1990 09 large circular array with one element driven.pdf

8/9/2019 1990 09 Large Circular Array with One Element Driven.pdf

1/11


2/11

KING. ARGE CIRCULAR ARRAY

1463

I m C b

h/4,Z)

- 3

Fig. 2.

The function Cb(h,z ) for kh = u/2

- 3

Fig. 3.

The function

S b ( h ,

z) for kh = u

where Q is a constant and

These have very different properties that are determined ini-

tially with the principal component of current, sin k(h

-

21) =

sin k h cos kz

-

os kh sin k z . Thus, the relevant quantities

are

where RI = d w , 2 =

d m .

he

real and imaginary parts of these

two

functions have been

evaluated, respectively, for h = X/4 nd XI2 with P I X as the

parameter. CP(0.25X, z ) is shown in Fig. 2, S , O S X , z ) is

shown in Fig.

3 .

It is seen that when P I X is sufficiently small

as when p = a , the following relations are well satisfied:

kh = ~ / 2 : e [CP(0.25X, z ) CP(0.25h, h)]

-

os kz,

The sums include element 1 and the n elements on each side

that contribute significantly to the parameter @ in (9). In gen-

eral, this is limited to elements for which

bli <

h . Note that

because of geometrical symmetry and the opposite signs of

the progressive phase differences on each side of element 1,

@

in

(9)

is real.

A similar study of the imaginary parts of C , ( h , z ) and

S,(h , z)-also illustrated in Figs. 2 and 3-shows that

ImCP(0.25h, z ) ImCP(0.25h, 0)cos ik z, (l la )

ImS,(OSh,

z )

-

mSP(0.5h, 0)cos ik z. (l lb )

The approximation is very good when kh 5 ~ / 2 ; hen kh =

T the approximate form vanishes at z = h , he exact form

does not. However, when the boundary condition

Z(h)

=

0

is enforced, the approximate form is quite acceptable. The

approximations (1 a), (1 b) suggest that, in general,

This suggests the approximation (12b)

~ m ) ~ / ) [ ~ k ? ) z ,’ ) - Ki:)(h, z’) ]dz’

-

m ) z ) @ ,

As

with

(9),

significant contributions to (12a) are limited to

the element 1 and its n close neighbors on each side for which

(9)

bl; < h .

l h


3/11

1464

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,

VOL.

38, NO.

9.

SEPTEMBER

1990

The indicated properties of the integral in (2) Suggest a

When (18) is substituted in (15), the result is

rearrangement of the integral equation

in

the form

1

2

( m )os kz' + ( m )in k lz'l

z ' (z ' )[Kk;)(z , z ' ) Kkm,'(h,

z')l

dz'

[C(m)coskz+ ~ ( m )inklzl

2

dm os kz' + Um) M m ) I R ~ , ( ~ , z ')

d z ' ,

(20a)

l

Z '(Z)\k '

1

or

-

U m )

=

C ( m) B ( m)

+

$

y ( m ) & m ) +p ) B : m )

TO

2

(13) + U ( m ) & m ) + M(m)Bim)

- C m ) B l m )

+

; m ) B m )

2

here

9

@Ob)

D m ) B : m )

+ ~ ( m ) ~ 3 [ 1

~ i 4 - 1

(m)-

TO Sh (m)(z ' )Ki:)(O,

')dz',

v' ) =

3 Z' ( z ' )Kk~' (h , ' )dz ' ,

M( ')= - Lh Z ( m ) ( z ' ) K E ) ( z ,')dz',

(14)

4?r

-h

where

B; , . . Bim

re the four integrals in (20a). The eval-

uation of these integrals is carried out in Appendix 11.

The equations in (19b) and (20b) can be solved for D m )

and U( ) in terms of the eight constants, A', , . . .

A i m ) ;

B; , .

.

,Bi' ;

the given voltage V m ) ,he contribution

M@)by the more distant elements, and the yet to be deter-

(15)

4?r

-h

h

(16)

and mined constant C( ') The results are

= ~ ( m )l i m ) + ; m )

:m )

+ M( m) ( m )

This sum is over the N 2n +1) elements for which b ~i

.

