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JANUARY2012 VOLUME60 NUMBER1 IETPAK (ISSN0018-926X) PAPERS Antennas MaximumGainofaLossyAntenna............................................................ A.ArbabiandS.Safavi-Naeini 2 ExperimentalValidationofPerformanceLimitsandDesignGuidelinesforSmallAntennas ................................ ........................... D.F.Sievenpiper,D.C.Dawson,M.M.Jacob,T.Kanar,S.Kim,J.Long,andR.G.Quarfoth 8 SubstrateIntegratedWaveguide(SIW)Leaky-WaveAntennaWithTransverseSlots ........................................ ...............................................................................................J.Liu,D.R.Jackson,andY.Long 20 SubwavelengthSubstrate-IntegratedFabry-PérotCavityAntennasUsingArtificialMagneticConductor.................. ....................................................................... Y.Sun,Z.N.Chen,Y.Zhang,H.Chen,andT.S.P.See 30 AReconfigurableWidebandandMultibandAntennaUsingDual-PatchElementsforCompactWirelessDevices........ ................ H.F.Abutarboush,R.Nilavalan,S.W.Cheung,K.M.Nasr,T.Peter,D.Budimir,andH.Al-Raweshidy 36 Frequency-ReconfigurableMonopoleAntennas............................................ A.TariqandH.Ghafouri-Shiraz 44 LowProfileFullyPlanarFoldedDipoleAntennaonaHighImpedanceSurface.............................................. ....................................................................A.Vallecchi,J.R.DeLuis,F.Capolino,andF.DeFlaviis 51 CrumplingofPIFATextileAntenna ...................................................................... Q.BaiandR.Langley 63 HigherOrderModeExcitationforHigh-GainBroadsideRadiationFromCylindricalDielectricResonatorAntennas ... ......................................................................... D.Guha,A.Banerjee,C.Kumar,andY.M.M.Antar 71 OntheCharacteristicsoftheHighlyDirectiveResonantCavityAntennaHavingMetalStripGratingSuperstrate........ .......................................................................................................A.ForoozeshandL.Shafai 78 Nature of Cross-Polarized Radiations from Probe-Fed Circular Microstrip Antennas and Their Suppression Using DifferentGeometriesofDefectedGroundStructure(DGS)........................................C.KumarandD.Guha 92 Single, Dual and Tri-Band-Notched Ultrawideband (UWB) Antennas Using Capacitively Loaded Loop (CLL) Resonators ............................................................................ C.-C.Lin,P.Jin,andR.W.Ziolkowski 102 LeakyWaveEnhancedFeedsforMultibeamReflectorstobeUsedforTelecomSatelliteBasedLinks .................... .............................................................................. A.Neto,M.Ettorre,G.Gerini,andP.DeMaagt 110 Arrays 94GHzSubstrateIntegratedMonopulseAntennaArray .................................. Y.J.Cheng,W.Hong,andK.Wu 121 ThePlanarUltrawidebandModularAntenna(PUMA)Array............................ S.S.HollandandM.N.Vouvakis 130 ATwo-ChannelTimeModulatedLinearArrayWithAdaptiveBeamforming ..................... Y.TongandA.Tennant 141 AperiodicArrayLayoutOptimizationbytheConstraintRelaxationApproach ............................................... .................................. T.N.Kaifas,D.G.Babas,G.S.Miaris,K.Siakavara,E.E.Vafiadis,andJ.N.Sahalos 148 TimeReversalBasedBroadbandSynthesisMethodforArbitrarilyStructuredBeam-SteeringArrays ..................... .......................................................................................D.Zhao,Y.Jin,B.-Z.Wang,andR.Zang 164 AGeneralizedHybridApproachfortheSynthesisofUniformAmplitudePencilBeamRing-Arrays ...................... .................................................................................................... O.M.BucciandD.Pinchera 174 PolarimetryWithPhasedArrayAntennas:TheoreticalFrameworkandDefinitions .......................................... .............................................................K.F.Warnick,M.V.Ivashina,S.J.Wijnholds,andR.Maaskant 184 AnAmplifyingReconfigurableReflectarrayAntenna ............................................ K.K.KishorandS.V.Hum 197 DesignofRetrodirectiveAntennaArraysforShort-RangeWirelessPowerTransmission .......... Y.LiandV.Jandhyala 206 (ContentsContinuedonp.1) (ContentsContinuedfromFrontCover) AnisotropicImpedanceSurfacesforLineartoCircularPolarizationConversion ............................................. ..................................................E.Doumanis,G.Goussetis,J.L.Gómez-Tornero,R.Cahill,andV.Fusco 212 Imaging andPropagation Transmitting-ModeTimeReversalImagingUsingMUSICAlgorithmforSurveillanceinWirelessSensorNetwork..... ........................................................................................... X.-F.Liu,B.-Z.Wang,andJ.L.-W.Li 220 UWBMicrowaveImagingofObjectsWithCanonicalShape................................................................... .................................................................... N.Ghavami,G.Tiberi,D.J.Edwards,andA.Monorchio 231 ExperimentalCharacterizationofanUWBPropagationChannelinUndergroundMines.................................... ............................................................................................ Y.Rissafi,L.Talbi,andM.Ghaddar 240 StatisticalPredictionofSiteDiversityGainonEarth-SpacePathsBasedonRadarMeasurementsintheUK............... .....................................................................................................C.NagarajaandI.E.Otung 247 WidebandCharacterizationofBackscatterChannels:DerivationsandTheoreticalBackground ............................ ......................................................................................D.Arnitz,U.Muehlmann,andK.Witrisal 257 CalibrationofElectricFieldSensorsOnboardtheResonanceSatellite ........................................................ ............................................ M.Sampl,W.Macher,C.Gruber,T.Oswald,H.O.Rucker,andM.Mogilevsky 267 NumericalandAnalyticalTechniques ASecond-OrderAsymptoticApproximationfortheSommerfeldHalf-SpaceProblem ..............................W.Lihh 274 DirectRationalFunctionFittingMethodforAccurateEvaluationofSommerfeldIntegralsinStratifiedMedia........... ........................................................................................................................T.N.Kaifas 282 IntegralEquationModelingofDoublyPeriodicStructuresWithanEfficientPMCHWTFormulation ..................... ...........................................................................................S.Nosal,P.Soudais,andJ.-J.Greffet 292 Reduced-Order Models of Finite Element Approximations of Electromagnetic Devices Exhibiting Statistical Variability .................................................................. P.Sumant,H.Wu,A.Cangellaris,andN.Aluru 301 SphericalADIFDTDMethodWithApplicationtoPropagationintheEarthIonosphereCavity ............................ ......................................................................................................D.L.PaulandC.J.Railton 310 AnalysisofDirectionalLoggingToolsinAnisotropicandMultieccentricCylindrically-LayeredEarthFormations ...... ...................................................................................... G.-S.Liu,F.L.Teixeira,andG.-J.Zhang 318 Analytic Transient Analysis of Radiation From Ellipsoidal Reflector Antennas for Impulse-Radiating Antennas Applications ............................................................ S.-C.Tuan,H.-T.Chou,K.-Y.Lu,andH.-H.Chou 328 An Analytic Solution of Transient Scattering From Perfectly Conducting Ellipsoidal Surfaces Illuminated by an ElectromagneticPlaneWave .......................................... H.-T.Chou,S.-C.Tuan,K.-Y.Lu,andH.-H.Chou 340 Green’sFunctionExtractionforInterfacesWithImpedanceBoundaryConditions............. E.SlobandK.Wapenaar 351 ElectromagneticFieldofaHorizontalInfinitelyLongWireOvertheDielectric-CoatedEarth .............................. ................................................................................................... Y.J.Zhi,K.Li,andY.T.Fang 360 DecomposableMediumConditionsinFour-DimensionalRepresentation ...... I.V.Lindell,L.Bergamin,andA.Favaro 367 COMMUNICATIONS NovelUHFRFIDTagAntennaforMetallicFoilPackages ..................................J.Ryoo,J.Choo,andH.Choo 377 DesignofaBroadbandAll-TextileSlottedPIFA ............P.J.Soh,G.A.E.Vandenbosch,S.L.Ooi,andN.H.M.Rais 379 AnUltrawideband(UWB)SlotlineAntennaUnderMultiple-ModeResonance .............................................. ......................................................................................... X.D.Huang,C.H.Cheng,andL.Zhu 385 ACompactHepta-BandLoop-InvertedFReconfigurableAntennaforMobilePhone........................................ ........................................................................ Y.Li,Z.Zhang,J.Zheng,Z.Feng,andM.F.Iskander 389 Hybrid Monopole-DRAs Using Hemispherical/Conical-Shaped Dielectric Ring Resonators: Improved Ultrawideband Designs .............................................................................. D.Guha,B.Gupta,andY.M.M.Antar 393 AHalfMaxwellFish-EyeLensAntennaBasedonGradient-IndexMetamaterials ........................................... .......................................................................................Z.L.Mei,J.Bai,T.M.Niu,andT.J.Cui 398 Dual-PolarizedPlanarFeedforLow-ProfileHemisphericalLuneburgLensAntennas ........ A.R.WeilyandN.Nikolic 402 TMScatteringbyPerfectlyConductingPolygonalCross-SectionCylinders:ANewSurfaceCurrentDensityExpansion RetaininguptotheSecond-OrderEdgeBehavior............................G.Coluccini,M.Lucido,andG.Panariello 407 AModificationoftheKummer’sMethodforEfficientComputationofthe2-Dand3-DGreen’sFunctionsfor1-DPeriodic Structures .........................................................................................................S.P.Skobelev 412 ASpatialBeamSplitterConsistingofaNear-ZeroRefractiveIndexMedium ................................................ ....................................................................................... R.-B.Hwang,N.-C.Hsu,andC.-Y.Chin 417 ExtendedMode-BasedBandwidthAnalysisforAsymmetricNear-FieldCommunicationSystems ....Y.TakandS.Nam 421 ERRATA Erratato“Three-DimensionalNear-FieldMicrowaveHolographyUsingReflectedandTransmittedSignals” ............ .................................................................R.K.Amineh,M.Ravan,A.Khalatpour,andN.K.Nikolova 425 ListofReviewersfor2011 ........................................................................................................ 426 IEEE ANTENNAS AND PROPAGATION SOCIETY AllmembersoftheIEEEareeligibleformembershipintheAntennasandPropagationSocietyandwillreceiveon-lineaccesstothisTRANSACTIONSthroughIEEEXploreuponpaymentoftheannualSociety membershipfeeof$24.00.PrintsubscriptionstothisTRANSACTIONSareavailabletoSocietymembersforanadditionalfeeof$36.00.Forinformationonjoining,writetotheIEEEattheaddressbelow. MembercopiesofTransactions/Journalsareforpersonaluseonly. ADMINISTRATIVECOMMITTEE M.SALAZARPALMA, President S.R.BEST, PresidentElect J.S.TYO, Secretary-Treasurer 2012 2013 2014 Y.M.M.ANTAR M.ANDO* R.D.NEVELS* J.T.BERNHARD* D.B.DAVIDSON S.MACI M.EL-SHENAWEE M.MARTINEZ-VÁZQUEZ S.K.RAO D.F.SIEVENPIPER M.W.SHIELDS HonoraryLifeMembers:R.C.HANSEN,W.R.STONE *PastPresident CommitteeChairsandRepresentatives AntennaMeasurements(AMTA):S.SCHNEIDER EuRAAPRepresentative:W.ROSSSTONE Publications:T.S.BIRD Antennas&WirelessPropagationLettersEditor-in-Chief: FellowsNominationsCommittee:L.C.KEMPEL Region8Representative: B.ARBESSER-RASTBURG G.LAZZI Finance:J.S.TYO Region9Representative:S.BARBIN AAAwpPpa-Slri/deUsd:RCSCoI.mJGopi.unCttaHMtiRoeIneSatTilOnEgDMsOCUSLooOmcUimetiytte(Ae:CEJ.SL):.YAO.UFN.PGETERSON GHIEoiEsltdEorRPiaernpe:rsessKLe.inaDtias.toiSvnTe::EPRH.AJN.MAILLOUX RSeengsioonrC10ouRnecpilr:esAen.tIa.tZivAeG:HJO.ULL-W,T..LSI.BIRD,M.W.SHIELDS AwardsCoordinator: C.A.BALANIS IEEEPublicRelationsRepresentative:W.R.STONE StandardsCommittee—Antennas: 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IEEETRANSACTIONSONANTENNASANDPROPAGATION Istheleadinginternationalengineeringjournalonthegeneraltopicsofelectromagnetics,antennasandwavepropagation.Thejournalisdevotedtoantennas,includinganalysis,design,development,measure- ment,andtesting;radiation,propagation,andtheinteractionofelectromagneticwaveswithdiscreteandcontinuousmedia;andapplicationsandsystemspertinenttoantennas,propagation,andsensing,such asappliedoptics,millimeter-andsub-millimeter-wavetechniques,antennasignalprocessingandcontrol,radioastronomy,andpropagationandradiationaspectsofterrestrialandspace-basedcommunication, includingwireless,mobile,satellite,andtelecommunications.AuthorcontributionsofrelevantfulllengthpapersandshorterCommunicationsarewelcomed. 