balanis_antenna theory review

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Antenna Theory: A Review

CONSTANTINE A. BALANIS, FELLOW, IEEEInvited Paper

Th ispape r s a tutorial on the theory of antennas, and it has beenwritten as an introduction for the nonspecialist and as a review forthe expert. The pap er t races the history of antennas and some ofthe most basic radiating elements, demonstrates the fundamentalprinciples of antenna radiation, reviews Maxwells equations andelectromagnetic boundary conditions, and outlines basic proce-dures and equations of radiation. Modeling of antenna sourceexcitation is illustrated, and antenna parameters and fisures-of-merit are reviewed. Finally, theorems, arraying principles, andadvanced asymptotic methods for antenna analysis and design aresummarized.

I. INTRODUCTIONFor wireless communication systems, the antenna is

one of the most critical components.A good design ofthe antenna can relax system requirements and improveoverall system performance.A typical example is TV forwhich the overall broadcast reception can be improved byutilizing a high performance antenna.A n antenna is thesystem component that is designed to radiate or receiveelectromagnetic waves. In other words, the antenna is theelectromagnetic transducer which is used to convert, in thetransmitting mode, guided waves within a transmission lineto radiated free-space waves or to convert, in the receivingmode, free-space wavesto guided waves. In a modernwireless system, the antenna must also act as a directionaldevice to optimize or accentuate the transmitted or receivedenergy in some d irections while suppressing it in others[l] .The antenna servesto a communication system the samepurpose that eyes and eyeglasses serve to a human.

The history of antennas[ 2 ] dates back to James ClerkMaxwell who unified the theories of electricity and mag-netism, and eloquently represented their relations througha set of profound equations best known asMaxwellsEquations. His work was first published in1873 [3]. He alsoshowed that light was electromagnetic and that both lightand electromagnetic waves travel by wave disturbances ofthe same speed. In1886, Professor Heinrich Rudolph Hertzdemonstrated the first wireless electromagnetic system. He

Manuscript received N ovember 26, 1990; revised March1, 1991.The author is with the Departmentof Electrical Eng ineering, 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 of4m a spark in the gap of a transmittingX/2 dipole whichwas then detected as a spark in the gap of a nearby loop.It was not until 19 01 that Guglielmo Marconi w as able tosend signals over large distances. He performed, in1901,the first transatlantic transmission from Poldhu in C ornwall,England, to St. Johns, Newfoundland.

His transmitting antenna consisted of50 vertical wiresin the form of a fan connected to ground through aspark transmitter. The wires were supported horizontallyby a guyed wire between two 60-m wooden poles. Thereceiving antenna at St. Johns was a 200-m wire pulledand supported by a kite. This was the dawn of the antennaera.

From Marconis inception through the1940s, antennatechnology wa s primarily centered on wire related radiatingelements and frequencies up to about UHF. It was notuntil World War I that modern antenna technology waslaunched and new elements (such as waveguide apertures,horns, reflectors, etc.) were primarily introduced. Muchof this work is captured in the book by Silver[4]. Acontributing factor to this new era was the invention ofmicrowave sources (such as the klystron and magnetron)with frequencies of1 GHz and above.

While World War I launched a new era in antennas,advances made in computer architecture and technologyduring the 1960~-1980s ave had a major impact onthe advance of modern antenna technology, and they areexpected to have an even greater influence on antennaengineering in the1990s and beyond. Beginning primarilyin the early 1960s, numerical methods were introducedthat allowed previously intractable complex antenna systemconfigurations 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, contributingsignificantly to the maturity of the antenna field. While

in the past antenna design may have been considered asecondary issue in overall system design, today it playsa critical role. In fact, many system successes rely onthe design and performance of the antenna.Also, while

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in the first half of this century antenna technology mayhave been considered almost a cut and try operation,today it is truly an engineering art. Analysis and designmethods are such that antenna system performance can bepredicted with remarkable accuracy. In fact, many antennadesigns proceed directly from the initial design stage tothe prototype without intermediate testing. The level ofconfidence has increased tremendously.