3 9 (21)

where

111. THEDISTRIBUTIONF

CURRENT

It follows from

(

13) that a first approximation of the current

01 = [A\ ( 1 - D i m ) )+B \ m ) A i m ) ] Q - l ,

is

1


4/11

KING: LARGE CIRCULAR ARRAY

1465

When this is introduced in ( 2 5 ) and the terms are re- 1 e - J k R ~ e - j k R 2

arranged,

Db(h,

Z ) = I COS 3kZ' 7

-)

2 dz'. 35)

-~i "'(sinklzI inkh)

+P;"(cos k z

-

os k h )

2

Pb

cos

Z -

cos - k h

1

+M"'[Np(cos kz - os k h )

+ NF cos l k z

and

IV. THECONTRIBUTION

ROM

THE MOREDISTANTLEMENTS

The contribution to the current by the more distant elements

is contained in the coefficient

M(m)

s defined in (16) with

(17).

The substitution of the current 27) into (16) involves

the following integrals:

It is

of

importance to study the properties of the four dis-

tributions 30)- 33) that combine to give the complete contri-

bution

of

the more distant elements. What is relevant is their

behavior as functions of the radial distance b with 0 5

)z

5

h .

This is carried out in detail in [6] with the help

of

numerous

three-dimensional graphs. These show that

for

all distribu-

tions a plane-wave character is approached in a remarkably

short radial distance- of the order of

b

= h

.

The longitudinal

variations at

b = h

are also investigated in [6]. They consist

of

a small decrease to

z

= f h rom a maximum at

z

= 0.

This slightly reduces the components (cos kz

-

coskh) and

(cos k z / 2

- o s k h / 2 )

in the ends near

z

= h-where they

are already small-in those few elements that are in the range

functions

S l ( z ) ,

S*(z ) ,C(z) and D(z) become constant in

z

at their respective values at

z

=

0.

It is concluded in

[6]

that a good approximation in all cases

with kh 5 .x is an approximate plane-wave behavior with the

amplitude and phase at z = 0. This value is chosen because all

currents vanish at z = h and the major coupling effect is in the

central half of each element. Thus, the relevant quantities in

determining the currents induced by the more distant elements

at b are

h

<

bli

<

4h.At all greater distances the amplitudes of the

Sl(0)

= S I

=

sinkhCb(h,

0)-

coskhSb(h, 0), 36)

D(0)E D = Db(h,

0 )

C O S ikhEb(h, 0).

39)

They are readily evaluated numerically. For greater simplic-

ity

and only a small decrease in accuracy that is limited to

elements in the range h

5

bli 5 4h, complete set of ap-

proximate formulas is derived in [6].

With

36)- 39),

(16) can be solved for

M( ')

o give

e-JkRj2

C ( Z ) =

(coskz' - oskh)---

dz'

When this is substituted in ( 2 7 ) , he current is

j 2 7 r ~ '

I2

( 3 2 ) r m (z ) = C0@(tn)

A m)

l h

=

Cb(h, 2 ) COSkhEb(h, Z ) ,

33)

1

2

where Cb(h, 2 ) and Sb(h,

z )

are defined in (7a) and (7b) and

=Db(h, Z) COS -khEb(h, Z ) ,

+

(Pkm'

+

F("N~')(coskz oskh)

e - j k R ~ e - J k R 2 ( 4 1 )

Eb(h, Z ) = lh7

-)

2 d z ' ,

34)

Note that the more distant elements contribute to the ampli-


5/11

1466 IEEE TRANSACTIONS ON ANT ENNAS AND PROPAGATION, VOL. 38. NO. 9, SEPTEMBER 1990

tudes of (cos kz cos kh) and (cos kz/2

cos

kh/2 ); they

have no effect on the terms sin k(h z 1 and sin k Iz in kh .

V. FINALORM

OR THE

CURRENT

In the derivation of the formula (41), the peaking property

of the real part of the kernel is used to obtain

Z‘”(Z‘)[Kk~)(Z,

z’) -

Kk:’(h, z ’ ) ] d z ’ Q ( ” ~ “ ’ ( z ) ,

(42)

where Q m ) is to be defined at the value of

z

for which the

kernel peaks. Since the current is shown to consist of four

different distributions, a more accurate procedure is to define

a constant Q m ) for each distribution at its maximum value.

