Theory of Unequally-spaced Arrays-hqC
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691
The outputs
of
the spaced oops are then sin
2 6 f K
The indicated bearing
d,
n-ill be given
b?-
cos 28 and cos M+h7sin
sin 20 K cos 28
COS
28
h- in
28
t an @
Let
28 2 ~ .
hen expressing all variables n terms
of
E
and
8
and rearranging, we get
If E is small,
A- cos 4e
2 E
1 A- sin 48
This
is
maximum \\--hen sin 4 6 = -K nd since K s
small em (K:2), which for
K=0.03
gives
;I
maximum
error of
0.85”.
ACKXOKLEDGMENT
The entire experimental program was carried out by
J .
Earnshaw while working as
a
summer student with
the Sati onal ResearchCouncil.
Theory
of
Unequally-Spaced Arrays*
Summary-Althoughecentlynequally-spacedrraysave
been shown to be useful, the theory has not been fully developed,
except or theuse of matrices, omputers, or theperturbation
method. This paper presents a newpproach to the nequally-spaced
array problem. It is based on the us e of Poisson’s s um formula and
the introduction of a new function, the “source position function.”
By appropriate transformation, the original radiation pattern
is
con-
verted into a series of integrals, each of which is equivalent to the
radiation romacontinuoussourcedistribution whose amplitude
andphasedistribution clearlyexhibit the effects of theunequal
spacings.
By this method, i t is possible to design unequally-spaced arrays
which produce a desired radiation pattern. This method
is
effective
in reating arrays of a arge number of elements, and unequally-
spaced arrays on a curved surface . Three examples are shown to
illustrate t he effectiveness of the method. The problem of sidelobe
reduction for the array of uniform amplitude, which was attacked
by Hamngton, is treate d by our method.
A
numerical example is
shown for 25-db sidelobe level. Also, the problem of secondary
beam suppression is attacked with the use of the Anger function.
The interesting problem of azimuthal frequency scanning by means
of an unequally-spaced circulararray salso shown, using he
method of stationary phase.
I . NTKOUCC-TION
u
T I L A F E 3 Y \-ears antennaarrays had
alwa).s
implied array s of equalspacing,simplJ-
because hose are he onl). caseswhich canbe
handledyonventionalmethodsnvolving
pol>--
nonlials. Schelkunoff’s heor\-
of
linear arraJ -s and the
Dolpf-Chebyshev arral- are examples [4]. Despite
its elegance and usefulness. the poll-nornial method has
1962. The work described
i n
this paper \ v a s sponsored by the
Receix-ed April 19, 1967; revised manuscript received June 11,
Cambridge Research 1.aboratories under Contract 10 4F191604)-
4098.
ton, Seattle, \Vash.
Department of Electrical Engineering, Cniversity of \Vashing-
three serious drawbacks. First, since the order of poly-
nomials ncreaseswith thenumb er of elements,com-
puta tion becomes more and more laborious for a arge
numberofelements.Thesecond is thefact hat his
pol~-nomial
method can be appliedonly to equall>.-
spaced arra>-s. The th ird is that th is Inethod can not be
used
for an arra?-
on
a
curved surface. The first is not
as serious
as the last two,
because fol a lax-ge number of
elements, hearr3J-s
can
often be approximated
bs-
a
continuous
sourcedistribution shown
bl- i’an
der
3Iaas
for a linear ar ray
[4].
[j] nd
b\. IG~udsen
or
a
c.ircular
arra .
[6]: but th ere s een~s toe no w a > - o treat
unequall\--spaced arra?-s
on ;I
line or on
a
curved
sur-
face
b>-
he po l~~non~ia lnethod.
The meth od described in this paper is quite effective
to treat un equall~.-sp aced arraJ .s ocated on
;I
line
or ;I
curve. Also, this method is effective for arr ays with a
large number of elements. In fact, this method provides
deeper understanding
oi
the relations bet\\-een discrete
arraJ-s and continuous arra).s. It must be pointed ou t
that hemethode~np lo)-ed in thispaper is different
f r o m th e work of Ksienski [ i ] on theequivalencebe-
tween continuousndiscreterra\‘s.isienski‘s
method is based on the assumption of zero radiation in
the invisible region and its extension or end-fire pa tter n,
while the method i n this paper has no such limitations.
