
`including TV, point-to-point communications, feeds for re-
`flectors and lenses, and many others. This class of antennas
`is discussed in more detail in this issue in “Frequency-
`independent antennas and broad-band derivatives thereof,”
`by P. E. Mayes.
`It was not until almost 20 years later that a fundamental
`new radiating element, that has received a lot of attention
`and many applications since its inception, was introduced.
`This occurred in the early 1970’s when the microstrip or
`patch antennas was reported [30], [37]-[44]. This element
`is simple, lightweight, inexpensive, low profile, and con-
`formal to the surface. Microstrip antennas and arrays can
`be flush-mounted to metallic or other existing surfaces.
`Operational disadvantages of microstrip antennas include
`low efficiency, narrow bandwidth, and low power handling
`capabilities. These antennas are discussed in more detail in
`this issue in “Microstrip antennas,” by D. M. Pozar. Major
`advances in millimeter wave antennas have been made in
`recent years, including integrated antennas where active and
`passive circuits are combined with the radiating elements
`in one compact unit (monolithic form). These antennas are
`discussed in this issue in “Millimeter wave antennas,” by
`F. K. Schwering.
`
`111. THEORY
`To analyze an antenna system, the sources of excitation
`are specified, and the objective is to find the electric and
`magnetic fields radiated by the elements. Once this is
`accomplished, a number of parameters and figures-of-merit
`that characterize the performance of the antenna system can
`be found. To design an antenna system, the characteristics
`of performance are specified, and the sources to satisfy
`the requirements are sought. In this paper, the analysis
`procedure is outlined. Synthesis procedures for pattern
`control of antenna arrays are discussed in this issue in
`“Basic array theory,” by W. H. Kummer and “Array pattern
`control and synthesis,” by R. C. Hansen. Theorems used in
`the solution of antenna problems are also discussed. Design
`and optimization procedures are presented in many of the
`other papers of this issue.
`
`A. Radiation Mechanisms and Current Distribution
`One of the most basic questions that may be asked
`concerning antennas is “how do they radiate?” A quali-
`tative understanding of the radiation mechanism may be
`obtained by considering a pulse source attached to an open-
`ended conducting wire, which may be connected to ground
`through a discrete load at its open end. When the wire is
`initially energized, the charges (free electrons) in the wire
`are set in motion by the electric lines of force created by the
`source. When charges are accelerated in the source-end of
`the wire and decelerated (negative acceleration with respect
`to original motion) during reflection from its ends, it is
`suggested that radiated fields are produced at each end and
`along the remaining part of the wire [45]. The acceleration
`of the charges is accomplished by the external source in
`which forces set the charges in motion and produce the
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`associated fields radiated. The deceleration of the charges
`at the end of the wire is accomplished by the internal
`(self) forces associated with the induced field due to the
`buildup of charge concentration at the ends of the wire.
`The internal forces receive energy from the charge buildup
`as its velocity is reduced to zero at the ends of the wire.
`Therefore, charge acceleration due to an exciting electric
`field and deceleration due to impedance discontinuities or
`smooth curves of the wire are mechanisms responsible
`for electromagnetic radiation. While both current density
`( J z ) and charge density (que) appear as source terms
`in Maxwell’s equations (see (lb) and (IC)), charge is
`viewed as a more fundamental quantity. Even though this
`interpretation of radiation is primarily used for transient
`radiation, it can be used to explain steady-state radiation
`[45]. Another qualitative description of antenna radiation
`can be found in [l, ch. 11.
`The previous explanation demonstrates the principles of
`radiation from a single wire. Let us now consider radiation
`and interference amplitude pattern (lobing) formation from
`a two-wire transmission line and antenna element, such as
`a linear dipole. We begin with the geometry of a lossless
`two-wire transmission line, as shown in Fig. l(a). The
`movement of the charges creates a traveling wave current
`of magnitude Io/2 along each of the wires. When the
`current arrives at the end of each of the wires, it undergoes
`a complete reflection (equal magnitude and 180’ phase
`reversal). The reflected traveling wave, when combined
`with the incident traveling wave, forms in each wire a
`pure standing wave pattern of sinusoidal form as shown in
`Fig. l(a). The current in each wire undergoes a 180” phase
`reversal between adjoining half periods. This is indicated
`in Fig. l(a) by the reversal of the arrow direction.