Specifically, let

l h

sink(h z’I)[Kk’’(z, z’) - Kk?’(h, z’)]dz’

l h

@“sink(h - zl), (43a)

(sinklz’l - inkh)[Kk:)(z,

z’)

- Kk:’(h, 2’)Idz’

f h

Qim)(sinklzI inkh), (43b)

l h ( c o s k z ’- oskh)[Kk~)(z,

’)

Kk:’(h, z’)]dz’

-

Qg)(coskZ

-

oskh), (44)

cos-kz -cos-kh . (45)

i 2

The parameter

aim)

efined for sin k(h - Z 1 is an adequate

approximation for sink

Iz

1-

sin kh . With these definitions,

(42) can be corrected to take account of the somewhat dif-

ferent

Q

functions for each component. Since the dominant

current in all cases is sink(h - lzl), it is appropriate to use

@im)s defined in (43a) as the function Q(”’) which is used

to determine all of the coefficients. The improved formula for

the current is

If desired, either the term sink(h - lzl)

or

the term

(sin klzl ink h) can be eliminated. Since P ( m ) A (m )

coskh, it follows that with sink(h - lzl)

-

Pp’( sin klzl

sin kh) = sin kh (cos kz os kh)

A(m)(

in

k

Z in kh) ,

(46a) becomes

sinklzl - inkh)

.i

Alternatively, use can be made of the identity - sin

k

Iz

1

in

kh)

=

-(cos kz

-

os kh) tan kh

+

[sin k(h Jzl)]/cos kh .

With this (46b) becomes

The three forms (46a)-(46c) are equivalent. The form (46c)

has the disadvantage that individual terms become infinite

when kh = a/2 .

VI. THE

PARAMETERS

The three parameters

Q(m )

hat occur in (46) are introduced

in

(43)-(45). Explicit definitions are

1 1

.

(cos

,kz

- cos ,kh)

.

Kk ’(h, z’)] d ~ ’ , (47)

X/4, kh 2 n/2; and

-

Kk:’(h, z’)] dz’, (48)


6/11

KING:

LARGE CIRCULAR A R R A Y 1467

2

1

q.$“’= (1 -cos ikh) I f h (cos ?kz’

-

cos

[Kk:’(O,

z’)

- Kji“,’(h,

z ’ ) ] dz’.

(49)

Note that these parameters are defined at the respective max-

imum of the particular distribution. This occurs at z

=

0

for

@?

and Qgwhen kh 5 7r. The explicit evaluations are

carried out in Appendix

111.

VI1 . ARBITRARILYRIVEN RRAYONEELEMENTXCITATION

The Nindividual driving voltages I/; and currents Z;(z), =

1, 2 , .

.

.

,N,

in an arbitrarily driven circular array are related

to the phase-sequence voltages

V ( m )

nd currents Z cm ) ( z )by

the formulas

I/. e j W i - l ) m / N v ( m ) ;

N - l

I

m

=O

N - l

=

e j 2 * ( ; - l ) m / N ~ ( m ) (

) ,

(50)

m=O

when referred to antenna 1. The inverse relations are

y m ) =

N -

5

- 2 * ( ; - l ) m / N v..,

i = l

N

1 is driven and all others are parasitic,

V ;

=

V I ,

=

1;

V ,

=

0,

=

2 , . . . , N In this case (51) gives V ( m )

=

V I / N

for all

m

and V “ ) / V I can be replaced by l /Nin (53a)-(53d).

When element 1 is driven with all others parasitic, it is

evident from (53) with = 2, 3 , . . . , N that there are sig-

nificant contributions from the terms sin k(h

zl)

and (sin

k J z J

inkh) to the currents in the parasitic elements when-

ever there are any elements near enough to the driven one

to contribute to

Q (m ) .

These terms have discontinuous deriva-

tives at

z

=

0,

which is incompatible with the requirement

from symmetry that dZ;(z)/dz

=

0 at z

=

0 when

V ;

= 0.

This defect is a consequence of the approximations involved

in the representation (9). The representation is simplified with

the form (46b) in which sin k(h

-

zl) does not appear. The

equivalent of (52)

for

an array based on (46b) instead of (46a)

and specialized to have only element 1 driven is

-t;(sinklzI inkh)

+f;(coskz -coskh)+h; -cos-kh)}, (54)

2

where

N - 1

*g’

y:”)

f;(cos kz -Coskh) +h; (52) hi = N - I

m =O

where

e

2 s ( i

-

) m / N

.

