691

The outputs of the spaced loops are then sin 2 6 f K

The indicated bearing d, n-ill be given b?- cos 28 and cos M+h7 sin

sin 20 K cos 28

COS 28 h- sin 28 tan@

Let 28 2 ~ . Then expressing all variables in 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 46= -K, and since K is 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 Sational Research Council.

Theory of Unequally-Spaced Arrays*

Summary-Although recently unequally-spaced arrays have been shown to be useful, the theory has not been fully developed, except for the use of matrices, computers, or the perturbation method. This paper presents a new approach to the unequally-spaced array problem. It is based on the use of Poisson’s sum 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 from a continuous source distribution whose amplitude and phase distribution clearly exhibit the effects of the unequal spacings.

By this method, it is possible to design unequally-spaced arrays which produce a desired radiation pattern. This method is effective in treating arrays of a large number of elements, and unequally- spaced arrays on a curved surface. Three examples are shown to illustrate the effectiveness of the method. The problem of sidelobe reduction for the array of uniform amplitude, which was attacked by Hamngton, is treated 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 circular array is also shown, using the method of stationary phase.

I. INTKOUCC-TION

u NTIL A FE3Y \-ears antenna arrays had alwa).s implied arrays of equal spacing, simplJ- because those are the onl). cases which can be

handled by conventional methods involving pol>-- nonlials. Schelkunoff’s theor\- 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 \vas 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 increases with the number of elements, com- putation becomes more and more laborious for a large number of elements. The second is the fact that this pol~-nomial method can be applied only to equall>.- spaced arra>-s. The third is that this Inethod cannot 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, the arr3J-s can often be approximated bs- a continuous source distribution shown bl- i’an der 3Iaas for a linear array [4]. [j]. and b\. IG~udsen for a c.ircular arra!. [6]: but there seen~s to be no w a > - to treat unequall\--spaced arra?-s on ;I line or on a curved sur- face b>- the pol~~non~ial Inethod.

The method described in this paper is quite effective to treat unequall~.-spaced arraJ.s located on ;I line or ;I

curve. Also, this method is effective for arrays 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 out that the method e~nplo)-ed in this paper is different from the work of Ksienski [ i ] on the equivalence be- tween continuous and discrete arra\‘s. lisienski‘s method is based on the assumption of zero radiation in the invisible region and its extension for end-fire pattern, while the method i n this paper has no such limitations. I t ma!- be added here that the other method of treating a11 array a sampled data s\-stem was proposed by Cheng and 3,121 [8], but this also applies onll- to equally- spaced arrays.

Although in recent \.ears unequally-spaced arrays have been shown to be useful, a theory has not been fully developed. For example, Unz [9] used a matrix form

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solution which requires the manipulation of matrices of order equal to the number of elements. King, Packard, and Thomas [ lo] , computed the pattern of the various trial sets of spacings, but no unified theory was given. Sandler [ l l expanded a term for each element in a series. Harrington [12], on the other hand, employed the perturbation technique to obtain reduced sidelobes for an array with uniform amplitude. Andreason [I31 employed a computer to show the various possibilities of unequally-spaced arrays. Also, the use of approximate integral techniques were reported recently [ E t - [26].

The method employed in this paper is different from any of the above works. The method is based on the use of Poisson’s sum formula [14] and the introduction of a new function, called the “Source Position Function.” By this method, it is now possible to design unequally- spaced arrays which produce a desired pattern char- acteristic. It is essentially a new approach to the array problem, and there are a number of possible cases in which this method may be applicable.

In this paper, three examples are shown to illustrate the effectiveness of this method. They are the sidelobe reduction problem, secondary beam suppression, and azimuthal frequency scanning antennas.

11. A N ARIL4I’ O F lv ISOTROPIC RXDIATOKS WITH ARBITRARY SPACING

Let us consider the radiation pattern due to an array of LV radiators as shown in Fig. 1. The radiation pattern is given by

~ ( 0 ) I,,+n sin 8 (1) n=l

where I, is the current in the nth element, and S, denotes the position of this element as measured from a reference point 0.

The first step of our new formulation is the trans- formation of the radiation pattern of (1) in the following manner.