`For the two-wire balanced (symmetrical) transmission
`line, the current in a half-period of one wire is of the
`same magnitude but 180’ out-of-phase from that in the
`corresponding half-period of the other wire. If, in addition,
`the spacing between the two wires is very small (s <<
`A), the fields radiated by the current of each wire are
`essentially canceled by those of the other. The net result
`is an almost ideal (and desired) nonradiating transmission
`line. As the section of the transmission line between 0 5
`e 5 X/4 begins to flare, as shown in Fig. l(b), the current
`distribution is essentially unaltered in form in each of the
`wires. However, because the two wires of the flared section
`are not necessarily close to each other, the fields radiated by
`one do not necessarily cancel those of the other. Therefore,
`there is net radiation by the system.
`Ultimately, the flared section of the transmission line can
`take the form shown in Fig. l(c). This is the geometry of the
`widely used X/2 dipole antenna. Because of the standing
`wave current pattern, it is also classified as a standing wave
`antenna. If e < A, the phase of the current standing wave
`pattern in each arm is the same throughout its length. In
`addition, it is oriented spatially in the same direction as
`that of the other arm, as shown in Fig. l(c). Thus the fields
`radiated by the two arms of the dipole (vertical parts of a
`flared transmission line) will primarily reinforce each other
`
`I 4
`
`e t 11
`
`Fig. 1. Radiation from a two-wire transmission line and linear
`dipole, (a) Two-wire transmission line. (b) Flared transmission line.
`(c) Linear dipole.
`
`in some directions and cancel each other toward others. This
`results in the formation of amplitude pattern lobes which
`are illustrated in the later part of this paper.
`
`B. Maxwell’s Equations, the Wave Equation,
`and Boundary Conditions
`An antenna configuration is an electromagnetic boundary-
`value problem. Therefore, the fields radiated must satisfy
`Maxwell’s equations which, for a lossless medium (c = 0)
`and time-harmonic fields (assuming an ejwt time conven-
`tion), can be written as [33]:
`
`In ( 1 a H l d ) both electric ( J ; ) and magnetic (M;) current
`densities, and electric (que) and magnetic (qum) charge
`densities are allowed to represent the sources of excitation.
`The respective current and charge densities are related by
`the continuity equations
`
`BALANIS: ANTENNA THEORY: A REVIEW
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`V .Mi = -jwq vm'
`(2b)
`Although magnetic sources are not physical, they are often
`introduced as electrical equivalents to facilitate solutions
`of physical boundary-value problems. In fact, for some
`configurations, both electric and magnetic equivalent cur-
`rent densities are used to represent actual antenna systems.
`For a metallic wire antenna, such as a dipole, an electric
`current density is used to represent the antenna. However,
`an aperture antenna, such as a waveguide or horn, can be
`represented by either an equivalent magnetic current density
`or by an equivalent electric current density or both. This
`will be demonstrated in Section IV-B.
`In addition
`to satisfying Maxwell's equations of
`( l a H l d ) , the fields radiated by the antenna must satisfy
`the boundary conditions of [33]
`-fi x Ed = M ,
`( 3 4
`fi x Hd = Js
`(3b)
`ii . ( € E d ) = qse
`( 3 4
`fi ' ( P H d ) = q s m .
`( 3 4
`The first two equations, (3a) and (3b), enforce the boundary
`conditions on the discontinuity of the tangential components
`of the electric and magnetic fields while (3c) and (3d)
`enforce the boundary conditions on the discontinuity of
`the normal components of the electric and magnetic flux
`densities. In (3a)43d), J , / M , and y s e / q s m represent,
`respectively, the electridmagnetic surface current densities
`and electric/magnetic surface charge densities. For time-
`harmonic fields, all four boundary conditions of (3a)-(3d)
`are not independent. The first two, (3a) and (3b), form an
`independent and sufficient set [33].
`the
`In addition to the boundary conditions of (3a)-(3d),
`solutions for the fields radiated by the antenna must also
`satisfy the radiation condition which requires in an infinite
`homogeneous medium that the waves travel outwardly from
`the source and vanish at infinity.
`Since (la) and (lb) are first-order coupled differential
`equations (each contains both electric and magnetic fields),
`it is often more desirable to uncouple the equations. When
`this is done, we obtain two nonhomogeneous vector wave
`equations; one for E and one for E. These are given by [33]
`V 2 E + p2E = -Vqve + V x M; + jwpJ;
`1
`V 2 H + P2H = -Vqvm - V x Ji + ~ w c M ; (4b)
`1
`CL
`where p2 = w2pe.