(5%)

9 (534

)

j 2 n ( i - l )m / N

s;

= v,’

a@

m=O

The self- and mutual admittances are

Z;(O)

j27r

= __ [lisinkh +f;(l -coskh)

- 1 v (m ) p (m )

m =O

(53b)

y l f =

= v;l

VI

l o

S

) j 2 * ( ; - l ) m / N ,

Qkm

A m )

When the distances between elements are all large enough so

that

bl; 2

for all values of i , the parameter

Pim)

educes

to @ s ,

he value for the isolated element. When this is true,

(55a) gives t ;

= 0, =

2 ,

3 , . . . N .

When there are closely

spaced elements with 61;

<

h,

f;

0,

so

that the term (sin

k l z ( inkh) with its discontinuity in slope at z = 0 remains

for

the parasitic elements. This

is

not correct. The current

induced by a driven element in an adjacent parasitic one is

similar to that in the driven one but it must have zero slope at

In order to improve the representation for the parasitic ele-

ments,

=

2,

3, . . ,N, the approximation (43b) can be rein-

troduced specifically for the term (sin

klzl -

sinkh) which

m=O

e 2u(i - I ) m / ”

(53c)

m

=O

(534 = o .

2 n ( i - I )m / N

The self-admittance is Y = Z I ( O ) / V ~ , he mutual admit-

tances are Y

=

Z;(0)/V1, i = 2 , . . . ,N . When only element


7/11

1468

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL.

38,

N O,

9,

SEPTEMBER

1990

generates unreal discontinuities at z = 0. Thus,

sin k Iz

in

kh

e j 2 s ( i - l ) m / N

x Re { S b , , ( h , )

-

S b , , ( h ,

h )

- inkh

Here

Rlj

= J(z

’ ) ~ b:;,

R l h ; = , / (h ’ ) ~ b:;. It

follows with ( S a ) that

;(sink(zI - inkh)

-

;(z) g;(h),

(57)

where

N - l

gi(z) i(h)

N-I (* i m ) ) -2 e j 2 s ( i - l )m l N

m =O

5

e j2ir (k- l )m/N

k = l k = N - n + )

‘Re{Sb~k(h,

> - S b l * ( h , h )

inkhEb,,(h, z ) -Eblc(h, h)l).

(584

When there are no near elements,

n

=

0,

and with

bl

I

=

a ,

@im) Qs, the value for the isolated element. It follows that

(58a), like (S a ) , vanishes. Thus

g;(z)

- g ; ( h ) =

N-’*;2Re{S,(h, z )

S , ( h , h )

-

inkh[E,(h,

z )

E , ( h , h ) ] ]

.

2 r ( j - l )m l N

=

0 ,

N - l

i = 2 , 3 , .

.

, N .

(58b)

In this case the entire term sink )z in

kh

disappears from

all parasitic elements. It remains only for the driven element

=

1,

where it

is

needed.

When n

0,

the values of

Qim)

or the

N

values of

m

are

not alike so that t ; in (S a ) and gi(z)

-

; ( h )

n

(58a) differ

from zero. Each value of

Qim)

ncludes the term k = 1

in

the sum in (58a) and this is combined with the

2n

terms for

which blk <

h .

The general nature of the difference between

the left and right sides of (57) can be seen from the leading

term k =

1.

For this, sin klzl

-

sinkh

is

replaced by Re

m =O

I I I

-Ot4 -0.2 0 0,2 0.4

Z / X

Fig.

4.

The functions @(sin

klzl -

inkh) and

Re { S , ( h , z) - S , ( h , h )

sin

kh[E,(h, ) - E , ( h , h ) ] } or kh = ?r and kh

=

*/2;

a X = 0.007022.

{ S , ( h , z ) - S S a ( h , )-sin k h [ E , ( h , z)-E,(h, h)]} /Qs . he

two functions are shown graphically in Fig. 4 with a / A =

0.007022 or ka = 0.044, kh = a and kh = a / 2 . It is seen

that the principal difference is a rounding of the sharp peak at

z

=0. When the contributions by the other terms-for which

the

Pim)

differ from

\ks

in that they depend on

bl;

and not

bll = a-are included, their effect is a broader rounding of

the peak at z =

0.

