Circular Arc Antennas

CIRCULAR ARC ANTENNAS

S. BALARAM RAO

COLLKOK OF EncjINEEKING, GuIN D Y , MaD R A S. '

{Received for publication December 1, 1955; after revision March 14, 1956)

AB STRACT. This paper deals with the general problem of a radiating element bent in the form of an arc of a circle assuming a sinusoidal current-distribution along the length of the arc. The far field in the plane of an isolated circular arc current-filament is derived.

This leads to the derivation of the radiation fields of two types of symmetrical composite arc antennas in the horizontal plane. The vertical radiation patterns of the composite types of arc antennas are also derived. In the experimental section of the paper, the horizontal patterns of the composite types of anteimas are verified for cases whore the radius of the arc is small compared to the wave length.

SECTION I I N T R O D U C T I O N

In this paper an attempt is made to determine the radiation fields of circular arc antennas. The present investigation appears to be the first general one into the problem of circular arc antennas.

The previous work on circular antennas has been restricted to closed circular loops and open circular loops. While there have been quite a few references to circular loop antennas in literature (Sherman, 1944 ; Moullin, 1946 ; Foster, 1944 ; Glinsky, 1947), the most general ones appear to be those of Sherman and of Glinsky. Sherman has investigated closed and open circular loops whose circum­

ferences are an integral number of wavelengths, assuming a sinusoidal distri­

bution of current. He has deduced the far field in the plane of the loops and along the axes of the loops. Glinsky takes into account the attenuation of the current along the loop in deriving the radiation field of the loop, the circum­

ference being of the order of half a wavelength.

By a circular arc antenna is meant a radiating element bent in the form of an arc of a circle. The radiation field due to such an arc in the plane containing the arc is derived. Neither the length of the arc nor the angle subtended by it at the centre is restricted.

These results lead to the derivation of the horizontal radiation patterns of two Bjrmmetrical composite circular arc antennas.

The radiation pattern of the composite antennas in a vertical plane contain­

ing the line of symmetry of the antennas, is also derived.

39

390

C ircular A rc A ntennas

391

The theoretical radiation patterjis in the horizontal plane o f the composite types o f antennas are verified experimentally for a few particular ceases where the radius o f the arc is small compared to the wavelength. The horizontal radia­

tion pattern o f a horizontal half-wave dipole has also been obtained to check the degree o f agreement between the observed and theoretical patterns under the experimental conditions. The experimental observations arc in fair agreement with the theoretical patterns.

In the last section o f the paper, a short account- o f the method o f evaluation o f the integrals concerned is given. The M. K . S. system of units is used.

SKCTION JI

H E B A 8 1 O V\ H C U 1. A H A K A N T 10 N N A

111 figure 1, A B is an arc o f a circle, centre O, radius a and subtending an angle /y at O. OA is along OX, The arc A B is the antenna and is fed at A by some method, so that assuming a sinusoidal distribution o f current along the arc, at B, there is a current node. The positive direction of (current at an intermediate point, Q, is for convenience, assumed to be anti clockwise. Our problem is to evaluate the radiation field at P, a distant point in the plane containing the arc A B , Since P is a very distant point, QP and OP can be considered parallel.

The current / at ^ on A B in terms o f the maximum value o f the current at a current antinode is given by

where

/ "Inali 2Tra}jr \ / = /,„siri ^ x j

= sin (A/)—Air)

^ = angle AOQ, A — ■

... (1)

A = wavelength o f the radiation in free space and p = angle AOB

392 S. Balaram Boo

The magnetic field at P due to a small element o f length adyjr o f the antenna at Q is given by

Iad}lr . Mt-ijc)

Binye ... (2)

where y = angle SQPy 8Q being the tangent at Q to the arc z ^ P Q

c = velocity o f electromagnetic radiation in free space w = angular frequency corresponding to the wavelength A

<!> =: angle X O P t = time

3 — V — 1

From the geometry o f figure 1, noting that OP ^ A B

y = 9 O °+ C ^ -0 ) ... (3)

z = Zo — a cos {\lf—(/>) ... (4)

where z^ = OP.

In equation (2) the z in the denominator can be substituted directly by 2, while the more refined substitution o f equation (4) should be carried out in the phase term in accordance with the practice in radiation theory.

