- - ---------------------

Journal of Research of the National Bureau of Standards-D. Radio Propagation

Vol. 63D, No.3, November- December 1959

Pattern Synthesis for Slotted-Cylinder Antennas *

James R. Wait and James Householder

(June ll , U)59 )

The radiation [rom a cyli nder excited by an array of axia l slots is discussed. A pro­cedure for synethesizing a given radiation pattern is developed with particular attention being paid to a Tchebyscheff-type pattern . Speci ry ing thc side lobe level and the width of the main beam, t he req uired sou rce distributions a re computed for a number of cases. The effect of u sing a finite number of slo t eleme nts to approximate the continuous so urce distribution is also considered.

1. Introduction

It is the purpose of this commullicaLiolJ to discu ss the synthesis of radiation pattel"11s for an tennas of slotted-cylinder type. ']' he particular model em­ployed is a m etallic cylinder of infinite length and perfect cOllductivit;y. The cylinder is to be energized by an array of slots of rectangular shape which arc orien ted in the aAia.l direction and dlspos~d around the circumference. EacJl slot is to be energized in sueh a way that lhe tangential field in th e slot has only a transyerse component. The radiation field of such a slot on an infinite cylinder is plane polarized. Furthermore, tbe idealization of an infinitely long cylinder is not oyerly restrictive since the surface currents excited by the axial slot arc very similar to what th ey would be on a finite cylinder. This con­jecture is substantiated by experim ental data on the conductance of axial slots on finite cylinders [1] .1

2 . Basic Pattern Function

Ohoosing a cylindrical coordinate system (p, 4>, z), the cylinder is defined by p= a. Since attention is only confined to the azimuthal or 4> behavior, it is only necessary to specify the 4> variation of the field in the axial slo t. For example, if the slot width extends from 4>1 to 4>2 , the azimuthal radiation pattern M* (x , 4» was shown to be given by [1]

1 "" ~ eimr / 2 cos m4> (4) - 4> ) M* (x, 4» =-. - ~ m H (Z)/( ) Jo ~2 [m , (1)

~7rX 7n=O m X

where H~;i'(X) is the derivative with respect to x of the Hankel function of order m, J o is the Bessel func­tion of order zero , k= 27r/wavelength, x= ka sin 0, ~o= l , ~m=2 (mrfO), and 0 is the angle sub tended by the cylinder or z axis and the direction to the observer (i . e., the usual polar angle in spherical coordinates).

-The research reported in this paper was sponsored, in part, by the U.S. Air Force Cambridge Research Center under Contract CSO a nd A 58- 40.

1 Figures in brackets indicate tbe literature referen ces at the end of tbis paper.

The above form for the paLtel"ll 1\11* ex, 4» is slrictly vali d only wilen t he field E ", in the slot is

(2)

wh er e f ' (ZI) is the voltage across the slot at Z=ZI. 'rhe fLclcl approaches infinity as the inverse sq uare root of the distance to the edge of the slot . Such a b ehavior is characteristic of the field in the vicinity of a perfectly conducting knife edge. _ The amplitude and normalized phase <i' [= phase lv1* (x, 4» -x cos 4>] of th e quantity M* (x, 4» is cal­culated for a series of values of x and 4>2- 4>1' These are listed in tables 1 and 2 for x equal to 3 and 5, respectively, and 4>2-4>1 varying hom 0° to 30°. For the smaller cylinder , it is seen that the effec t of widening tbe slot (for a given voltage) is small. For the larger cylinders, the effect is becoming more significant although if the angular slot width is also less than 10°, the pattern is indistinguishable from that of zero width.

It should be emphasized that these E-plane patterns characterize only the azimuthal dependence of the radiation field for a given value of x( = ka sin 0) . The elevation patterns or FI plane pattern of such an axial slo t requires that the longitudinal distribution of voltage along the slo t be specified [1].

3 . General Array Synthesis

By superimposing the patterns of individual aAial slots arranged circumferentially around the cylinder various forms of patterns can be obtained. Th~ problem sometimes arises that a pattern i specified and the manner of excitation must b e determined. This is described as synthesis. Some elegant pro­cedures have been developed for linear arrays and these have been described extensively in the li tera­ture [2 through 5]. When the antenllas arc disposed around circular arcs , the techniques [6] are not so straightforward, particularly if difl"raetion is involved. An example falling in this category will be considered in what follows .

