1ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityLecture 33Antenna Arrays and Phase ArraysIn this lecture you will learn:• Antenna arrays• Gain and radiation pattern for antenna arrays• Antenna beam steering with phase array antennasECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityTwo Hertzian Dipoles – A Two Element Arrayyz()()0300ˆhrdIzrJrrrr−=δx0hr1hr0Jr1Jr()()1311ˆhrdIzrJrrrr−=δ()()[]()()()φθφθθπηθθπηθ,,, sin4ˆ sin4ˆ 1010.ˆ01.ˆ000.ˆ1.ˆ0FreIIeIIerdIkjeIeIerkdjrEhrkjhrkjrkjohrkjhrkjrkjoffΕ=⎥⎦⎤⎢⎣⎡+=+=−−rrrrrrrOne can write the E-field in the far-field as a superposition of the E-fields produced by all the elements in the array:Consider first an array of just two Hertzian dipoles:Element Factor Array FactorEach antenna in the array is an “element” of the array• The “element factor” is just the E-field produced by the first element if it were sitting at the origin• The “array factor” captures all the interference effects2ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn N-Element Antenna Array - IConsider an N-element antenna array where the elements are: - all identical (all loops, or all Hertzian dipoles, or all Half-wave dipoles, etc) - all oriented in the same way- but with possibly different current phasorsLet the current phasor of the m-th antenna be ImLet the position vector of the m-th antenna bemhryxmhrzmI0I1I1−NIzOne can write the E-field in the far-field as a superposition of the E-fields produced by all the elements in the array:()()()()φθφθφθ,,, ,, 10.ˆ0FreIIrrENmhrkjmffmΕ=∑Ε=−=rrrrrECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn N-Element Antenna Array - II() ()()()φθφθ,,, 10.ˆ0FreIIrrENmhrkjmffmΕ=∑Ε=−=rrrrrr()()()()222,, ofpart angular4ˆ.,,φθφθπφθFr,rPrtrSG⎥⎦⎤⎢⎣⎡Ε∝=rrr()()()max,,,φθφθφθGGp =Gain:Pattern:yxmhrzmI0I1I1−NIz3ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn N-Element Hertzian Dipole Phase ArrayxzyN-element Hertzian dipole arrayaαjeα2je()α1−Nje• Current magnitude is the same for all the dipoles• Current phase difference between successive dipoles is α1αjmmeII=+1()() ()()() ()()210cossin210cossin.ˆ10.ˆ0,,0∑=⇒∑=∑=−=+−=−=NmmakjNmmakjmjhrkjNmhrkjmeFeeeeIIFmαφθφθαφθφθrr() ()()() ()()⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+=φθαφθαcossin21sincossin2sin22kakaNArray Factor:Element Factor:() ()rkjoerdIkjr−=Εθπηθφθsin4ˆ ,,0rECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityxzyN-element Hertzian dipole phase array antennaaαjeα2je()α1−Nje1() ()() ()()() ()()⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+=φθαφθαθφθcossin21sincossin2sinsin1,2222kakaNNpPattern:Coming from the element factorarray factor() () ( )222,sin1,φθθφθFNp =An N-Element Hertzian Dipole Phase ArrayNdIkPo2012πη=()()() ( )222,sin234ˆ.,,φθθπφθFNrPrtrSG ==rrComing from the element factorarray factor4ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn N-Element Hertzian Dipole Phase Array: MaximaLets look at radiation in the x-y plane:φa()φcosayxRadiation going in the φdirection from adjacent elements will add in-phase, and one will have a big maximum in the radiation pattern, provided: (){KK,3,2,1,0 2cos=±=+nnakπαφ()φπθ,2=pφ2420ππα=−==kaN()2,2φπθ=F2420ππα=−==kaN(degrees) φnullssmallmaxima(side lobes)bigmaxima(main lobes)main lobesside lobesECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn N-Element Hertzian Dipole Phase Array: Maximaφa()φcosayxCondition for a big maximum in the x-yplane:(){KK,3,2,1,0 2cos=±=+nnakπαφAt a big maximum the value of the array factor is:()()()()()2222max cos21sincos2sin,2NkakaNF=⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+==φαφαφπθAnd the value of the antenna gain is:()2,2φπθ=F2420ππα=−==kaN(degrees) φnullssmallmaxima(side lobes)bigmaxima(main lobes)() () ( )NGFNG23,2,sin23,max22=⎟⎠⎞⎜⎝⎛=⇒=φπθφθθφθ5ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn N-Element Hertzian Dipole Phase Array: NullsφayxRadiation in the x-y plane:()()()()()⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+==φαφαφπθcos21sincos2sin,2222kakaNFThe array factor will give a null in the radiation pattern provided: ()φcosa(){KK,3,2,,0 2cosNNNnNnak≠±=+παφ()2,2φπθ=F2420ππα=−==kaN(degrees) φnullssmallmaxima(side lobes)bigmaxima(main lobes)ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn N-Element Hertzian Dipole Phase Array: Width of Big Maxima()2,2φπθ=F2420ππα=−==kaNφφayxφ∆Angular width of a lobe associated with a big maxima:For a big maxima:() { ,3,2,1,022cos KK±±±===+ nNnNnakππαφAt the nulls nearest to the above maxima we must have: 122cosNnNak±=+⎟⎠⎞⎜⎝⎛∆+παφφWhich can be solved for ∆φ– and for N large (N >> 1) one gets approximately:() sin14φπφkaN≈∆6ECE 303 – Fall 2007 – Farhan Rana – Cornell Universityxzyaka = παjeα2je()α1−NjePhase difference between successive elements is αN = 2 N = 10()φθ,p()φθ,p1N-element Hertzian dipole arrayAn N-Element Hertzian Dipole Phase Array: ExamplesECE 303 – Fall 2007 – Farhan Rana – Cornell Universityxzyaka = π / 2αjeα2je()α1−NjePhase difference between successive elements is αN = 2 N = 10()φθ,p()φθ,p1N-element Hertzian dipole arrayAn N-Element Hertzian Dipole Phase Array: Examples7ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityAn (N+1)-Element Hertzian Dipole Binomial Arrayxzy(N+1)-element Hertzian dipole arraya• Current phase is the same for all the dipoles• Currents magnitudes in the dipoles follow a binomial distribution()!!!0mNmNIIm−=()()() ()()() ()() ()⎥⎦⎤⎢⎣⎡=+=⇒∑−=∑===φθφθφθφθφθcossin2cos21,!!!,222cossin20cossin.ˆ0.ˆ00akeFemNmNeeIIFNNNakjNmmakjhrkjNmhrkjmmrrArray Factor:Element Factor:() ()rkjoerdIkjr−=Εθπηθφθsin4ˆ ,,0rπλ=⇒=kaa2ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityLets look at radiation in the x-y plane:φa()φcosayxRadiation going in the φdirection from adjacent elements will add in-phase, and one will have a big maximum in the radiation pattern, provided: (){()nnnakπφππφ±=⇒=±=cos2,3,2,1,0 2cosKKAn (N+1)-Element Hertzian Dipole Binomial Array: Maxima and Nulls() ()⎥⎦⎤⎢⎣⎡==φπφπθcos2cos2,2222NNF()