I t
ma -
be added here that the o ther metho d of treatin g
a11 array asampleddatas\-stem
was
proposed by
Cheng and
3,121 [8],
but this also applies onll- to equally-
spaced arrays.
Although in recent.earsunequally-spaced rrays
have been shown to be useful,a theory has notbeen ful ly
developed.Forexample,Unz [9] used
a
matrix orm
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solution which requ ires the manipula tion of matrices of
order equal to
the
number of eleme nts. King, Pack ard,
and Thomas [ lo] , comput ed the pattern of the various
trial sets of spacing s, but no unified theory
was
given.
Sandler [ l l expanded a term foreachelement in a
series. Harrington[12], on theotherhand,employed
the perturbation technique to obtain reduced sidelobes
for an arra y with uniform amplitude. Andreason [I31
employed
a
computer to show the various possibilities
of unequally-spaced arrays.Also, th e use of approximate
integral techniqueswere reported recently
[ E t - [26] .
The method employed in this paper is different from
an y of t he abov e works. The method isbased
on
the
use of Poisson’s sum formula [14] and th e introdu ction
of a new function, called the “Source Position Function.”
By this m ethod, it is
now
possible to design unequally-
spacedarrayswhichproduce a desired patternchar-
act eris tic. It is essentially a new approach to the array
problem, and here are a number of possiblecases n
which this method may be applicable.
In this paper, three examples are shown to illustrate
the effectiveness of this method. They are the sidelobe
reductionproblem,secondarybeamsuppression,and
azimuthal frequency scanning antennas.
11. A N ARIL4I’ O F v ISOTROPIC
RXDIATOKS
WITH ARBIT RARYPACING
Let us consider the radiation pattern due to an array
of LV radiators as shown n Fig. 1. The radiatio n pattern
is given by
~ ( 0 )
I,,+n
sin
8
(1)
n=l
where
I,
is the current n the nth elemen t, and
,
denotes
th e position of this elementas measured froma reference
point 0.
Th e first st ep of our new formulatio n s he rans-
format ion of the radi atio n pat tern f (1) in the following
manner.
Let us rewrite
(1)
as follows:
E ( 0 ) c ( 4 .
n=l
Now, the Poisson’s sum formula will be applied to (2).
Th e Poisson’s sum formula is [I41
5 J-If(r)ej?mr*dv. ( 3 )
n=-m
Thus, (2) becomes
E(0)
2
”f(r)ejz”“”de
(4)
m=-w
where the limit f the integrat ion s from 0 to N because
the radiationE ( @ s th e finite sum and
(v)
vanishes for
v N .
t may be noted that the same result
(4) canbeobtainedby mploying heDiracdelta
functions, and this is shown in the Ap pendix. The l imit
of in tegrat ion in (4) is not the only choice. In fa ct, any
range which covers all th e integers from 1 to
LV
may be
used. Thus, (4) may also be written as
where 0 1.
Th e second ste p in our formulation s the introduction
of a new function, which we call the “source position
function.” Let us define the “source position function”
bY
s
This functiongives the position of the nth elemen t when
v 12. Thus
+e may also consider in (6) as
a
function of
s.
Thus
’ “ S)
(8)
and
a
We maycall the functionV(S) “source number function”
because this yields the numbering
of
each element when
is at the co rre ct osition of this e lemen t.
0
Fig. 1-Radiation
from
an array
of
unequal spacings.
Fig. 2-Source position function
or
source
number function V= V(S) .
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1962
693
The next step is the change
gration in (4) or (5) from v t o S.
of thevariable of inte-where
]Ye obtain
SAV
ka sin 0,
?rnsr
(10) x =x(y) normalizedourceositionunction
- l < x < $ l ,
NOW or our linear arraq. problem, conside ring the ex-
pression n
(I) ,
we write
where
I ,
I ( s J A,e-j n.
(13)
is the amplitude of the current in the nth element
and is the phaseof the current, and
4 s)
is a function
whichyields
at
and herefore his may be con-
sidered as an envelo pe of the amp litu de
of
each current.
(s)
is
a function which yields $n at Thus
-4
n (14)
Eqs. (11) and (12) clearly show the physical signifi-
cance of our formulation. 14,'e no te th at ( 12 ) is the radia-
tion patt ern of a continuous source distribution whose
amplitude is
(15)
and whose phase distribution is
Therefore, our formulation is n essence the transforma -
tion of the unequal ly-spaced arraq; into an equi vale nt
continuous source distribution.