Let us rewrite (1) as follows:

E(0) c f ( 4 . n=l

Now, the Poisson’s sum formula will be applied to (2). The Poisson’s sum formula is [I41

5 f(.) 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 of the integration is from 0 to N because the radiation E(@ is the finite sum and f(v) vanishes for v<O and v > N . I t may be noted that the same result

(4) can be obtained by employing the Dirac delta functions, and this is shown in the Appendix. The limit of integration in (4) is not the only choice. In fact, any range which covers all the integers from 1 to LV may be used. Thus, (4) may also be written as

J

where 0 1. The second step in our formulation is the introduction

of a new function, which we call the “source position function.” Let us define the “source position function”

bY

s

This function gives the position of the nth element when v 12. Thus

!+‘e may also consider in (6) as a function of s. Thus

E’ “(S) (8)

and

a

We may call the function V(S) “source number function” because this yields the numbering of each element when

is at the correct position of this element.

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 to S.

of the variable of inte- where ]Ye obtain

SAV

ka sin 0,

&?rnsr (10) x =x(y) normalized source position function - l < x < $ l ,

NOW for our linear arraq. problem, considering the ex- pression in ( I ) , we write

where

I , I ( sJ A,e-j$n. (13)

is the amplitude of the current in the nth element and is the phase of the current, and -4 (s) is a function which yields at and therefore this may be con- sidered as an envelope of the amplitude of each current. $(s) is a function which yields $n a t Thus

-4

#n (14)

Eqs. (1 1) and (12) clearly show the physical signifi- cance of our formulation. 14,'e note that (12) is the radia- tion pattern of a continuous source distribution whose amplitude is

(15)

and whose phase distribution is

Therefore, our formulation is in essence the transforma- tion of the unequally-spaced arraq; into an equivalent continuous source distribution.

Eq. (11) is an infinite series form, but th is is not a serious disadvantage at all, because this is an extremely rapidly convergent series. This may be recognized from (12). Yv'e notice 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 near 8=0, and E+1 and E-1 are small corrections, while the rest of the terms are neg- ligibly small. This point will be more fully demonstrated in later sections bq- actual examples.

I t is more convenient t o rewrite (12) by means of normalized variables as follows:

y normalized source number function - l < y < + l .

Thus, the actual position of the nth element is

If X is odd, N = 2 X + 1

If X is even, N = 2 X

(1/2) y n for 0,

2 l f

1, 5 2 , M. (21)

The total length of the array is not 2a, bu 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(-I74-X4j.

Fig. 4-Sormalized source position function. X is even ( N = l O j . The total length 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 not be an odd function of x.

The factor l)m(x-l) in (17) is simply 1 for :V odd, but 1)" for N even. This is caused by the fact that the phase center for N even is at the midpoint between two elements. RIathematically, this is due to the shift of by the amount of

111. LINEAR ARRAY OF EQUAL SPACING

Although our purpose is to investigate unequallJ7- spaced arrays, it is instructive to see what our formula- tion corresponds to for the case of the array of equal spacing. In this case, in

x.

Thus

1 1

2 -1

E,(zi) J A ( 2 3 )

We note that

is the radiation from a continuous source with amplitude distribution A and phase distribution $(x). More- over,

E,(zJ) EO(U

Thus, E,(u) is the same radiation pattern as Eo except that the origin of u is shifted by msrN.

This point may be more clearly demonstrated by a simple case of constant amplitude with no phase varia- tion. In this case

The radiation pattern for this case using ordinary array theory is [3]

sin (+iVkd sin 0)

Y sin ($kd sin 0) EA(u) ( 2 7)

Noting that in this case

(28)

we write (25) as

sin

Comparing (29) with we note that our formulation is in essence the expression of the total radiation field

(29) in a series, each term of which is (sin pattern except that the origin is shifted by rnnlV. In other words, we replace the linear array by a series of continuous source distribution, each of which has such a phase variation that the peak occurs a t This is illus- trated in Fig. 5.