The widespread interest in antennas is reflected by thelarge number of books written on the subject [5]. Thesehave been classified under four categories: Fundamental,Handbooks, Measurements, and Specialized. In English,in all four categories, there were 4 books published inthe 1940s, 9 in the 1950s, 17 in the 1960s,20 in the1970s, 69 in the 1 9 8 0 ~ ~nd 1 already in the 1990s.This is an outstanding collection of books, and it reflectsthe popularity of the antenna subject, especially since the1950s. Because of space limitations, only a partial list isincluded here [l ], [4], [6]-[32]. Some of these books arenow out of print.

In this paper, the basic theory of antenna analysis, the pa-rameters and figures-of-merit used to characterize antennaperformance, and significant advances made in the last threedecades that have contributed to the maturity of the fieldwill be outlined. It will be concluded with a discussionof challenging opportunities for the future. Some of thematerial in this paper has been borrowed from the authorstextbooks on antennas[11 and advanced electromagnetics[331.

11. ANTENNA LEMENTS

Prior to World War I1 most antenna elements were ofthe wire type (long wires, dipoles, helices, rhombuses,fans, etc.), and they were used either as single elementsor in arrays. During and after World War 11, many otherradiators, some of which may have been known for some

time and others of which were relatively new, were putinto service. This created a need for better understandingand optimization of their radiation characteristics. Manyof these antennas were of the aperture type (such asopen-ended waveguides, slots, horns, reflectors, lenses,and others), and they have been used for communication,radar, remote sensing, and deep space applications both onairborne and earth based platforms. Many of these operatein the microwave region. In this issue, reflector antennasare discussed in The current state of the reflector antennaart-Entering the 1990s, by W. V. T. Rusch.

Prio r to the 19 5 0 ~ ~ntennas with broadband patternand impedance characteristics had bandwidths not muchgreater than about 2: l. In the 19 50s, a breakthrough inantenna 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 specifiedby angles instead of linear dimensions, they have ideallyan infinite bandwidth. Therefore, they are referred to asfrequency independent. These antennas are primarily usedin 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 antennasis discussed in more detail in this issue in Frequency-independent antennas and broad-band derivatives thereof,by P. E. Mayes.

It was not until almost20 years later that a fundamentalnew radiating element, that has received a lot of attentionand many applications since its inception, was introduced.

This occurred in the early 19 70s when themicrostrip orpatch antennas was reported [30], [37]-[44]. This elementis simple, lightweight, inexpensive, low profile, and con-formal to the surface. Microstrip antennas and arrays canbe flush-mounted to metallic or other existing surfaces.Operational disadvantages of microstrip antennas includelow efficiency, narrow bandwidth, and low power handlingcapabilities. These antennas are discussed in more detail inthis issue in Microstrip antennas, byD. M. Pozar. Majoradvances in millimeter wave antennas have been made inrecent years, including integrated anten nas where active andpassive circuits are combined with the radiating elementsin one compact unit (monolithic form). These antennas arediscussed in this issue in Millimeter wave antennas, byF. K. Schwering.

111. THEORYTo analyze an antenna system, the sources of excitation

are specified, and the objective is to find the electric andmagnetic fields radiated by the elements. Once this isaccomp lished, a number of parameters and figures-of-meritthat characterize the performanceof the antenna system canbe found. To design an antenna system, the characteristicsof performance are specified, and the sources to satisfythe requirements are sought. In this paper, the analysisprocedure is outlined. Synthesis procedures for patterncontrol of antenna arrays are discussed in this issue inBasic array theory, byW. H. Kummer and Array patterncontrol and synthesis, by R.C. Hansen. Theorems used inthe solution of antenna problems a re also discussed. Designand optimization procedures are presented in many of theother papers of this issue.

