An approximate expression for the radiation from spherically capped conical antennas is derived by the Fourier-Laméeigen-function method. Radiation patterns have been calculated for antennas with flare angle of π/6 and various lengths

King, Ronold

Papas, Charles H.

Caltech Authors - Main

PROCEEDINGS OF THE I.R.E.
Radiation from Wide-Angle Conical Antennas
Fed by a Coaxial Line*
CHARLES H. PAPASt, ASSOCIATE, IRE, AND RONOLD KINGt, SENIOR MEMBER, IRE
Zling
Summary-An approximate expression for the radiation from
spherically capped conical antennas is derived by the Fourier-Lame
eigen-function method. Radiation patterns have been calculated for
antennas with flare angle of r/6 and various lengths.
I. INTRODUCTION
I N A PREVIOUS PAPER' the input impedances of
spherically capped conical antennas fed by a co-
axial line were calculated for large flare angles. By
an extension of this earlier work an expression for the
electromagnetic field of an antenna of arbitrary length
a and flare angle Go (see Fig. 1) is obtained. It is sup-
SPHERICAL CAP-
,ANTENNA
COAXIAL LINE-
Ir I
Fig. 1-Spherically capped conical antenna fed by coaxial feed
line. Characteristic impedance of line equals characteristic im-
pedance of antenna and flare angle is large, i.e., Oo>.7r/6.
posed that the characteristic impedance of the coaxial
line
P2
Zline = 60 In
P1
where P2 and pi are respectively the radii of its outer and
inner conductors, is equal to the characteristic imped-
ance of the antenna
* Decimal classification: R120XR326.612. Original manuscript
received by the Institute, January 16, 1950.
The research reported in this paper was made possible through
support extended Cruft Laboratory, Harvard University, jointly by
the Navy Department (Office of Naval Research), the Signal Corps
of the U. S. Army, and the U. S. Air Force, under ONR Contract
N5ori-76, T. 0. 1.
t Cruft Laboratory, Harvard University, Cambridge, Mass.
1 C. H. Papas and R. King, "Input impedance of wide-angle coni-
cal antennas fed by a coaxial line," PROC. I.R.E., vol. 37, pp. 1269-
1271; November, 1949.
Oo
Zo = 60 In cot 2
From symmetry considerations it is known that the only
nonvanishing components of the electromagnetic field
in the antenna region p2_r<a, Oo:!7Ir/2, and the
radiation region r > a, 0 <0 .7r/2 are Ho, Eo, Er. Assum-
ing that only the TEM mode exists in the antenna
region, and expanding the field in the radiation region
in an infinite series of eigen-functions, the radiation field
is determined by forcing the annihilation of Eo over the
spherical cap and the continuity of Eo over the mathe-
matical surface r=a, 0o<0<r/2.
II. FORMULATION OF THE SOLUTION
In the antenna region for large flare angles, i.e.,
00. 7r/6, only the TEM mode need be considered. The
0 component of the electric field for this TEM mode is
given by the following expression:'
E r 6) =e ikr - feikr
Eo (r, 0)= ,/
rsi (1)
where a and : are constants, k = 2r . free-space wave-
length.
In the radiation region, it is easily shown that2
Eo(r, 0) = i1/ -EAn[ hn-1(2)(kr)
n
-- h(2)(kr) P,,,(cos 0),kr (2)
where An's are constants, hn(')(x) are the spherical Han-
kel functions of the second kind, Pn'(cos 0) = -d/dOPn
(cos 6) are the associated Legendre polynomials, and
the summation is over odd integers.
Since Eo(a, 0) must vanish over the spherical cap,
from (2) it follows that
0 = E An (hn1(2)(ka) - -h (2)(ka)) P.1(cos 0) (3)
n=l1 ka
for 0.6<0go. To match Es across the mouth of the an-
tenna, i.e., across the mathematical surface, r = a,
00.0_ _7r/2, it is necessary to satisfy the relation
,e *ka- fe6ia
a sin 0
-
i A, hn_1(2)(ka) - k n(2)(ka) Pnl(cost) (4)
n=l ka
2 C. H. Papas and R. King, "Input Impedance of Wide-Angle
Conical Antennas," Cruft Laboratory Technical Report No. 52, De-
cember 1, 1948.
1951 49
PROCEEDINGS OF THE I.R.E.
obtained by equating (1) and (2) for r=a. Multiplying
both sides of (3) and (4) by Pn'(cos 0) sin 0 and integrat-
ing with respect to 0 over their respective intervals of
validity, it follows from the orthogonality property of
the associated Legendre polynomials that
aoe- ika _fgeika\(aeika I3eikc~) P7(cos 00)
a
r ~
~nn(n + 1
iA n hn-(2)(ka) - p hn (2) (ka) n+ (5)
Solving (5) for the An's and then substituting into
(2), the 0 component of the electric field in the radia-
tion region is obtained
I II. RADIATION PATTERNS
For given values of the flare angle Oo, the antenna
