HW11 - _. r ~jk'r1 —jk'r2 (a) Et=E1+xE2+E3=2Eerk +Eoer...

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Unformatted text preview: _. r ~jk'r1 —jk'r2 (a) Et=E1+xE2+E3=2Eerk +Eoer +E°er 1 2 Where the center element is placed at the-origin. For far-field obserizations r1 :r—dcosfi r2 2, 7, + dcos 0} for phase variations r1 '2 7'2 2 'r for amplitude variations and , ' ' eflkr 2 ejlcd cos 9 —jkd cos 9 Et — E0 7, { + e ‘ } 1 e—jkr 1 . - . 2 I 2 _ ejkdcose —Jkdcosfi I E0 I 1' { [1 + 2(. + e ) e—jkr = E0 T {2[1 + cos(kd cos Thus the array factor is equal to ‘AF(0) = 2[1 + cos(kdcos 0)] = 40032 fl cos 9 l - which in normalized form can also be written} as AF(0)n = l + cos(kd cos 6) = 2 0032 cos 6) “flggdmwwm mm w w ‘ “mam a 4 I, -_~rm.<s__.mw.._~._i-a4aflui “a.” “A (b) The nulls of the pattern can be found using either of the above forms for the (c) array factor. For example one form the other form AF(9) = 1 + cos(kd cos 9“) = 0 2cos2 cos 0“) = 0 cos(kdcos 9n) = —1 3d cos 0,, = cos—1(0) = 2—”, kdcos 9n = cos—1(—l) = n1r,n = :l:1;!:3, . .. ' ‘ n = i1, :l:3, 9n = cos—1(nA/2d),n = :l:1, i3, :l:5, . . . which are hf ‘identical form. Therefore both forms yield the same results. Thus for d = /\/4 0n = cos‘1 = cos“1(2'n), n = :tl, i3, . . . => 2d d=V4 . Similarly the maxima of the pattern can be found using either of the two forms for ‘the array factor. For example (d) one form other form AF(9) = 2coszv<k2—d cos 6m) 2 cos cos 9",) = i1 AF(9) = 1 + cos(k,dcos (9m) = 2 cos(kdcos 9m) = 1 2 d , kdcos 9m = cos"1(1) = 2m7r, k? cos 9m = cos‘1(d:1)‘= m7r, m=0,i1,..., m=0,:l:1,... = '—1 7L" '" _ _ d 6m cos (d),m—O,:l:1,i2,... which are of identical form. Therefore both yield the same results.’ Thus for d‘='A/4. - ' r = ' = '_1, = 0m=cos‘1(4m), m=0,:1:1,i2’_,{m 0- 90- COS (0) em = cos-1,m = 0, :|:1,:|:2, . . ., m = 5&1: 91 = cos—1(4) => Does not exist. The same is true for other values of m (i.e, m = i2, i3, .. Therefore theyonly maximum occurs at 0 :90”; Computer Program Directivity I When J: /\/4 A1719): l-lcos2 cos 9) = 4cos2 (1 cos 9) . ‘ 2 4 Un = D. = 1.4384 = 1.5787 dB 9n = cos‘1(n)\(2d)), n = :l:1, :l:3, . . . Prob1em 6.1(d) Input parameters: The 1ower bound of theta in degrees = 0 The upper bound of theta in degrees = 180 The 1ower bound of phi in degrees = 0 The upper bound of phi in degrees = 360 Radiated power (watts) = 8.736 Directivity (dimension1ess = 1.4384 Directivity (dB) = 1.5787 Page 1 _—12£=—1222—1E=123. (a) 6n — cos :lsz) cos (i31) cos (in3),n , , ,... n7EN=3 2 48.19° _ . _ —1 _ = n_1._ 6’1—cos (i3) {131.810 72:2: (92:00?1 (i§)=does not exist I (b) 0m = cos“1 (2E1?) = cos-1(i2m),m = 0, 1,2, . . .. 