`For a radiation problem, the first step is to represent the
`antenna excitation by its source, represented in (4a) and (4b)
`by the current density Ji or Mi or both, having taken into
`account the boundary conditions. This will be demonstrated
`in Section IV. The next step is to solve (4a) and (4b) for
`E and a. This is a difficult step, and it usually involves
`an integral with a complicated intergrand. This procedure
`is represented in Fig. 2 as Path 1.
`To reduce the complexity of the problem, it is a common
`practice to break the procedure into two steps. This is
`
`(4a)
`
`Integration Radiated
`Sources
`(G)Tth43
`
`Fig. 2. Procedure to solve antenna radiation.
`
`represented in Fig. 2 by Path 2. The first step involves an
`integration while the second involves a differentiation. To
`accomplish this, auxiliary vector potentials are introduced.
`The most commonly used potentials are A (magnetic vector
`potential) and F (electric vector potential). Although the
`electric and magnetic field intensities (E and H ) represent
`physically measurable quantities, for most engineers the
`vector potentials are strictly mathematical tools. The Hertz
`vector potentials II, and IIh make up another possible pair.
`The Hertz vector potential II, is analogous to A and I I h
`is analogous to F [l], [33]. In the solution of a problem,
`only one set, A and F or II, and IIh, is required. In the
`first step of the Path 2 solution, the vector potentials A and
`F (or 11, and IIh ) are found, once the sources J; and/or
`M ; are specified. This step involves an integration but one
`which is not as difficult as the integration of Path 1. The
`next step of the Path 2 solution is to find the fields E and
`E, from the vector potentials A and F (or II, and U,).
`This step involves differentiation. The equations that are
`essential for the solution of Path 2 will be outlined next
`using the vector potentials A and F. The derivation can be
`found in many books, such as [l], [4], [7], [18], [19], [21],
`[33], and others.
`
`C. The Vector Potentials A and F
`In step one of Path 2, once the sources Ji and Mi
`are specified, the vector potentials A and F are related,
`respectively, to Ji and Mi by
`V2A + p2A = -pJi
`(5a)
`V 2 F + p2F = -€Mi.
`(5b)
`If the current densities are distributed over a surface S,
`such as that of a perfect conductor immersed in an infinite
`homogeneous medium, the solutions of (5a) and (5b) can
`be written, by referring to Fig. 3(a), as
`
`where R is the distance from any point on the source to
`the observation point. For the solutions of (6a) and (6b), J;
`and Mi in (5a) and (5b) are replaced by J, and M , which
`have the units of A/m and V/m, respectively. If the current
`densities are distributed over a volume, the surface integrals
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`and
`
`The forms of (5a)-(5b) and (7a)-(7b) are based on
`choosing the relationships between A & 4 and F & 4 of
`
`which each is known as the Lorentz condition or gauge (&
`and
`are scalar functions of position usually referred to
`as scalar potentials.) The choices of (7c) and (7d) were
`made to reduce (5a), (5b), (7a), and (7b) to their simplest
`forms; however, they are not the only possible choices.
`In (7a) and (7b), the terms within the first brackets are the
`fields due to the vector potential A (and as a consequence
`due to electric current density) while the terms within the
`second brackets are the fields due to the vector potential
`F (and as a consequence due to magnetic current density).
`If either of the sources (electric or magnetic) do not exist,
`the corresponding electric and magnetic fields (and vector
`potentials A or F) are set to zero. It is evident from (7a)
`and (7b) that superposition is used when both electric and
`magnetic sources are needed to represent the source of
`radiation.
`
`E. Field Regions
`The space surrounding an antenna is usually subdi-
`vided into three regions: the reactive near-field region,
`the radiating near-field (Fresnel) region, and the far-field
`(Fraunhofer} region. These regions are so designated to
`identify the field structure in each. Although no abrupt
`changes in the field configurations are noted as the bound-
`aries are crossed, there are distinct differences among them
`[l], [ll]. The boundaries separating these regions are not
`unique, although various criteria have been established and
`are commonly used to identify the regions. The following
`definitions in quotations are from [46].
`The reactive near-field region is defined as “that region
`of the field immediately surrounding the antenna wherein
`the reactive field predominates.” For most antennas, the
`outer boundary of this region is commonly taken to exist
`at a distance R < 0 . 6 2 d m from the antenna, where
`X is the wavelength and D is the largest dimension of the
`antenna.