A . Radiation Mechanisms and Current Distribution

One of the most basic questions that may be askedconcerning antennas is how do they radiate? A quali-tative understanding of the radiation mechanism may beobtained by considering a pulse source attached to an open-ended conducting wire, which may be connected to groundthrough a discrete load at its open end. When the wire isinitially energized, the charges (free electrons) in the wireare set in motion by the electric lines of force created by thesource. When charges are accelerated in the source-end ofthe wire and decelerated (negative acceleration w ith respectto original motion) during reflection from its ends, it issuggested that radiated fields are produced at each end andalong the remaining part of the wire [45]. The accelerationof the charges is accomplished by the external source inwhich forces set the charges in motion and produce the

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associated fields radiated. The deceleration of the chargesat the end of the wire is accomplished by the internal(self) forces associated with the induced field dueto thebuildup of charge concentration at the ends of the wire.The internal forces receive energy from the charge buildupas its velocity is reducedto zero at the ends of the wire.Therefore, charge acceleration dueto an exciting electricfield and deceleration dueto impedance discontinuities or

smooth curves of the wire are mechanisms responsiblefor electromagnetic radiation. While both current density( J z ) and charge density q u e ) appear as source termsin Maxwells equations (see ( lb ) and (IC)), charge isviewed as a more fundamental quantity. Even though thisinterpretation of radiation is primarily used for transientradiation, it can be usedto explain steady-state radiation[45]. Another qualitative description of antenna radiationcan be found in [ l, ch.11.

The previous explanation demonstrates the principles ofradiation from a single wire. Letus now consider radiationand interference amplitude pattern (lobing) formation froma two-wire transmission line and antenna element, such asa linear dipole. We begin with the geometry of a losslesstwo-wire transmission line, as shown in Fig. l(a). Themovement of the charges creates a traveling wave currentof magnitude I o / 2 along each of the wires. When thecurrent arrives at the end of each of the wires, it undergoesa complete reflection (equal magnitude and 180 phasereversal). The reflected traveling wave, when combinedwith the incident traveling wave, forms in each wire apure standing wave pattern of sinusoidal form as shown inFig. l(a ). The current in each wire undergoes a 180 phasereversal between adjoining half periods. This is indicatedin Fig. l(a) by the reversal of the arrow direction.

For the two-wire balanced (symmetrical) transmissionline, the current in a half-period of one wireis of thesame magnitude but 180 out-of-phase from that in thecorresponding half-period of the other wire. If, in addition,

the spacing between the two wires is very smalls


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V . M i = - j w q v m ' (2b)

Although magnetic sources are not physical, they are oftenintroduced aselectrical equivalents to facilitate solutionsof physical boundary-value problems. In fact, for someconfigurations, 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 electriccurrent density is used to represent the antenna. However,an aperture antenna, such as a waveguideor horn, can berepresented by either an equivalent mag netic current densityor by an equivalent electric current density or both. Thiswill 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 satisfythe boundary conditions of [33]

-fi x Ed = M ,

fi x Hd = J s

ii . E d )= q s ef i ( P H d ) = q s m .

( 3 4(3b)

( 3 4( 3 4

The first two equations, (3a) and (3b), enforce the boundaryconditions on the discontinuity of the tangential compo nentsof the electric and magnetic fields while (3c) and (3d)enforce the boundary conditions on the discontinuity ofthe normal components of the electric and magnetic fluxdensities. In (3a)43d),J , / M , and y s e / q s m represent,respectively, the electridmagnetic surface current densitiesand 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 anindependent and sufficient set [33].

In addition to the boundary cond itions of (3a)-(3d), thesolutions for the fields radiated by the antenna must alsosatisfy the radiation condition which requires in an infinitehomogeneous medium that the waves travel outwardly fromthe source and vanish at infinity.

Since (la) and (lb) are first-order coupled differentialequations (each contains both electric and magnetic fields),it is often more desirable to uncouple the equations. Whenthis is done, we obtain two nonhomogeneous vector waveequations; one forE and one forE . These are given by [33]

V 2 E + p2E = -Vqve + V x M ; wpJ; (4a)11

CLV 2 H +P2H = -Vqvm - V x Ji +~ w c M ; 4b)

where p2 = w 2 p e .For a radiation problem, the first step is to represent the

antenna excitation by its source, represented in (4a) and (4b)by the current densityJi or Mi or both, having taken intoaccount the boundary conditions. This will be demonstrated

in Section IV. The next step is to solve (4a) and (4b) forE and a. his is a difficult step, and it usually involvesan integral with a complicated intergrand. This procedureis represented inFig. 2 as Path 1.

