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Unformatted text preview: Problem 7.2
Design a linear array of isotropic elements placed along the zaxis 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°212=121=19E =>z.=
8=60°; 1p=q21=lri ‘3)
e=lzo°= 1]» = wa= atg)='—g—=> 2.: mm:
“emfore the array factor of (75) Can be written as AF: (2 —z.)(z — 2;.)(523) =(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: Owl0:. 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 zaxis 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” (78“) ' 4% 13054) 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 ‘ l3654 )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. 76. 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. [iillama  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 (11370 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. PVQ Problem 7.10 Design, using the WoodwardLawson method, a linesource 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 03460 4 14.313 0.2100 5 o 0 ‘5 ISO. 0 The Pottern 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 rueroalAFcell 0° 50" 90° Bo" "0°
Obsevuo‘tion Angie Sweat9.25) 7.12. In targetsearch, groundingmapping 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 farzone 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 farﬁeld 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 linesource, using the WoodwardLawson 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)={ 0342 cscce) 20°\<6.g€o° 0 ElSewhere
ﬂue required parameters for a Woodmarol linesource 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 linesource, using the WoodwardLawson 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=93° Gm: SFC6=le=1 The computed poifern using Equation (7—!8) is shown in 133397—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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 Spring '08
 Balanis

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