1- Consider an array of six elements with element spacing d = 3 λ/8. a) Assuming all elements are driven uniformly (same phase and amplitude), calculate the null beamwidth. b) If the direction of maximum radiation is desired to be at 30 o from the array broadside direction, specify the phase distribution. c) Specify the phase distribution for achieving an end-fire radiation and calculate the null beamwidth in this case. 5- Four isotropic sources are placed along the z-axis as shown below. Assuming that the amplitudes of elements #1 and #2 are +1, and the amplitudes of #3 and #4 are -1, find: a) the array factor in simplified form b) the nulls when d = λ 2 . a) b) 1- Give the array factor for the following identical isotropic antennas with N and d. 3- Design a 7-element array along the x-axis. Specifically, determine the interelement phase shift α and the element center-to-center spacing d to point the main beam at θ =25 ° , φ =10 ° and provide the widest possible beamwidth. Ψ=x kdsincosθφα +⇒= 0 kd sin25cos10 ° °+⇒=− αα 0.4162 kd nulls at 7Ψ nnπ2 nn π 2 =±, = 1,2, ⋅⋅⋅⇒Ψnull =±7 , = 1,2,3,4,5,6 α kd2π n kd d λ α −=−7 ( =⇒= 1) 0.634 ⇒= 0.1 ⇒=− 0.264 2- Two-element uniform array of isotropic sources, positioned along the z-axis λ 4 apart is seen in the figure below. Give the array factor for this array. Find the interelement phase shift, α , so that the maximum of the array factor occurs along θ =0 ° (end-fire array). Nkd sin (cosθ − 1) 1 1 sin[]kd (cosθ − 1) ()AF =2 =,Let 2x =− kd (cosθ 1) n N kd 2 kd sin (cosθ− 1) sin (cos θ − 1) 2 2 1sin2x 12sin x cos x ()AF== == cos xkd cos(cos[][]θ −= 1) cos kd cos θ + α ) n 2 sinx 2 sin x π π maxatθ=⇒=−0 α ( since kd = ) 2 2 1- Aralarındaki faz farkı π 2 olan iki tane izotropik anten y ekseninde y =± λ 4 noktalarına dizilmi ştir. Hangi φ açılarında bu anten dizsinin elektrik alanı sıfır olur? 1- Two isotropic antennas are placed on the y-axis, at y =± λ 4 with an intrinsic phase shift of π 2 . Given that the azimuthal angle, φ , is defined as the angle from the x-axis, for what values of φ are there nulls for the electric field pattern in the xy plane? 2π λ sin[ 1 N (α+ kd cos θ ) ] kd==πθ,() AF =2 , θφ =−→π λ2n N sin[] ( α+ kd cos θ ) 2 sin[]1 N (α+ kd sin) φ sin([]π + πφ sin) π AF ()φ =2 =2 ,π sin φ =→==°° φ sin−1 1 30,150 n 1 π π 2 Nsin([]2 α+ kd sin) φ 2sin([]4 + 2 sin) φ 2 1- k or β= 2 π λ is called the Phase Constant . sin(nψ 2) The formula was first introduced by Schelkunoff . sin(ψ 2) The phase change along a distance of a quarter wavelength is π 2 . sin(nψ 2) As the sign of the formula changes from positive to negative, the phase of sin(ψ 2) the pattern changes by π . sin(nψ 2) The maximum value of ) is n . sin(ψ 2) sin(nψ 2) The maximum value of occurs in any direction for which ψ = 0. sin(ψ 2) For a Broadside Array: the phase difference, α between elements of a “Broadside Array” is α = 0 . in general, as the element spacing is increased, the main lobe beamwidth is decreased. when the element spacing is d ≥ λ grating lobes are introduced as the number of elements increases, the main lobe beamwidth is decreased. For an End Fire Array: the array radiates in the direction along the line of the array . the phase difference, α between elements of a “Endfire Array” is α = ± kd . as d changes α must change to keep the main beam of the array in the same direction. when the element spacing is d ≥ 0.5 λ grating lobes are introduced. to increase the directivity one can increase the phase difference, α between elements. 2- 5 tane izotropik antenden olu şan anten dizisi aşağıda gösterilmektedir. a) antenler arasındaki faz farkı α ’yı bulun. b) Radyasyonun θ =110 ° ’de maksimum yapması için gereken d’yi λ cinsinden bulun. c) N=5 için AF n (Ψ ) grafi ğini çizin. d) Radyasyonun olaca ğı ba şlangıç ve biti ş Ψ açılarını bulun. e) Radyasyonu kutupsal koordinatlarda çizin. 