303

</>'-</>1

0

</>= 0 0. 959 10 . 956 20 . 946 30 . 927 40 . 895

50 .859 60 .825 70 . 794 80 . 747 90 . 664

100 . 561 110 . 482 120 . 470 130 . 444 140 . 349

150 .206 160 . 155 170 . 259 180 . 312

TABLE l'

(From reference [7])

IM *(x, </» 1

20 30 0 ---

0.957 0.955 6.9 . 953 . 948 7. 1 .937 . 926 7. 7 . 909 . 888 8.4 .869 .838 8.9

.824 .783 8.8

. 782 . 731 8.0

. 743 . 684 7. 4

. 691 . 626 7.8

. 609 .545 7. 6

.512 . 455 3. 2

.445 . 395 - 7.0

. 428 . 379 - 18. 1

. 404 . 358 - 23.5

. 320 . 283 - 28.7

.187 .106 - 43.5

. 142 . 126 - 105.0

. 237 . 211 - 137. 0

. 285 . 254 - 142.1

a All angles are shown in degrees.

</>'- </>1

0

IM* (x, <!o) 1

TABLE 2'

(From reference [7])

x= 5

10 20 30 0

<l> (deg)

20

5.7 5.9 6.6 7.4 8.1

8.2 7. 9 8. 0 8.8 9.0

5.0 - 5.1

- 15.4 - 20.4 - 23.9

- 40.0 - 101. 7 - 133.5 - 138. 5

<l> (d eg)

10 20

-------------------</>= 0 0. 981 0.980 0.979 0.979 4.9 4.5 2.8

10 . 977 . 976 . 970 .960 5.1 4.7 3.0 20 . 966 . 963 . 945 . 918 5.1 5. I 3.5 30 . 951 . 944 . 908 .855 5.8 5.5 4.1 40 . 936 . 924 .865 . 782 6.1 5.9 5.0

50 .914 .899 . 815 . 701 7. 0 6.9 6.4 60 . 865 . 844 . 745 . 610 7. 9 7.9 8.0 70 .804 . 781 . 672 . 527 7. 0 7. 1 7. 9 80 . 757 . 732 . 615 . 462 5.5 5.8 7.6 90 . 681 .656 . 540 . 392 5.0 5.4 8.0

100 . 570 . 548 . 447 . 318 -0.6 0 3.4 110 .513 . 493 . 400 .282 - 11.3 - 10.5 -6.2 120 . 452 . 434 . 350 . 245 - 17.5 - 16.6 - 11.6 130 .318 . 306 . 246 . 172 - 32.7 - 31.8 -26.4 140 . 288 . 277 .224 . 158 -68.0 - 67. 0 -61.4

150 . 293 . 281 . 228 . 162 - 82. 8 -81. 7 - 75.7 160 . 147 . 141 . 114 .082 - 99.7 -98.8 - 92. 6 170 . 142 . 137 . 112 . 080 - 205.2 - 204.0 - 198.0 180 .241 .232 . 189 . 135 -219. 3 - 218. 2 - 212.1

• All angles are sh own in degrees.

30

4.2 4.5 5.2 6.2 7.2

7.7 8.0 8.7

10.2 11. 0

7. 5 -2.0

-11.7 - 16.1 - 19.2

- 35. 2 -97. 0

- 128. 6 - 133. 6

30 --

0.2 . 5

1.2 2.3 3.8

6. 2 8. 7

10.1 11. 6 13. 7

10.5 2.7

- 1.3 - 15. 3 - 49. 8

-63. 5 - 77.0

- 185.7 - 199. 6

rrhe cylinder is now to have P aJo..ial slots equi­spaced around a circumference of the cylinder as illustrated in figures 1a and lb. The individual slot elements are centered at

2'1T'p ¢= ¢p=pwherep= O, 1,2 ... P - l.

The quantity L(cf>p) is used to describe the relative excitation of each element. Therefore, the resultant azimuthal pattern of the array is

1 P - l M (cp) = "2 2: L(¢p)M*(cf> - cf>p) (3a)

'IT' p - O, 1,2 ...

