theory of unequally-spaced arrays-hqc

8/20/2019 Theory of Unequally-spaced Arrays-hqC

1/12

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

AlultIXDoM1a1UfIX Ra


2/12

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) .



3/12

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 .



4/12

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



5/12

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.



6/12

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)



7/12

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

atf2.fl


8/12

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)



9/12

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.



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.



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


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

theory of unequally-spaced arrays-hqc

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