Antenna Handbook VOLUMEIFUNDAMENTALSAND MATHEMATICALTECHNIQUES Editedby Y.T.La Electromagnetics Laboratory Department of Electrical and Computer Engineering University of Illinois- Urbana s.W.Lee Electromagnetics Laboratory Department of Electrical and Computer Engineering University of Illinois- Urbana -,'.,'r;-,-,:;_,CT::DI Br-' -"..".. -' ""1 m t,.. Basics1-25 into the space. If the antenna is lossy, the radiated power P3 is only a portion of P2 Theratio (57) iscalledtheradiationefficiency. Three Gains in State (k, U) As indicated in (39) the antenna radiates in two orthogonal polarizations (u: v). Alternatively, its polarization can be represented by (within a phase factor)a single unitaryvectorUinthemannerdescribedin(46)and(47).Thentheradiation intensityinwattspersteradianinstate(k, U)is J(k)=IA(kW= IA(k, uW+IA(k, vW(58) whichincludesradiationinall(both)polarizations.The radiatedpowerP3 bythe antennainspaceis P3 = Jll dOJ211 dcp[J(k) sin 0] o0 (59) The general definitionof again(dimensionless)inthedirectionkandforall polarizationsis G(k)= 4n J(k)= 4n IA(k)12= 4n intensityof theantennaindirectionk(60) Pn Pn areferencepower wherePn isareferencepower.Depending onPmtherearethree commonlyused gains: (a)realizedgainGI(k)ifPn =PI=powerincidentattheantenna, (b)gainG2(k)ifPn =P2 =poweracceptedbytheantenna, (c)directivityD(k)ifPn =P3 =powerradiatedbytheantenna. Therelationsamongthethreegainsare Thethreegainsaregraphicallyillustratedinthelowerhalfof Fig.14. Three Gains in State (k, u) (61) The antenna hasapolarizationU asdefined in(47).Indefining gains wemay specifyapreferredpolarizationu.Thentheradiationintensityinstate(k, u)is J(k, u)= IA(k, uW= IA(k),u*12(62) inwattspersteradian.Theratio 1-26 Fundamentals and Mathematical Techniques

..",.PJ ---.P 2... 'RADIATION INTENSITY IN(k, u) IA(k,uW-IA(k)ou'12 PMISMATCH.PL EFFICIENCY.P3 POWERS:INCIdENTACCEPTEDRADIATED G(k) - g(k,u) + g(k, v) IA(k)12-IA(k,u)12+ IA(k,vW RADIATION INTENSITY IN(k, U) POLARIZATION EFFICIENCY IUou'I2 Fig,14,Sixdefinitionsof gain(abmeansat= b).(CourtesyG.A.Deschamps) =J(k, u)=lu.*12 PJ(k)u (63) is called the polarization efficiency in state (k, u) for an antenna that has polarization U.Withrespecttostate(k, u)wedefineagain 4.rr4.rr I12 g(k, u)=P J(k, u)=PA(k)'u*=pG(k) nn. =4.rr intensityof theantennainstate(k, u) areferencepower (64) Again,depending onthe referencepowerPmthereare three gains g .. g2,and d, whichareanalogoustothecasesassociatedwith(60).Thesethreegainsare illustratedintheupperhalf of Fig.14. The gaindefinedin(64)iscalled the partial gainforaspecific polarization u. The(total)gainG(k)in(60)isthesumof partialgainsforanytwoorthogonal polarizations: whichfollowsfrom(58). Peak Gains G(k)=g(k, u)+ g(k, v) Allofthesixgainsdependontheobservationdirectionk=(J, f/J).In applicationsweoftenusethemaximumofagainasafunctionofk.Thus, correspondingto(60),wedefineapeak gainforallpolarizationsby G= max G(k)=IA(kW n (65) Basics1-27 In a similar manner we define a peak gain for a specific polarization u.When a gain isgivenwithoutaspecified observation direction,itiscustomarilyassumed to be thepeakgain. Gain-Related Terms Whendefiningantennatermsrelatedtogain,wecanuseanyone of thesix definitions of gain. If the radiation in all polarizations is of interest, we use G(k) in (60).If the radiation in a specific polarization isof interest, weuse g(k, u) in(62). Withthisunderstanding weuseG(k)torepresentatypicalgaininthe discusston .below. An isotropic radiator isahypothetical lossless antenna having equal radiation intensity inall directions,i.e., J(k)==a constant.If an input power Pn were fedto anisotropicradiator,itsradiationintensityinwattsper steradianwouldbe (66) Here Pn can be anyone of three powers explained at the beginning of this section. Intermsot- (66)thegaindefinitionin(60)hasthefollowinginterpretation: intensityof theantennaindirectionk = ~ - - ~ - - ~ ~ ~ - - ~ - - ~ - - ~ ~ ~ ~ - - - - - - - - - -intensityof anisotropicradiatorfedbythesamepower (67) WeoftenexpressthedimensionlessGbyitsdecibel(dB)value:1OloglOG. Sometimes we write dB as dBi, where the letter "i" emphasizes that the gain is over anisotropicradiator. ConsideranantennawithgainG(k)fedbyaninputpowerPn.Itsradiation intensity J(k) wouldbe the same asthat foran isotropic radiator if the latter were fedwithaninputpowergiveninwattsby EIRP= PnG(k)(68) whereEIRPstandsforequivalent(effective)isotropicallyradiatedpower.We oftenexpressEIRPbylOloglOPnGindecibelsreferredto1W(dBW). Whenanantennaisusedforreceiving,afigureofmeritoftheantennais definedby peakgainof theantenna - = - - ~ - - - - - = = ~ - - - - - - - - - - - - - - -Tanoisetemperatureof theantenna G (69) Ta,whichisusuallygiveninkelvins(K),isdiscussedinChapter2. 1-28Fundamentals and Mathematical Techniques 11.Pattern Approximation by (cos 6)9 The far-field patterns of many aperture-type antennas have two characteristics: (a)Asinglemajorlobeexistsintheforwardhalf-spacez>0andthelobe maximum is in the +z direction.(b) The radiation in the backward half-space z < 0 i.snegligible.For thesetypesof antennastheirfarfieldscanbeapproximatedby simpleanalyticalfunctionsdescribedinthissection.For anx-polarizedantenna, such as the flangedwaveguide shown in Fig.15, wehave (for 0 O.Note that F,isnot a goodapproximationof Fforz 0),whichdoublestheelectriccurrentin(51)andcontainsnomagnetic current.Thefieldproducedby(2Js,0)intheabsenceof ~ ,isF2,whichisthe secondapproximationof Fforz>o. (c)Magneticcurrentalone(Fig.lOd).Anotherequivalentsourceis(0, 2Ks) , whichdoublesthemagneticcurrentin(51)andcontainsnoelectriccurrent.The fieldproduced by it in the absence of ~ lisF3,which is the third approximation of F forz>O. Version(a)aboveissometimesknownastheStratton-Chu formula.Field Fl istheaverageofF2andF3Kirchhoff'sapproximationhasbeenappliedto transmission through aperture problems in which the aperture is not necessarily in a planarscreen,suchasradiationfromanopen-endedwaveguide. 