Fig. 5-The radiation from an array of equal spacing.

sin (a) Array Factor

N sin? N

sin (b) Eo (c) E-1

sin (u IVT) 1Vn

Iv . SIDELOBE REDUCTION O F A LINEAR ARRAY

OF 'CTNIFORM AMPLITUDE

In order to illustrate the use of our formulation, let us consider the problem attacked by Harrington The problem is to reduce the sidelobe level of the radia- tion from a linear array with equal amplitude by means of the unequal spacings.

Since we are interested in the range near 0, Eo should give a good approximation to the total field. Thus, considering A (x) 1 and $(x) =0, we have

But this is exactly the same as the radiation from a continuous source distribution with amplitude variation (dy ldx ) . Thus, the technique for the continuous source distribution can be directly applicable to this case.

In this paper Taylor's method for a line source is employed

In the case of Taylor's design, the solution is obtained as follows: Let

dY f(x).

ax

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Then, we write the solution in a series form:

+Q f(x) --Iqe-7*rr 1 x (33)

Y=-Q

Thus,

and -4, is given by

-4 Eo(qa). (36)

Since the main beam is in the broadside direction 0 0, and the radiation pattern is 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)

fi [l (:)?I p= 1 24,

p= fI 1 1 ( 3 1 For a detailed account of how (37) is obtained, the 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

Thus for 25 db, we get

A' 1.29177~'. (40)

For the choice of Q, the reader should refer to previous papers [15], [If]. Here, 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 (32)

The denominator normalizes such that y ( l j 1. For our case, this normalization is automatically satis- fied. Thus, noting thatf(xj is even, we obtain from (33)

From (43), the position of each antenna is accurately determined. The position of the nth element is given by (19). In order to compute 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 computer was used to find x,, for a given In order to check the validity of our method, we corn- pute the pattern of our array.

The radiation pattern for N odd is given by

where

And for LV even,

where

In Fig. 6 the radiation pattern from the array of 21 elements is shown. I t may be noted that, near ZL =0, the radiation pattern is very close to the designed pattern. The bearnwidth and the sidelobes are about the same as that of Ta>.lor's design for a line source. However, as ZL.

increases beyond 77r, the sidelobes start going up. This is due to the effect of the other terms E,,,, m # O . In general, for the main contribution comes from EP1 and thus the assumption (30) E(zi) =Eo(z~) is no longer true. 4s will be shown in Section I:, the radia- tion pattern near can be treated using EP1.

In Fig. 7 the radiation pattern from the array of 20 elements is shown. It exhibits almost the same charac- 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 ka.

Thus, if 2a 1OX and ~\'=20, the average spacing is X/2, but the position of each element must be calculated b!- (19), and the over-all length a(x,-x-,) is 9.05X in this example, which is smaller than 1OX.

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696 IRE TRANSACTIONS O N ANTELVATAS AND PROPAGATION

0

-30

-40

Fig. &Radiation fron an unequally-spaced array of 21 elements with uniform amplitude. The designed sidelobe level is 25 db.

db

Fig. 7-Radiation from an unequally-spaced array of 20 elements v i th uniform amolitude. The designed sidelobe lex-el is 2.5 db.

Fig. 8-Suppression of secondary beam near =20r. N=20 and 2iVA1=5. (a) O<u<13s. (b) 1 5 ~ < u < 3 0 ~ .

V. SUPPRESSION OF SECONDARY BEAM Then, (47) becomes

In this section, we consider the problem which was E-l(u) e-j2.VAl sin

attacked by King, Packard and Thomas [lo]. The -1

problem is to suppress the secondary mainlobe by means of unequal spacings. A s lAle-j?-VAl sin

The secondary main beam occurs at -mnN in 2 -1

(18). Thus, let us consider the case m = -1. Iiear NT, the radiation pattern is approximated by

(46) This becomes

Let us consider the following simple form for dy/dx. Jz (Z ) ='Anger function defined by Jahnke, et al. [20] I /T& cos sin x)dx. ( 5

A' cos n.x' and let A' I ' '48) Extensive tables of the Anger function for order and argument ranging 10 t o are available [19]. Thus it

Thus, is possible to calculate (51).

A1 x sin

n.

For small .41, we can approximate (52) by

E-l(U) J(u,s)-.%7(2NA 1). (53)

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697

In Fig. 8 the radiation 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. Thus, the approximation (53) will become closer to the actual radiation pattern as LV increases.