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DigitalObjectIdentifier10.1109/TAP.2011.2181917 2 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.60,NO.1,JANUARY2012 Maximum Gain of a Lossy Antenna Amir Arbabi,StudentMember,IEEE, and SafieddinSafavi-Naeini,Member,IEEE Abstract—Anupperboundontheachievablegainofalossyan- or,ashasrecentlybeenreported,byexploitingengineeredma- tennaisderived.Thislimitdependsontheantennasizeandanewly terialsasapartofthematchingcircuit[8].Furthermore,inmil- defined“lossmeritfactor,”whichisshowntobeameasureofthe limeterwave,Tetrahertz,andopticalfrequencyantennas,radi- antenna material loss. The derived limit extends the well-known ationefficiencyreductionduetotheantennalossesismorere- upperlimitontheratioofantennadirectivitytoitsqualityfactor tothelossyantennas.Optimalantennacurrentdistributionforthe strictivethanthebandwidth.Basedonthesefacts,thestudyof maximumgainisfound,andtheradiationpatternandantennaef- the material loss and optimizing the antenna current distribu- ficiencyarealsopresented. tion in a way to minimize its material loss and maximize the gainallowustooptimizetheantennastructureundermorere- Index Terms—Antenna maximum gain, lossy antenna, quality factor,smallantenna. alisticconditions.Toourknowledge,thisinvestigationhasnot been performed yet. Although [9] considered the effect of en- ergylossduetosurroundingmediumofanantenna,theantenna I. INTRODUCTION itselfwasstillassumedtobelosslessandenclosedinalossless sphere.Therefore,theantennamateriallossthatresultedfrom ahighreactivenearfieldwasnotconsidered. A N ANTENNA is an essential component of a wireless In this paper, the effect of the antenna material loss on the system.Small-sizeantennaswithoptimalgainandband- maximumachievablegainisanalyzed.Tothisend,theantenna widthareonhighdemandforcompactpower-efficientportable isassumedtobeenclosedinavirtualsphereanditsfieldsout- radios.However,thereisawell-knowntradeoffbetweenthean- side the sphere are expanded in terms of orthogonal vectorial tenna gain, its size, and its achievable bandwidth. It is shown spherical modes whose coefficients are related to the current thatarbitraryhighgaincanbeachievedfromanarbitrarysmall distributioninsidethesphere.Itisshownthateachofthevec- perfectelectricconductor(PEC)sphereifthecurrentdistribu- torial spherical modes has its own radiation efficiency which tiononthesphereisselectedinaproperway[1].Chuderived dependsontheantennasizeandanewlydefinedmaterialloss an upper limit on the antenna directivity and ratio of antenna meritfactor.Itisobservedthatthemodeefficiencies,exceptfor directivitytoqualityfactorforomnidirectionalantennaswitha afewlowestordermodes,areverysmall.Thisresultisaman- finitenumberofvectorialsphericalmodes[2].Indeed,regard- ifestation of the finite rank of the free space Green’s function lessofsize,thereisnolimitonthegain ofa losslessantenna, whosemoregeneralformisdiscussedin[10]. but for achieving high gain, the current amplitude on the an- tenna surface should be large. Large current amplitude on the II. PROBLEMDESCRIPTION antenna generates large reactive stored energy around the an- tenna.Thisreactiveenergyincreasestheantennaqualityfactor. Itis assumed thatthe antenna radiatesinfreespace and the FollowingChu,severalresearcherstriedtoquantifythetradeoff smallestspherethatenclosestheantennahasaradiusof .The between the antenna directivity and the large reactive energy coordinatesystemischoseninawaythattheoriginisthecenter storedaroundtheantenna[1],[3]–[6].In[5]and[6],itisshown ofthesphereandits axisisorientedintheantenna’smaximum thatthereisalimitontheratioofdirectivitytoqualityfactorof radiationdirection.Itisalsoassumedthattheantennaisfedbya a general lossless antenna. In all of these studies, the antenna lumpedsourceand,therefore,thedirectradiationoftheapplied isassumedtobelosslessanditsqualityfactorisregardedasa source is ignored. The electric and magnetic field satisfy the measureoftheantennainputimpedancebandwidth.Although Maxwell’sequationsintheentirespace alargequalityfactorrepresentshighreactiveenergyandsmall bandwidth,therelationbetweentheconventionalqualityfactor (1a) andbandwidthbecomeslessusefulforlowandmoderatevalues (1b) ofqualityfactor[7].Inaddition,low- antennascanbematched toadesiredimpedanceoverabandwidthwiderthanwhatisde- where istheequivalentcurrentdensity terminedbytheirqualityfactorsbyusinglumped,quasilumped (2) ManuscriptreceivedJanuary30,2011;revisedMay01,2011;acceptedJuly 02,2011.DateofpublicationSeptember15,2011;dateofcurrentversionJan- and ispositiondependent.Itshould benoticed uary05,2012. Thisworkwassupportedby theNationalScienceandEngi- neeringCouncil(NSERC)ofCanadaandResearchInMotion(RIM). thattheconductivityofthematerialcanbeconsideredasapart