Eq. (11)
is
an infiniteseriesform, but
t h i s
is not
a
serious disadvantage at all,ecause this is an extremely
rapidly convergent series. This may be recognized from
(12) .
Yve not ice that near
8 = 0 ,
the main contribution
comes from the source distribution such that the phase
-27nira(s)
is small. For example , if is zero, En
is
the main contribution near8 = 0 , and E+1and E-1 are
small corrections, while the rest of the terms are neg-
ligibly small. This poin t will be more fully demons trated
in late r sec tions bq- actua l examples.
I t ismoreconvenient t o rewrite (12) bymeans of
normalized variables as follows:
y normalized source number function
- l < y < + l .
Thu s, th e ac tua l position of t he nth element is
If X
s
odd,
N = 2 X + 1
I f
X
s even, N = 2 X
(1/2)
y n
for 0,
2 l f
1, 5 2 , M.
(21)
The tot al leng th of the arra y is not 2a, b u t
Lo
.cy-.%I>],
which is smaller than
2a.
See Figs.
3
and
4.
2
I.'ig.
3--Xormalized source position function. N
is
odd (:V=9). The
total
length is u ( - I 7 4 - X 4 j .
Fig. 4-Sormalized source position function. X is
even
( N = l O j .
The t otal le ngth is
u ( X s - X - ~ j .
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It must be noted that the only requirement for the
function y=y(x) is and -1, and that
y(x)
need no t be an odd func tion of
x.
The factor l)m(x-l)n (17) is simply 1 for :V odd,
but
1)"
for
N
even. This is caused by the fact that
the phase centerfor
N
even is at the midpoint between
two elements. RIathematically, this is due to th e sh if t
of by t he a mo un t f
111. LINEAR R R A YOF EQUAL
SPACING
Althoughourpurpose s o nves tiga te unequallJ7-
spaced ar rays, it s instructive to see wh at our for mula-
tion corresponds t o for the case of the ar ray of eq ual
spacing. In this case, in
x.
Thus
1 1
2 -1
E,(zi) J
A
( 2 3 )
We note that
is the radiation from continuous source with amplitude
istribution A andphasedistribution (x). More-
over,
E,(zJ) EO(U
Thus, E,(u) is the same radiation pattern as Eo except
th a t th e rigin of
u
s shifted by
msrN.
This point may be more clearly demonstrated by a
simple case of c onstant amplitud e with no phase varia-
tion. In this case
The radiation patternfor this case using ordinary array
theory is [ 3 ]
sin
(+iVkd
sin
0)
Y
sin
( kd
sin 0)
E A ( u ) ( 27)
Noting that n this case
(28)
e
write
( 2 5 )
as
sin
Comparing
29)
with we note that our formulation
is
in essence the expression of t he total rad iat ion field
(29) in a series, each te rm of which is (sin pa tte rn
exc ept tha t the rigin is shifted bynnlV. In other words,
we replace the linea r array by a series of continuous
sourcedistribution,each of which hassuch a phase
varia tion hat he peak occurs a t Thi s is illus-
tra ted in Fig.
5.
Fig.
5-The radiation
from
an arrayof equa l spacing.
sin
(a)
Array
Factor
N
sin?
N
sin
(b) Eo (c)
E-1
sin
(u
IVT)
1Vn
Iv . SIDELOBE
REDUCTIONF
A LINEAR RRAY
OF 'CTNIFORM AMPLITUDE
In order to illus trate the use of our formulation, let
us consider the problem attacked by Harrington
The problem
s
to red uce the idelobe level of the radia -
tion from a linear array with equal amplitude by mean
of
the unequal spacings.
Since we are interested in the range near 0, Eo
shouldgive
a
good approximation to he ot al field.
Thus, considering
A
x) 1 and
(x) =0, we
have
But his
is
exactly he same as the radiation from a
continuous source distribution with amplitude variatio
( d y l d x ) .
Thus , th e tec hnique for the continuous source
distribution can be directly applicable to thisase.
In hispaperTaylor'smeth od for a inesource s
employed
In thecase of Taylor's design, the solution is obtaine
as follows: Let
dY
f x).
a x
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Then,
we
write the solution in a series form:
+Q
f x)
--Iqe-7*rr
1 x ( 3 3 )
Y= -Q
Thus,
and -4 s given
b y
-4 Eo(qa). ( 3 6 )
Since the main beam
is
in the broadside direction 0,
and the radiation pattern s symmetric about
B
=0,
f ( x )
is an even function of
x.