To reduce the complexity of the problem, itis a commonpractice to break the procedure into two steps. This is

S o u r ces I n t eg r a t i o n R ad ia t ed

G ) T t h 4 3

Fig. 2. Procedure to solve antenna radiation.

represented inFig. 2 by Path 2. The first step involves anintegration while the second involves a differentiation. Toaccomplish this, auxiliary vector potentials are introduced.The most commonly used potentials areA (magnetic vectorpotential) and F (electric vector potential). Although theelectric and magnetic field intensities( E and H ) representphysically measurable quantities, for most engineers thevector potentials are strictly mathematical tools. The Hertzvector potentialsII and IIh make up another possible pair.The Hertz vector potentialII is analogous toA and I I his analogous toF [l], [33]. In the solution of a problem,

only one set, A and F or II and IIh is required. In thefirst step of the Path 2 solution, the vector potentialsA andF (or 11, and IIh ) are found, once the sourcesJ ; and/orM ; are specified. This step involves an integration but onewhich is not as difficult as the integration of Path1. Thenext step of the Path 2 solutionis to find the fieldsE andE , from the vector potentialsA and F (or II and U,).This step involves differentiation. The equations that areessential for the solution of Path 2 will be outlined nextusing the v ector potentialsA and F . The derivation can befound 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 Miare specified, the vector potentialsA and F are related,respectively, to Ji and Mi by

V2A + p 2 A = - p J iV 2 F + p2F = - M i .

(5a)(5b)

If the current densities are distributed over a surfaceS ,such as that of a perfect conductor immersed in an infinitehomogeneous medium, the solutions of (5a) and (5b) canbe written, by referring to Fig. 3(a), as

where R is the distance from any point on the source tothe observation point. For the solutions of (6a) and (6b),J ;and Mi in (5a) and (5b) are replaced byJ , and M , whichhave the units of A/m and V/m, respectively. If the currentdensities are distributed over a volume, the surface integrals

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and

-Y

/ I

(b)

Fig. 3.far-field radiation.

Coordinate system arrangements for (a) near-field and (b)

of (6a) and (6b) are replaced by volume integrals. Whenthe currents are distributed over a thin wire, the surfaceintegrals of (6a) and (6b) can be approximated by lineintegrals [l], [33].

The most difficult step in the solution of Path2 isthe evaluation of the integrals of (6a) and (6b). Formost practical antenna geometries, these integrals cannotbe evaluated in closed form. Usually approximationsare made and/or numerical techniques are employed.With todays computer technology, numerical evaluationof integrals is a much simpler, more convenient,andmore efficient procedure than in the past. Computationalelectromagnetics for antenna radiation are discussed in thisissue in Low-frequency computational electromagneticsfor antenna analysis, by E. K. Miller and G . J .Burke.

D . The Electric and Magn etic Fields E and HOnce the vector potentialsA and F have been found by

(6a) and (6b), the second step in the solution of Path2 isto find the electric and magnetic fieldsE and E . This is

accomplished by using the equations [l],[33]

E = E A + E F1

w w- j w A - j - - - V ( V, A )

Th e forms of (5a)-(5b) and (7a)-(7b) are based onchoosing the relationships betweenA & 4 and F & 4 of

which each is known as theLorentz condition or gauge (&and are scalar functions of position usually referred toas scalar potentials.) The choices of (7c) and (7d) weremade to reduce (5a), (5b), (7a), and (7b) to their simplestforms; however, they are not the only possible choices.

In (7a) and (7b), the terms w ithin the first brackets are thefields due to the vector potentialA (and as a consequencedue to electric current density) while the terms within thesecond brackets are the fields due to the vector potentialF (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 vectorpotentials A or F) re set to zero. It is evident from (7a)and (7b) that superposition is used when both electric andmagnetic sources are needed to represent the source ofradiation.