The authors are with the Department of Electrical and Computer Engi- of .Withthisassumption,thecurrentisnonzeroonlyoverthe neering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: regionoccupiedbytheantenna,whichisenclosedbythevirtual [email protected];[email protected]). DigitalObjectIdentifier10.1109/TAP.2011.2167934 sphere.Outsideofthissphere,fieldsaresolutiontosourcefree 0018-926X/$26.00©2011IEEE ARBABIANDSAFAVI-NAEINI:MAXIMUMGAINOFALOSSYANTENNA 3 Maxwell’sequationsandcanbeexpandedintermsofvectorial notedthatalthough and constituteasetofmutually sphericalharmonics[11] orthogonalvectorialfunctions,theydonotformacompleteset asshownin(8a)and(8b)atthebottomofthepage. Thetotalradiated power bythe antennacanbe foundusing (3a)and(3b)withintegrationofthePoynting’svector (9) (3a) because of the orthogonality of the modes, the total radiated poweristhe superpositionofthepowerradiatedbyeachindi- vidualmode.Maximumradiationintensity,aswasassumedat (3b) thebeginning,isinthe 0directionandcanbefoundfrom (3a)inthefar-fieldregionas where is the spherical Hankel function of the second order, arethesphericalharmonics (4) and isadifferentialoperatordefinedas (10) (5) Nowthemateriallosswillbeconsidered.Thedissipativeloss oftheantennaresultsfromtheantennamateriallossandisgiven The and in (3a) and (3b) are coefficients of the by and modes,respectively.Thesecoefficientscanbefoundby projectingthevolumecurrentdensityonanorthogonalvectorial set.If representsthespherevolume,then(seeAppendixA) (11) (6a) Itwillbeshownthatforagivenantennasize,theantennagain (6b) onlydependsonadimensionlessparameterdefinedas where and are given in (41a) and (41b) in Ap- (12) pendixA.Itcanbeverifiedthat and aremutually orthogonal which is referred to as “loss merit factor.” Using this new pa- (7a) rameter,materiallossbecomes (7b) (13) (7c) is an “effective loss merit factor” for the antenna and is defined according to (13). This parameter is equal to the loss where isKronecker’sdeltafunctionwhichisnonzeroonly merit factor of the antenna material if the antenna is made of when and, in this case, it is equal to , and onlyonetypeofmaterial.Foraninhomogeneousantenna,the arenormsof and ,respectively.Thesenorms effectivelossmeritfactorissmallerthanthelargestmeritfactor aregivenin(8a)and(8b)andareindependentof .Itshouldbe ofitsdifferentconstituents. (8a) (8b) 4 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.60,NO.1,JANUARY2012 Thevolumecurrentdensity canbewrittenasthesummation andbydefining of two parts: Its projection on the space spanned by and andapartwhichisorthogonaltothisspace,thatis (20a) (14) (20b) where isorthogonalto and Equation(19)canberewrittenas (15) and and areprojectionsof onnormalized and andaregivenby (21) (16a) it can be seen from (21) that and are efficiencies of eachindividual and modes.Finally,theantenna (16b) gaincanbefoundtobegivenby(22),shownatthebottomof the page. isthenonradiatingpartoftheequivalentcurrentdensity.The III. OPTIMIZATIONOFTHEGAIN fields generated by vanish identically at any point outside thespherevolume .Thenonradiatingcurrentdensity nei- In(22),thetermsinthenumeratorandthedenominatorareall therexcites nor modesoutsidethesphere.Furtherdis- positiveand isonlypresentinthedenominator.Tomaximize cussiononnonradiatingcurrentdensitiescanbefoundin[12]. thefraction,thedenominatorcanbeminimizedindependently From(14)and(15),thenormof canbewrittenas by setting the nonradiating current density to zero. As was mentioned in Section II, does not radiate and according to (18), it adds to the dissipated power; therefore, the optimum antennashouldhave .Similarreasoningleadsto (17) (23) pluggingtheleft-handsideof(17)in(13)gives Bydefining (24a) (24b) (24c) (18) (24d) Theantennatotalinputpoweristhesummationoftheradi- atedanddissipatedpower Equation(22)canberewrittenas (25) Becauseofthesymmetryof(25)withrespectto and and (19) with respect to and , equating partial derivatives of the (22) ARBABIANDSAFAVI-NAEINI:MAXIMUMGAINOFALOSSYANTENNA 5 tozerowillresultinsimilarequationsandidenticaloptimal valuesforthesecoefficients (26a) (26b) Furthermore,thenumeratorismaximizedbyrequiring (27) using(26)and(27),(25)issimplifiedas (28) Fig.1. Modeefficienciesfor(cid:0)(cid:2)((cid:0) )and(cid:0)(cid:3)((cid:0) )modesasafunction