Thus
A, - A p q
and rewriting
(35),
we
obtain
sin
.LL
Eo(2L)
i l
p =
1
24,
p=I 1
3 1
For
a detailedaccount of how (37) is obtained, he
reader should refer t o previous papers [15], [17].
Let
us
consider the example of 25-db sidelobe level.
lye let
The sidelobe ratio
is
given by
20 log
jcosh
A )
db.
( 3
Th us for
25
db , we get
A
1.29177~'.
(40)
For the choice of Q, the reader
should
refer to previous
papers [15], [If ]. He re , we choose Q =4 Thus, using
(36)-(38), we obtain
d o
1
d l 0.22974
A 2 0.00537
0.00662
0.0049. (41)
?Tow,
y(x)
is obtained from ( 3 2 )
The denom inato r normalizes such hat
y ( l j 1.
For our case, this normalization is automatically satis-
fied. Thus , no ti ng th at f(x j s even, we obtain from (33)
From (43), he position of each ant enn a is accurate ly
deter mined. The position of the nth ele men t s given by
(19). Inordero ompute
x,=x(y,,), (43)
must be
solved for
as a
function
of
Some elaborate*method
of inverting
(43)
may be employed
[18],
but for our pur-
pose, the comp uter was used t o find
x,,
or a given
In order to che ck the val idity of our meth od, we corn-
pute the pattern
of
our array.
The ra diation pattern for
N
odd is given by
where
And for
LV
even,
where
In Fig.6 he adiationpattern rom hearray of 21
elements
is
shown.
I t
may be noted that, near
L =0,
the
radiation pattern is very close to th e designed pa ttern .
The bearnwidth and the idelobes are about the same as
tha t
of
Ta>.lor's design for a
line
source. However, as
ZL.
increases beyond 77r, the sidelobes s ta rt going up. This
is due t o the effect
of
theother erms E,,,,
m # O .
In
general, for the main contribution comes
from
EP1
nd thus the assumption (30)
E(zi)= E o ( z ~ )s
no longer true. 4 s will be shown
i n
Section
I:,
the radia-
tion pattern near can be treated using
EP1.
In Fig.
7
the radia tion patte rn from the array of 20
elements is shown. It exhibits almost the same cha rac-
teristic
as
the array of 21 elements. I n general, the array
of
even numbers of elements requires smaller over-all
length for the same radiation characteristics.
The visible region
is
- k a u k a .
Thus,
if
2a
1OX
and
~\ '=20 ,
the average spacing s
X/2,
ut the osition
of
each element must
be
calculated
b -
(19), and the over-all length
a(x,-x-,)
is 9.05X i n
this example, which
is
smaller than
1OX.
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IRERANS ACTI ONS
O N
ANTELVATASANDROPAGATION
0
-30
-40
&Radiation fron an unequally-spaced array of
21
elements with
uniform amplitude.The designed idelobe evel s
25
db.
db
7-Radiation from an unequally-spaced array
of
20 elements
vi th uniform amolitude.T h e designed sidelobe lex-el is 2.5 db.
Fig. 8-Suppression
of
secondary beam near
= 2 0 r .
N = 2 0 and
2iVA1=5. (a)
O < u < 1 3 s .
(b) 1 5 ~ < u < 3 0 ~ .
V.
SUPPRESSIONF SECONDARY
EAM
Then, (47) becomes
Inhis ection, weonsider the robl em whichwas
E-l(u) e-j2.VAl
sin
by King,ackardndhomas
[lo].
T h e -1
s to uppress he econdarymainlobeby
of unequalpacings. A s Ale-j?-VAl sin
The secondary main beam occurs at -mnN in
2 -1
(18). Thu s, et us onsider the case m = 1. Iiear
N T , the radiation pattern is approximated by
(46) This becomes
us consider th e following simple orm for
dy/dx. J z ( Z )
'Anger fun ct ion defined by Jahnke, et al. [20]
I / T & cos sin x)dx. ( 5
A'
cos
n.x' and
let A' I ' '48) Extensive ables
of
th e Anger function for order and
argument ranging 10 t o are available [19]. Thus t
s, is possible toalculate (51).
A1
x sin
n.