FI GURE lao The axial slot­ted-cylinder array.

Observer

FIGURE lb. C1'OSS section of Ihe cylinder.

where M*(cf> - cf>v) is the pattern of the individual elements of tbe array. (The factor 1/2'1T' is included for convenience in what follows.) Now when P is sufficiently large, M(cf» can be approximated by an integral so that p is then regarded as a continuous variable. This leads to

(3b)

It is now assumed that the excitation function L( cf>p) is expressible as a Fomier series in the manner

+ '" L(cf>p) = ~ Lne1n</>p (4)

n=- co

where the sum is over all integral valu es of n (in­cluding positive and negative integers). It is also assumed that the width of the individual slots is very SITI'111 and therefore

which follows directly from eq (1). Inserting these expreSSIOns for L(cf>p) and M*(cf>- cf>p) into eq(3b) leads to

(6)

Having the pattern expressed in this form enables a synthesis procedmc to be directly applied. Because of orthogonality

(7)

which, when inserted into eq(4), enables L(cf>p) to be determined and is the excitation required to produce the pattern M (cf». In a formal sense, any pattern M(cf» could be specified, however directive, and a corresponding function L( cf>p) could be determined. From a practical standpoint, there is a limitation

304

I

~ I

wh en M(q:,) becomes very directive since then L (q:,p) van e very r apidly in both amplitude and phase and lead to a complicated procedure for feeding Lh e array. Furthermore, small errors in tb e excitation of tlw elements lead to a large degradation of the pattern .

A different approach to the synthesis problem is to utilize t he remarkable properties of the Tchebyscheff polynomials. These polynomials are particularly appropriate when the pattern is to have side lobes of specified and equal amplitude. As shown by Dolph [2] and others [3 , 4, 5, 6], the resultant design is an optimum one for a linear array of discrete elements. In the case of a circular array, a similar procedure can be adopted as follows :

The T chebyscheff polynomial of order N is defined by

TN(z) = cos (N arc cos z), z ~l ,

= cosh (N arc cosh z), z> 1. (S)

In order to u tilize the optimum properties of the poly­nomial, the following transformation is introduced

z= a cos cp + b,

where a and b ar e constan ts. Now cp = O and z=zo are to correspond to tb e direction of the main beam. In fact , as cp varies from 0 Lo 271", Z is to vary from Zo to - 1 back to zo0 Therefore

z + 1 a=_o_ and 2

As shown by Duhamel [5], N

b=Zo- l . 2

TN(a cos q:,+ b) = ~C;: cos nq:" n=O

(9)

where C;: are functions of a and b. For example, when N = 4, they are given by

C~= I - Sb2+ Sb4+ 24a2b2 +3a4-4a2

C1 = - 16ab+ 32ab3+ 24a3b

C~= - 4a2+ 24a2b2+ 4a4

C~= Sa3b

C!= a4

and for N = 6, they are given by

C8 = - 1 + ISb2- 4Sb4+ 32b6+ 9a2- 144b2a2

(10)

The nex t step is to equate the Tchebyscheff pattern to the general form of the slotted cylinder pattern given by eq (6) . Thus

This enables the coefficients L in to be specified in terms of t he coefficients Ct: as follows

EmLm= i 7l"xe-im7r/2Ct/.H W,' (x) for m ~N, = 0 for m> N. (12)

The corresponding excita tion is then given by

'" L (q:,p) = ~EmLm cos mq:,p' (13)

m= O

It is thus possible to produce a TchebyscheIf type of pat tern from a dis tribution of ax"ial slots around a circular cylinder. As shown below, ei ther the side­lobe level or the first null can be specified by a proper choice of Z00 For a constan t side-lobe level, the beam width is decreased by increasing the order of the polynomial.

The nulls of TN(Z) occur wh en

( 2k- l) arc cos Zk= 2N 7r, k= l , 2,3 . . . N

and the nulls in th e cp domain arc thus

[COS [ (2 k- 1) 2~ J-b]

q:,k=±arc cos a , k= 1,2 ... N .

