Balanis-Ch07-Exp

Balanis-Ch07-Exp - Problem 7.2 Design a linear array of...

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Unformatted text preview: Problem 7.2 Design a linear array of isotropic elements placed along the z-axis such that the zeros of the array factor occur at 0 = 0°, 60°, and 120°. Assume that the elements are space M4 apart and that the progressive phase shift between them is 0°. a) Find the required number of elements. b) Determine their excitation coefficients. c) Write the array factor. d) Plot the array factor pattern to verify the validity of the design. 7—2. Q. For (3:0 and d:7\/4 «12 = kdcose+e = gCQfi) 0039 = %oose which reduces “to 6=0°21|2=1|21=19E =>z.= 8=60°; 1p=q21=lri ‘3) e=lzo°= 1]» = wa= at-g)='—g—=> 2.: mm: “emfore the array factor of (7-5) Can be written as AF: (2 —z.)(z — 2;.)(5-23) =(E—jX-‘t' *f‘liCHJDXE — 'FLEU’J‘ U .2 13+ (H551); —(fi+3)z*+z3 j AF: (Medan/1+ (er32)e5m553z + (er323 e583“) 2* + (1323 Anather Form is AF = (I)é5"/1+(1.W31)é°'?553‘£ + (|.t732)é' AF: Owl-0:. a + a3 Four (N=4) elements are 'rEqu‘weol. b. The excioflon Coefiicients are equal 'to a¢= c.)e5“’z=é5'rh= mew/2 =1 Ha a: = tuna.) 630'7553 = I.|732../_o.?553 as, = Ctflsz) ejs'r'sqr. (L731) 6712's“: Hulda“? = Will—2.52.6 04. = l = téo 3.2.526 z*+ (1)723 a" + 01.23 C. The army fucker is given Ly an} of the two above forms. Problem 7.5 Determine the current distribution and the approximate radiation pattern of a line—source placed along the z-axis whose desired radiation pattern is symmetrical about 8 = m’2, and whose shape factor is given by: 5m): {1 40°46$I4o° 0 elsewhere 40°é64l40° => £5 £6 e 2%" Using E“tuna” (7-8“) ' 4% 1-3054) sjé k/cr3054) for ‘ka=0, and Mafia/>40". Thus k ’ ’ 1w) = m [ 51n(7%)/6%:fi)] and 1 SF(9)d 1' SFCGh : % {Si [flaw + L30154fl " 31%"(039 ‘ l-3654 )1} Fur i=5/\ cmd. fl=lo?\, the normali3ed current distribution is shown In Fig, Page.) and the patterns in Figure, P7565)- 1 . 0.8 Line Source L 1: 51) Line Samoa (1: to 7K.) 0.6 0.4 0.2 Normafli eoi Current I (£0 —1 0 , 1 2 3 4 E! Source Posnion 2/1. 0.6 0.4 . 0 4 1 14 1 1 F13 . (b) 0 2° 0 60 Bo Obgepvgotionm Angmlje G (dejTEeSJ Problem 7.6 Determine the current distribution and the approximate radiation pattern of a line—source placed along the z- axis whose desired radiation pattern is symmetrical about 8 = 1:12, and whose shape factor is given by: SF(8) = 1 60° 5 9 51200 SF(9) = 0 elsewhere Let I = 5)\. and 10%. Compare the reconstructed patterns with the desired one. 7-6. U898 Equation (’7—8a), 42/2 $5 5 Ira/2 for 'i'lz=0 and 1200’s “30°. Thus Itz’)= 9%[Sini%’)/(%’)] and Sled tsFiela =TIT.{81[,%“CCOSB+ 'EH “ Si. [ii-llama - i1] } For ll: 51 and fl=t0)\, ‘t‘ne normlisecl current distribution i‘s Shown in Fla. P76 (0s) and fine patterns in Fijure P’Zé (b). 1 0.3 Line Sources (1-1370 Line Source “'MA) 0.8 0.4 0.2 Normaliaeol Current lav ‘ —5 ' —4 *3 —2 -1 o 1 2 0.3 0.6 0.4 o \A‘vfi 0 2o 40 so so 100 120 140 160 0 obSerl/Q‘HO n Angle GCdeflreu) Fig. PV-Q Problem 7.10 Design, using the Woodward-Lawson method, a line-source ofl = 57L whose space factor pattern is given by SF(9) = sin} (0) 0° < 0 < 180°. Determine the current distribution and compare the reconstructed pattern with the desired pattern. 7'10. For a desired pattern of SR6) = sin’fe), 0°s 0 e 180" the desireei excitation Coefficients and other pommetem for [=5A are listed below. M 9m (deg) am C= SFC6=9m) m 9Mqu aw. = 8F (6 =6.) 0 ?0° 1.0000 1 7346" 0. ?406 -1 Iol.54 0.?406 2 63 42° 0. trifle —2 “8.58 0 .