8.Scattering by an Obstacle Consider the scattering problem sketched inFig.11.The obstacle (scatterer) is illuminatedbyanincidentplanewaveFigivenby (52) where kl=wavevectorwithmagnitudek=wv',UEandpointinginthedirectionof propagationofFi Z=y-l =VflJt: Ul=aunitaryvectorwhichdescribesthepolarizationofFiandisorthogonal tokl C=amplitudeofFiin(watt)1I2(meter)-' WesaythatFiisinstate(k"ud.Inthepresenceof theobstaclethetotalfield 2-22 p \ Y. SOURCE'l.k \1 Fundamentals and Mathematical Techniques . N Fig.11.We are interested in the scattered far field at r with state (k2' U2)when an obstacle is illuminatedbyanincidentplanewaveFiwithstate(kb u,). everywhere is the sum of Fi and a scattered fieldF.At a far-field observation point r,fieldFisrepresentedbyasphericalwave {E(r)}....., }e-ikr, H(r)y'Yk2 xA(k2)r (53) The distance rismeasured froma reference point 0in the vicinity of the obstacle. Vector k2has a magnitude equal to kand isinthe direction of Or. The amplitude vectorA(k2)givenin(watt) 112maybedecomposedintotwoorthogonal components,asdiscussedinSection7,Chapter1.Letusconcentrateona particular component of A(k2)withpolarizationU2,namely,AuI.Weintroduce thenotation(afterG.A.Deschamps) A(2, 1)=A(k2)uI(54) whichrepresents the scatteringamplitude instate2 (direction k2andpolarization U2)due to anincident plane waveinstate 1.Using this notation, wewillintroduce somedefinitionsandtheoremsforthescatteringprobleminFig.11. Bistatic Cross Section Thebistaticcrosssection(BCS)insquaremetersfromstate1 tostate2is definedby (55 a) Theorems and Formulas2-23 431(intensityof scatteredfieldinstate2) = - - - - ~ ~ ~ ~ ~ ~ ~ - - - - - - - - - - ~ - - ~ - -powerdensityof incidentplanewaveinstate1 (56a) 431(intensityof scatteredfieldinstate1') = - - - - ~ ~ ~ ~ ~ ~ ~ - - ~ - - - - - - ~ - - ~ ~ powerdensityof incidentplanewaveinstate1 (56b) WewillconsidertwoexamplesofReS.(a)Forasmoothconductorwhose dimension is large in terms of wavelength and is illuminated by a linearly polarized incident field,its ReS is approximately independent of polarization and is given by (57a) where R land R2are two principal radii of curvature of the conducting body at the specular pointP(Fig.12).The point Pisdetermined bythe relationinwhichthe surface normal N is in the opposite direction ofkl. We assume that there is only one such specular point. If the conductor is a sphere of radius a, the use of (57a) leads to ReS:::7ra2,awell-knownresult.(b)Forarectangularconductingplateof dimension a xb in the z =0 plane, the ReS is again approximately independent of polarizationandisgivenby Res - 1(kb0 sin a Sinf3)2 =- acos0----3laf3 where (00,d,wehave e(xy)= y (Z 1-) 112 cos(.!!. x) ,{3cdc (70a) h(x,y)=(Z k/{3)-1 Z xe(70b) where Z=y-I=()11f:)"2 {3=k[l- (n/kc)21'/2(71) Asforthecharacteristicimpedance,thereareatleastfourcommonlyused definitions,namely, ZcI=Zk/{3 Zc2= 2(d/c)ZcI Zc3=Zc4= (n/2)(d/c)ZcI (72a) (72b) (72c) (72d) Thus there are at least four different ways to define the (modal) voltage and current intherectangularwaveguide,correspondingtothefourchoicesofZcin(72). OnceachoiceofZcismade,wefindthe(total)voltageandcurrentatthe referenceplanez=0aregivenby Theorems ana Formulas v =V++V- =VZc(a+ b) 1= r+ r=ffc(a- b) Theinputimpedanceof theantennaisdefinedby 1V Zin= Yin= I Therelationsamongr, Ziri'andYinare Zin=Yc=1+ r ZcYin1 - r r= (Zin/Zc)- 1= 1- (YinIYc) (Zin/ZJ+11+(YinIYc) 2-29 (73) (74) (75) (76) (77) It is clear from(76) that antenna impedance Zin also depends on the choice of Zc. Thepowertransmitted(radiated)fromthesourceintothefreespaceviathe antennaisgivenby p, =Re{ JJ(ExH*).ZdxdY} =lal2 - Ibl2 =la12(1- Ir12) = Re{VI*} (78a) (78b) (79) Thereflectedpower backtowardthesource(duetothemismatchof ZcandZin) isIb12. 10.Three Ideal Sources for Transmitting Antennas Asdiscussedintheprevioussection,anantennaasaone-portdevicecanbe describedbyeither waveamplitudes(a, b),orbycircuitparameters(V, I).When the antenna isused fortransmitting,a sourceisconnected tothetransmissionline (orfeedwaveguide).FollowingDeschamps[6],wewillintroducethreeprincipal idealsources: (a)unitamplitudesource(definedbya=1 (watt)1/2) (b)unitvoltagesource(V=1 volt) (c)unitcurrentsource(I=1ampere) Their graphical representation is shown inFig.16.Note that the dot ineach source indicatesthepropagationdirectionoftheincidentwaveleavingthedot,the positive voltage terminal at the dot, or the current out of the dot. The term "ideal" isusedbecausethosesourceshavespecialinternalimpedanceZ.nviz., 2-30Fundamentals and Mathematical Techniques a a-I 8 ~ --V-I 1-1 b-r bc Fig.16.Threeidealsourcesfortransmittingantennas. (a)Zs=Zcfortheamplitudesource,inwhichZcisthecharacteristic impedanceofthetransmissionline.Thusthesourceismatchedtothe transmission line, and there isno reflection at the source-transmission-line junction.(However,themismatchattheantenna-transmission-line junc-tionmaystillexist;Zcmayor maynotbeequaltoZin') (b)Zs=0 for the voltage source so that there is no internal voltage drop within thesource. (c)Zs-+00forthecurrent sourcesothat thetotal sourcecurrent entersthe transmissionline. In Fig.16 weusethe zigzag line, straight line,and gap inside the circles to indicate thesourceimpedances. For atransmittingantenna anyone of thethree ideal sources maybe used as theexcitation.Foreachcase,welistinTable1theinternalfeedwaveguide quantities (V,1,a,b)and the external radiation quantities (radiated fieldF and its powerPI)'HereF stands forvector fields(E, H).The subscripts (1,2,3)areused toidentifyexcitationsduetoaunit(amplitude,voltage,current)source.For example, atisthe value of a when the excitation isa unit amplitude source, and F3 isthevalueofFwhentheexcitationisaunitcurrentsource.Thethreefields (Ft, F2,F,,)inTable1arerelatedby (80) F=nc13F= ~ V Z .