(14)

On the other hand, the cen ter of the lobes occur at q:, = q:, ~ which is a solution of

These are given by

[ k7l" ] cos - - b

cp~= ± arc cos ~ . (15)

When the ratio of the side lobe to the main beam is B then

+ 240b4a2- 1Sa4+ IS0b2a4+.10a6 since (16)

C~=36ba- 192b3a+ 192b5a- 144ba3+ 4S0b3a3+ 120ba5

cg= 9a2 - 144b2a2+ 240b4a2 - 24a4+ 240b2a4+ 15a6

cg= - 4Sba3+ 160b3a3+ 60ba5

C!= - 6a4+ 60b2a4+ 6a6

C~= 12ba'

C~= a6.

The quantity B thus specifies Zo which in turn deter­mines a, b, q:,k and q:,~ for a given value of N . The required excitation L (q:, p) actually is a continuous function. From a practical standpoint, the number of elemen ts must be finite. It would be expected, however , if the separation between the elements is small compared to the wavelength, the actual pat-

305

tern would not difl'er appreciably from tbo Tcheby­scheff pattern. Tbis conj ecture is verified by actu­ally computing tho pattern for a fini te number of equispaced slots disposed around the cylinder. For example, the polynomial TN (a cos 4> + b) is to be com­pared with the function

with P b eing a fini te in teger. In this case, the spac­ing between the axial slots is 27ra jP. In spite of the fact that the spacing is finite , the M (4)) pattern is quite close to t he ideal TN pattern as indicated in the following r esults.

1.0

4. Presentation of Numerical Results

The theoretical basis of the synthesis procedure has been outlined in the previous section. H ere numerical results are presented for certain specific cases which illustrate the interrelation between the parameters of the problem .

'1'he basic patterns used ar e illustrated in figure 2a and 2b . They arc derived from the T4 and T& polynomials, respectively. '1'he quantity B, which is the (vol tage) ratio of the major lobe to the minor lobe, is taken as 5, 10, 15, and 20 for the two cases. It is seen that for a given polynomial the beam is broadened as the side-lobe level is reduced . Conse­quen tly, if for a given side-lobe level, one wishes to

BEAM WIDTH FIRST NULL

Values of ¢ for T4~zJ =0.5 Values of ¢ fo r T4~Z) = 0

CD

~

f-~

,. CD

N

'" f-

08

0.6

0 .4

0.2

0.8

0 .6

0 .4

0 .2

B 5

10 15 20

2¢ B 47.30· 5 53 .90· 10 57. 38· 15 59 .68· 20

N o 4 Fa cosq,t b

Figure 20

¢ 39 .27" 47.33· 52 . 10· 55.48·

BEAM WIDTH FIRST NULL T (z )

Valuesaf¢for T=0.5 T (z)

Volues of ¢for T =0

B 5

10 15 20

2 1> B 31.76· 5

36.38· 10 38.86· 15 40. 50· 20

N 0 6 zoo cos q,+ b

Figure 2b

¢ 26.38· 31.95· 35.30· 37.69·

o 40 60 OL--L __ ~~~~ __ ~~L--L~~~L--L~~~~-L~~~~~~~~

80 100 20

q, (DEGREES )

FIGURE 2. Optimum patterns based on T chebyscheiJ polynomials T N(Z).

306

1400 1400

24 1200 1200

20 1000 1000

" ~ -e-:J

800 :J -"

800 ~ PHASE-

~,,-- ~ W 0

; ::> I w r-

600 <f> 2 12 , I <! I " "- <!

"-0 w

600 <f>

" :r: "-

- AMPLITUDE

400 400

200 ---- - -------,.--/,,-

200

20 20 40 60 80 100 120 140 160 180

~p • AZIMUTH DEGREES ¢p. AZI MUTH DEGREES

28 1400 70 1400

24 1200 60 1200

20 1000 50 1000

~ -" ~ ::J 16

-e-800 :J 800 :;

~ , - - - "- W "-

0 0 ( 0

:::> I ::> ~ r- AMPUTUOE PHAS~~ ___ ) w t:: AMPLITUDE PHASE I

2 12 600 <f> 600 " <! I

" it ,.,-- - _\..._-I , " a.