E . Field Regions

The space surrounding an antenna is usually subdi-vided into three regions: thereactive near-field region,the radiating near-field (Fresnel) region, and the far-field(Fraunhofer} region. These regions areso designated toidentify the field structure in each. Although no abruptchanges in the field configurations are noted as the bound-aries are crossed, there are distinct differences among them[ l ] , [ l l ] . The boundaries separating these regions are notunique, although various criteria have been established andare commonly used to identify the regions. The followingdefinitions in quotations are from [46].

The reactive near-field region is defined as that regionof the field immediately surrounding the antenna whereinthe reactive field predominates. For most antennas, theouter boundary of this region is commonly taken to existat a distance R < 0 . 6 2 d m rom the antenna, whereX is the wavelength andD is the largest dimension of theantenna.

The radiating near-field (Fresnel} region is defined asthat region of the field of an antenna between the reactivenear-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 distanceR over which this region exists is0 . 6 2 d m 5 R < 2D2/X (providedD s large comparedto the wavelength). This criterion is based on a maximumphase error of7r/8 radians (22.5) [l] , [ l l ] . n this region

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the field pattern is, in general, a function of the radialdistance and the radial field component may be appreciable.

The fur-field (Fruunhofer) region is defined as thatregion of the field of an antenna where the angular fielddistribution is essentially independent of the distance fromthe antenna. In this region, the real partof the powerdensity is dominant. The radial distanceR over which thisregion exists isR 2 2 D 2 / 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 of7r /8 radians (22.5') [l] , [l l ] , [M I. In this region, the fieldcompon ents are essentially transverseto the radial distance,and the angular distribution is independent of the radialdistance. The evaluation of the integrals in (6a) and (6b)varies according to the region where the observations aremade. The evaluation is most difficult in the reactive near-field region. As the observation point is moved radiallyoutward, 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 isusually the regionof most applications.

Let us outline the procedure for the evaluation of integralsfor the potentialsA and F as the observation point is movedto the far-field region. This will be done for the surfaceintegrals of (6a) and (6b).As the observation point is movedradially outward, the distanceR can be approximated andleads to a simplification in the evaluation of the integral.

In the far-field (Fraunhofer) region, the radial distanceRof Fig. 3(a) can be approximated by [l], [33]

N- {r ' cos$ for phase terms ( 8 4r for amplitude terms. (8b)

Graphically, the approximation of (sa)is illustrated in Fig.3(b) where the radial vectorsR and r are parallel to eachother. Although such a relation is strictly valid only atinfinity, it becomes more accurate as the observation pointis moved outward at radial distances exceeding2D2/ A.Since the far-field region extends at radial distances ofR 2D2/A, the approximation of (sa) for the radialdistance R leads to phase errors which do not exceedT / 8radians (22.5'). It has been shown that such phase errorsdo not have a pronounced effect on the variations of thefar-field amplitude patterns (at least at parts of the patternsin which the relative field strength is not lower than about25 dB) [l], [ l l ] .

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

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 s the intrinsic impedance of the medium 7 =fi hile N O ,N +, and Le, L+ are the spherical0 and 4components ofN and L rom (sa) and (loa). In antennatheory, the spherical coordinate system is the most widelyused system.

By examining (lla)412c), it is apparent that

E8 21 vH+ ( 1 3 4

E+ N -THO. (13b)The relations of (13a) and (13b) indicate that in the far-field region the fields radiated by an antenna and observedin a small neighborhood on the surface of a large radiussphere have all the attributesof a plane wave whereby thecorresponding electric and magnetic fields are orthogonalto each other and to the radial direction.

To use the above procedure, the sources representingthe physical antenna structure must radiate into an infinitehomogeneous medium. If that is not the case, then theproblem must be reduced further (e.g., though the useofa theorem, such as the image theorem) until the sourcesradiate into an infinite homogeneous medium. This againwill be demonstrated in SectionIV-A for the analysis of

the aperture antenna.

IV. ANTENNAOURCE ODELINGThe first step in the analysis of the fields radiated by

an antenna is the specification of the sources to representthe antenna. Here we will present two examples of sourcemodeling; one for a thin wire antenna (such a dipole) andthe other for an aperture antenna (such as a waveguide).These are two distinct examp les each with a different sourcemodeling; the wire requires an electric current densitywhile the aperture is represented by an equivalent magneticcurrent density.