Carrying out the substitution o f equations (3) and (4) in (2) we have dH ^ 2Az, sin(A/i—A}Jr) cos(}Jf—<j!>)e L e e ... (5) To obtain the field at P due to the entire antenna we have to integrate the expression between the limits 0 and ^ o f ijr. Calling the total field we have

dH

] 3 2^ - sin (Afi - A f ) cos - ^4)c^“ ^ ^ di/f ... (6) The evaluation o f the integral is rather involved and the method o f evalua­

tion is given in the final section.

The value o f the field is obtained aa V i « W A ) -

L J

H , = j

X [ A ooa { n + l ) { f i - ^ ) - A c o s ^ y ? c o s ( n - f l ) ^ - ( w - f l ) s i n . d / ? 8 i n ( n + l ) ^ J

+ j ^ . ( l _ c o 8 ^ / ? ) ... (7)

Giretdar Arc Antennas 393

where j

n=sany positive integer including zero,

Ji(A)y Jn(A) etc are Bessal’s function o f argument A and order 1, n, etc.

In figure 2, we have the circular arc radiator which can be considered as the complement o f the antenna o f figure 1. It carries a current that is the “ return”

for that o f figure 1. A ' B' h the antenna fed from A'. The nomenclature o f the different points in figure 2 is the same as in figure 1 except for the primes.

The current at an intermediate point Q' on the antenna is given by equation, (1), the positive direction being anti-clockwisi along the arc. The distant point at which the field is considered is P as before.

Fig. 2. The complementary basic circular arc anienna.

From the geometry o f the figure we have / = 9O°-(^4'+0) z’ =

The field at P due to the entire antenna A' B' is given by

= f 3 cos sin ( A p - A f ) d

— ^ c o s (^ 4 f)]

dir

On evaluating the integral we have,

N " ,•« M c o s (» + l){^ + /^ ) A ^ - { n + i r ^

nmO

- A COB A fi COB ( « + l ) ? > + ( » + l ) sin sin

( S )

394 8. Bcdaram Rao

In the equations (7) and (8), some terms become indeterminate when A = ( w + l ) . These can be evaluated using the usual methods o f evaluating indeterminate forms.

SECTION III

H O K I Z O N T A L R A D I A T I O N P A T T E R N O F T H E C O M P O S I T E CU R C U L A R A R C A N T E N N A- ^ T Y P E I

This type o f antenna is illustrated in figure 3. The radiation pattern in the plane containing the antenna (referred to as the horizontal plane) can now be deduced from the equations (7) and (8).

Fij?. 3. The composito cireular arc antenna— Type I. (For the horizontal pattern)

A B and A ' B' are the antenna elements. The transmission lino feeds the antenna at A and A\ (This feed arrangement means that if at any point in A B , the instantaneous direction o f current is counter-clockwise, then in A ' B' at the corresponding point the direction is also counter-clockwise. (See section IT.).

R and are the centres pf the arcs. O is the centre pf R R\ and is the origin.

The radiation pattern o f A B is known with respect to its centre R, taking RO as the initial line. The required change in the azimuth o f P, to make equa­

tion (7) apply to this case, is obtained substituting 90®+^ iot ^ in equation (7). P is the point at which the field is being considered. The necessary change in the phase term is effected by expressing R P in terms o f OP and other related parameters. From the geometry o f the figure, noting that P is a distant point, we have

R P = Zq—u sin (f>

(9)

since R P ^ P U ^ O P - O U = Zo a sin 0.

The Zq o f equation (7) has to be replaced by Zq— a sin <l> (in the phase term only) and <f> o f equation (7) replaced by 9 0 °+ ^ . The Zq o f equation (7) in the amplitude term need not be changed in conformity with the practice in antenna theory since the effect on the amplitude is relatively insignificant. Carrying out these

Circular Arc Antennas ₃₉₅

substitutions and tho necessary simplifications we have, for the field at P due to A B ,

H i = j c “ “ j (1 - cos A^\

9>1,3,5 .