<! <! I ( I

400 I / 4 00

- - -- ..,.. ---/ ,. , --I / 200 ~

.-200 ---

0 0 0 20 40 60 eo 100 120 140 160 180 20 40 60 eo 100 14 0 180

¢p. AZIMUTH DEGREES ¢p . AZIMU TH DEGREES

35 1400 1400

30 1200 60 1200

25 1000 50 1000 r --- r----

~ ___ AMPUTlJD£ PHASE I -" I

r~-J " -e- AMPLITUOE PHASE I -" :J 20 ~ :J I

BOO~ 800 --' - 40 _.J ,----- - --' W I "-

W I "-

0 0 I 0 :::> I 0 I r- -----_./ ~ ::>

- --- - ---/ w

::J 15 600 " f- 30 600 ~ "- ~ ::J

" I I "- I

, if

<! I "- " ./ <! / 10 --- 20 ~-------/ 400

r 400 -- _/ - -- - -- - -- -

200 10 200

0 0 0 20 40 60 80 100 120 140 160 0

¢p. AZIMUT H DEGREES ¢p. AZIMUTH DEGREES

]<'raUHES 3 an d 4. Excitation required for a Tchebpchej)' pattern.

307

56 ITT I 1400 70 1400

48 1200 60 N"4 8"20 '" 3J 1200

Figure 60

40 - 1000 50 1000

-~ -~

-~ -~

~ -<>--<>- :J -<>-

-32 800 ::; -40 800 ::; W

_____ AMPLITUDE PHA;E~ __ "- w "-0

0 '" -- 0 ::l ,-

'" => PHASE _____ • ./ ~ '= I ,n f-- ___ - AMPLITUDE

a' 24 / 600 " ~ 30 600 '" I ./ I

" .--- - -- ---" Cl. " ----- Cl.

'" /' '" ,-/-/ /

/

16 _/ 4 00 20 -/ 400 ----- -----/ ---- -----------200 10 200

0 0 0 0 20 40 60 SO "'" 120 140 160 180 0 80 100 120 140 160 IBO

<i>p . AZIMUTH DEGREES <i>p. AZIMUTH DEGREES

70 1400 140 14 00

60 1200 120 1200

50 1000 100 1000

:J> -~ ~ Q

::J -<>- -' -S-- 40 BOO ::J -80 800 :J

w ~ w "-0 -- '" ,- 0 ::J ( W => W f-- if> '= 60 PHASE> __ )

if> :::; 30 AMPLITUDE PHASE~ I 600 '" .~ AMPLITUOE 600 '" Cl. I Cl. I

" /~-- _../ Cl. ::> Cl.

'" « I' / /

20 I 400 4 0 / 400

/

..- - ----" ,..----~/ /'

/ / /'

10 --- 200 20 200

0 0 0 20 4 0 60 SO 100 120 140 160 180 0 140 160 IBO

<i>p. AZIMUTH DEGREES <i>p. AZIMUTH DEGREES

140 1400 1400

120 1200 120 1200

100 1000 100 1000 --- I ----~ / ~

~ ___ AMPLITUDE PHASE~ __ / -~ -S- I -~ -<>- :J AMPLITUDE PHASE~ __ / -S-

-80 - --- - 800 ::; -60 BOO ::J w I' "- W I "-0 1 0 '" I

0 ::J W => W f-- 1 800 :a '= /

/ 600 if>

~ 60 .,-------' a' 60 '" I - I

" / a. ;:; / a. « '" I / I

40 -------/ 4 00 40 --_./ 4 00

---/----20 200 20 200

0 0 0 0 20 40 60 180 0 20 40 60 80 100 120 100 t&l

<i>p. AZIMUTH DEGREES <i>p. AZIMUTH DEGREES

FIGU RES 5 and 6. Excitation required for a TchebyscheiJ pattern.

308

~ ~ ~ -- ----------------

140 1400 200 2800

---120 PHAS£~/ 1200 240 2400

/ AMPLfTUOE

r---- .... , - 1000 200 2000 I

I -~ ~ ;' :!l- -~

" ~ -' :!l--' " 160 1600 800 :; -'

w "- w 15 0 0

0 :> ~ I w l- i!! :::; 60 / 600 if> :::; 1200 <!