Thus, using (51) or (53), i t is possible to predict the behavior of the secondary beam. For example, by look- ing at the tables of the Anger function, we can choose the argument such that the peak is below a certain level.

\,‘I. LTNEQLAI,I.I--SPACED -ARRAYS OK A C L - R ~ E D St-RFACE

The formulation presented in this paper is also appli- cable to the more general problem of unequal]>--spaced arra)rs on a curved surface.

Let us consider the field 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 the contribu- tions from each element. Thus

(54) 7l=l

where I(s,) is the current located at and G(r , X,) is the appropriate Green’s function.

KOW, we can transform this series in the same manner.

e(r) 2 JoA”I(xn)G(rl s)ei*mrrdr, (55) m=-a

Thus, we note that we converted our array problem to the problem of continuous source distribution, whose amplitude is multiplied by dv/ds and whose phase is modified by 2 - m ~ c ( s ) .

It may be noted that this formulation is quite general and this is applicable not o n l ~ . to the radiation pattern problem, but also the field at any observation point.

1.1 I. CIRCVLAR ,ARRAYS V,-ITH UKEQITAL SPACING A N D

ITS FREQYEKCY SC:.IKNING CHARACTERISTICS

I n this section, we consider an interesting problem of frequency- scanning antennas i n an azi~nuthal direction. Let us suppose that an arm>- of antennas is located on a circle, and excited by a single slow waveguide. The spacings between adjacent elements are so distributed that, at any frequenc~-, only a few of the elements are excited in phase and that, at different frequenq-, the elements a t different locations become i n phase. (See Fig. 10.) This problem can be analyzed by our method.

Let us consider a case of the radiation from a circular arra?- in free space. I n the spherical coordinate sy-stem ( r , e , q5), the antennas are located at

FIELD

I.

Fig. 9-Field due to an array on curved surface.

f ,

f r

Fig. 10-Azimuthal frequency scanning by a circular array with unequal spacings. The guide wavelength is a t f i , X, at fc, and

atf2.fl<fc<f2.

n= X . Now, the Green’s function for the radiation pattern i n the direction (e, due to the source distribution at (0, n/2, is given b)-

Thus, (56) becomes

where

This source position function is chosen so that p ranges from t o 27r as varies from 0 to 2 ~ . Let us consider the case of constant amplitude for the source, but assume that each element is excited b)- a slow wave- guide with the azimuthal propagation constant T h u s

(59) 1(4,,) e-lCa60

where

I n this case, the radiation pattern i n the plane of the arraJ- (8 is

E(+) Em(+) , z ka m=-s

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IRE TRANSACTIOAY$ OlV ASTEXA'AS A X D PROPAGATION Xovember

We note here that for the case of equal spacing, p In order to obtain a real stationary point, let us choose and

and

We notice that this is the same situation treated by Knudsen [6j.

Now, let us consider (60). In general, i t is difficult to obtain an explicit expression for the radiation pattern. However, we can obtain some useful information from (60) by applying the method of stationary phase. Of course the validity of this method depends on ka. For large ka, this should become closer to the actual radia- tion pattern.

Let us consider the stationary phase point. For Eo(+), we get

which yields

Now, if the arra). is excited by a slow waveguide,

Therefore, there is 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

ka 2

I t may be noted that when the right-hand side vanishes and therefore + = + s = ~ . Thus, this is one of the stationary points. There may be other stationary points in the visible region. But since we desire only one main beam, we desire only one stationary point. The constant -41 must satisfy certain conditions. First, p must be a single valued function of and therefore dp/d$o must always be positive. Thus,

which yields

1.

As an example of p , let us take where

2 p Alsin--,

dP 90 1+-cos-> 0,

d90 2 2

The second requirement for -41 is that in our problem we desire only one stationary phase point from (70). This requires that the magnitude of the slope of the right- hand side at +s =T is greater than the magnitude of the slope of the left-hand side at Thus we get

From (72) and (73), we get

(74)

Let us now evaluate (60) by the method of stationary phase. Then we get

which is similar to the one used for linear array. Thus, I t is expected that the stationary point is at when

(66) becomes 6 =T. If we desire a sharp beam at this point, f " ( + o ) must be small. In fact, by the proper choice of AI, this can be zero. This requires that

sin (68) A 1 - - - 4 - . ka

ka (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 , vs 3 0 ~ , and ‘-1 as given by (77). I t ma)‘ be noted that the antennas i n the region 12Oo<+0<24O0 radiate the rays in the real angle of and the other antennas radiate in the invisible region.