For small .41, we can appr oxim ate (52)
by
E-l U)
J(u,s)-.%7(2NA
1) . (53)
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697
In Fig. 8 the radiatio n pattern from 20 elements with
uniform amplitude
is
shown together with the approxi-
mations (51) and (53). I t is seen that
(51)
is almost
identical to the pattern, but
(53)
shows a slight differ-
ence, as may be expected. However? for a large number.
of elements,
- 4 1
becomes smaller for
a
given argument
2:VAI. Thu s, t he a ppro ximatio n (53) will become closer
to the actual radiation pattern as Vincreases.
Th us , using
51)
or (53),
i t
is possible to predi ct the
behavior of the secondary beam. For example, by look-
ing at the ta bles
of
the Anger function, we can choose
the argument such that the peaks below a certain evel.
\,‘I.
LTNEQLAI,I.I--SPACEDARRAYS OK A CL -R~EDt-RFACE
The formulation presented in this paper is also appli-
cable to th e more general problem of unequ al]>--spaced
arra)rs on a curved surface.
Let us consider the ield due to an arrav with unequal
spacing which is located on a curve as shown i n Fig. 9.
In general. the field is given by a sum of th e con trib u-
tions from each element. Thus
(54)
l=l
where
I(s,)
is the current located at and G ( r ,
X,)
is
the appropria te Green’s function.
KOW,we can transform this series in the same m anner.
e(r)
2
oA”I(xn)G(rl
)ei*mrrdr,
(55)
m=-a
Thus, we note that we converted our array problem to
the problem of continuoussourcedistribution, whose
amp litu de is multiplied b y d v / d s and whose phase is
modified by
2 - m ~ c ( s ) .
I t maybe noted that this formula tion s quite general
and this is applicable not o n l ~ .o the radiation pattern
problem , but also the field a t any observation point.
1.1
I . CIRCVLAR
A R R A Y S
V,-ITH
U K E Q I T A LSPACINGN D
ITS FREQYEKCY
C:.IKNINGHARACTERISTICS
I n
this section, we consider an interesting problem of
frequency- scanning anten nas i n an azi~nuthal direction.
Let us suppos e that an arm>- of ant enn as is located on
a circle, an d excited b y
a
singleslowwaveguide. The
spacings between adjacent elements are
so
distributed
tha t, at an y freq uen c~- , only few of the elements are
excited
i n
phaseand that, at different frequenq-, he
elements a t different ocationsbecome i n phase. See
Fig.
10.)
This problem can be analyzed by our method.
Let us consider a case of th e rad ia tio n from a circular
arra?- in free space. I n the spheri cal coordin ate sy-stem
( r , e q5), the ntennas re ocated t
F I E L D
I.
Fig. 9-Field d u e
to
an arr ay on curved surface.
f ,
f r
Fig.
10-Azimuthal frequency canning by a circulararraywith
unequal spacings. T h e guide wavelength is a t i , X, a t fc a n d
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IRE TRANSACTIOAY
OlV
ASTEXA'AS A X D PROPAGATION
Xovember
We note here that for the case of equa l spacing, p In order to obtain a real stationary point, let u s choose
and
and
Wenotice that his is thesamesituation reatedby
Knudsen [6 j .
Now, let us consider
(60).
In general,
i t
is difficult to
obtain an explicit expression for the radiatio n patte rn.
However, we can obtain some useful information from
(60) byapplying hemethod of stat ionaryphase. Of
course the validity of this method depends on k a . For
large k a , this shou ld become closer to the a ctu al ra dia-
tionpattern.
Let us consider the stationary phase point. Foro(+),
we get
which yields
Now,
if th e arra). is excited by
a
slow waveguide,
Therefore, here s no real angle of 6 0 for which (64)
holds. In other words, the main beam corresponding to
this stationary point
is
in an invisible region, and the
radiation in a real angle may be small. For E,,,, the
stationary point 0 +s is given by
Then, (68) becomes
sin cos
A 1
k a 2
I t may be noted that when the right-hand side
vanishes and herefore
+ = + s = ~ .
Thus , this is one of
the stationary points. There may be other stationary
points in the visible region. But sincee desire only one
main beam, we desire only one stationary point. The
constant -41 must atisfycertainconditions.First, p
must be a single valued function of and here fore
d p / d o must always be positive. Thus,
which yields
1.