(1+ Zin)13F 31- ratt2(Zcatt (81) whereat=1(watt)t/2,V2 =1volt,and13=1 ampere. 11.Three Ideal Meters for Receiving Antennas When anantenna isused forreceiving,the source isreplaced bya receiver or, for our present purpose, a meter.Again, followingDeschamps we introduce three principalidealmeters: (a)Amplitudemeter,whichmeasures theincomingtraveling-waveamplitude batareferenceplanez= 0,andhasinternalsourceimpedanceZs= Zc suchthatthemeterismatchedtothefeedwaveguide.(Theantenna impedance Zin mayor may not be matched to Zc-)Note that b is a complex number,includingbothmagnitudeandphaseinformation. Theorems and-Formulas2-31 Table 1.Relationships for Transmitting Antennas ~ AmplitudeVoltageCurrent Quantity V (volt)v'2,.( 1 + r)a IV2 =1 Zin/3 1(ampere)VY;(1- r)alYinV2 13=1 a (watt) 1/2 al=1 1vv:.(Yin) TYc1 + Yc V2 1(Zin). Tv'2,.ZC+ 1h b (watt)112 fal ~VY;(1- i;) V2 ~vz:: (Zin- 1) 1 2cZ3 c Pt (watt)(1- If12)a12 V/Re YinI/ReZin FieldFIF2F3 Source Zc (matched)o(short)00(open) impedance (b)Voltmeter,whichmeasuresthevoltageVat areferenceplanez=0,and has an infinite internal source impedance (Zs -+00)such that Vis the open-circuitvoltage. (c)Ammeter, which measures the current 1 at a reference plane z =0, and has azerointernalsourceimpedance(Zs=0)suchthat1 istheshort-circuit current. The graphical representation of meters is given inFig.17.Note our convention that the circles represent sources, while squares represent meters (compare Figs.16 and17). When the receiving antenna isilluminated byan incident fieldwemay connect anyof thethreeidealmetersinFig.17toitsfeedwaveguide,correspondingto matched-load,open-circuit,andshort-circuitsituations.Table2liststheinternal feedwaveguide quantities (V, I, a,b).The subscripts (4,5,6) are usedto identify theuseof an(amplitudemeter,voltmeter,ammeter).For the same incidentfield thethreesituationsarerelatedby (82) a-O 8-..-.. ----v 1=0 v-O a b b c Fig.17.Three ideal meters for receiving antennas. (a) Amplitude. (b) Voltage. (c) Current. 2-32Fundamentals and Mathematical Techniques Table 2.Relationships for Receiving Antennas ~ AmplitudeVoltmeterAmmeter Quantity V (volt)VZcb4 Vs 0 1 (ampere)-ffcb4 0 16 a (wattt2 0 1 TffcVs 1' TVZch b (watt) 112b4 1 Tffcvs 1 -Tffc/6 Source impedanceZc (matched)00(open)o (short) (83) Commonly,Vsisknownastheopen-circuitvoltage,and16astheshort-circuit currentof areceivingantenna. 12.ReciprocitybetweenAntenna Transmitting and Receiving The reciprocity theorem ina circuit iswellknown.Its application to antennas, however,isnotsimpleforthereasonexplainedbelow.Ifviewedfromthe transmissionline,theantenna looks likea circuit element whosetransmitting and receiving properties are describablebytwo(complex) numbers:(a, b)or (V, 1),as discussedinSection9.Outsidetheantennainthefree-spaceregion,eitherthe radiatedfieldofthetransmittingantennaortheincidentfieldonthereceiving antenna ismore complex.They are vector fieldscharacterized bypolarization and spatialvariation,whicharenotdescribablebycircuitquantities.Hencethe reciprocity for an antenna cannot be simply stated bythe usual exchange of sources andmeters.Inthissectionwewillgivetworeciprocityrelationsforantennas. Reciprocity Involving General Incident Fields Consider the transmitting situation in Fig.18a, where the antenna is excited by a unit amplitude source with al= 1 (watt)1/2.The radiated fieldis FI= (EJ, "d. In thereceivingsituation(Fig.18b)thesameantennaisconnectedtoanamplitude meter withmatchedimpedance(sothat a4= 0),and isilluminated byan incident fieldF4=(E4' "4).Thenareciprocitystates[6] (84) Theorem. and-Formulaa a z-o I I I 0)-1I 81 Ib) I 2-33 b . Fig.18.An antenna intransmitting excited witha unit-amplitude source, and inreceiving connectedwithanamplitudemeter.(a)Transmitting.(b)Receiving. Here S isanarbitrary closed surface whichencloses theantenna but excludes the source of F4. Its outward unitnormal is N.It is emphasized that F4isthe incident field that would exist in the absence of the antenna in Fig.18b, and does not include thescatteredfieldF;'fromtheantenna.Itcanbeshown,however,that(84) remainsvalidifF4isreplacedbyF4+ F;',becausethecrossflux(F;'SF.)=O. Reciprocity Involving Plane Waves Inmanyapplicationsweareinterestedinaspecialcaseof Fig.18,namely, theradiatedfieldF.inthetransmittingsituationisknowninthefar-fieldzone (Fig.19a) ,andtheincidentfieldF4inthereceivingsituationisaplanewave (Fig.19b).WeexpresstheradiatedfieldF.by 1:-0 I I I I 0)-1I e: I b) I I I aI F={E1(r)}_{ V Z ~ ( k )} e-ikr, H.(r)y'VkxA(k)r (85) b Fig.19.Same antenna asFig.18except incidentfieldF4isa plane wave.(a) Transmitting. (b)Receiving. 2-34Fundamentals and Mathematical Techniques whereZ= y-l = (pJe) 1/2.WerepresenttheincidentfieldF4by (86) whose amplitude isC in(watt)I12/meter, propagation vector (-k), and polarization vectoru*.Thenareciprocityrelationstates[6] b4 = -j). ( ~ )[A(k)u*] (87) where)' ='brlk isthe free-space wavelengtn.If the open-circuit voltage and short-circuit current inthe receiving situation are of interest, wemayuse(87)or (84)in conjunctionwith(82)and(83).Theinterpretationof(87)isasfollows:The receivedamplitudeb4 ofanantennaunderthematchedconditionduetothe incidence of a plane wave in state (-k, u*) is proportional to the antenna radiation far-fieldamplitudeinstate(k, u).