In all cases the slope of the current has the

correct zero value at the center of all parasitic elements when

(57) with (58a) or (58b) is substituted in (54). The resulting

improved form is

, = 2 , 3 , . . .,N, (59a)

where

g ; ( z )

- gj(h) is given in (58a),

f;

nd hi in

55b)

nd

(5%). The mutual admittances are

Ylj = ;(O) - g ; ( h )

zi o)I j 2 =-0 [

+ j( 1 - oskh) + h i 1

-

COS - k h

.

(59b)

( 3

A

comparison of (59b) with (55d) indicates that g;(O) - g ; ( h )

replaces

t;

sinkh

.

These terms are significantly different.

However, the principal contributions to the mutual admittances

come from the terms f (1- cos

k h )

+

h;(

1- cos kh / 2 ) .

VIII.

CLOSELY

PACEDLEMENTS

When all elements are closely spaced, the phase sequences

include the cage antenna with m = 0 (all elements driven


8/11

KING: ARGE CIRCULAR ARRAY 1469

in

phase) and various forms of multiconductor transmission

lines when

m

> 0. These are readily illustrated with N = 4

which includes the four-element cage antenna when

m = 0

and two types of transmission line with m = 1 and 2 m 3

interchanges the currents in elements 2 and 4). The sum in the

definitions of the eight constants A (m ) nd B(“‘)becomes

N 4

C e j 2 r ( i - l ) m / N = x e j ( r / 2 ) ( i - I ) m . (60)

It follows from Appendix

I

that Ay’

=

(4j/9(O)) Si 2 kh,

A?’ = (4jj9‘O)) Cin 2kh, A?’ = (4j/9(0))[2Sikh/2 + Si

3kh/2],

440

= (8j /9(0))Sikh;AI” = AI )

=

0 ,

=

1, 2, 3, 4. The constants B(m)

n

Appendix I1 are

i = l

r = l

PF) = -( 1/3) COS kh,

E‘:)

= PE) = P‘3’

-

( 1/3) sin kh,

p g )

=PE)

=p ( 3 )= 0; A ( l ) = A 2)= a 3) (2/3) cos kh.

Since there are no distant elements ( b >.h) , the coefficient

F ( m )= 0 for all m.With the above values for Pim), ;”,

P E ) ,

A(m)

nd

9im),

46c) reduces to

p z ) = j2nV“’__ sink(h

-

z l ) .

oq l) coskh ’

Here Z:” is the characteristic impedance of the two conductor

transmission line. Similarly

The current in (62) is that of a center-driven two-wire line

with open ends and spacing

bl3.

When driven in the phase

sequence m = 1, the diagonal pairs are two-wire lines with

each in the neutral plane of the other. The current in (63) is

connected in parallel.

2Cin2kh)

1

2

- coskh[Si4kh si2kh] 9 (61b) that of a four-wire line with the diagonally opposite conductors

Cinkh)

1 . 1

2 2

- sin -kh[Si3kh Sikh ]

IX.

THE SOLATED

LEMENT

The brief study of four closely spaced elements shows that

the quite complicated general formula (46c) reduces exactly

to the familiar relations for nonradiating multiconductor trans-

mission lines. It is appropriate to specialize the general for-

mula (46a) to the single center-driven element for comparison

with measured data. The current in the isolated element is

+

F)

coskz

-

oskh)

+ (-)DPD

(cos ? k z -cos -kh)}

.

(64)

9 s 2

The admittance is

+

(

y )1 - coskh)

+ y )1 -co skkh )}. (65)

When kh = n/2 and a / h = 0.007022, @S = 9 c = 6.19,

@ D

=

6.21 and

(9.00 -j7. 62)c oskz +j2.68(1

-

inklzl)

V


9/11

1470

k h : < , 4 0.007022

kz

IEEE

TRANSACTIONS

ON ANT E NNAS AND

PROPAGATION. VOL.

38,

NO.

9,

SEPTEMBER 1990

I i ( z ) / V ,

1

( z ) / V

m A / V )

Fig.

5 .