As anxample of p , letsakehere
2
p Alsin--,
dP 90
1+-cos-> 0 ,
d90 2
2
Th e second requirement for
41
is that in o ur problem e
desire only one stationary phase point from (70). This
requ ires that the magn itude of th e slope of the right-
hand side at +s
= T
s greater than the magnitudeof the
slope of the left-hand side
at
Th us we get
From (72) and (73) , we get
(74)
Let us now evaluate 60) by the methodof statio nary
phase. Then we get
which is similar to the one used for linear array. Th u s , I t is expected that the stationary point ist when
(66)
becomes
6 T Ifwe desire a sharp beam at this point,
f ( + o )
must be small. In fact, by the proper choice of A I , this
can be zero. This requires that
sin (68) A 1 - - 4 - .
k a
k a
(77)
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699
I n Fig. 11, the rays from the stationarJ - points are
shown for the case of ka l o r , v s 3 0 ~ ,nd ‘-1 as given
by
(77) .
It ma)‘ be noted that the antennasn the region
12Oo-
be noted that our formulation
is
particularlL- suited for
the arraJ- s excited
by
a traveling waveguide as shown
below.
-4. -neq~la l ly -Spa .ced
lot
A r r a y s
o n
T.V’acegz~ide
Let us consider
a
slot arm)- on a waveguide. Let us
simplify the situation by assuming that each slot radi-
ator has the sa me inte nsity, b ut its hase
is
determined
b5- the phase velocity of t he waveguide.
Thus, the current n the fzth slot is given by
I , e- lBSn
( 8 3 )
where s the propagat ion constant of the waveguide
and is the dist ance along the guide. The n
(18)
be-
comes
A s
he frequency varies, the stationary phase point shifts.
Thi s ,.ields an azimuth frequency scannin g antenna .
In Fig. 12 (next page), the frequency scanning char-
acteri stics of a circular array with unequal spacings are
shown for the case of
ka
lor , and
Y, 30n
a t
f = f c .
. I 1
is chosen to be 1.25. p s is assumed to be inde-
pendent of frequency , and thus, p,=p, , . The radiation
pattern is calculated from
n=l
where
the radiation pattern
is
approximated bl?
Let us consider the ran ge of zt which is near ?to . Then,
But this
is
exactly the same as the radiation from an
amplitudemodulat ed raveling -wave ntenna, whose
peak
is
a t and whose source amplitude distribution
is T h u s , the echnique used i n Section
I V
directly applicable.
AlultIXDoM1a1UfIX Ra
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8/20/2019 Theory of Unequally-spaced Arrays-hqC
10/12
Fig. 12-Frequency scanning of a
circular
array. (a) f=0.6 fc, the peak
wlue=O.i2.
(b)f=0.8fc, the peak value=0.87. f = f c ,
the peak
value=1.00. (d) f = l . l fc,
th e peak value=
1.07. (e)
f= . 2 f C , he peak
value=0.97.
AlultIXDoM1a1UfIX Ra
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8/20/2019 Theory of Unequally-spaced Arrays-hqC
11/12
If th e 11-aveguide s ;I fast v-aveguirle, -
[21].
and Goldstone and
Oliner (221. n fact , hi s unequall --spaced ar ra \- ma -
providenothereans of producing leak --wave
antenn;Ls.
If th e waveguide is a slow wave, ka and the peak
is in the invisibleregion. Thus,
this
ma\ - be useful t o
obtainn endfire amplitudemodulatedlow-wave
antenna.
B. Frequency Scannizg -4nte-nnas
I t is known tha t when an arm\- is excited b\v an ap-
propriate slow waveguide, the frequency scanning an-
tennamay be obtained
[ Z ] .
However, in practical
cases, the mpeda nce of thewaveguidevaries as the
frequency is varied. nparticular,when hebeam is
direc ted broadside, all the reflections from each element
add i n phase and the radiation pattern deteriorates.
If
the u nequa llp-spaced array is used effectively, thi s
imped ance problem may be considerablJ- reduced.
The frequency scanning antennas obtained when the
phasevelocitJ- is slow and in
(84j
is such hat
( 1 4 0
-nmLV.T)
s very small. For example, f nz
1
is take n,
the radi atio n in the visible region is approximated by
where
As
the frequency varies, varies and this produce s the
frequent . scanning. Eq. (86j is i n the same orm as
( 4 7 ) ,
and a similar technique ma>-be emplo)-ed and the im-
proved impedance characteristics ma\- e obtained.