<i>)

where

—A cos A p cos (g + 1 )^ — ((7+ I ) sin Afi sin (g + l)^ ]

_|_ jiP [A sin ( p + l ) ( / f _0) + ^ cos A/) sin

—( P + l ) sin cos (p -f 1)^]| ... (1 0) q = any positive odd integ^

X> = any positive even integer

Jp(A)y Jq{A) etc. are Bessel’s functions.

In a similar fashion the field due to B ' in the juxta-position given in figure 2, can be obtained from equation (8) by substituting —(90''—0) for 0 and a sin ^ for in that equation. Carrying out these substitutions and the neces­

sary simplifications, we have for the field at P due to 4 ' J5', H i = j e M * - c" - C ^)[ j ( 1 - coaA^)

f.4 cos {q-\-l)(<f>+fi)—A cos Ajt cos {q+\.)<j>

* ^... —(<74-1) sin A p sm (q-[-\)<i>]

4- V ' 4®*” y p ( A ) —Jp^i(A) M a i n ( p 4 - l ) ( 0 4 A ) - ^ co8^/?sin (p + l)? i j>=0,2... \r \ !

—( p +1) sin 4/7 c o s(p + l)^ ]| ... (1 1) Tho total field at P is given by the addition o f and and we have after

some simplification _

H = H i + H i

X [jA cos (A sin cos (g4-l)^{cos ( g +1)^—cos A ^}

4- sin (A sin sin (?4 -l)?i{(?4 -l) sin A / f-A sin (g41)A>]

4. [•^p(A)—Jp+2(A)] r gjjj gjjj gjn ( p 4 J) <!>{ cos A fi— cos (p 4 1 )^ } 4 * - ( p 4 l ) ®

11-0,2,1....

+ C08 (p 4 -l) COS (4 sin ^S){4 sin (p 4 1 )^ —(j^4"l) * ^ * * * ••• (12)

396 8. Bcdaram Rao

The symmetry o f the pattern as defined by equation (12) about the axis OX is apparent on substitution—^ for ^ when the value o f H remains unaltered both with regard to the amplitude and phase. On substituting 180®— for the quadrature terms remain unaltered while the in-phase terms reverse in sign.

This does not alter the amplitude o f the field which is the square root o f the sum o f the squares o f the quadrature terms and the in-phase terms.

SECTION IV

H O R I Z O N T A L R A D I A T I O N P A T T E R N O F C O M P O S I T E A R C A N T E N N A - — T Y P E II

This type o f the arc antenna is illustrated in figure 4 and is made up o f the two basic types. The transmission line feeds at A and A \ The positive direc­

tion o f current at any two corresponding points o f the antenna have already been mentioned in the section on the basic circular arc antennas. (See section II).

Fig. 4. The composite circular arc antenna— Type II. (For the horizontal pattern) The field at the distant point P due to A B and A 'B ' is obtained by adding the fields due to A B and A 'B ' separately as given by equations (7) and (8).

Performing the addition

^ j f j » j ^ ( 1 - cos Afi)

I A

nrnm

3^ ^ (n + l)^ [c08 (n + 1 )^ —cos Afi] | ... (13)

SECTION V

V E R T I C A L R A D I A T I O N P A T T E R N O P T H E C O M P O S I T E C I R C U L A R A R C A N T E N N A — T Y P E I

, , Figure 5 illustrates the composite circular arc antenna Type I. The field pattern in the plane X O Z is required, X O Z being referred to as the vertical plane.

Circular Arc AiUennas ₃₉₇

g and are two corresponding points on the two arcs, A B and A 'B ' o f the antenna. The positive directions o f current at Q and Q' are from Q to 5 and from Q 'to A ' respectively (See section II). The tangents at Q and Q’ are inclined to the X axis at angles o f on either side o f OX.

B'

F ig , 5. T lio com p os ite circular arc antenna— T y p o I. (F or the vortical pattern).

Take small elements o f the antenna, ad\jr,about Qand Q*and split them into components along O X and O Y respectively. Let us write down the elementary fields in the plane X O Z , due to the two components each at Qand Q' o f the ele­

ments, o f the antenna.