length a, and the operating angular frequency &(=kc
where c is the velocity of light in vacuum), the radia-
tion pattern of the antenna can be computed from (8).
When ka<<l, only the first terms in the series need be
considered and consequently
M(0)= sin 0, (9)
which is to be expected for the radiation pattern of an
infinitesimal vertical dipole on a perfectly conducting
horizontal plane. As the parameter ka increases, the
X0 2n± 1 l (2) (kr) - (n/kr) h, (2) (kr)
E(r, 6) =K>J Pn(cos 0)Pn1(COS )n + n hk-1 (2)(ka) - (n/ka)kh,(2)(ka)
where
K =
cae-I ka
= Ieia)
At distances from the antenna large compared to
a wavelength, i.e., kr-*oo, the expression [h. 1(2) (kr)
-(n/kr)hn(2)(kr)] approaches (i)ne-ikr/kr. Hence for
kr oo
departure from (9) becomes more pronounced as is seen
from the polar plots of Fig. 2 for the case Oo = r/6.
ACKNOWLEDGMENT
The authors gratefully acknowledge the aid of Miss
J. Klimas who performed all the computations.
e-ikr oo 2n + I in
Eo(r, 6) = K s I P,n(COS 60))Pn'(cos 6) n + 1
r, n'~~~~~+ n Iz,-1(2)(ka) - (n/ka)1z,(2)(ka)
The expression (7) yields the 0 component of the electric
field along the circle r, 0 when kr>>1. Dividing (7) by
(Eo(r, r/2) yields the normalized radiation distribution
function
R(0) = Ee(r, 6)
Eo (r, j)
00
~~~2n +1I
E P,n(cos 60)Pn'(cos 0) - in[hn1(2)(ka) - (n/ka)hn(2)(ka)]-1
n=l n2+ n
00
~~2n + 1EI P.(cos 00)P 1'(0) 2 + in [hnk1 (2) (ka) - (n/ka) hn (2) (ka) ]-
n=l n2+ n
R(6) is a complex function of 0 whose modulus M(0)
gives the distribution of the absolute value of the
normalized electric field over a hemisphere of very large
radius. Explicitly,
M(0) =
X, Pn(COS 6O)Pl'(cOs 0) in [hnl(2)(ka) - (n/ka)hn (2)(ka)]-l
n-I n2 + n
| Pn (COS 60)Pn'(0) in[hn-1(2)(ka) - (n/ka)Ikn(2)(ka)]-1 1
n=l + In
(6)
(7)
(8)
50 JTanuary
PROCEEDINGS OF THE I.R.E.
(a) (b) (e) (f)
0.25 0.50 0.75 1.0 1.25
30° 0° 10° 20° 30°
(c) (d) (g) (h)
Fig. 2-Plots of the absolute values of the far-zone electric field as a function of the zenithal angle 0 for various values of ka and
flare angle 6o ka- 2rx length of antenna in free-space wavelengths.
Angular Jitter in Conventional Conical-Scanning,
Automatic-Tracking Radar Systems*
CHARLES E. BROCKNERt, SENIOR MEMBER, IRE
Summary-Four sources of angular jitter in a conventional coni-
cal-scanning, automatic-tracking radar are discussed and each is
expressed in terms of range and the several system parameters. The
total-angular-jitter-versus-range curve is shown to have a charac-
teristic shape exhibiting a certain range interval of optimum tracking.
The importance of beamwidth in determining the magnitude of
angular jitter is also stressed.
INTRODUCTION
N THE CONVENTIONAL fire control system
(Fig. 1), present target position data obtained from
the fire control director are used in the computer,
to calculate target rates and accelerations. These tar-
get rates and accelerations are in turn used to predict
the future target position, which is required in the solu-
tion of the fire control problem.
* Decimal classification: R116XR537. Original manuscript re-
ceived by the Institute, March 17, 1950.
t Sperry Gyroscope Company, Great Neck, L. I., N. Y.
In present-day practice an automatic-tracking radar
is used to control the director, and this tracking method
yields data which include random deviations from the
true target position. These perturbations are called
"jitter." Jitter is a matter of consequence because the
computer, in computing rates and accelerations from
the radar data, greatly amplifies these random devia-
tions with the result that dispersions in the gun (or
missile) orders are many times the values of the original
perturbations.
With the development of modern high-altitude, high-
speed, highly maneuverable targets, the computer's
task has been made more difficult on two accounts: (a)
The computer smoothing time has been shortened be-
cause the targets are faster and more maneuverable, and
(b) the time required for projectile flight has been
lengthened because longer range missiles are used. Since
the amplification of the tracking perturbations is pro-
511951
0° 10° 20°


Radiation from Wide-Angle Conical Antennas Fed by a Coaxial Line

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Radiation from Wide-Angle Conical Antennas Fed by a Coaxial Line

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