A . Wire Source Modeling

Let us assume that the wire antenna is a dipole, as shownin Fig. 4. If the wire has circular cross section with radiusa , the electric current density induced on the surface ofthe wire will be symmetrical about the circumference (no4 variations). If the wire is also very thinU


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X I

Fig. 4. Dipole geometryfor electromagnetic wave radiation.

the antenna is a current along the axis of the wire. Thiscurrent must vanish at the ends of the wire, and for very thinwires assumes a sinusoidal distribution. For a center-feddipole, this excitation is often represented by

6,10 sin [p t /2 ) )] 0 5 Z 5 e l ( 1 4 46,Io sin [ / 3 ( ( l / 2 )+ z)] - l / 2 5 z 5 0. (14b)

No m agnetic source representation is necessary for this typeof an antenna. Th e fields radiated by the antenna can now be

found by first determining the potentialA using (6a), wherethe surface integral is reducedto a line integral, and in turnthe fields are determined by (7a) and (7b). For observationsin the far-field region, the approximations of(Sa) and (8b)can be used. When this is done, it can be shown by alsousing (9)-(12c) that the electric and magnetic fields can bewritten as

I,= {

EBH+

To illustrate the field variations of (lsa), a three-dimensional graph of the normalized field amplitude patternfor a half-wavelength e = X/2) dipole is plotted inFig. 5 using software from [47]. A 90 angular section ofthe pattern has been omittedto illustrate the figure-eightelevation plane pattern variation. As the length of the wireincreases, the pattern becomes narrower. When the lengthexceeds one wavelength e> A), sidelobes are introducedinto 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 antennaby the Field Equivalence Principle (FEP), also referredto as Huygens Principle [l], [lS], [19], [21], [33]. Withthis method, the actual antenna is replaced by equivalentsources that, externallyto a closed surface enclosing the

actual antenna, produce the same fields as those radiatedby the actual antenna. This procedure is analogous to theThevenin Equivalent of circuit analysis which produces thesame response, to an external load, as the actual circuit.

The FE P requires that first an imaginary su rfaceis chosenwhich encloses the actual antenna. This is shown dashed

A 1 2 d i p o l e

Fig. 5.dipole.

Three-dimensional amplitude radiation patternof a X / 2

in Fig. 6(a). Once the imaginary closed surface is chosen,one of the equivalents of the FEP requires that the volumewithin the closed surface be replaced by a perfect electricconductor (PE C) and an equivalent magnetic current densitysource placed over the entire surface of the conductor, asshown in Fig. 6(b).To insure equivalence, the magnetic

current density source over the closed surface mu st be equalto

where i is a unit vector normalto the surface andE,, E ,are the electric and magnetic fields produced by the actualantenna on the chosen closed surface. Therefore,to formJ , by (16) andM , y (17), the tangential components ofthe magnetic and electric fields due to the actual antennamust be known over the chosen imaginary closed surface.Since J , of (16) is tangentto the PEC, it is shorted out andit produces no radiation. Therefore, using the equivalent ofFig. 6(b), J , does not contribute and it can be setto zero,as shown in Fig. 6(b). The critical step here is to choosethe imaginary closed surfaceso that M , an be formedeverywhere on it. Often the closed surface is chosenSOthat M , s finite over a limited area of the surface and zeroelsewhere.

The equivalent problem now isto determine the fieldsradiated by a magnetic current density nextto a PEC.The procedure outlined in Section111 cannot be usedhere, because the source does not radiate into an infinitehomogeneous medium (the PEC is present). Therefore,a further equivalent reduction of the problem must beperformed. It may seem that the equivalent problem ofFig. 6(b) is no simpler than the actual problem. However,in some cases, it can be solved exactly while in others it

suggests useful approximations. This will be demonstratedby the example that follows,Another equivalence of the FEP requires that the volume

within the closed surface be replaced by a perfect magneticconductor (PMC) with equivalent electric and magneticcurrent density sources over its entire closed surface, as