= 3 sin f eMt-tjc) d\]r _{(1 4 )}

dH . =

3

dH^ = 3 sin ijr d ^

ZAZ ^{( 1 6 )}

dH^ = —j — cos ^ dxjf ^{( 1 7 )}

where dHi and dH^ are respectively the fields due to the components along O Y due to the elements o f the antenna at Q and Q\

dH^ and dH^ are respectively the fields due to the components along OX due to the elements o f the anteima at Q and Q\

398 8, Balaram Rao

z ™ QP = O'P, P being the point ^vhe^e the field is being considered.

0 ^ angle XOPy the elevation o f P , The other symbols have already been explained.

The value o f dH^ is negative because o f the direction o f current at Q* relative to that at Q,

The total field at P due to the small elements o f the antenna at Q and Q' is given by

dH = dH^ + dH^ + dH^ + dH^

= j sin ^

Az ^{... (18)}

Now 0) = QP = O P —OM = Zg—a sin ^ cos 0 where Zg = OP and QM is perpendicular to OP,

1 = sin (Afi — A\Jr) Making these substitutions in (18) we have and,

... (19)

... {•)

dH = j sin (Afi - A f ) sin f e H “““ ® ) # ... (20)

A Z g

The total field at P is given by p

H == ^dH

= I J sin ^ e

AZn d f

This integration though involved can be performed and we have fin a%

3 )| ji(*)(l—cos Afi)

~ S fcos ( ? + ! ) / ? - cos ^ /f]

l>-0,8....

(21)

\\ here k A cos 0

angle o f elevation 6 is involved in the pattern only as the argument o f the Bessel function i.e. A cos l>utting ~ () for 0 docs not alter A cos 0 and hence the field pattern as defined by equation (21) is symmetrical about OX

On substituting 18O“- - 0 for 0. A cos 0 becomes negative. We have as a property o f Bassels functions.

^ Jn(-^k), if n is even

^mi^) = «^n( — ^)j if is odd

This means that alJ the ciuadraturo terms reverse in sign. But this does not alter the amplitude of the field which is the square root of the sum of the squares o f the quadiature terms and the in-phase terms respectively.

S E C T I O N VI

V E R T I C A L R A D I A T I O N PA T T E E N OF T H E C O M P O S 1 ^J’ E C I R C U L A R A R C A N T E N N A ~T Y P E 11

Figure 6 illustrates the circular arc antenna—^t ype II. The far field at any point P in the plane X O Z is required.

Circular A rc Antennas 3 9 9

Fig. 6. The composite circular arc aiiionna— Tyj^e II. (For the vertical pattern).J Following arguments identical with that given in the previous section, the differential field at P due to differential elements ad^Jr o f the antenna at Q and Q' is given by

d H ^ j Ifi^

A X ^Bin{Afi-A>Jr) cos if c"+ " <>) d f

The total field at P due to the entire antenna is given by

4 0 0 iS. B alarom R ao

H

e 0

j sin (A ^ —Aijr) eos i/re

jw

(

$ ) '

On performing the integration we have.

" "){j (1 - cos A p) H = j

Az„

+ Y ^ j \ A . ] } (24)

where k — A cos 0

In a manner identical to that used in the previous section it can be seen that the pattern as defined by (24) is symmetrical as far as the amplitude is con­

cerned about the axes OX and OZ. Ecpiation (24) for the value o f ^ = 0, will agree with the value of equation (L‘l) for the value (j> 0, as both equations for these two particular values define the field at any point along the axis OX.

SECTION VII

E X P E R I M E N T A L V E R 1 F 1 CJ A T I O N

The horizontal relative radiation intensity patterns o f the composite types o f circular arc antennas have been obtained experimentally in three cases where the values o f A were small. The relative intensities at different azimuths are with resjDect to the intensity at the azimuth e(]ual to 0°. The type and particulars o f the antenna under test are given in the figures concerned. Since the patterns are symmetrii^al, only the values o f the relative intensities for one quadrant are given.

The observed relative intensity pattern o f a linear half wave dipole has been obtained under the same conditions under which the other patterns were obtained.

This was to chock, under the conditions o f the experiment, the extent o f agreement between the observed values o f the relative intensities and the theoretical values for an antenna whose radiation pattern can be considered to be well established.

This also helped to decide that the observed relative field intensities were reliable and that they had not been vitiated by extraneous factors like haphazard reflections, pick up from the ammeter leads, etc.