‘T6‘?8 3 55_,3° 0.5120 -3 126.87 0.5120 4 35,870 0-3460 -4 14.313 0.2100 5 o 0 ‘5 ISO. 0 The Potter-n Computed. Using these Parameters and Equation C748) - is shown plotted in Fig. Pmor- '7. u. . Itis Identical tom obtained Usihj discrete array modeling. LO _. .. _ _ - Desired S tdh $25.11ch 01:52 b. N=1i(d=qlz) SPACE FALToRISFtOll [tqu rue-roalAFcell 0° 50" 90° Bo" |"|0° Obsevuo‘tion Angie Sweat-9.25) 7.12. In target-search, grounding-mapping radars, and in airport beacons it is desirable to have the echo power received from a target, of constant cross section, to be independent of its range R. Generally. the far-zone field radiated by an antenna is given by Imam 6! = C0 R where C0 is a constant. According to the geometry of the figure R = hfsin (6) = h csc (l9) / / //' / //.‘./ For a constant value of 9b, the radiated field expression reduces to Ina¢=afl=qmm R R A constant value of field strength can be maintained provided the radar is flying at a constant altitude It and the far-field antenna pattern is equal to f(3) = Czcsc(9) This is referred to as a cosecant pattern, and it is used to compensate for the range variations. For very narrow beatn antennas, the total pattern is approximately equal to the space or array factor. Design a line-source, using the Woodward-Lawson method, whose space factor is given by 0.342 050(3), 20° 5 3 S 60° 0 elsewhere |flRfl¢=%fl=Q SF(0) = { Plot the synthesized pattern for I = 20A, and compare it with the desired pattern. 7'12 Since "the dew'lrecl pattern ('8 Olden b y :1» SF(6)={ 0-342 cscce) 20°\<6.g€o° 0 ElSewhere flue required parameters for a Woodmarol line-source design of 9.=20)\ are listed balsa) “1 9n G193) an: -‘- SFC9= an.) m Gamay.) am =8F(e=6m) 0 ‘20. o 1 37. I3 0 -1 3287 o 2 842.6 0 —2 35.14 0 3 31.37 0 -3 98.63 0 4- ’78.46 0 ‘4 Iol.54 0 5 75.52 0 -5 104.48 0 6 7254 O ‘6 [07.46 0 7 69. 51 0 -"l ll0.4.9 0 8 66.42 0 -8 “3.58 o 9 63.26 0 ‘9 H6. ‘74 0 lo 60. 0.3949 -I0 120. 0 H 56.63 0.4095 -I t 123.37 0 I2. 53, 13 0.4275 ~12 12687 0 13 4?.46 0.4500 r[3 [30,54 0 14 1155‘] 0. 4189 44 I34.43 0 I5 4L4! 0.5m —l5 L385? 0 I6 85.87 0.5'700 -|6 I43.13 0 17 31.79 0.6492. ~17 I432! 0 is 25.34 oneae -I8 154.|6 0 19 IBJ‘? 0 -I‘3 HSLBI 0 20 o 0 -20 I80 0 The pattern comptrted using these parameters and Equwtion (7—!8) is shown plotted in Fig. P742, LO 7212 CCont'd) -— -- - 'Des'wecl Slantkesfied (Q = :01) 0.5 space FACTOR 1 men ' ' fl Io Obsaumfion Angie. Oqurees) ( Fig. P'].I=-> 110° 7.14. For some radar search applications, it is more desirable to have an antenna which has a square beam for 0 S 6 S 90. a cosecant pattern for 90 E 6 5 6m. and it is zero elsewhere. Design a line-source, using the Woodward-Lawson method, with a space factor of 1 15° 5 B < 20° SF(6) = 0.342 c'sc(0) 20" S 0 S 60° 0 elsewhere Plot the reconstructed pattern for l = 20A, and compare it with the desired pattern. For the desired Pattern of _ .1. 15‘ $ 6 $20“ 3H8) ' { 0.842 CSCCG) 20' S. e 5 60° 0 Elsewhere time required parameters for [1:201 are identical 'l‘o those listed in the Solution of Wob- '7. 12 execp‘l‘ that for m=I9 “H’th Should be equal to W=l3 9m=|9-|3° Gm: SFC6=le=1 The computed poi-fern using Equation (7—!8) is shown in 1333-97—14. LO _ _ _ _ __ "Desi “A ..._.__. gain United (0 = 2012) SPACE FACToR Ichoil |o° 5c“ m° mo ( _ Pr], (:3) Close: nation Angie. Qtdaancs) l'lO° ...
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This note was uploaded on 11/01/2009 for the course EEE 443 taught by Professor Balanis during the Spring '08 term at ASU.

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Balanis-Ch07-Exp - Problem 7.2 Design a linear array of...

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