`The radiating near-field (Fresnel} region is defined as
`“that region of the field of an antenna between the reactive
`near-field region and the far-field region wherein radiation
`fields predominate and wherein the angular field distribu-
`tion is dependent upon the distance from the antenna.”
`The radial distance R over which this region exists is
`0 . 6 2 d m 5 R < 2D2/X (provided D is large compared
`to the wavelength). This criterion is based on a maximum
`phase error of 7r/8 radians (22.5’) [l], [ll]. In this region
`
`11
`
`-Y
`
`/
`
`I
`
`(b)
`Fig. 3.
`Coordinate system arrangements for (a) near-field and (b)
`far-field radiation.
`
`of (6a) and (6b) are replaced by volume integrals. When
`the currents are distributed over a thin wire, the surface
`integrals of (6a) and (6b) can be approximated by line
`integrals [l], [33].
`The most difficult step in the solution of Path 2 is
`the evaluation of the integrals of (6a) and (6b). For
`most practical antenna geometries, these integrals cannot
`be evaluated in closed form. Usually approximations
`are made and/or numerical techniques are employed.
`With today’s computer technology, numerical evaluation
`of integrals is a much simpler, more convenient, and
`more efficient procedure than in the past. Computational
`electromagnetics for antenna radiation are discussed in this
`issue in “Low-frequency computational electromagnetics
`for antenna analysis,” by E. K. Miller and G . J.
`Burke.
`
`D. The Electric and Magnetic Fields E and H
`Once the vector potentials A and F have been found by
`(6a) and (6b), the second step in the solution of Path 2 is
`to find the electric and magnetic fields E and E. This is
`accomplished by using the equations [l], [33]
`
`E = E A + E F
`1
`- j w A - j - - - V ( V , A )
`w w
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`the field pattern is, in general, a function of the radial
`distance and the radial field component may be appreciable.
`The fur-field (Fruunhofer) region is defined as "that
`region of the field of an antenna where the angular field
`distribution is essentially independent of the distance from
`the antenna." In this region, the real part of the power
`density is dominant. The radial distance R over which this
`region exists is R 2 2D2/A (provided D is large compared
`to the wavelength). The outer boundary is ideally at infinity.
`This criterion is also based on a maximum phase error of
`7r/8 radians (22.5') [l], [ll], [MI. In this region, the field
`components are essentially transverse to the radial distance,
`and the angular distribution is independent of the radial
`distance. The evaluation of the integrals in (6a) and (6b)
`varies according to the region where the observations are
`made. The evaluation is most difficult in the reactive near-
`field region. As the observation point is moved radially
`outward, approximations can be made in the integrands of
`(6a) and (6b) to reduce the complexity of the evaluation.
`The evaluation is easiest in the far-field region, and that is
`usually the region of most applications.
`Let us outline the procedure for the evaluation of integrals
`for the potentials A and F as the observation point is moved
`to the far-field region. This will be done for the surface
`integrals of (6a) and (6b). As the observation point is moved
`radially outward, the distance R can be approximated and
`leads to a simplification in the evaluation of the integral.
`In the far-field (Fraunhofer) region, the radial distance R
`of Fig. 3(a) can be approximated by [l], [33]
`
`{ r - r' cos$
`
`N-
`
`r
`
`for phase terms
`for amplitude terms.
`
`( 8 4
`(8b)
`
`With the approximations of (9)-(lOa), the spherical com-
`ponents of the electric and magnetic fields of (7a) and (7b)
`can be written in scalar form as [l], [33]
`Er N 0
`
`( W
`
`where 7 is the intrinsic impedance of the medium (7 =
`fi while NO, N+, and Le, L+ are the spherical 0 and 4
`components of N and L from (sa) and (loa). In antenna
`theory, the spherical coordinate system is the most widely
`used system.
`By examining (lla)412c), it is apparent that
`E8 21 vH+
`(134
`E+ N -THO.
`(13b)
`The relations of (13a) and (13b) indicate that in the far-
`field region the fields radiated by an antenna and observed
`in a small neighborhood on the surface of a large radius
`sphere have all the attributes of a plane wave whereby the
`corresponding electric and magnetic fields are orthogonal
`to each other and to the radial direction.
`To use the above procedure, the sources representing
`the physical antenna structure must radiate into an infinite
`homogeneous medium. If that is not the case, then the
`problem must be reduced further (e.g., though the use of
`a theorem, such as the image theorem) until the sources
`radiate into an infinite homogeneous medium. This again
`will be demonstrated in Section IV-A for the analysis of
`the aperture antenna.