The antenna under test was used as the receiving antenna and the relative intensity pattern was obtained by rotating the receiving aerial. The output from the aerial was rectified using a crystal diode and the detector current indicated on a micro<*ammeter placed some distance away.

Circukr Arc Antenmy, 401

The frequency used was 246Mc/s. The l.ransmittor was of the U.S. Navy ype OWS-52244 with a rated output of 25 watts. The distance between the ifiUismitting aerial and the receiving aerial was about 50 yards. The trans­

mitting aerial was arranged to radiate horizontally polarised waves with a fiurly sharp beam to minimise the effects of haphazard r(‘fleotions.

The figures 7, 8 and 9 give the theoretical and observed relative intensities for the composite circular antennas. The details pertaining to the antenna of each figure are given along with the figure concerned. The theoretical pattern is shown by a continuous lino while the observed values are marked by crosses.

10® 20"

80" OT

Fig. 7* Horizontal relative radiation ^)3attem Fig. 8. Horizontal relative radiation pattern of the composite circular arc antenna of Iho composite circular arc antenna

—Type I. Theoretical and ^{observed — Type} I^: Theoretical and observed values (.4 = 0.5, /? = ^r). values (A — 1. /2,

X Observed values

—- Theoretical pattern

402 S. Balaram Rao

Figures 10 gives the relative intensity pattern of a half wave dipole and the observed values of the relative intensities. The agreement with the theorettical values can be seen from the figures.

10" ^{20" 30"}

40°

60°

60®

70" 0*2

80" 0-1

90*

Fig. 9. Horizontal relative radiation pattern Fig, 10. Horizontal relative radiation pattern of the composite circular arc antenna of horizontal half wave linear dipole :

—Type I I : Theoretical and observed Theoretical and observed values.

V a l u e S . ( ^ I , ^ = 7 T /2 ).

X Observed values

— Theoretical values

The actus.1 micro-ammeter readings are given in tho tables that follow,

Circvlar Arc Antennas 403

TABLE I

Composite circular are antenna—-Type I

A = 0.5, /? == 7T(See figure 7)

A zim u th M icro-am ­ meter read­

ing in

iia

Observed relative intensity

Theoretical relatiN^o intensity

0 4 8 .0 1 . 00 1 .0 0

lO 4 4 .0 0 .9 2 0 .9 8

20 4 4 .0 0.9 2 0 .9 3

30 3 9 .5 0 .8 2 0 ,8 4

40 3 9 .0 0 .8 1 0 .7 4

50 2 8 .0 0 .5 8 0 .6 2

60 1 8 .0 0 .3 8 0 .5 0

70 9 . 5 0.20 ^{0 .3 5}

80 4 . 0 0 .0 8 0 .1 8

90 0 0 .0 0 0 .0 8

TABLE n

Composite circular arc antenna-—Type I

A

— =

A zim u th Micro am- Observed meter read- relative

ing in gr/ intensity

Theoretical relative intensity

0 ^{4 2 .5 1 .0 0 1 .0 0}

lO 4 1 .0 0 .9 6 0 .9 9

20 ^{3 6 .0 0 .8 5 0 .9 4}

30 3 8 .0 0 .8 9 0.88

40 3 3 .0 0 .7 8 0 .8 0

50 2 2 .5 0 .5 3 0.68

60 1 1 . 5 0 .2 8 0 .5 2

70 4 . 5 0.11 ^{0 .3 8}

80 0 . 5 0.01 ^{0 .2 9}

90 3 . 5 0 .0 8 0 .2 4

404 S . B a la ra m R a o

T A B L E m

C o m p o s it e c ir c u la r a r c a n te n n a -—T y p e IT

- - --- -

A = \, p =

-6, „ ___ ____

.Azimuth in degrees

Micro-am- . motor road- ..

ing in

Observed relative intensity

Theoretical relative „„

intensity

0 4 9 .5 1 .0 0 1 .0 0

10 4 6 .0 0 .9 3 0 .9 8

20 4 5 .0 0 .9 1 0 .9 4

30 4 1 .0 0 .8 3 0 .8 7

40 3 5 .5 0 .7 2 0 .7 8

50 2 6 .0 0 .5 3 0 .6 6

60 2 0 .0 0 .4 0 0 .5 8

70 1 3 .5 0 .2 7 0 .4 8

80 1 0 .0 0 .2 0 0 .4 2

90 6 .5 0 .1 3 0 .4 0

T A B L E I V (S e e fig u r e 10 )