<! CL CL I "

I

" CL <! CL

<! I I

40 i 400 00 800 / ,

I I

200 40 400

a a a 180 20 40 60 eo 100 120 140 160

¢p ' AZIM UTH DEGREES 4>p ' AZIMUTH DEGREES

175 1400 " 1400

ISO 1200 30 1200

12.5 AMPLITUDE 1000 25 1000 AMPLITUDE

~ ~ r ~ -<>- ~

-' :!l- :; -&-

10.0 PHASE -' -20 - PHASE_ / BOO :;

W "- W ~-~- "-0 0 g 0 :> w I i!! l- I-

ii' 7.5 600 if> fil 15 I 600 <! I <!

" I

" ...---- I CL CL

<! '" /' I

5.0 4 00 10 / 4 00

2.5 - 200 200

/' /' -----

a c a a 20 40 60 80 100 120 140 160 a 20 40 60 00 100 120 lAO 160 100

<Pp. AZIMUTH DEGREES <Pp . AZIMUTH DEGREES

2B 1400 7.0 1400

,..- -24 1200 60 I 1200

I I ,- ____ ..J

20 1000 50 ,

1000

1: AMPLITUOE PHASE~ ___ I

-~ ~ PHASE~ __ J

-~ -'

I ' :!l- :; AMPU TUOE /' ~ 16 BOO -' - 40

, BOO

J I I W /_---/ L ____ , ~ W I ~ 0 :> 0 ,.------/ l- i!! :> w :::; I \ I-

30 I if>

CL 12 I \

600 <t fil J 600 '" " I I

<t / ~-- CL " -_/ CL

<t

I /' /'

~ 400 20 /' 400 ------- -------200 10 200

a a 0 a a 20 40 60 BO 100 120 lAO 160 180 a 20 40 60 80 100 120 140 160 100

<Pp . AZIMUTH DEGREES <Pp. AZIMUTH DEGREES

~ FIG U RE S 7 and 8. Excitation requiTed for a Tchebyscheff pattern.

~

I 309

I

---------

<-~

"" W 0 ;:: ~ Il.

" <l

6C~-

140

'00

80

00

4 0

I

I 20 ---

20 40

70~

6t 50 I

60

-----, , I ,

/ I

80 100 120

¢p. AZIMUTH DEGREES

____ AMPLITUDE

4>p. AZIMUTH DEGREES

20

4>p. AZIMUTH DEGREES

1600

PHASE 1400

1200

, I 1000 ~a. I -e-,

:;

800 ~ W if> <t

iE 600

400

200

140 160 180

T 1400

1200

1000

-~ -s-

800 :; "-0 W

~ 600 if> <t I "-

lOO 200

0 140 160 180

1400

1200

1000

-~

~ 800 -'

"-0

, W "- if>

600 <t I "-

400

200

280 ~

24 0

200

-~

~ 160

W 0 ::0 f-~ 120

" <t

80

40

0 0 20 40 60 80

'"p' AZIMUTH

7l I

60

50

-~ -s-:J

40 .____AMPL ITUOE

W 0 ::0 >--~ 30 "-

" <t

20 -~ /' ..--10 -~ ---~ -0

0 20 40 60 80

9p . AZIMUTH

140

120

100 AMPLITUDE

~ :J -80

w 0 -::0 -f- I ~ 60 "-

" /

" --:' <t .--'

4 0 ..- ..------

20

0 0

¢p. AZIMUTH

FI GU RES 9 and 10. Excitation requi red fo r a T chebyscheff pattern

310

100 120 140 160

DEGREES

PHASE

-,-/

/ ~

100 120 140 160

DEGREES

/ I I ,-____ ..1

PHASE ( '1 I

~

( I

/ ~ -

DEGREES

1"0'

r400

---i 2000

' 1600

- 800

l I~O

180

1400

, 1200

- 1000

-" Cl--' "-0

W if>

" I "-

"­o w

1 600 1f1 it

1

1400

{OO 180

400

~-1200

1000

:;,e-800 ::;

"-0 w if>

600 " I "-

400

200

l I

,

"

