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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
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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
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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-
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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)
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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
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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
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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
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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
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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
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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.