`
`IV. ANTENNA SOURCE MODELING
`The first step in the analysis of the fields radiated by
`an antenna is the specification of the sources to represent
`the antenna. Here we will present two examples of source
`modeling; one for a thin wire antenna (such a dipole) and
`the other for an aperture antenna (such as a waveguide).
`These are two distinct examples each with a different source
`modeling; the wire requires an electric current density
`while the aperture is represented by an equivalent magnetic
`current density.
`
`A. Wire Source Modeling
`Let us assume that the wire antenna is a dipole, as shown
`in Fig. 4. If the wire has circular cross section with radius
`a, the electric current density induced on the surface of
`the wire will be symmetrical about the circumference (no
`4 variations). If the wire is also very thin ( U << A), it is
`common to assume that the excitation source representing
`
`Graphically, the approximation of (sa) is illustrated in Fig.
`3(b) where the radial vectors R and r are parallel to each
`other. Although such a relation is strictly valid only at
`infinity, it becomes more accurate as the observation point
`is moved outward at radial distances exceeding 2D2/A.
`Since the far-field region extends at radial distances of
`R 2 2D2/A, the approximation of (sa) for the radial
`distance R leads to phase errors which do not exceed T / 8
`radians (22.5'). It has been shown that such phase errors
`do not have a pronounced effect on the variations of the
`far-field amplitude patterns (at least at parts of the patterns
`in which the relative field strength is not lower than about
`25 dB) [l], [ll].
`Using the approximations of (sa) and (8b) for observa-
`tions in the far-field region, the integrals of (6a) and (6b)
`can be reduced to
`
`12
`
`PROCEEDINGS OF THE IEEE, VOL. 80, NO. 1, JANUARY 1992
`
`Authorized licensed use limited to: Fish & Richardson PC. Downloaded on February 17,2021 at 15:11:19 UTC from IEEE Xplore. Restrictions apply.
`
`6
`
`

`

`X
`
`I
`
`Fig. 4. Dipole geometry for electromagnetic wave radiation.
`
`- A 1 2
`
`dipole
`
`the antenna is a current along the axis of the wire. This
`current must vanish at the ends of the wire, and for very thin
`wires assumes a sinusoidal distribution. For a center-fed
`dipole, this excitation is often represented by
`I,= {
`6,10 sin [p((t/2) - z’)] 0 5 Z’ 5 el2
`(144
`6,Io sin [/3((l/2) + z’)]
`- l / 2 5 z’ 5 0.
`(14b)
`No magnetic source representation is necessary for this type
`of an antenna. The fields radiated by the antenna can now be
`found by first determining the potential A using (6a), where
`the surface integral is reduced to a line integral, and in turn
`the fields are determined by (7a) and (7b). For observations
`in the far-field region, the approximations of (Sa) and (8b)
`can be used. When this is done, it can be shown by also
`using (9)-(12c) that the electric and magnetic fields can be
`written as
`
`EB
`H+ 2 - v
`To illustrate the field variations of (lsa), a three-
`dimensional graph of the normalized field amplitude pattern
`for a half-wavelength (e = X/2) dipole is plotted in Fig.
`5 using software from [47]. A 90’ angular section of
`the pattern has been omitted to illustrate the figure-eight
`elevation plane pattern variation. As the length of the wire
`increases, the pattern becomes narrower. When the length
`exceeds one wavelength (e > A), sidelobes are introduced
`into the elevation plane pattern [l].
`
`B. Aperture Source Modeling
`To analyze aperture antennas, the most often used proce-
`dure is to model the source representing the actual antenna
`by the Field Equivalence Principle (FEP), also referred
`to as Huygen’s Principle [l], [lS], [19], [21], [33]. With
`this method, the actual antenna is replaced by equivalent
`sources that, externally to a closed surface enclosing the
`actual antenna, produce the same fields as those radiated
`by the actual antenna. This procedure is analogous to the
`Thevenin Equivalent of circuit analysis which produces the
`same response, to an external load, as the actual circuit.
`The FEP requires that first an imaginary surface is chosen
`which encloses the actual antenna. This is shown dashed
`
`Three-dimensional amplitude radiation pattern of a X / 2
`
`Fig. 5.
`dipole.
`
`in Fig. 6(a). Once the imaginary closed surface is chosen,
`one of the equivalents of the FEP requires that the volume
`within the closed surface be replaced b