[Rememberthat(-k, u*)and(k, u)havethe samepolarization.] Antenna Effective Length Consider the receiving situation in Fig.19b, where the antenna is connected to amatchedmeter andisilluminatedbyanincidentplanewaveF4definedin(86). Theeffectivelengthhisdefinedbytherelation (88) whereV 5istheopen-circuitvoltageof thetransmissionlineatareferenceplane z = 0, and E4(r= 0) isthe electric field at a reference point 0of the incident plane wavegivenin(86).From(88),(87),and(82)weconcludethat h=vf( 1 + i:)( -j :J A(k) (89) whereZ= (pJe) 1/2.Clearlyhisacomplexvectorinmeters.Therelationin(89) relatestheeffectivelengthhandthefar-fieldamplitudeA(k)of anantenna. Transmitting Field in Terms ofEffective Length Excited bya unit amplitude source (a=1), the transmitted far fieldElis given in(85).Nowifthesameantenna isexcitedbyaunitcurrentsource(I =1),the correspondingtransmittedfarfieldE3isthengivenby 1(Z.)e-jkr E3(r)- 2VZcZ 1+i:A(k) -,-, (90) wherewehaveusedtheformula(81)inrelatingElandE3.ReplacingA byhin accordancewith(89),werewrite(90)as Theorems andbrmulas whichmaybeusedasanalternativedefinitionof theeffectivelengthh. Receiving Cross Section 2-35 (91) Consider the receiving situation in Fig.19b, where the antenna is connected to amatchedamplitudemeter.Theincidentplanewaveisgivenin(86),with amplitudeC and state (-k, u*).The received waveamplitude isb4,givenin(81) . . Then wedefinethereceivingcrosssection(effectivearea)a instate(k, u)of the ,antennaby a(k, u)= \bC4\2 receivedpowerof theantennaundermatchedcondition =(92) powerdensityof incidentplanewaveinstate(-k, u*) insquaremeters.Makinguseof(87),wehave (93) Apartial gainof theantenna instate(k, u)isdefinedby(Section10,Chapter 1) g.(k, u)= 4n \ d.A(k).u* \2(94) From(93)and(94)wehave 4n g.(k, u)=;e a(k, u)(95) where a;..=l2/4n issometimes known asthe receiving cross sectionof the fictitious isotropic radiator.As discussed in Section 10, Chapter 1, the partial gain g, in a pre-ferredpolarizationuisrelated tothe(total)gainG,inall(both)polarizationsby (96) where U describes the polarization of theantenna indirection k.Substituting (96) in(95)gives (97) whichrelatesthereceivingcrosssectionandthegainofanantenna.Thefactor 2-36Fundamentals and Mathematical Techniques (98) is called the polarization efficiency (or polarization mismatch/actor). We emphasize that unitary vector u*(not u) describes the polarization of the incoming plane wave in direction (-k), while U describes the polarization of the receiving antenna inthe outgoing direction(+k).For example,letk=z andU =(i - jY)/V2foraright-handcircularlypolarizedantenna.If theincomingplanewaveisalsoright-hand circularlypolarized,wehaveu*= (i + jY)/V2andp= 1. 13.The Radar Equation and Friis Transmission Formula TheconfigurationofabistaticradarissketchedinFig.20.Theobstacle (target)inthevicinityofpoint0isilluminatedbyanincidentwavefromthe transmittingantennaatpoint1.Apartof thescatteredenergyisreceivedbya receivingantennaatpoint2.BothdistancesR IandR2arelargeintermsof wavelengthsothatthescattererisinthefar-fieldzonesoftheantennas.The problemathandistodeterminethepowerratioP 2/ PI,where PI=powerincidentfromthegeneratortothetransmittingantenna, P2 =powerreceivedbythereceiverviathereceivingantenna. Tothisendweneedtodefinetheantennasandtheobstaclemoreprecisely,as below. (a)In the vicinity of point 0, the radiated fieldof the transmitting antenna isin R 1 - DISTANCE ill R2 ~DISTANCE 20 TRANSMITTING ANTENNA Fig.20.Abistaticradar. RECEIVING ANTENNA RECEIVER -Theorems2-37 the directionklandhasapolarizationdescribedbyaunitaryvector VI'Inother words,the state of the transmitting antenna at point 0 is(kl' VI)' The gain of the transmitting antenna in the direction kl isGI(kl), which is related to the directivity DI(kl)by (99) Here1] Iisthetransmittingantennaefficiencyaccountingfortheconductorand dielectriclosses,andr Iisthereflectioncoefficientaccounting fortheimpedance , mismatchbetweenthetransmitting antenna anditsfeed(Section10,Chapter 1). #(b) If the receiving antenna were used for transmitting (Fig.21), the state of its radiated fieldat point 0 would be (k2' V2), and its gain would be G2(k2). A relation similarto(99)holdsforG2 andD2. (c)Duetotheilluminationfromthetransmittingantennatheobstacle producesascatteredfieldatpoint2,withstate(ko=-k2,Vo).Itsbistaticcross sectionfromstate(kb VI)tostate(ko, Vo)isdenotedbytheBCS. Withtheabovedescriptionwemaynowcalculatevariouspowerquantities. ThepowerdensityWoof theradiatedfieldatpoint0fromthetransmittingan-tennais (100) inwattsper squaremeter.The powerdensityW2 of the scattered fieldatpoint 2 fromtheobstacleis ""-2 Fig.21.Thereceivingantennaof Fig.20isnowusedfortransmitting. 2-38Fundamentals and Mathematical Techniques (101) The received power P2bythereceiver at point 2 via the receiving antenna may be calculatedfrom(92)and(97)withtheresult A.2 P2 = W2 4.7lIUoU212G2(k2) (102) Combining (100)through(102),wehavethepowerratioforthebistaticradar in Fig.20,namely, (103) whichisknownastheradar(range)equation. Next,wewillconsidera specialcaseof (103).Letusremovetheobstaclein Fig.21,and studythe direct power transmission frompoint 1 to point 2 (Fig.22). Makinguseof thefollowingrelations: Un=UI, BCS=4.7lRl then(103)becomestheFriistransmissionformulaforthefar-fieldtransmission betweentwoantennas,namely, RECEIVER Fig.22.Powertransmissionbetweentwoantennas. Theorems anclFormulas2-39 (104) The factor (A./4.71R)2is called the free-spaceloss factor,and it accounts forthe loss duetothesphericalspreadof thetransmittedfield.Thepolarizationefficiency (105) accounts for the loss due to polarization mismatch between the transmitting and the receivingantennas.If bothantennashavethesamepolarization,UI =Uland ,p=1.(TheconjugateonUlisduetothefactthatU2 referstoapropagation directionoppositethatof UI.)Notethesymmetrybetween1 and2intheright-handsideof(104),implyingthatthesameformulaappliesiftherolesofthe receivingandtransmittingantennasareinterchangedinFig.22. 