Current on a half-wave dipole, as given by

(66).

k h = n . =

0 007022

n

2

k z f

n

4

-

I i k ) / V , 1; ( z ) / V

m A / V )

Fig. 6.

Current on a full-wave dipole, as given by

68).

The admittance and impedance are

Y =

9.46

-

4.10 mS; Z

=

89.0 +j38.6

0.

(67)

The complete distribution of the current

in

the form I,(z) =

I:(z)

+

I:(z) together with its component parts is shown

in Fig.

5.

Also represented is the measured current. The agree-

ment is seen to be very good.

When kh

=

?r

and a / h

=

0.007022,

Qs

=

5.69,

Qc

=

7.49, QD

=

7.12; the detailed formula for the current is

-

2.92 sink Iz

I

+ (0.025 + 0.539)( 1

+

coskz)

= {

+

(0.858+ 0.463)cos

The admittance and impedance are

Y =

0.91 + 1.54 mS;

=

284.4

-

481.3

R.

(69)

The complete distribution and its component parts are

shown in Fig. 6 together with the measured values. The agree-

ment is good. Both the maximum amplitude of the current

and the admittance are quite accurately given. However, the

measured curve approaches z =

0

more steeply than can be

represented precisely by the simple trigonometric functions.

X.

CONCLUSION

A systematic solution of the

N

coupled integral equations of

a circular array with one element driven has been carried out

with the method of symmetrical components.

A

careful study

of mutual interactions shows that the near elements (bl;

< h )

and the more distant elements

(bl; )

must be treated sep-

arately since they contribute differently to the currents and

the self- and mutual admittances. The usual discontinuities at

the centers of parasitic elements introduced by the method

of

symmetrical components have been removed.

APPENDIX

EVALUATIONF THE CONSTANTS( )

The constants

Ai ,

. .

,Ai '

are defined

in

(19) with (14).

They all involve the imaginary part of the kernel,

Ror

=

dz-. (70)

For

the 2n

+

1 near elements,

byj <

h2 so that (sin kRoj)l

Roi (sin kz')lz'. The following integrals are involved:

I = 2 1

coskz'-- sin kz'

dz' =

.Ikh?

u

=

Si2kh,

Z

d u = Cin2kh,

2 = 2

~ dz'

=

lh

1

sinkz'

dzl

= 4

f k h

cos2

U

sin

U

I 3

= 2 1 cos ?kzT7

du

1

3

k h

sin U

+

sin 3u

du

=

Si -kh +Si -kh ,

(73)

=.I 2 2

1 4 - 2

.Ih

z/ =

2 i k h

du =2Sikh. (74)

With these integrals

-n

N

n

(77)

m :

N

n


10/11

KING:

LARGE CIRCULAR

ARRAY

147 1

APPENDIX

near-range difference kernel is involved. This is

EVALUATIONF THE CONSTANTS

m )

Kkz ’ zm,

z’ ) -Kkz’(h, z’)

The four constants B ( m ) re defined in (20) with (15). They

-

all involve the real part

of

the near-range kernel,

n + l

coskRIhi

e j 2 r i - l ) m l N p

(79)

where

zm

locates the maximum value of the particular current-

Kk?)(h,

z’) =

Rlhi ’ distribution function and

i=N-n+l

where

The following integrals occur:

Ro; = dt‘

+

b:i, RIh; = d ( h

’ ~

6:;.

86)

RI,,; = J(h

’ ) ~ b:;

and R2h; =

J ( h

+Z’)’ +

b:;.

The following integrals are involved:

kh

5

n/2

2h

-

In G s ( h ) ,

b

4h 1

[ bli 2

-

os kh

In -

Cin4kh sinkh Si4k h,

(80)

where

Gs(h)

=

-Cin 2kh sc kh {Si 2kh cos kh

+ $

Si 2kh

2Cin2kh)

-

[Si4kh Sikh] cos2kh

1

2

-coskh[Si4kh -2Si2kh1,

dz’

93 =

h

os Zkz‘-

CoskRlh;

Rlhr

h

1

cos -kh In -(Cin 3kh

+

Cin kh)

2 [ bli 2

(82)

1 1

2 2

-

sin -kh[Si 3kh Sik h],

+ [In2

-

(Cin4kh Cin2kh)sin2kh]},

(87c)

(h

-

X/4 ’)’

+

b:;

(h X/4 -

‘) ‘ +

b

h

82s =

L

ink(h -

z’l)

cos kd (h

-

’ ) ~ b:i

dz’, kh /2

n(2kh

-

a

4kh

-

K

cos kRlh; 4h

dz’

-

n Cin2kh.