The frequency scanning characteristic is also obvious from (68). L k frequency varies, v 3 and ka vary. Thus (70) becomes

Fig. 11-The rays from the stationary points for an azimuth frequency scanning antenna.

Let us write

(79)

where p , , is the relative phase velocity at a center fre- quencyf, and p , is the phase velocity at the other fre- quency f. Then

sin ($s $1

and

ka p‘, P s

The position of each element is given by (69)

.I-.

I t shows that, over a frkquency range of 0.6f, to 1.2fc? almost 170-degree scanning is accomplished.

It may be noted that, as the frequent?, varies, the other stationar)- phase point ma)’ appear in the visible region, a case which must be careiully studied. Also, no consideration is gil-en to the problem of sidelobes in this paper.

Since the main purpose of this paper is to develop the theor). of unequally-spaced arra>’s and to show its effec- tiveness, the detailed stud!, of azimuth frequency scanning antennas m i l l be discussed in a separate report.

1’1 I. OTHER A 4 ~ ~ ~ ~ ~ . \ ~ ~ ~ ) ~ ~

There are a number of possible other cases in which this new formulation may be applicable. I n this section, a few cases of interest will be briefly discussed. I t ma>- be noted that our formulation is particularlL- suited for the arraJ-s excited by a traveling waveguide as shown below.

-4. Z-neq~lally-Spa.ced Slot Arrays on 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 same intensity, but its phase is determined b5- the phase velocity of the waveguide.

Thus, the current in the fzth slot is given by

I , e-lBSn (83)

where 0 is the propagation constant of the waveguide and is the distance along the guide. Then (18) be- comes

A s the frequency varies, the stationary phase point shifts. This ,.ields an azimuth frequency scanning antenna.

In Fig. 12 (next page), the frequency scanning char- acteristics of a circular array with unequal spacings are shown for the case of ka l o r , M 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 range of zt which is near ?to. Then,

But this is exactly the same as the radiation from an amplitude modulated traveling-wave antenna, whose peak is at and whose source amplitude distribution is T h u s , the technique used in Section IV directly applicable.

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700

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, the peak value= 1.07. (e) f= 1.2fC, the peak value=0.97.

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If the 11-aveguide is ;I fast v-aveguirle, <Ra and the peak occurs i n the ITisible region a t 0,. where = k a sin O p . This is the same situatioll as the leal;!. wave antenna discussed b\- Hone>- [21]. and Goldstone and Oliner (221. I n fact, this unequall!--spaced arra\- ma!- provide another means of producing leak!--wave antenn;Ls.

If the waveguide is a slow wave, ka and the peak is in the invisible region. Thus, this ma\- be useful to obtain an endfire amplitude modulated slow-wave antenna.

B. Frequency Scannizg -4nte-nnas

I t is known that when an arm\- is excited b\v an ap- propriate slow waveguide, the frequency scanning an- tenna may be obtained [ Z ] . However, in practical cases, the impedance of the waveguide varies as the frequency is varied. In particular, when the beam is directed broadside, all the reflections from each element add in phase and the radiation pattern deteriorates.

If the unequallp-spaced array is used effectively, this impedance problem may be considerablJ- reduced.

The frequency scanning antenna is obtained when the phase velocitJ- is slow and in (84j is such that (140 -nmLV.T) is very small. For example, i f nz 1 is taken, the radiation in the visible region is approximated by

where

As the frequency varies, varies and this produces the frequent!. scanning. Eq. (86j is i n the same form as (47), and a similar technique ma>- be emplo)-ed and the im- proved impedance characteristics ma\- be obtained.

The stud>- on the above topics are under w a \ - and the result will be discussed i n a separate report.