~

I

I

I

~

---- --------

22

20

18

16

14 -S-

12 12

N 10

" f- 8

6

4

2

0

2 2

20

18

16

~ 14

I:::;; 12

N 10

Ul f- 8

6

4

2

0

0

0 20 4 0 6 0 80

8 " 20 , , '5 ,

Figure 11 0

I N"6 , 8 "20 , , " 8 , P" 18 ond 36

L Figure l i b

100 12 0 140 160 180

FIGURE 11. ' M (rp) , com pared with , T N(Z) , .

r educe the beam width, a higher order polynomial mus t be employed. This fac t is illustrated by noting that the beam width is narrower for th e T6 poly­nomial than for the T4 polynomial for a given value of B.

The curves shown in fi gure 3a, 3b, 3c to figure lOa, lOb , 10c, inclusive, are plo ts of th e amplitude and phase of the excitation function L (¢p) . The abscis­sas are the azimuthal cOOl'dinates ¢ v expressed in degrees . On each figure is listed th e r evelant value of th e order N of the Tchebyscheff polynomial, the lobe ratio B and cylinder parameter x.

311

I t should be no ted that the quantity x , which is ka sin (), is fixed for each set of curves . In the equa­torial plane (i.e. , () = 90 0 ) x becomes equal to k a which is the circumference of the cylinder in wave-lengths. v

In fi gure 7a (for x= 3) it is noted that the excita­tion amplitude is varying in a very pronounced manner . H ere, although the diameter of the cyl­inder is only about one-half wavelength, the m ain beam width is less than 32°. In figure 7b (for x= 5) and fi gure 7c (for x = 8), the oscillatory na ture of the

22

20

18

16

~ 14

I:>' 12

- 10 N

1-"

6

4

2

0 0

22

20

18

16

~ 14

I:>' 12

;::; 10

1-'"

6

______ / M (¢ J/ ASSUMING U ¢p !=O WHEN ¢p ~ 80.

N = 4 8= 20 x= 5

Figure 120

/ ~mm ~~=~ ~· ~~ .- .. '-, "---- - ......... :'-. / / '-.", _____ - M(¢ ! ASSUMING U ¢p!=O WHEN ¢p ~ 80·

-

....:..~

20

~ AND WI TH SLOTS SPACED 10' APART

"'" " ':~

"

40

\, ,. "-

~~-'-~=--'--:1:--=-~,","=--='::"'-"-:== __ -.::..,.,....

60 80 100

N=6 8=20 x=5

Figure 12b

/M(¢J/ ASSUMING LI¢pJ=O WHEN ¢p ~ 80·

AND WITH SLOTS SPACED 20· APART

~ . . ~/M(¢J/ ASSUMING U¢p!=O WHEN ¢p ~ 80'

\. AND WITH SLOTS SPACED /0· APART ,\

\\ ,. \\

'\ "'-.:::-. .... ,

........ ~ ... ---..:;.-..:: ::~:--- -

FIGURE 12 . Pattern colllpa1'ison for vaTious aS8umpli ons.

excitation is less evident. The following sequence of figures for the larger B values show a similar trend.

In nearly every case shown in figure 3a to figure 10c, the phase of the excitation increases as ¢v ranges from 0° to 180°. When the amplitude is rapidly varying such as in figure 7a the phase in­creases almost abruptly by 7r radians (or 180°) near each minimum in the amplitude. There are several seemingly anomalous cases (fig. 7c, for ex­ample) where the 7r radian phase jump is down rather than up.

5. Effect of Using Discrete Sources

The excitation functions presented above are really descriptions of continuous source distribu­tions. In a practical scheme i t is usually necessary to approximate these by an array of discrete sources. As mentioned above, this is to be accomplished by an array of thin axial slo ts disposed uniformly around the circumference of the cylinder. In fi gure ll a is shown the pattern 1M (¢) I calculated from eq

312

j F

i I

r

(17) lI sing 36 10Ls to approximate the continuous ollL'ce distribution given by figure 6b. 'I'hese

calculated poin ts lie virtually on top of the ideal T chebyschcII (T4) pattern. The apparent coinci­dence of the calculated and the ideal patterns is not surp rising since the separation between the eliscrete sou rces is only ;i2 of a wavelength, and thus the excitation is virtually a continuous one. A similar se t of calculations are shown in figme 11 b where t ho source distribution of figure 10c is approAlmated by both 18 and 36 discrete clements. There no\v appears to be some notable departllL'es from the idea l Tchebyseheff pattern, particularly in the case for 18 clemen ts where the side lobes are incr eased. In both cases, however, the main beam is preserved. In these two cases, the separation between the ele­ments is % and % of a wavelength, so the degradation of pattern is not unexpected .