83)

“ = L h X

b

ii

[

k2b;;

]

- I n

[ I

os 2kh

The approximate formulas are obtained after bl; has been ne-

lent for

b l l = a

and other elements for which kbl;

< 1.

For

elements in the range

1

< kb l i < kh, the accurate integrals

1

glected in all terms except those that become infinite at

z’

= h ,

since in the near range b:;

< h 2 .

This approximation is excel-

-

[Cin (2kh

-

a + Cin n]

-

Si 2kh + cos 2kh [Cin (4kh

-

n)

- Cin (2kh

- a +

Si 4kh - Si 2kh]

1

2

1

1

2 2

must be evaluated.

The final formulas are

= -sin2kh[2ln2+Si(4kh -a )

APPENDIX

11

(88b)

Si(2kh n) Cin4kh +Cin2kh ],

(84)

kh

5 n

-2ln- +Gc(h) ,

EVALUATION

F

THE PARAMETERSm )

The three parameters

Qr),

km ,and Q r ’ are defined in

(47)-(49). They include contributions from the 2n + 1 ele-

2h

b

;

ents that are within the range bli < h . The real part

of

the


11/11

1472

IEEE TRANSACTlONS ON ANTENNAS AND PROPAGATION. VOL.

8,

NO. 9, SEPTEMBER 199

where

Gc(h) = -Cin2kh + ( 1 -coskh)-’

. [2 Cin kh +

Cin4kh Cin 2kh] cos kh

i Si4kh sink h, (89c)

) dz’

OS ikz‘ OS

$

kh

8D

=

L

1

-cos ikh

Rlhi

n

i

n

2h

b

i

2 1n GD(h),

(90a)

(90b) kh

5

T .

Note that, for the isolated element, @S

=

dls, kh 5 ~ / 2 ;

@S = 8 2 ~ 3 h > ~ / 2 ;@c

=

8 ~ ,h

S T ;

@ D

=

80,

REFERENCES

here

G D ( ~ ) 1 -cos $kh)-’[i cos ikh(Cin3kh

+ 5

Cin kh

-

2 Cin 2kh)

-

Cin ikh

-

Cin ikh

-

sin ikh( Si3k h Sikh)].

(90c)

The approximate formulas involve the neglect of b in

the arguments of the trigonometric terms. This is a good

approximation when kbli

< 1. For

elements in the range

1

5

kbl;

5

kh, the accurate integrals must be evaluated. With

the integrals (87)-(90) the are

n

@ i m ) = C e j 2 ~ i - l ) m j N d 1s;

kh 5 T/2, (91)

i

[ I ]

R.

W .

P. King, “Supergain antennas and the Yagi and circular arrays,”

IEEE

Trans. Antennas Propagat., vol. 31, pp. 178-186,Feb. 1989.

[2] R. W . P. King, R.

B.

Mack, and S . S. andler, Arrays of Cylindrical

Dipoles.

[3] R. B.

Mack,

“ A

study

of

circular arrays, Parts

1-6,”

Cruft Lab.,

Harvard Univ., Cambridge,

MA ,

Reps. 381-386, 1963.

[4]

R.W .

P. King, “Linear arrays: Currents, impedances and fields,

I,”

IRE Tmns. Antennas Propagat., vol. AP-7, pp. S440-S451, Dec.

1959.

J . D.

illman,

Jr., The Theory and Design of Circular Antenna

Arrays. Univ. Tennessee Experiment Station, 1966.

R.

W .

P. King, “Electric fields and vector potentials

of

thin cylindrical

antennas,”

IEEE Trans. Antennas P ropugat.,

pp. 1456-1461, his

issue.

London; New York: Cambridge Univ. Press, 1968.

[5]

[6]

I

Ronold

W. P.King A’3O-SM’43-F’53-LF’71),or a

photograph and

bi-

ography please see page 846 of the June 1990 ssue of this TRANSACTIONS.

1990 09 large circular array with one element driven.pdf

Documents