2- A 5-element array of isotropic sources is shown below. a) Find the interelement phase shift α . b) Find the value of d λ if the main beam is directed in the direction θ =110 ° . c) Sketch AF n (Ψ ) for N=5. d) Find the visible range in Ψ . e) Sketch the polar plot of the radiation. απkd θ π kd θ ππ2 d π d λ =Ψ=4, cos +=⇒ 4 0 cos =−⇒ 4λ cos110 °=−⇒= 4 0.365 kdααπ kd π π π ππ −+≤Ψ≤ + →−0.73 +≤Ψ≤4 0.73 + 4 →− 0.48 ≤Ψ≤ 0.98 sin(5Ψ 2) sin5 (kd cosθ + π ) 2 4 N=⇒5 AF () Ψ= → AF ()θ = n 5 sin(Ψ 2) n 5 sin1 (kd cos θ + π 2 4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 3- Design a 7-element, uniformly excited, equally spaced linear array along the z-axis. Select the element spacing d and linear phasing α such that the beam width is as small as possible and also so that no part of a grating lobe appears in the visible region. Provide the angles where pattern nulls occur, and provide computed-generated polar plots of the patterns when o the main beam at broadside ( θο=90 ). 1sin(7Ψ 2) 126π N=→=7 AF ,α = 0, kd =→= d λ 7 sin(Ψ 2) 7 7 7Ψ Ψ 7 Ψ m nullsatsin= 0&sin ≠→== 0N mπ , m 1,2,3,,,& ≠ 1,2,3, 2 22 7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 3- Design a 7-element, uniformly excited, equally spaced linear array along the z-axis. Select the element spacing d and linear phasing α such that the beam width is as small as possible and also so that no part of a grating lobe appears in the visible region. Provide the angles where pattern nulls occur, and provide computed-generated polar plots of the patterns when o the main beam at ( θο=80 ). 1 sin(7Ψ 2) N=→7 AF = , Ψ= kd cos80 °+=→=−α 0 α kd cos80 °=− 0.1736 kd 7 sin(Ψ 2) 12 π −0.1736kdkd −≥− →≤ d 0.73λ →=− α 0.1736 kd =− 0.797 7 visible range 0 ≤≤θπ →−kd + α ≤Ψ≤ kd + α →−5.38 ≤Ψ≤ 3.79 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 3- Design a 4-element uniformly excited, equally spaced linear array along the z-axis. The main beam maximum should point to θ = 75 o. Find the d and α such that the beam width is as large as possible and that full main lobe should be visible. Sketch the polar plot of the radiation pattern. Ψ=kdcosθα +→= 0 kd cos75 °+→=− αα kd cos75 °=− 0.31. kd π kd−0.31 kd =⇒= kd 0.725π ⇒= d 0.3623 λα ⇒=− 0.31(0.725) π =− 0.225 π = 40.43 ° 2 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 6- Array factor of a 5-element uniformly excited, equally spaced linear array is drawn below. Find α , the progressive phase shift and d, the inter-element spacing for a pattern with exactly 3 side-lobes and the main beam at θ = 45 a and a null in the back lobe ( θ =180 a ). kd Ψ=kdcos45a +=→=−α 0 α kd cos45 °=− . 2 8πkd 8 π kd 0.94 π α −=−kd ⇒− −=− kd →= kd0.937π →= d 0.47, λα =−=− =− 0.66 π 5 2 5 2 2 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 3- Determine the progressive phase shift α for a 5-element array equally spaced with d = λ 2 along the z-axis so that the main beam occurs at θ0 = π 4 . Sketch the polar plot. 3- z-ekseninde yarım dalga boyu aralıkla dizili 5-elemanlı bir anten dizisinin ana huzmesinin θ0 = π 4 ’da gerçekle şmesi için faz farkı α ’yı bulun. Dizi faktörünü kutupsal çizin. λ kd 2πλ π d=, Ψ= kd cosπ +=⇒=−ααα 0 kd cos π →=− =− =− =− 0.87 π 2 4 4 2λ 22 2 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 3- Design a 7-element array along the z-axis consisting of λ/2 wave dipoles collinear with the z-axis. Specifically, determine the interelement phase shift α and the dipole center-to-center spacing d to point the main beam at θ =5 ° and to provide the widest possible beamwidth. Ψ=kdcosθα +=⇒=− 0 α kd cos θ =− kd cos 5 ° since we want a wide beamwidth, the dipo les should be as close together as possi ble. ⇒=dλ 2 (any closer thay would touch) so kd =→=−°−π α kd cos 5 π −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 3- Consider an array consisting of two isotropic radiators positioned symmetrically about z=0 along the z-axis. The elements have uniform amplitude and progressive phase shift α.