of(cid:2)for(cid:3)(cid:4) (cid:4)10and(cid:5) (cid:4)(cid:5)(cid:6) . Finally, and give (29) and (30a) (30b) IV. RESULTSANDDISCUSSION Plugginginnormsof and from(8)into(20),the modeefficienciesarefoundas Fig.2. Maximumantennagainasafunctionof(cid:3)(cid:4)fordifferentvaluesof(cid:5) (31a) for(cid:3)(cid:4)between0.05and0.5.ThedashedlinecurveshowsHarrington’snormal gain(cid:6) (cid:4)(cid:7)(cid:3)(cid:4)(cid:8) (cid:9)(cid:10)(cid:3)(cid:4). (31b) with are propagating modes and other modes are in cutoff. Our results show that the efficiency of each individual Theseequationsshowthatthemodeefficienciesand,therefore, sphericalmodeisindependentof .Thestep-likedependency themaximumgainareonlyafunctionof andtheantenna of efficiencies on can be regarded as a cutoff, but the cutoff size. For copper, silver, and gold, the loss merit factor ( ) at edgeisnotonlyafunctionoftheantennasizebutalsoantenna microwavefrequenciesisontheorderof andforadielectric material(“effectivelossmeritfactor”).Forexample,inFig.1, material with a permittivity of 10 and a loss tangent of modeswith canbeassumedtobepropagatingmodes, isabout5 .Fig.1showsthemodeefficiencies andothermodeshavesmallefficiencyandcanbeignored. asafunctionof for 10and .Ascanbeseen Figs.2and 3showthe maximum antennagain for different fromthisfigure, and showstep-likebehaviorandare values of . The dashed line curve shows Harrington’s almostequaltoeachotherforthesamevalueof .Thisstep-like “normal”gaindefinedas .Itisclear behavior is reminiscent of the assumptions made by Chu and thatforalargeandmoderatevalueof ,themaximumgain Harringtonforomn-directionalanddirectionalantennas[1],[2]. is larger than what Harrington found using only modes with For a directional antenna, based on the mode impedance for .Thislargergaincomeswithhigherqualityfactorand eachofsphericalmodes,Harringtonassumedthatonlymodes smallerinputimpedancebandwidth. 6 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.60,NO.1,JANUARY2012 Fig.3. Maximumantennagainasafunctionof(cid:0)(cid:2)fordifferentvaluesof(cid:3) Fig.5. Efficiencyoftheantennawithmaximumgainasafunctionof(cid:0)(cid:2)for for(cid:0)(cid:2)between0.5and5.ThedashedlinecurveshowsHarrington’snormal differentvaluesoftheantennaeffectivelossmeritfactor((cid:3) ). gain(cid:4) (cid:0)(cid:2)(cid:0)(cid:2)(cid:3) (cid:4)(cid:5)(cid:0)(cid:2). V. CONCLUSION In this paper, a new fundamental limit on antenna gain was introduced. It was shown that although a lossless antenna re- gardless of its size can have arbitrarily high gain, the antenna dissipativelosslimitsthegainforarealantenna.Themaximum gainisafunctionoftheantennasizeanditsmateriallossmerit factor. General plots for maximum gain for different antenna sizeandlossmeritfactorwereprovided. APPENDIX SPHERICALHARMONICSCOEFFICIENTSOFTHEFIELDOFA VOLUMECURRENTDENSITY Coefficientsoffieldexpansionintermsofvectorialspherical harmonicscanbefoundfromvolumecurrentandchargedensity as [11] Fig.4. Radiationpatternoftheantennawithmaximumgainwith(cid:5)(cid:2) (cid:0) (cid:5) and(cid:3) (cid:0) (cid:6)(cid:7) . (33a) Using(14),(16),and(30),theequivalentcurrentdistribution (33b) insidethesphereforanantennawithmaximumgainisgivenby where is the speed of light in vacuum and is the volume chargedensityrelatedtocurrentdensitybythecontinuityequa- tion (32) (34) Fig.4showstheradiationpatternofanantennawiththemax- after substituting from (34) into (33a), the first term in the imumgainfor and .Since and right-handsidecanbewrittenas are almost the same, the radiation patterns in different cutting planes containing the axis are similar. As can be seen from thisfigure,thesidelobelevelisabout20dB.Fig.5showsthe efficiency of the antenna with maximum gain as a function of itssizefordifferentvaluesoftheeffectivelossmeritfactor. ARBABIANDSAFAVI-NAEINI:MAXIMUMGAINOFALOSSYANTENNA 7 (42b) (35) REFERENCES thesurfaceintegraliszerobecause isassumedtobeenclosed [1] R.F.Harrington,TimeHarmonicElectromagneticFields. NewYork: inthevolume .Theright-handsideof(33b)canbesimplified McGraw-Hill,1961. as [2] L.J.Chu,“Physicallimitationsofomni-directionalantennas,”J.Appl. Phys.,vol.19,pp.1163–1175,Dec.1948. [3] H.A.Wheeler,“Smallantennas,”IEEETrans.AntennasPropag.,vol. AP-23,no.4,pp.462–469,Jul.1975. [4] R.C.Hansen,“Fundamentallimitationsinantennas,”Proc.IEEE,vol. 69,no.2,pp.170–182,Feb.1981. [5] R.L.Fante,“Maximumpossiblegainforanarbitraryidealantenna