BALANIS: ANTENNA THEORY: A REVIEW 13



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(c)Fig. 6. Electric and magnetic conduc tor field equivalents for radi-ation. (a) Actual antenna system. (b) Electric conductor equivalent.(c) Magnetic conductor equivalent.

shown in Fig. 6(c). A PMC, although nonphysical, isoften used as an equivalent medium and requires thatthe tangential components of the magnetic field vanishon its surface. To insure equivalence, the electric andmagnetic current densities must be equal to (16) and (17).Therefore, to form J , by (16) and M , y (17), thetangential components of the magnetic and electric fieldsradiated by the antenna must be known. SinceM s f (17)is tangent to the PMC, it is shorted out and it produces noradiation. Therefore, using the equivalent of Fig. 6(c),M,does not contribute and it can be set to zero, as shown inFig. 6(c).

The choice of whether to use the magnetic currentdensity equivalent of (17) and Fig. 6(b) or the electriccurrent density equivalent of(16) and Fig. 6(c) as the

FEP equivalent is left to the investigator. However, ajudicious choice can significantly simplify the analysis. Thisis demonstrated by the example that follows. Once the FEPequivalent choice is made, the next step is to findA orP sing (6a) or (6b) and then to find the fields using (7a)and (7b). For far-field observations, the approximations of(9 H1 3b ) can be used in l ieuof (6a)-(7b).

To demonstrate the procedure, let us assume that an

open-ended rectangular waveguide aperture mounted on aninfinite planar PE C radiates on a semi-infinite homogeneo usmedium, as shown in Fig. 7(a). Let us assume that thefields in the waveguide aperture are thoseof the dominantTElo mode. Hence, the tangential electric field over thez-y plane is

iLYEo O S ( ~ X ) - ~ / 2 5 ~ / 2 (18a)E , = { -b / 2 5 y 5 b / 2

0 elsewhere over the PEC. (18b)

By adopting the FEP equivalentof Fig. 6(b), an imaginaryclosed surface is chosen to coincide with the infinite PEC(z-y plane) and covers also the waveguide aperture. Theimaginary closed surfaceis chosen to coincide with thez-y

plane because the tangential compo nents of the electricfield, and thus the equivalent magnetic current density, arenonzero only in the aperture. Using (17) and (18a)-(18b),we can write

M , -6 x E ,~L,EOCOS( ( T / U) ~ ) ~ / 2 2 5 ~ / 2

- b / 2 5 y 5 b / 2 (19a){ o elsewhere over the PEC.

(19b)

The equivalent problem to solve now is that shown in Fig.6(b); i.e., the magnetic current density of (19a) and (19b)next to the PEC . Using im age theory, the equivalentof Fig.6(b) is replaced by a magnetic current density of twice thestrength of (19a) radiating into an infinite homogeneousmedium, as shown in Fig. 7(b). Now using (9)-(12c),the far-field electric and magnetic fields radiated by thewaveguide can be written as[ l ]

E, N H , : 0 . ,

Gob)7r c o s ( X ) sin(Y)

Eo N - - C s i n 42 X ) 2 ) 2 YT c o s ( X ) sin(Y)

E+ N - - C C O S O C O S ~ (20c)2 ( X ) * T / 2 ) 2 Y

EeH+ N +-rl

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Antenna

Generator

Receiver

IvgJ

Lr

b

( c )

Fig. 9. Thevenin equivalentsfor transmitting and receiving an-tennas. (a) Antenna system.(b) Thevenin equivalent: transmitting.( c ) Thevenin equivalent: receiving.

where E and H are the fields radiated by the antenna(* indicates complex conjugate). The real partis usuallyreferred to as radiation density.

Radiation intensity U is defined as the power radiatedfrom an antenna per unit solid angle (steredian). Theradiation intensity is usually determined in the far field,and it is related to the real part of the power density by

U = r2sr (23)

where T is the spherical radial distance.Beamwidth is defined as the angular separation between

two directions in which the radiation intensity is identical,with no other intermediate pointsof the same value. Whenthe intensity is one-half of the maximum, it is referred toas halfpower beamwidth.