On examin ing some of the excitaLion funclions (for example, fig. 6b), i t is seen LhaL on Lhe rea r half of tbe cylinder, the ampli tude IL (c/>p ) I is less Lhan 10 p ercent of tbe maximum aL c/> p= O. As a matter of interest, Lhe funcLion M (c/» \v as eompuLed usi ng eq (17) again, buL now approximaLing L(c/>p) by discrete sources located only 011 a fronL porLion of the c~'lindel' (i. e., ic/>" I < 80°) . S li ch result s arc shown in fi gufe 12a fot' ang ular inLervl'l,ls bel ween the slots of ] 0 ° and 20°. 'I'he compuLed patLern 111 (c/» bears only a sligltt resemblance Lo l he ideal 1'4 patLeI'H wh ich would be closel:,' simulaLed if Lhe slots extended right around to Lhe back of the cylinder. Similar calculations are shown in 6gure 12b where the exciLation function SJIOWll ill figure lOc is approximated by discrete so urces on Li te fronL porLion of Lhe cylinder. H ere again Lhe paLlel"ll is considerably degraded from Li te ideal 1'6 paLLern al though the in crease oJ the side-lobe level may be tolerated for certain applicatioas.

6. Concluding Remarks

It can be seen from the above curves Lhat, ill principle, any pattern may oe obLained if the exei-

313

La Lion function or source distribution is speci fled in a certain way. Unfortunately, if nar row main beams are desired, the cylinder must be suffieienLly large, oLherwise very complicated excitaLion func­Lions arc required.

It appears LhaL ll sually Lhe continuo us exc iLaLion or source funcLion L (c/>,,) Cim be approximated by discrete sources (i.e., narrow axial sloLs) if Lhe epa­ration is less than abo ut A/8 . Howcver , the rapidly varying excitation f unctions associaLed wiLh smaJl cylinders and narrow beams wou ld require smaller spacing between the source elemenls.

7. References

[1] J . R . Wait, Electromagnetic radiation from cylin drical structures (P ergamon Press, 1959) ; and J . R . \Vait, Radiation characteristics of ax ial slots on a conductin g cylinder, Wireless Eng. 32 316 (1955) .

[2] C. L. Dolph, A current di stribution for broads ide arrays which optimizes the relat ionship between beam width and side lobe level, Proc. IRE 34 335 (1946).

l3] H . J. Riblet, DiscLl ssion of Dolph's Paper , Proc . IRE 35 4 9 (1947) .

[4] G. Sinclair and F. V. Cairns, Optimum pat(,erns [0" a rrays of non- isotrop ic sources, IRE Trans. PGAP- 1, 50 (1952).

[5] R . H . Duhamel, Optimum patterns for end-fire arrays, hoc. IRE U 652 (1953) .

[6] R. H. Duhamel, Pattern synthesis for anten na arrays, Tech.llept. No. 16, Elec. Eng. Research Lab. Univ. of Ill. (May 1952) .

[7] J . R . Wa it and P . Wheelon , Further numerical results for the patterns of s lotted-cylinder antennas, Rept. on Contract AF04 (645)- 56 prepared by P.KC. Corp. uncler th e direction of D . G. Burkhard for the Martin Co. Denver Di v. (June 1958) .

Xotc added in proof: A closely related problem has been studied by Ado lf Giger

in On the Constrwtion oj Antenna A rray with Presc1'ibed I?adiation Plltterns, doctoral t hesis, T echn ischen lIochschu le, Zuri ch, 1956. This wa::; brought to om attention r ecently by Prof. Fran z Tank.

BO U LDER , COLO. (Paper 63D3- 27)