14.Noise Temperature or an Antenna For a high-resolutionantenna a crucialfactorthatlimitsitsability to detecta weaksignalistheantennanoise,whichisthesubjectof thepresentsection. Antenna Noise Temperature The receiving antenna sketched in Fig.23receives the desired signal as wellas noise.Wedenotetheavailablenoisepoweratterminall:byPa inwatts.It isa common practice to express Pa in terms of an (effective) antenna noise temperature Tainkelvinsviatherelation E' RECEIVER TRANSMISSION LINE E I I I I IANTENNA I I Fig.23.Areceivingantennareceivesthermalnoisesfromexternalnoiseemitters. 2-40Fundamentals and Mathematical Techniques (106) where k'=Boltzmann'sconstant=1.38x10-23 11K !l.f =bandwidthof theantennareceivingsysteminhertz Theinterpretationof(106)isthatantennanoisepowerPa isequaltothatof a matched resistor whose physical temperature isTa.Hereafter we willuse Pa and 'Fa interchangeably.Clearly,'Fadepends on the antenna receiving characteristics and externalnoiseemitter. Brightness Temperature ofan Emitter Every object with its physical temperature above the absolute zero (0 K)is an emitter of thermal energy inthe formof electromagnetic waves.Withinanarrow frequencyband the amount of energy radiated in direction k and polarization u or simplyinstate(k, u)isproportionaltoaparametercalledthebrightness temperatureTb(k, u).ThelatterisrelatedtothephysicaltemperatureT;,of the emitterviatherelation (107) wherethe dimensionless parameter e(k, u)iscalled the emissivityin state(k, u)of theemitter.Todeterminee(k, u)weilluminatetheemitter byanincidentplane waveinstate(-k, u*).SeeFig.24.Then incidentpowerabsorbedbyemitter e(k, u)= incidentpowerinterceptedbyemitter Foralargeemitteranapproximateformulaforcalculatingitsemissivityis e(k, u)==1- If(k, u)i2(108) Heref(k, u)isthereflectioncoefficientatthesurfaceof theemitter.Sincethere aretwoindependentpolarizationswecandefinetwoindependent(partial) brightness temperatures Tb(k, u)and Tb(k, v)in a given direction k, for each choice of orthogonal,unitary,polarizationvectors(u, v).Acommonchoiceof(u, v)is u=horizontalpolarization,v= verticalpolarization(109) Thesumofthetwo(partial)brightnesstemperaturesgivesthetotalbrightness temperature (110) whichisproportionaltothetotalenergyradiatedbytheemitter indirectionkin Theorems and Formulas2-41 ~ ~PLANE /WAVE Fig.24.To determine the emissivity (k, u)of a noiseemitter weilluminate the emitter by anincidentplanewaveof state(-k, u*). bothpolarizations.Ablackbodyabsorbsalltheincomingenergyimpingingonit (aperfectabsorber).Itsemissivityisunity(aperfectemitter),and'I2Tb(k)= Tb(k, u)=Tb(k, v)=1'pforanydirectionk. Calculation ofAntenlUl Noise Temperature The power received bya receiving antenna can be traced to three sources:the desiredsignal,interferencefromothercoherentradiators,andincoherentnoise fromnoise emitters. Fig.25shows some important noise emitters inthe free-space environment.Letusconcentrate on atypicalnoiseemitter (No.1inFig.23).Its contribution to the antenna noise temperature can be calculated fromthe following formula: where 1JlfJ21f Ta=4Jl0sinOdO0dlj>[Tb(-k,u*)d(k,u)+Tb(-k,v*)d(k,v)](111) (u, v)= unitaryvectorsdescribingtwoorthogonalpolarizations d(k, u)=(partial)directivityof theantennainstate(k, u) Tb( -k, u*)=(partial)brightnesstemperatureof theemitterinstate(-k, u*). Notethatstates(k, u)and(-k, u*)havetheoppositeduections butthesamepolarization If there ismore than one noise emitter in space, the superposition principle applies tothecalculationofTa.Thisissobecausenoiseemittersareincoherentandthe superpositionofpowers(temperatures)ispermissible.Foranidealizedomni-directional antenna, which radiates equally in both polarizations, we have d(k, u)= d(k, v)=112,and 2-42Fundamentals and Mathematical Techniques AMPLIAER NOISE GROUND THERMAL NOISE I -tl RADIO STARS TRANMISSION-LINE NOISE DISCHARGE LIGHTS IGNITIONNOISE TV STATION Fig.25.Importantnoiseemittersforareceivingantenna.(CourtesyY.T.Lo) 1JllJ2.:rr To= TaO= 8.1l0sin 0 dO0d [Tb( -k, u*)+Tb( -k, v*)](112) Fig_26presents some typical values of TaOof an omnidirectional antenna, together withthenoisetemperatureofatypicalreceiver. Noise Power at the Receiver's Terminal CorrespondingtotheantennanoisetemperatureTain(111),theavailable noise power at terminal in Fig.23 is given by Pa according to (106). This incoming powerpropagatesthroughthetransmissionline,andarrivesatthereceiver's terminal withitsvalueinwattsequalto P'a= k' +(1-1])Tol where 1]=powertransmissionefficiencybetweenterminalsandTo=physicaltemperatureofthetransmissionline (113) Theorems anclf.ormulas 60 50 :.: 40 ! 30

,20 Q I ...' 10 0 -10 10 IATMOSPHERIC SUMMER:2200-2400 HOURS.NYC Z WINTER:0800-1200 HOURS.NYC t;/ i'. .....,/r--V ""- URBAN '-.HUMAN-MADE

'-* >-D1.5D1.5 R, Z;Rr . -*kL 1 so that sin (kL cos lJ)/kL cos lJ-+ I. 3-16Fundamentals and Mathematical Techniques analysis.InChart 6,results arepresented foranarbitrary-lengthdipoleand fora half-wavelengthdipole(L=lI2). The Traveling-Wave Antenna Thetraveling-waveantenna[8,10,13,15,16]isdefinedasanantennawhose conductors support a current distribution described by a simple traveling wave.In comparisontothedipoleantenna,whichsupportstwocounter-directedtraveling waves givingrise to the sinusoidal (standing-wave) distribution,the traveling-wave antenna supports a single wave.A simple example of a traveling-wave antenna and itsattendant characteristics are presented inChart 7.For this antenna the current distributionisgivenby I(z)=I(.,e-jkz, where10isitsamplitudeandkthefree-spacewavenumber. Theimportantfeatureofthetraveling-waveantennaisthatthemaximum radiation is neither end-fire (8 = 0)nor broadside (8= n/2).Rather it occurs at an intermediate angleand creates a singlemajor