An isotropic radiator is defined as a hypothetical, loss-less antenna having equal radiation intensity in all direc-tions. Although such an antenna is an idealization, it isoften used a s a convenient reference to express the directiveproperties of actual antennas. Its radiation densitySro andintensity U0 are defined, respectively, as

PrU0 =?T

where P, represents the power radiated by the antenna.Directivity is one of the most important figures-of-merit

that describes the performanceof an antenna. It is definedas the ratio of the radiation intensity in a given direction

from the antenna to the radiation intensity averaged overall directions. Using (24b), it can be written as

where U ( B , 4 )is the radiation intensity in the directionB d and Pr is the radiated power. If the direction is notspecified, it implies the directionof maximum radiation

intensity (maximum directivity) expressed as

The directivity is an indicator of the relative directionalproperties of the antenna.As defined in (25) and (25a),the directional propertiesof the antenna in question arecompared to those of an isotropic radiator. Sometimes thiscomparison is made relative to another standard radiatorwhose directivity (relative to an isotropic radiator) is known(like a X/2 dipole or a standard gain horn). This allows usto relate the directivity of the element in question to thatof an isotropic radiator by simply adding (if expressed indecibels) the relative directivities of one element to another.This procedure is analogous to that used to determine the

overall gain of cascaded amplifiers.To demonstrate the significanceof directivity, let us

examine the directivity of a half-wavelength dipolee =X/2) approximated as

where 19 is measured from the axis along the lengthof the dipole. The values represented by (26) andthose of an isotropic source ( D = 1) are plottedthree-dimensionally inFig. 10. At each point, only thelargest value of the two directivities is plotted.It isapparent that when s i n - ( l / m ) = 57.44 5 B _122.56, the dipole radiator has greater directivity (greaterintensity concentration) in those directions than that of anisotropic source. Outside this range of angles, the isotropicradiator has higher directivity (more intense radiation). Themaximum directivity of the dipole (relativeto the isotropicradiator) occ urs whenB = 7r/2, and it is 1.67 (or 2.23 dB ).It simply means that at = 90 the dipole radiation is 2.23dB more intense than that of the isotropic radiator (withthe same radiated power).

Gain is probably the most important figure-of-merit of anantenna. It is defined as the ratio of the radiation intensityin a given direction, to the radiation intensity that would beobtained if the power accepted by the antenna were radiatedisotropically. Using (24b), it can be written as

(27)u(e,4> 4Tu(B,4)

U , Pa

where Pa is the accepted (input) power to the antenna. Ifthe direction is not specified, it implies the direction ofmaximum radiation (maximum gain). In simplest terms,the main difference between the definitions of directivity,as given by (25), and gain, as given by (27), is that thedirectivity is based on the radiated power while the gain

16 PROCEEDINGSOF THE IEEE, VOL. 80, NO. 1, JANUARY 1992



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\

Fig. 10. Three-dimensional directivity pattern for a X / 2 dipole and an isotropic source.

is based on the accepted (input) power. Since all of theaccepted (input) power is not radiated (because of losses),the two are related by

where e, is the radiation efficiency of the antenna as definedby (21). By using (25)-(28) the gain can be expressed a s

For a lossless antenna, its gain is equal to its directivity.

complex vector quantity represented by

Vector effective length height) h, for an antenna is a

M O4 = he(0 , 4) + % A J ( ~ ,) (30)and it is related to the far zone electric fieldE radiated bythe antenna, with currentI i n in its terminals, by [48]

E = EO + ,E+ = -jq-hee-j .I i n (30a)The effective height represents the antenna in the transmit-ting and receiving modes, and it is particularly useful inrelating the open-circuit voltageV,, of receiving antennas.This relation can be expressed as

4 ~ r

V,, = he (31)

where

V,,incident electric field,

he vector effective length.

open-circuit voltage at antenna terminals,

In. (31) V,, can be thought as the voltage induced in a linearatlienna of lengthe = he when he and are linearlypolarized [21].

For the dipole antenna of Fig. 4 with length e

balanis_antenna theory review

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