0: 00 = cos—1(0) = 90° 1: 01 = cos‘1(:l:2) = does not exist m m ( (C) eh 2 2 [90° ~ cos‘1 = 2 [90° — cos—1 (1.3:;(20J ' = 2[90° ~ cos-1(o.295)] = 2(90° — 72.83°) = 2‘(17.17°) = 34.34° or __ 7 _1 2.782 _1 2.782 9;; — [cos (cos 6'0 — m) — cosh (cos 60 + 00:900 ' V N=3,d=/\/2 [ _1( 2 X 2.782) _1(2 X 2.782)] = cos —————————— — cos —— ‘ 67r 67r . = [cos—1(—-O.295) — cos—1(O.295)] ' =17.1 —— 2. 2: .° 0 68 7 83 3434 CO“ fiecao VM (d) D0 = 2N<§> = 2(3) "Do :QRRI #- =41“: cl (6) sin cos 0 + ,8)] sin cos 6') (Ann = i__2_______l N=3 _ ____2_____ N Sin [éwdcosg +36” =0 3sin (1 cos 0) d=’A/2 2 _o_l _ o -(A_F_)0=.0°._l_ '1 __ (AF)n(0 _ o )_ 3(AF)n(0— 90 )=> (AF)0=900 _ 3 _ 201ogm _ 9.54 dB. or approximately: W sine?" cos 0) N sin(377r cos 0) (AF)" = 3sin(§ cos 6) ‘7 _3?7'c039 ' o 2 ‘ o i W = 3 = 0.2122 = —13.46 dB ' AFN? = 0 ? §;(AF)”(9 = 0 ) => (AF)n(6 = 90°) 371‘ ' Prob1em 6.8(d) Input parameters: The 1ower bound of theta in degrees = 0 The upper bound of theta in degrees = 180 The 1ower bound of phi in degrees = 0 The upper bound of phi in degrees = 360 Output parameters: Radiated power (watts) = 4.2004 Directivity (dimension1ess = 2.9917 Directivity (dB) = 4.7592 Page 1 AU 9 < ¢ < 0 ( Y H. V fiO C @ Placing one element at the origin and the other at d distance above it, the array factor is equal to . = 1 + ej(l<:dcosa+fi) .1 . .1 ' 2 1%(kdcosow) [e_J§(kd°°59+fl-) -+e+’§("d°°59+’6)] = 6 ____~_.________.____.._ > 2 I .1 _ I. . AF(9) = 26350;de 6+3) cos (kd cos 0 + ,8)] which in normalized form can be written as a. fi=~kd=—3—7E (5) = 3. b. For d = /\/4, C. (AF)n = cos [gums a + m] A 4 2 (AF)n —— cos [ghos 0 — 1)] (AF) = 1 = cos (cos 0m — 1)] : em = 0? 17.] max (AF)n = 0.707 = cos [2(005 0n — 1)] => %(cos 0,, = cos—1 (0.707) + Zr— For+7r/4 ,=> cosflh — l = 1 => 0050;; = 2 => 0h =’cos“1(2) 4 _ => Does not exist 2 —§For—7r/4» =>cos0h—1 = ——1 => cos 1% = 0 = 6h 2 cos—1(0) = 90° = g radians A 47f 4 ' 91 = —— = — :: _ = , and o elm-627‘ (102 7r 1 273 1 049 dB . , _ 2 Z _ Computer ProEaIn (U — cos [4 (cos0 iDo = 1.9945 = 2.9984 dB Prob1em 6.9(c) Input parameters: The 1ower bound of theta in degrees = 0 The upper bound of theta in degrees 2 180 The 1ower bound of phi in degrees = 0 The upper bound of phi in degrees = 360 Output parameters: Radiated power (watts) = 6.3005 Directivity (dimension1ess) = 1.9945 Directivity (d3) = 2.9984 Page 1 ...
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HW11 - _. r ~jk'r1 —jk'r2 (a) Et=E1+xE2+E3=2Eerk +Eoer...

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