I S . COKCLESIOS

A new approach to the arra>- problem is shown? which is particularll- suited for unequally-spaced arra5.s with a large number of elements which m a ~ r be located 011 a line or 011 a curve.

I t is shown that an unequall>--spaced arra!- of uni- form amplitude w i t h an!' desired siclelobe level m a y be designed, using our method. .Also, the secontl;u-!- beam suppression and the azimuth frequencl- scanning circu- lar array was discussed to show the effectiverless of the method. -Other applications including unequall\--spaced arrav on a traveling-wave waveguide, the amplitude modulated antennas, the leak\--wave antennas, a11d the frequencJ- scanning linear antennas are discussed.

'I'hen, expanding i n a range from E to ~+i\r, we get

But the sulnnlation

T h u s ,

And, therefore,

I t may be noted that in essence this is the method used b\- Iinudsen [6] for his stud!- on circular arrays of equal spacings.

REFERE~TES [l] S. Schell:unoR, "-1 mathematical theory of linear arrays,''

Bell. Tech. J., vol. 22, p. 80; January, 19-13. [2] S. Sil\-er, "liicrowave antenna theory and design," McGraw-

[3] J. I). Iirans, "Xnte~was," RlcGraw-Hill Book Co., Inc., New l I i l l Book Co., Inc.. Sew I'ork, S. Y., ch. 9; 1949.

[4j I I . Jasik, ";intentla Engineering Handbook,' RIcGraw-Hill Yorl;, Y., ch. 1; 1950.

[j. G. J . Van der l laas, "A simplilied calculation for Dolph-Tche- Book Co., lnc., Sew York, N. Y., ch. 2; 1961.

l)>-schefl arra\.s," J . d p p l . I'kys., vol. 25, pp. 121-124; January, 1951.

[6; L. Iinudsen, "Radiation from ring quasi-arrays,' IRE

July, 1956. TIGAS. os A S T E ~ S A S ASD I'IWPAGATIOX, -4P-4, p. 452;

lisiemki, "Equixdence betwecn continuous and discrete radiatingarrays," J . Plzys., 39, p. 335; February, 1961. I ) . I<. Cheng and hl. 'r. 1Ia, "A nen- mathematical approach fnr linear arra!' analysis," I R E TRANS. O S A S T E N X ~ S AKD

[9] H. L-nz, "Linear arra>-s with arbitrarily distributed elements," I'KOPAGATIOX, \~oL XI'-8, p. 255; May, 1960.

222-223: hiarch, 1960. Il<F; TIUXS. ON .ASTEXSAS ASD PROPAGATIOS, vol. AP-8, pp.

[ lo] D. King, R. F. Packard, and R. E;. Thomas, "L-nequally spaced, broad-band antenna arrays," I R E TRASS. ON ANTESSAS ASD I'ROPAGATIOS, vol. AP-8, pp. 380-385; Jull-, 1960.

[ l l j Sandler, "Some equi\.alence bet\veen equally and unequally spaced arral-s," II<E 'I'KAXS. O S ~ \ S T E S S . ~ S AXD PROPAGATIOS, vol, -11'4, p. 496; September, 1960.

[I?] R. F,. Hyrington. "Sidelobe reduction by nonuniform element

.1P-Y, p. 187; RIarch, 1961. spacmg, Il<E 'rRAXS. .L\STEXSAS AXD PROPAGATIOS, vol.

[13] G. ,Andreasen, -Linear arra\-s with variable interelement spacinga," I R E T I M X S . OK ~\KTI :SSA ASD I'KOPAGATIOS, .A1'-10: pp. 137-1431 hIarch, 1962.

[14] S. hIorse and 11. Feshbach. "klerhods of Theoretical Phvs- KS," LIcGratv-Hill Book co., S e n l-ork, Y., p. 166; 1953.

Authorized licensed use limited to: IEEE Xplore. Downloaded on May 10,2010 at 19:00:07 UTC from IEEE Xplore. Restrictions apply.