conicallobe independent of antenna lengthandtiltedinthedirectionof wavetravelwithminorlobesdeterminedby length.An approximation for the angle of the main lobe from the end-fire direction [13,16]isgivenby8m =cos-'[1- 0.371/(Ul)]. The Small Loop Antenna The smallloop antenna [8,9,10,11,13,14,15] of radius a,carrying auniform currentI,canbeanalyzedusingthetechniquespresentedpreviously.The characteristicsforthesmalllooparepresentedinChart8. ThestrikingsimilaritybetweentheexpressionsinChart8 forthesmallloop andthosefortheshortdipoleinChart4isimmediatelyevident.Clearly,the electriccurrentIonadipoleoflengthLgeneratesfieldswhichareobviously related to those of a loop of radius a carrying anelectric current I.Sincethe fields radiated by such a loop are identical with those radiated bya magnetic dipole [1]if wheremisthemagneticcurrent,wecanobtainthefieldsduetoasmallcurrent loop from those of a short electric dipole using duality. The procedure is as follows: (a)The originalproblemisalinear electric current Iof lengthLina medium withconstitutiveparameters Eand Jl.Obtain thesolutiontothis problem. (b)Bytheduality of Chart 3,construct the solution of the problem of a linear magneticcurrentmandlengthLinamediumwithconstitutiveparam-etersEand Jl.Thisisobtainedfromthesolutionin(a)abovebysetting I,=VEfJlm= jkl(na2)fL,E,(r)=VJlfEH2(r),andH,(r)= Notethatthemagneticcurrentisspatiallyorthogonaltotheplaneof the loop. (c)ReplacemLbyjWJl/(na2)toprovidethesolutionforthesmallloop radiatingelement. Techniques for"1.ow-Frequency Problems Chart 6.Characteristics for a Dipole with Sinusoidal Current Distribution I(z)10 sin ko(Ll2 - Izl),-Ll2 zLl2 Arbitrary-LengthDipole Az(r)I(e-jkr cos [(kLl2) cos 0]- cos (kLl2) ,21frksin20 Er(r) Hr(r) Pr[1l,12] 8.e-jkr cos [(kLl2) 0]- cos (kLl2) ly E2:rrrsm 0 .i...le-jkr cos [(kLl2) cos 0]- cos (kLl2) 'I" 1 02:rrrsin 0 (if/(? in {sin kL [Si(kt.) - Y2Si(2kL)] where + (1+ coskL)[ln(kLy) - Ci(kL)] - COS2kL [In (2kLy)- Ci(2kL)]} far zone 3-17 Si(x)= JX sinx dx, ox Iny= 0.5772 Er(r) Pr Rr D Half-WavelengthDipole(L=A12) le-jkr cos [(:rr/2) cos 0] o2:rcrk sin20 8.e-jkr cos [(:rr:2) cos 0] 1YE J2:rrrsm 0 fifM- Ci(2:rc)]= 0.194Y,u/d(,z [In (2:rcy)- Ci(2:rc)]= 0.194Y,u/E(Rr = 73ginfreespace) 4cos2[(:rr/2) cos 0]=1644 cos2[(:rr/2) cos 0] sin20[ln(2:rry)-Ci(2:rr)]sin20 4 In (2:rry)- Ci (2:rr)=1.644 3-18Fundamentals and Mathematical Techniques Chart 7.Characteristics for the Traveling-Wave Antenna I(z) loe-;kz,0 z L (l L) e-;kr e-;k(U2)(; -0056) sin [(kLl2)(1- cos 8)] o43fT.(kLI2)(1- cos 8) 0"(l L).8 e-;kr-;k(LI2)(1-OO56) sin [(kLI2)(1- cos 8)] IWP.0sm43fTe(kL/2)(1_cos 8) .i. 'k (l L).8 e-;kr-;k(U2)(1-OO56) sin [(kLl2)(1- cos 8)] TI0sm43fTe(kLI2)(1- cos 8) {In(2Jly) -1 + In (2L1,t)- Ci(2kL) + {In(2Jly) -1 + In (2L1,t)- Ci(2kL) +The Perfect Ground Plane Theperfectgroundplane,i.e.,theplanar,perfectlyconducting surfacewith conductivity a 00,is often found in the environment of antenna structures. Since the previous characterizations have been for antennas inhomogeneous media with realEand11,theintroductionofconductingbodiessuchasinfiniteplaneswill requireamodification. Theboundaryconditionsatanyperfectlyconducting surfacerequirethatthe tangential electric field satisfy, with n(r) the outward normal to the surface and r on thesurface, n(r)xE(r)= 0(33) andthatthemagneticfieldsatisfy n(r) H(r)= 0(34) Theexclusionof fieldsfromtheinterior of thesurfaceleadstothegeneration of surfacechargeandcurrentdensitiesgivenby n(r) E(r)=es(r)/E n(r)xH(r)=Jir) (35) Techniques foi Low-Frequency Problema Chart 8.Characteristics of the Small Loop Antenna /(cp) A(r) E(r) H(r) Er(r) Hr(r) Pr Rr GD(), cp) D r=a,()=n/2, -jkr (k1 ) r (lna2) cos () ear+ ? A e2J -jkr (k1 ) -8(/na2)sin()- k- - - "2 r,;- 4/2na4 y"Ek -6-4nrrr

1.5 sin2() 1.5 3-19 Anarbitrarilyorientedantennalocatedaboveaperfectlyconductingground planemustgenerate avanishingelectric fieldat the surfaceinorder to satisfythe boundaryconditionontheelectricfield.Suchaconditioncanbesatisfiedby creating anequivalent problemwheretheground planeisremovedandanimage sourceisintroducedtoproduce,whencombinedwiththeoriginalsource,a vanishingtangentialelectricfieldatthelocationoftheplane.Theappropriate imagesforelectric andmagnetic currents areshowninFig.2.Acloseinspection will show that the tangential electric and normal magnetic fields vanish at the plane surfaceandthatthefieldsareidenticalintheupperhalf-space. 3-20 a b Fundamentals and Mathematical Techniques IMAGE SOURCE Fig.2.Equivalentsourcedistributionsforsourcesaboveperfectlyconductingground planes.(a)Originalproblem.(b)Equivalentproblem. Thetheoryofimagescanbeappliedtoconfigurationsotherthaninfinite planes.Infact,asourceinthepresenceof anysurfacecomposedof intersecting planescanberepresentedintermsof thesourceandmultipleimages[1, 8].This conceptappliestobothexteriorandinterior(e.g.,waveguide)problems. An example of theuseof images isprovided bythemonopole antenna driven againstaperfectly conducting groundplane.The fieldsradiated bythemonopole into the upper half-space are identical with those of the dipole. The radiated power iscontained only inthe upper half-space and thus is only half of that for the dipole antenna.Hencewecanimmediatelyinfer that theinput voltagetothemonopole need onlybeone-half thatof thedipoletoproducethesamefieldsintheupper half-space.Thus Pr = /flo2 4 ~([sin (kL)][Si(kL)- 1/ZSi(2kL)] + [1- cos (kL)][ln (kLy)- Ci(kL)]- [cos (kL)l[ln (2kLy)- Ci(2kL)]} TheinputresistanceisgivenbyRr=Prllo2,andforaquarter-wavelength monopole Rr= / f 8 ~{In(2Jly)- Ci(2Jl)}= 36.5n Techniques forLow-Frequency Problems3-21 The Rectangular-Aperture Antenna The rectangular-aperture antenna [1,3,8,9,10,11,13,15], shown in Fig. 3, has