with specified quality factor,” IEEE Trans. Antennas Propag., vol. AP-40,no.12,pp.1586–1588,Dec.1992. [6] W. Geyi, “Physical limitations of antenna,” IEEE Trans. Antennas Propag.,vol.51,no.8,pp.2116–2123,Aug.2003. (36) [7] S.R.Best,“TheFosterreactancetheoremandqualityfactorforan- tennas,”IEEEAntennasWirelessPropag.Lett.,vol.3,pp.306–309, 2004. and with similar reasoning, the surface integral has vanished. [8] R.W.ZiolkowskiandA.Erentok,“Metamaterial-basedefficientelec- Using(35)and(36),(33a)and(33b)canberewrittenas tricallysmallantennas,”IEEETrans.AntennasPropag.,vol.54,no.7, pp.2113–2130,Jul.2006. [9] A. Karlsson, “Physical limitations of antennas in a lossy medium,” IEEETrans.AntennasPropag.,vol.52,no.8,pp.2027–2033,Aug. 2004. [10] D.A.B.Miller,“Fundamentallimitforopticalcomponenets,”J.Opt. Soc.Amer.B,vol.24,no.10,Oct.2007. [11] J.D.Jackson,ClassicalElectrodynamics,3rded. NewYork:Wiley, 1999. (37a) [12] A.J.DevaneyandE.Wolf,“Radiatingandnonradiatingclassicalcur- rentdistributionsandthefieldstheygenerate,”Phys.Rev.D,vol.8,no. 4,pp.1044–1047,1973. (37b) AmirArbabi(S’06)wasborninMalayer,Iran,in 1984.HereceivedtheB.Sc.degreeinelectricalengi- Usingthefollowingidentities: neeringfromtheUniversityofTehran,Tehran,Iran, in2006,theM.Sc.degreeinelectricalengineering (38) from the University of Waterloo, Waterloo, ON, Canada,in2009,andiscurrentlypursuingthePh.D. degreeinelectricalengineeringattheUniversityof and Illinois,Urbana-Champaign. HehasbeenaResearchAssistantwiththePho- (39) tonicSystemsLaboratory,UniversityofIllinois,Ur- bana-Champaign,since2009.Hisresearchinterests includeplasmonicandnanophotonicdevices,fastnumericalmethodsforsimu- Equations(37a)and(37b)arefurthersimplifiedas lationanddesignofnovelphotonicdevices,andfundamentalfeaturesandlimits inopticsandelectromagnetics. Mr.ArbabiwasarecipientoftheOntarioGraduateScholarshipandthePres- ident’sGraduateScholarshipwhileattheUniversityofWaterloo.Heisastu- dentmemberoftheOpticalSocietyofAmericaandareviewerfortheIEEE PHOTONICSJOURNAL. (40a) (40b) SafieddinSafavi-Naeini(M’00)wasborninGach- saran,Iran,in1951.HereceivedtheB.Sc.degreein electricalengineeringfromtheUniversityofTehran, andbydefining Tehran, Iran, in 1974 and the M.Sc. and Ph.D. degreesinelectricalengineeringfromtheUniversity of Illinois, Urbana-Champaign, in 1975 and 1979, respectively. HejoinedtheDepartmentofElectricalandCom- puterEngineering,UniversityofTehran,asanAs- (41a) sistantProfessorin1980andbecameanAssociate Professor in 1988. Since 2002, he has been a Full ProfessorintheDepartmentofElectricalandComputerEngineering,Univer- (41b) sityofWaterloo,Waterloo,ON,Canada.Hisresearchinterestsandactivities includenumericalelectromagneticsappliedtoradio-frequency/microwave/mil- limeter-wavesystemsandcircuits,antennasandpropagation,wirelesscommu- equations(40a)and(40b)become nicationsystems,very-high-speeddigitalcircuits,andopticalcommunication systems.Hehasbeenascientificandtechnicalconsultanttoanumberofna- (42a) tional and international telecommunication industrial and research organiza- tionssince1980. 8 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.60,NO.1,JANUARY2012 Experimental Validation of Performance Limits and Design Guidelines for Small Antennas DanielF. Sievenpiper,Fellow,IEEE, DavidC. Dawson,Member,IEEE, MinuM. Jacob,StudentMember,IEEE, Tumay Kanar,StudentMember,IEEE, SanghoonKim, Jiang Long,and RyanG.Quarfoth,StudentMember,IEEE Abstract—Thetheoreticallimitforsmallantennaperformance papersaswellasothersthatfollowedhaveprovidedtheoretical thatwasderiveddecadesagobyWheelerandChugovernsdesign guidelinesforotheraspectsofsmallantennadesign.Ingeneral, tradeoffsforsize,bandwidth,andefficiency.Theoreticalguidelines thebestperformancewillbeachievedifthedielectricconstant have also been derived for other details of small antenna design isaslowaspossible,iftheaspectratioisclosetounity,andif such as permittivity, aspect ratio, and even the nature of the in- ternalstructureoftheantenna.Inthispaper,weextractandan- the internal structure of the antenna is such that the fields fill alyze experimental performance data from a large body of pub- the minimum size enclosing sphere with the greatest possible lished designs to establish several facts that have not previously uniformity. beendemonstrated:(1)Thetheoreticalperformancelimitforsize, Alongwiththeworkthathasbeendonetodeveloptheoretical bandwidth, and efficiency are validated by all available experi- limits,alargeamountofefforthasbeenputintodevelopingspe- mental evidence. (2) Although derived for electrically small an- tennas, the same theoretical limit is also generally a good design cificantennadesignsinanattempttooptimizetherelationship ruleforantennasthatarenotelectricallysmall.