228 IRE TRANSACTIONS O N ANTEXNAS AND PROPAGATION March

the primary factor in determining the patterns, and thatquoted a t variance in a later publication. The formulas
fair predictions can be made by use of the calculated wererederivedbyWebsterandtheauthor,working
results. separately and independently, and Carter’s results wcre
Plans for this table originated during an investigation obtained in each case. These results were specialized to
of the directional errorsof airborne Adcock antennas by the case of a stub directly on the cylinder surface-
R. E. Websterandtheauthorin 1953. Inorderto
estimatethe effect of the fuselage,somerough and
ACKNOWLEDGMENT
laborious calculations were made with a desk calculator
andinterpolationgraphsconstructedfromexisting The author is indebted to D. Stickler for a further
tables. Although these calculations did show the gross check on the derivations. The experimental results were
effects, they were too inexact to warrant publication. obtained at the inception of the program in 1953, and
The subsequent establishmentof a computation labora- the assistance of B. Murphy is gratefully acknowledged.
t o r y a t T h Ohio
e State University and installationof a n This table would have been impossible without the ex-
IBM 650 computer made a more exact and extended cellent cooperation of The Ohio State University Nu-
table possible. merical Computation Laboratory and especially of Dr.
Before proceeding with the calculation it was neces- R. Reeves,whowrotethemachineprogramforthe
sary t o check Carter’s original results, which had been computation.

Linear Arrays with Arbitrarily Distributed Elements*

H. UNZf

Summary-A linear array with general arbitrarily distributed 4, it can

Since F(4) is a periodic function with respect to
elements is discussed. A matrix relationship is found between the beexpandedinacomplexFourierserieswiththe
elements of the array and its far-zone pattern. The lower bound of
the stored energy and the Q factor of the array are found. A figure coefficients:
of merit for the array is d e h e d . L
fn = AJ~(KZ~. (4)

T HE far-zone pattern of a linear array with arbi-

trarily distributed similar elements is given by:

~ ( $ 1=
L
A4Le*x2 sin 6. (1)
b=O

Eq. (4)gives a direct relationship between the currents

in the elements of the array, their distribution along the
1=0
axis of the array, and the coefficients of the complex
Fourier expansion of the radiation pattern.
Using the Jacobi expansion1 Since (4) must hold for every n, it may be written in

z
i, sin 4 =
nm
-
E ein+J n ( z ) , (2)
a matrix form:
fn] = [ ~ n l ] ~ r ] . (5)
we get In case we want to geta prescribed radiation pattern
from L + l arbitrarily distributed radiators, we have to
find the inverse matrix [Jnl]-I; but then we can use only
n=-w z=o the first L+1 coefficientsfn of the Fourier series.
Expansion (2) is not the only possible one. Other ex-
* Manuscript received by the PGAP, March 24, 1939. This work pansions in terms of Legendre polynomials and Gegen-
was supported by the U. S. Navy at the University of California, bauer polynomials might be found and similar identities
Berkeley,under contract No.N7onr-29329. The material wasin- might be derived.2
cluded in a thesis submitted in partial satisfaction of the require-
ments for the Ph.D. degree
- at theUniversity of California, Berkeley,
November, 1956.
j Elec. Engrg. Dept., University of Kansas, Lawrence, Kans. * H. Unz, “Linear Arrays with -4rbitrarily Distributed Elements,”
1 G. N. Watson. “4 Treatise on the Theowof Bessel Functions,” Electronics Res. Lab., Univ. of California, Berkeley, Rept. Ser. No.
Cambridge University Press, Cambridge, Eig.; 1952. 60, Issue No. 168; November 2, 1956. (Navy Contract N7onr-29329 )
1960 d[’nz:
r r a gLs i n e a r with
Elentents
Distributed
A.rbitrari1y 203

The total radiated power of the linear array with ar-

bitrarily distributed similar elements is given b y ?

I I
where I ( xi - .xm ) is the interaction coeflicient, and
may be found for basic array (isotropic radiators) and
for dipole elements array in Fig. 1. In case of a basic
array with multiples of half wavelength distribution of
the radiators we get from (6) and Fig. 1:

however, this is not a physically realizable array.

T h e gain of the array may be shown? to be Fig. 1-Interaction coefficients.

L L
A/d,*
T h e suggestion is made for the use of arbitrarily dis-
tributed elements in linear arrays. A general theory has
been given? in order to analyze the performanceof such
L O m=@ arrays, as well as to compare them.
Taking the stored energy 1: i n the immediate neigh- In the arrays with equally-spaced elements, we say
borhood of the radiators only,we can define theQ factor that each element has one degree of freedom, i.e., its
of the array complexamplitude. A lineararraywith L equally-
spaced elements has L degrees of freedom, since we can
1- match L coefficients of the Fourier series. By taking the
Q =z w-.
P general case of arrays with arbitrarily distributed ele-
ments, we add to each element another degree of free-
It may be shown? t h a t we get: dom, i.e., its position along the axis of the array. Al-
though there are certain restrictions on the position of
the element along the axis of the array,? the array with
arbitrarilydistributedelementshasmoredegrees of
freedom than a similar array with equally-spaced ele-
1=0 m=O ments. Therefore the array with arbitrarily distributed
elements needs, in general, fewer elements in order t o
where Q‘ and G I are the Q factor and the gain of each achieve the same performanceas an array with equally-
element of the array by itself. spaced elements. I n case we want to take the same num-
T h e “figure of merit” 4 of the arras. may be defined ber of elements, the performanceof the array with arbi-
from (8) and (10) to be trarily distributed elements can be made better. Using
L the above suggestions, the designer of arrays will have
more latitude in his work in order to achieve the re-
E=( quired pattern and performance of the array. Examples
of design may be found in the original paper.?
I 1=0 I Theaboveapproachmaybeextendedtotwoand
three dimensional a r r a ~ s . ~
In general we are interested i n making 6 small. It may
be shown t h a t in case of-4z = Const, we get the minimum
value of 6,and a1was.s E > 1. In case of super-gain array, H. Unz, “Multi-Dimensional Lattice A k r a y sw i t h Arbitrarily
4 will be very large and the array will not be effective Berlteley,
Distributed Elements,” ElectronicsRes. Lab., ITniv. of California,
Rept. Ser. No. 60, Issue No. 172; December 19, 1956. (Navy
(the gain per unit of stored energy will be very small.) Contract No. N7onr-29529.)