The stud >- n the above topics are under
w a \ -
and the
result will be discussed
i n
a separate report.
I S .
COKCLESIOS
A new approac h to the arra>-roblem is shown? which
is partic ularll- suited for unequa lly-spaced arra5.s with
a large number of elements which m a ~ r
e
located 011 a
line or 011a curve.
I t is show n ha t an unequall>--spaced arra - of uni-
form amplitude w i t h an ' desired siclelobe level m a y be
des igned, using our metho d. .Also, the secontl;u- - beam
suppression and the azimuth frequencl- scanning circu-
lar array was discussed to show th e effectiverless of t he
method. -Other applications including unequall\--spaced
arr av on a raveling-wavewaveguide, heamplitude
modulated antennas, the leak\--wave antennas, a11d the
frequencJ- scanning linear antennas are discussed.
'I'hen, expanding i n a ange rom E to ~+i\r,
we get
But the sulnnlation
T h u s ,
And, therefore,
I t
may be noted th a t in essence this is the meth od used
b \ - Iinudsen
[6]
fo r his stud - on circular arraysof equal
spacings.
R E F E R E ~ T E S
[ l ]
S.
Schell:unoR,
-1
mathematical heory
of
lineararrays,''
Bell . Tech. J., ol. 22, . 80; January, 19-13.
[2]S.Sil\-er, "liicrowave antenna heory and design," McGraw-
[3] J .
I). Iirans,
"Xnte~was,"RlcGraw-Hill Book
Co.,
Inc.,New
l I i l l Book
Co.
Inc.. S e w I'ork, S. . ,ch.
9;
1949.
[4j I I . Jas ik, ";intentla EngineeringHandbook,' RIcGraw-Hill
Yorl;, Y.,
ch .
1; 950.
[j. G. J . Van der l laas , A simplilied calculation
for
Dolph-Tche-
Book Co., lnc., Sew York, N. Y. h.
2;
961.
l)>-schefl arra\.s," . d pp l . I ' ky s .,
vol.
25, pp. 121-124; January,
1951.
[6; L. Iinudsen,Radiat ion from ringuasi-arrays,'
I R E
July, 1956.
TIGAS. os
A S T E ~ S A S
ASD I 'IWPAGATIOX,
-4P-4, p. 452;
lisiemki, Equixdencebetwecn ontinuous nddiscrete
radiatingarrays," J .
Plzys.,
39, p. 335; February ,961.
I ) . I-swith arbitrarily distributed elements,"
I KOPAGATIOX,
~oL
I'-8,
p.
255;
May,
1960.
222-223: hiarch , 1960.
I l
-
8/20/2019 Theory of Unequally-spaced Arrays-hqC
12/12
702
IRERANS ACTI ONS
O N SNTER’:ITAS AhTD
PROPAGATION
November
[ l j ] T. T. Taylor, “Design of line-source antennas for narrow beam-
width nd low sidelobes,” IRE
TRANS.
K ASTESSAS A K D
[16] “Design of circular apertur es for narrow beamwidth and
PROPAGATIOX,
ol.
AP-3,
pp. 16-28; January, 1955.
low sidelobes,” IRE TRANS. N ~ T E I V K A S ND PROPAGATIOK,
vol.
AP-8,
pp. 17-22; January, 1960.
[17] A. shimaru and G. Held, “Analysis and synthesis of radiation
patterns fromcircular apertures,”
Canad.
J . Phy s . , vol. 38,
pp. 78-99; January, 1960.
[18]
Morse
and Feshbach,
op. cit.,
p. 411.
[19] G. Bernard and A. Ishimaru, “Tables of the Anger and Lom-
mel-{Veber Functions,’‘niversity of \{-ashington Press,
[20] Jaknke,
F.
Emde, and
F.
Losch, “Tables
of
Higher Func-
Seattle; 1962.
tlons,McGraw-HillBook Co. New York, N.
Y . ,
p. 251;
1960.
R. C. Honey, “A flush-mounted eaky-wave antenna with pre-
The
dictable patterns,” IRE TR~SS.
S
ANTEXNAS ND
PROPAGA
TION, vol. AP-7, pp. 320-328; October, 1959.
[22] L. Goldstone and
A . , A
Oliner, “Leaky-waveantennas
I:
rectangular waveguides, I R E
TRAXS.
N ANTEXXAS N D
[23] A. shimaru and
H. S.
Tuan, “Frequency scanning antennas, ”
I R E
TRAXS.