(3)Thetheoretical between size and bandwidth. In the 64 years since Wheeler’s predictionsfortheperformanceduetodesignfactorssuchasper- first paper, thousands of new antenna designs have been pub- mittivity,aspectratio,andtheinternalstructureoftheantennaare lished, and each year we continue to see new publications ex- alsosupportedbytheexperimentalevidence.Thedesignsthathave ploringeveryconceivablearrangementofmetalshapesanddi- thehighestperformancearethosethatinvolvethelowestpermit- tivity,haveanaspectratioclosetounity,andforwhichthefields electricregions.However,manyofthesedesignshavesub-op- filltheminimumsizeenclosingspherewiththegreatestuniformity. timumperformance,andcouldhavebeenpredictedtoperform Thisworkthusvalidatestheestablishedtheoreticaldesignguide- poorly if the theoretical design guidelines were more clearly lines. understood from the start. In addition, many antenna designs Index Terms—Bandwidth, dielectric resonator antenna, effi- are proposed for which performance is overestimated, such as ciency, fractal, metamaterial, planar antenna, quality factor, slot by ignoring losses or incorrectly calculating the true electrical antenna,smallantenna. sizeoftheantenna.Thischallengesnotonlyantennaengineers, whomustaddresstheseunphysicalperformanceclaims,butalso system engineers,whoenduprelyingonperformancemetrics I. INTRODUCTION that are ultimately unachievable. These issues may be caused inpartbecausethetheoreticaldesignguidelinesarenotwidely S MALLantennashavebeenanimportanttopicofresearch understood, and in fact have never been rigorously validated for many decades, and interest in the field is increasing experimentally. with the development of new systems that require broadband Unfortunately, it is impossible to experimentally “prove” a antennas with a small form factor. The analysis of small an- physicaltheory—itcanonlybedisprovenbycontradictoryex- tennasisgenerallyconsideredtohavebegunwith theworkof perimentalevidence.Nonetheless,atheorythathasbeentested Wheeler[1]andChu[2],whoestablishedthetheoreticallimits extensivelyandfoundtobetrueinalltestsisgenerallyaccepted thatshowhowelectricalsizeandbandwidtharerelated.Since ascorrect,atleastuntilcontradictoryevidenceisfound.Inthe thisearlywork,numerousauthorshaverevisitedthesetheories, fieldofsmallantennas,therehavebeenmanyattemptstoopti- andhavesuggestedfurtherrefinements.Althoughslightlymore mizeantennadesignstogetascloseaspossibletothetheoretical accurate,allofthesenewtheoriessharethesamebasicconclu- limits. However, each of these antennas represents a local op- sionsestablishedinthe1940s—thatsizecanonlybereducedat timization,andineachcaseitispossiblethattheauthorshave theexpenseofbandwidthorefficiency.Furthermore,theearly simplynotchosenthebestdesign,andperhapsanotheronemay befoundthatcouldexceedthetheoreticallimit.Thepurposeof thispaperistosystematicallyextractexperimentalresultsfrom a sufficiently large sample of existing designs to demonstrate Manuscript received February 10, 2011; revised May 17, 2011; accepted July12,2011.DateofpublicationSeptember15,2011;dateofcurrentversion that the Wheeler-Chu limit is valid and correct across a broad January 05, 2012. This work was supported by SPAWAR under Contract rangeofelectricalsizesandbandwidths.Wefurthermoreshow N66001-03-2-8938. thatthedesignguidelinesthathavebeenestablishedforpermit- D.F.Sievenpiper,M.M.Jacob,T.Kanar,S.Kim,J.Long,andR.G.Quarfoth areallwiththeUniversityofCaliforniaSanDiego,LaJolla,CA92093USA tivity,aspectratio,andtheinternalstructureoftheantennaare (e-mail:[email protected]). alsosupportedbytheexperimentalevidence. D. C. Dawson is with SPAWAR Systems Center Pacific, San Diego, CA Thispaperestablishesseveralimportantfacts:(1)Whenthe 92152USA(e-mail:[email protected]). DigitalObjectIdentifier10.1109/TAP.2011.2167938 sizeandefficiencyarecorrectlycalculated,themeasuredband- 0018-926X/$26.00©2011IEEE

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