1 h a lf- w a v e d ip o le , ( D ip o le p e r p e n d ic u la r t o 0®

Azimuth

in degrees Micro-am­

meter read­

ing in fia

Observed relative intehsity

Theoretical relative intensity

0 2 4 .5 1 .0 0 1 .0 0

10 2 3 .5 0 .9 6 0 .9 8

20 2 1 .5 0 .8 8 0 .9 1

30 1 8 .0 0 .7 3 0 .8 2

40 1 5 .0 0 .6 1 0 .7 0

f 5 0 1 1 .0 0 .4 5 0 .5 5

60 6 .0 0 .2 4 0 .4 2

- 70 4 . 0 9 ,1 6 0 .2 9

80 2 -0 0 .0 8 or; 16

90 0 0 ,0 0 0 ,0 0

Circular Arc Antermas 405

SECTION v n i

Method op Evaluation op the Inteoualb

The integrals to be evaluated in the previous sections o f the paper are o f the following form.

I sin

{A/i

Aijr)

In the expression (25) all the constant factor* have been omitted.

The main difficulty in the integration o f (25) is the presence o f the factor cot (iM-rt which cannot be directly integrated. This difficulty is got over by expanding cmCiH-#) using the well known Jourier Bessel expansions, namely,

n^g>

cjAeotirl^+f) =, j^(^A) + 2 ' y ' cos n

71 ■ 1

Once this substitution is made, the terms o f the resulting series are integrable term by them after judicious grouping o f the terms. A few salient steps in the process are given below.

/ ( ^ ) = sin {A fi—Axir) cos co»(\^+f)

= \ [sin -4 y5 r+ ^ + 0 )+ sin ^o(^)

cos n{i/r-\-^)]

n«i

= 4 jQ(A)[Bhi{A^—AtJr-\-^+^)+iiia(Afi—A\jr—iJr—^)]

n« J9 '

-f- ^ j*^J\‘^\9iu[Aj3—Ax/f-\~(n-[-i){xJr+(/>)\+iini[A/^-~AiJr—{n—l)(xJr+ip)]

7 > « 1

+ s in [ilyff— (w+l )( v! ^+^) ] +sm[ - 4/ y— 1) ( ^+^) ] }

/ (^ ) can now be regrouped as follows noting that the (n-\-2y^ term has some ol the terms in common with the term and taking care to include terms which may be left out o f the regrouping.

M ) = i lA fi-A i/ r+ (7 t+ m + < f> n

4

Expression (26) can now be integrated term by term and the limits inserted to obtain the results given in the previous sections*

406 S. Bahrain Boo

This method o f evaluating the integral has been used by Moullin (1949).

It should be noted that with this method there is no restriction on the values o f fi and A.

It should be noted that when A is an integer, one term each o f the equations (7), (8), (12), (21) and (24) become indeterminate. These terms can be evaluated using the usual methods o f evaluating indeterminate forms.

A C K N O W L E D G M E N T S

The author is indebted to the University o f Madras for permission to publish this paper, the material o f which formed part o f the thesis submitted to the Uni­

versity in 1954 for the Master’s degree. The author thanks Dr. K . Sukumaran, Ex-Hcad o f the Department o f Electrical and Telecommunication Engineering, College o f Engineering, Madras, under whose guidance this work was carried out. The author is grateful to Prof. K . S. Hegde, the present Head o f the Depart­

ment o f Telecommunication Engineering, College o f Engineering, Madras, for his suggestions and criticisms during the course o f the investigation.

K E F E K E N C E S Foster. D., 1944, Proc. Inst; Radio Engra., 82, 603-607.

Glinsky. G., 1947, J. Appl. Phya. 18, 638-644.

Moullin. E. B., 1946, Proc. Inat. Elec. Engra., 98 pi. I ll, 345-351.

” (1940), “Radio Aerials” Oxford Clarendon Press, 223.

Sherman. J. B., 1944, Proc. Inat. Radio Engra., 82, 534-537.