702 IRE TRANSACTIONS ON SNTER’:ITAS AhTD PROPAGATION November

[ l j ] T. T. Taylor, “Design of line-source antennas for narrow beam- width and low sidelobes,” IRE TRANS. OK ASTESSAS AKD

[16] “Design of circular apertures for narrow beamwidth and PROPAGATIOX, vol. AP-3, pp. 16-28; January, 1955.

low sidelobes,” IRE TRANS. ON ~ T E I V K A S AND PROPAGATIOK, vol. AP-8, pp. 17-22; January, 1960.

[17] A. Ishimaru and G. Held, “Analysis and synthesis of radiation patterns from circular apertures,” Canad. J . Phys., 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,’‘ University of \{-ashington Press,

[20] Jaknke, F. Emde, and F. Losch, “Tables of Higher Func- Seattle; 1962.

tlons, McGraw-Hill Book Co., New York, N. Y. , p. 251; 1960. R. C. Honey, “A flush-mounted leaky-wave antenna with pre-

The

dictable patterns,” IRE TR~SS. O S ANTEXNAS AND PROPAGA- TION, vol. AP-7, pp. 320-328; October, 1959.

[22] L. Goldstone and A.,A Oliner, “Leaky-wave antennas I: rectangular waveguides, IRE TRAXS. ON ANTEXXAS S N D

[23] A. Ishimaru and H. S. Tuan, “Frequency scanning antennas,” IRE TRAXS. OX ANTENSAS AXD PROPAGATION, vol. AP-IO, pp. 140-150; March, 1962. Y . T. Lo, “A spacing weighted antenna array,” 1962 IRE INTERKATIOSAL COSVENTION RECORD, pt. 1, p. 191.

[ X ] X. L. Maffett, “Array factors with nonuniform spacing param- eter,“ IRE TRASS. ON ASTENNAS ASD PROPAGATIOS, vol.AP-IO,

[26] J. L. Yen and J. L. Chow, “On Large Non-uniformly Spaced pp. 131-136; March, 1962.

Arrays,” presented at the Copenhagen Symposium on Electro-

25-30,1962. magnetic Theory and Antennas, Copenhagen, Denmark; 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 current along a dipole antenna driven by a two-wire transmission line has been studied experimentally. I t was found that an unbalanced component of current on the line signiiicantly influences the measured distributions of current along antennas of shorter lengths. A quantitative study was made by de- composing the currents into symmetric and antisymmetric parts. The associated unbalance in the transverse field distribution was measured by a field probe and correlated with the ratio of the amplitudes of the symmetric to antisymmetric components of cur- rent in the transmission line and the antenna.

I. INTRODUCTIOS AND DESCRIPTION OF APPARATL-s

w HEN a symmetric dipole antenna is center- driven from a transmission line, the distribu- tion of current along the antenna is 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 the line. I t is the purpose of this paper to study the effect of unbalanced currents on the distribution of current along and the measured imped- ance of a symmetrical 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 measurements is shown in Fig. 1. A cylindrical dipole antenna made of $-in brass tubing was center-driven by a two-wire transmission line about 5 wavelengths long with a spacing of in

1962. This research was supported through Contract KO. NONR Received April 20, 1962; revised manuscript received July 23,

Office of Naval Research. 1866(32) between Harvard University, Cambridge, Mass., and the

Mass. t Gordon McKay Laboratory, Harvard University, Cambridge,

between the centers of the wires. The length 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 the antenna was slotted to permit the use of a movable shielded-loop current probe which was 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 probe was moved by pulling the cable by means of a carriage on the rack located be)-ond the end of the two-wire line. The nylon thread passed over a pulley beyond the end of the antenna and was kept taut by a weight. The coaxial output of the generator was converted to a balanced two-wire trans- mission line with a balun. The generator was located on a different floor t o avoid possible stray fields that might excite an unwanted current on the two-wire line or the dipole antenna or both.

A movable charge probe to 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. In the experiment the line stretchers and stubs on the balun were adjusted SO that the field distribution measured by the charge probe was sl-mmetric with respect to the neutral plane of the transmission line. One set of meas- ured results is shown in Fig. 2.

In some of the measurements of the current along the antenna, the probe carriage was driven by a synchronous motor and a pen recorder was used. Owing to the lag in the response of the latter, the motion had to be so slow that the measurements for each of the longer antennas required a rather long time. a consequence, the

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