manyfeaturesincommonwiththeelectricdipole.Inthefollowingwewillillus-tratetheuseof equivalenceandimagestoprovideaconvenientmechanismfor evaluating itsfields.In each casethe electric-field distributionintheaperturewill be assumed to be known. Then the fields produced by the aperture can be obtained usingequivalence,images,andthefieldrepresentationspresentedearlier. Foranassumedfielddistributionintheapertureinaperfectlyconducting plane, the pictorial representation of Fig.4a is appropriate.Using equivalence, the ,situationinFig.4bappliesandthefieldsinthehalf-spacez>0areidentical with those inthe original problem but zero for z < O.In this figure Ja =ixHand K"=-i xE.As a result of the vanishing fields for z < 0, the perfectly conducting planeiscompletedthroughtheapertureasshowninFig.4c.Usingimages,the plane can be removed and the situation in Fig. 4d holds. Note that the fields are the correctfieldsforthehalf-spacez>0andnotforz. R yyz Fig.6,continued. Techniques for Low-Frequency Problems3-31 Impedance:-3-dB Gain: Resistive atBandwidth dBabove Pattern TypeConllguratlonJr,R(O)PercentIsotropicDipolePolarizationNumber IIy Slot Inlarge -@tt ground plane Lx 350702.140hE L-A/2......I-!/d-29 Vertical yx full-wave -N-. loop 45133.141hF D-A/',; D/d-36 Helical over IIy reflector screen, ~ . tube 6Along 13020010.148eire.G coded Into 6 turns A/4 apart IIy ~ Rhombic ~16000 R,.,x 60010016.7414.5hH L-9A 1-9A/2 Parabolic IIy with folded ~ - . dipole feed 3003014.7412.5hH (A/2) D-5A/2 y Hom, #-. coaxial feed 5015.14hH3513 L-3A RLf 1-3A II G y I ~....... -f-II yII .. ,. YYII H ~- -Fig.6,continued. 3-32-/ / a/ / Fundamentals and Mathematical Techniques Fig.7.Imagesforimperfectgroundanalysis. / / / / / b z z Fig.8.Plane-wavereflectionataninterface. Techniques for Low-Frequency Problems3-33 radiated fieldEl(1, cp)due to the given antenna is computed for the case when it is locatedinaninfinitehomogeneousmediumatitsoriginallocation.Thenthe reflectedfieldF R ( (1,cp)iscalculatedfrom (54) whereEr( (1,cp)isthefieldduetotheimageoftheoriginalsourceinaperfect ground plane and p isthe unit vector normal to the plane of incidence (p =z x .at). Finally,thetotalfieldE( (1,cp)isthesumofthetwocontributions: (55) Chart11illustrates the steps and theresults of a procedure forevaluating the fieldsof averticalandhorizontalshort dipolecarryingaconstant current 10The procedure foranalyzing an arbitrary antenna,thoughmore involved, can proceed inasimilarmanner. TheFresnelreflectioncoefficientsarestrictlyvalidonlyforplanewavesand are thereforenot rigorously valid forantennas near ground planes [13,23,24,25]. An-ays Thesubjectof antennaarrayshasbeenextensivelydocumentedandwillbe discussedinChapters 13,14,and 17.Previous workisdescribed inthe writings of Bach and Hansen [26],Kraus [14], Jasik [10],Stutzman and Thiele [13], Weeks [9], andMa[27].Here wewillmerely showtherelationship of someaspects of array analysistopreviouslydiscussedsubjectsinthischapter. In the following wewillpresent expressions whichare useful for expressing the fieldsduetoanarrayofidenticalantennaelements,eachhavinganidentical current distribution differing by,at most, a complex scaling constant. The similarity of current distributionimpliesidenticalorientation of theelements.The fielddue toanelectric current source J(r) canbe evaluated usingthe equations inCharts1 and2. Some elements inan antenna array are shown inFig.9.Using element number 1asthereferenceelementinanN-elementarrayweestablishourcoordinate systemdefinitionssothatalinear shiftof thenthelementwillcauseittobeco-incidentwithelement1.Inthiscasesuchashiftbetweencoincidentpointsis r n- rlor r' n- r' I.The relationshipbetweenthe element current distributions is givenby (56) Themagneticvectorpotentialinthefarzoneforthenthelementis . .1e-jkr f.., An(r)= aneJkr'(r"- rl)4- -- Jt(r')eJkr.r dv' :ITrv (57) or equivalently, Chart 11.Evaluation of Field Components for a Short Dipole of Constant Current in Presence of a f4'initely Conducting Ground z z II to'P.o.y tl' P.o'(] 10 J( VerticalHOrizontal StepI. The field,due to aprimary source infree space,isevaluated: -jk(r-h cos 8) Er( 0, 91)=0jWlJ.o/oL sin 0 e4.7lr Ay= loLEr(O,91)=jWlJ.rlxtxA .e-jk(r-h cos 8) = }wlJ.o/oLA(-coS91cP - cosOsm916) .7lr Step 2.The field,due to the image source,is evaluated: ..e-jk(r+"oos8) E/(O, 91)= 6}wlJ.o/oLsmO e-jk(r+h cos 8) E/(O,91)= jWlJ.o/oLA.....(cos 914>+ cosOsin916) Step 3.Evaluate the ground-reflected fieldusing(54): ER(O,91)= RyE/(O,91)ER(O,91)= RyE/(O,91)+ (Rh- Ry)(E/(O,91)'44> e-jk(r+h cos 8) = jWlJ.oloLA.....(-Rh'COS 91 4>+ RycosOsin 916) Step 4.Evaluatethe total fieldusing(55): -jkr. E( 0, 91)= 6 jWlJ.o/oL sin 0 _e - (elkh cos 8 + Re -jkh cos 8) 4.7lrv -jkr E(O, 91)= jWIJ.I..JoL e4.7lr[-4> cos 0 (ejkhOOS8+ Rhe-jkhCOs8) - OcosOsin91(ejkhCOS8- Rve-jkhCoS8)] Techniques for Low-Frequency Problems Fig.9.Antennaarraygeometry. (58) Thefar-zonemagneticvectorpotentialforthearrayisthen N A(r)= At(r)L aneikr'(r"- rl) (59) n=t wherethesummation,denotedby [(8, cp),isreferredtoasthearray [actor: N [(8,cp)=L aneju'(r" - rl) (60) n=t This factorization separates the element contribution or element pattern At(r) from thearray pattern [(8, cp),whichdepends onlyon therelativesourcestrengths and 3-36Fundamentals and Mathematical Techniques locations.Thisfactoringisinvolvedinaprocedurereferredtoaspattern multiplication. Theelectricandmagneticfieldsinthefarzonecanthenbecalculatedusing (16)and(17)inChart2: E(r)=jwp,1(0, cp)rxrxA(r) H(r)=-jk 1(0, cp)rxA(r) (61) Chart12presents theradiation fieldsforN-elementarrays of uniform current (Io)andequallyphaseddipolesofspacingd.Thecomputationsliavebeen performed using the preceding equations for two cases,namely, arrays of collinear elements and parallelelements. The