X ANTENSAS AXD PROPAGATION,ol.
AP-IO,
pp. 140-150; March, 1962.
Y .
T.
Lo,
“A
spacingweighted antennaarray,” 1962 IR E
INTERKATIOSAL OSVENTION RECORD, pt., p. 191.
[ X ] X. L. Maffett, “Array factors with nonuniform spacing param-
eter,“ IRE
TRASS.
N ASTENNAS ASD PROPAGATIOS,ol.AP-IO,
[26] J. L. Yen and
J.
L. Chow, “On Large Non-uniformly Spaced
pp. 131-136; March, 1962.
Arrays, ” presented at th e Copenhagen Symposium on Electro-
25-30,1962.
magnetic Theor y and Antennas, Copenhagen, Denmar k; June
PROPAGATIOK,
X-01.
AP-7, pp. 309-319; October, 1959.
Effect
of
an Unbalance
on
the Current
Along
a
Dipole
Antenna*
Summary-The effects of an unbalanced component
of
current
on
the distribution of the curre nt along
a
dipole antenna driven by
a two-wire transmission ine
has
beenstudied experimentally. It
was found hat an unbalanced component
of
current
on
the ine
signiiicantly influences the measured distributions of current along
antennas of shorter lengths. A quantitative study was made by de-
composing thecurrents ntosymmetricandantisymmetricparts.
The associated unbalance in the transve rse field distribution was
measured by a field probeandcorrelated with
the
ratio
of the
amplitudes of the symmetric to antisymmetric components of cur-
rent in the transmission ine and the antenna.
I . INTRODUCTIOS
ND
DE SC R I PT I ONF
APPARATL-s
w
E N a ymmetricdipoleantenna is center-
driven rom a transmission ine, hedistribu-
tion of curren t along the antennas significantly
affected
if
the line is unbalanced. The apparent admit-
tance of the antenna as determined from measurements
made along the line also depends on the degree of bal-
ance maintained on he ine. I t is the purpose of th is
paper to study theeffect of unbalanced currents on the
distr ibution of current along and the measured imped-
ance of a symmetrica l dipole when center-driven from
a two-wire line that may be unbalanced in varying de-
grees by asymmetrical excitation.
A general arrangement for the measurementss shown
in Fig.
1.
A cylindrical dipole antenna made f $-in brass
tubingwascenter-drivenbya wo-wire ransmission
line about 5 wavelengths ongwithaspacing of in
1962. This research wassupported hroughContract KO. NONR
Received April 20, 1962; revised manuscript received J u l y 23,
Office of Naval Research.
1866(32) between Harvard Univers ity, Cambridge, Mass., and the
Mass.
t
Gordon McKay Laboratory, Harvard University, Cambridge,
between the centers of th e wires. The l eng th of the an -
tenna was variable step-wise from
h 0.05X
t o h 9%h;
2h is
the distance between the tips of the antenna. The
right half of t he antenna was s lott ed to permi t theuse
of
a
movableshielded-loopcurrentprobewhichwas
held between a 1/16-in-diameter Microdot coaxial cable
and a thin n .lon thread. The Microdot cable with an
additional brass shield passed through the $-in tubings
which constituted both one-half of the antenna and one
conductor of the line. The p robe was moved by pulling
the cable by means of a carriage on the rack ocated
be)-ond the end of the two-wire line. The nylon thread
passed over
a
pulley beyond the end of the antenna an
was kept taut by a weight. The coaxial output of t he
generator was converted to a balanced two-wire trans-
mission line with a balun. The generator was located on
a different floor to avoid possible stray fields tha t might
excite an unwanted current on the two-wire line or the
dipole antenna or both.
A
movable charge probe t o measure the electric field
along
a
path above and
at
right angles to the two-wire
line was used to check the balance of the line. I n th e
experiment the line stretchers and stubs on the balun
wereadjusted
SO
tha t the field distributionmeasured
by the charge probe was sl-mmetric with respect o the
neutral plane of the transmission line. One set of meas-
ured results is shown in Fig.
2.
I n some of the measurements of the cur ren t along the
antenna, the probe carriage
as
driven by
a
synchronous
motor and a pen recorder was used. Owing to the lag in
the response of the latter, the motion h ad to be o slow
that the measurements for each of the longer antennas
requireda ather ong ime. a consequence, the