procedureusedinderiving the directive gain forshortelements(kL~1)followsthatin[26].Resultsobtainedusingthe expressions fordirective gainat broadside(0=nl2, cp=n/2) indecibelsrelative to a single isotropic sourcearepresented versus element spacing d in Fig.10 fora collinear arrayand inFig.11fortheparallelarray.The number of elements Nis aparameter. SomereferencedataforpracticalarraysisalsopresentedinFig.12[21]. Inthepreviousdiscussionthecurrentdistributiononeachelementwas assumed to be known and unperturbed by the presence of the other elements in the array.For adequate interelement spacing thisisavalidassumption,but forclose spacingtheinteractionof theelementscancauseperturbationsof theelemental currentdistributionsand,of course,theinputimpedancefortheelements.The rigorous treatment of these effects requires a consistent treatment of the boundary-valueproblems,asdiscussedlaterinthischapter. 3.Integral Equations in Antenna Analysis Inpreviousdiscussionsthecurrentdistributionsonradiatingelementswere assumedtobeknown.This,however,isgenerallynotthecasesincetheprecise description of the current distribution on a metallic structure such as an antenna in thepresenceofanexcitingsource,suchasavoltagegeneratoratitsterminals, involves the solution of a complicated boundary-value problem.In the following we willdescribetheintegralequationswhichcanbesolvedfortheunknownsource distributionsinducedbyspecifiedexcitations,andinPart2wewillconsiderthe numericalsolutionof theseequationsandsomeassociatedissues. Perfectly Conducting Wires and Bodies Herewewillfocusour attentiononradiating structures composedof perfect electricconductorsoverwhichtheboundaryconditionsgivenby(33),(34),and (35)musthold.Furthermore,wewilldevoteourattentionmainlytowirestruc-tures,withconductingbodiestouchedonbriefly. Inordertofacilitateensuingdiscussionsconcerningintegralequations, questions concerning validity of each specific equation, the existence or uniqueness of solutions,and various featuresof the limiting process whichreduce the integral representationsforradiatedfieldstointegralequationsforunknownsourcedis-Chart 12.Radiation Fields for Vertical Collinear and Parallel Arrays of Short Dipoles Supporting Equal Amplitude and Equal Phase Currents z z JIC.~II .1(1IIII~ l I til til x J( Collinear Parallel Elemental Vector Potential e-ikr sin [(kLl2) cos 0] AI(r)= Z 4Jlr(/nL)(kLl2) cos 0 N f(O.rp)=2:eik(II-I)tlcosO= eilk(N-I)dI2ICososin[(kNdI2)cosO] 11=1sin [(kdI2) cosO] Er(r)= 6jw,u(/nL)sinOe-ikrsin[(kLl2)cosO] 4Jlr(kLl2) COS 0 Xeilk(N-I)dI2Icoso sin [(kNdI2) cos 0] sin [(kdI2) cos 0] Go(O.rp)= sin20 sin2[ (kNdI2) cos 0]/sin2[ (kdI2) cos (J] N-I e-ikr sin [(kLl2) cosO] A1(r)=Z 4Jlr(/nL)(kLl2) cos 0 Array Factor (all= 1) f(O. rp)= eilk(N-I)dI2IsinOsinq> sin [(kNdI2) sin Osinrp] sin [(kdI2) sin 0 sin rp] Far-Zone Field Er(r)=6jw,u(/nL) sin 0 e-ikr sin [(kLl2) cos 0] 4Jlr(kLl2) cos 0 xeik(N-I)(dI2)sinOsinq> sin [(kNdI2) sin Osin rp] sin [(kdI2) sin 0 sin rp] Directive Gain Go(O. rp)= sin20 sin2[(kNdI2) sin 0 sin rp]/sin2[(kdl2) sin 0 sin rp] N-I II 2N/3+ 42:(N - m)[(sinmkd)/(mkd)3- (cosmkd)/(mkdf]2N/3 + 22:(N - m)({[(mkdf - 1]sinmkd}/(mkd)3 + (cosmkd)/(mkd)2) m=!m=1 3-38-14 12 / II ~ '/ !/iI '/ IV/, I / Ii 2 IlL ~ o o If /L ~ ::...-0.2 Fundamentals and Mathematical Techniques ./ ........... V --~ V ""' ...... L """ N-;7 /' V -......... ....... ./V ............ """ V ~ V --...... ./ " ........ L V ~ V ~...... L ~ V )/ ~ -~--... V V V --./ V V V ~ -............. ./ ~ .J' / V ~ V/ V V ~ 2 ~ V lL "" V 0.40.60.81.01.21.41.6 ELEMENT SPACING (d/A)-WAVELENGTHS Fig.10.Gainof acollineararrayof shortdipolesrelativetoisotropicsource. tributionsarenecessarilyglossedover.Forfurtherinformationthereaderis referredtoStratton[2],Silver[3],andPoggioandMiller[7]. The fieldequations inChart 1 are a convenient point tobegin the construction of theintegralequations.The electric-fieldrepresentationswillbeusedsince,for antennas,thedriving sourceismosteasily specifiedinterms of voltageor electric field.Later, themagnetic-fieldrepresentation willbe discussedinrelationtolarge conductingsurfaces. The representations inChart 1 are for the fieldsdue to volumetric distributions of sources.For electriccurrentsourcesconstrainedtoasurfaceS(whichmaybe consideredtobetheboundaryofVinChart1)theintegralrepresentationsare simplymodifiedinthatthevolumedensitiesbecomesurfacedensitiesandthe volume integral becomes a surface integral over S.The integral representations for theelectricfieldduetoelectricsourcesoverSareshowninChart13. Thegeneralboundary-valueproblemof determiningthecurrentdistribution onaperfectelectricconductingsurfaceisapproachedusinganintegralequation. TheboundaryconditiononSisstatedas n(r)xEt(r)=0,rES(62) Techniques fOF- Low-Frequency Problems 16 -~ ~ \ / /" / ~ -~ " ,r ",. ~ \' N-Y ./.JII / 14 / [!Y / ",. ~ \1 V/ lY /' ~ ' V j / .-L 1/1 ~ / y / \ ~ V' ,/ ./ "'"'-./ I II / ~ '/ ', \-V /~./ II Vi / Y ~ \ 1\~ /..,.. VII 1/ / '..... ~ ~ VIIJ ~ , ~V ./ ............ VI 1/ 2/ / "" ~ il J .;V ........ ~ ..... r// ~ / 2 j ~ 1/ '/ ./ . ~ o o0.20.40.60.81.01.2 ELEMENT SPACING (d/}..)-WAVELENGTHS .JII " / ./ ..,.. ." V ~ 1.4 Fig.ll.Gainof equiphasedshort-dipolearrayrelativetoisotropicsource. 3-39 . with n(r) the outwardly pointing normal to Sand E,(r), the total electric field at the surface.Thetotalfieldiscomposedof anincidentor drivenportionEi(r)anda portiongeneratedbytheinducedsurfacesourcesJ.\.(r')andpAr')referredtoas E(r).Theboundary conditionthenrequires n(r)xE(r)= -n(r)xEi(r) ,(63) ThefieldcomponentE(r)isgivenbytheintegralrepresentationsinChart5but withtheobservationpointronthesurface.Theprocedureoftakingthe observation point to the surface must be performed delicately due to the singularity in