end-fire radiation from planar and large cylindrical … · endfire radiation from planar and large...
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AFCflL-71-0322 UNCLASSIFIED
^
END-FIRE RADIATION FROM
PLANAR AND LARGE CYLINDRICAL ARRAYS
Giorgio V. Borgiotti Quirino Balzano
RAYTHEON CO. MISSILE SYSTEMS DIVISION BEDFORD LABORATORIES
Hartwell Rd. Bedford, Massachusetts 01730
CONTRACT NO. F19628-70-C-0226 / PROJECT NO. 4600
TASK NO..
WORK UNIT NO..
460010
46001001
SCIENTIFIC REPORT NO. 2 MAY 1971
CONTRACT MONITOR: Allan C. Schell Microwave Physics Laboratory
This document has been approved for publ'c release and sale; its distribution is unlimited.
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AIR FORCE CAMBRIDGE RESEARCH LABORATORIES AIR FORCE SYSTEMS COMMAND
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I. ORIGIN A YIN6 ACTIVITY 'Corporar* wii<
Raytheon Missile Systems Division Radar Laboratory Bedford, Massachusetts 01730
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1. REPORT TITLE
ENDFIRE RADIATION FROM PLANAR AND LARGE CYLINDRICAL ARRAYS
4. oescmPTive NOTES r*Vp« »' »P»«
Scientific Interim •mf inchiai** rfafaaj
I. AUTNORW (L»tl nmm: Hral MUM, InltlU)
Borgiotti, Giorgio V. Balzano, Quirino
«. REPORT DATE
May 1971 7a- TOTAL NO. Or PACKS
27 7». NO. OP nmws
10 •a. CONTRACT OR GRANT NO.
F19628-70-C-0226 ft. RROJBCT NO.
4600-10-01 e DoD Element: 62702F
* DoD Subelement 674600
tRTSJ
BR-6410 Scientific Report No. 2
• ft. OTHCR RfPOAT HO(S) (Any oth; number» tfiar may ft« aas/jnad tftMr report)
AFCRL-71-0322 10. AVAILABILITY/LIMITATION NOTICES
This document has been approved for public release and sale; its distribution
is unlimited. II. SUPPLEMENTARY NOTES
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12. SPONSORING MILITARY ACTIVITY Air Force Cambridge Research Laboratories (LZ) L. G, Hanscom Field Bedford. Massachusetts 01730
II. ABSTRACT
The analysis and the design of the elements of a large array of circular apertures on a triangular grid is approached by modeling the antenna as an infinite structure rotationally symmetric and periodic along the cylinder axis. Because of this particular symmetry every possible excitation is the super- position, with suitable weights, of a set of fundamental excitations having uniform magnitude and linear phase progression in the azimuthai direction and in the direction of the cylinder axis ("eigenexcitations"). Thus, by invoking super- position the electromagnetic analysis of the array is reduced to the solutions of the simpler boundary value problems pertinent to the set of eigenexcitations. This is done by expanding the field in normal modes in the region exterior to the cylinder and in the waveguides feeding the apertures, followed by a field matching at the cylinder surface (obtained approximately through Galerkin' s method). The realized gain pattern of the radiators can be modified to a con- siderable extent by using an "element pattern shaping network" (in the radiator waveguides), serving the purpose of matching the array for a selected eigenexcitation. Criteria for the network design are given. A series of numerical examples illustrates the technique and shows that a "flat" element pattern can be thus obtained with a gain fall off with respect to the peak of less than 6 db at 80 degrees.
1473 UNCLASSIFIED Security Classification
" .'. I 1 ■ <JWM1 ji , ■m.jy.in,.mnmjt*!iH;jn
UNCLASSIFIE D_ Security Classification
T* KEV WOfiOi
Phased Arrays Endfire Arrays Conformal Arrays
LINK *
HOLE WT
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noLC
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UNCLASSIFIED Security Classification
wJWaL_,.:ujf|iapju.iJHi **•- MJJLH-* ■,~MJJUI ~^^*--—-TV-^L-r^-rcw^-«-.''
AFCRL. \^H UNCLASSIFIED
t
Details of fflus^SM«^^
END-FIRE RADIATION FROM
PLANAR AND LARGE CYLINDRICAL ARRAYS
Giorgio V. Borgiotti Quirino Balzano
RAYTHEON CO. MISSILE SYSTEMS DIVISION BEDFORD LABORATORIES
Hartwell Rd. Bedford, Massachusetts 01730
CONTRACT NO. F19628-70-C-0226 PROJECT NO.
TASK NO.
UNIT NO.
SCIENTIFIC REPORT NO. 1 MAY 1971
CONTRACT MONITOR: DR. ALLAN SCHELL USAF ELECTRONIC SYSTEM DIVISION
DISTRIBUTION
Prepared for
AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE OF AEROSPACE RESEARCH
UNITED STATES AIR FORCE Bedford, Massachusetts 01730
UNCLASSIFIED
ABSTRACT
\ Endure radiation in planar finite arrays of circular apertures in aper-
tures in triangular arrangement is investigated by using a technique con-
sisting of using the results obtained via the usual infinite array model as a
zeroth order approximation of a perturbation procedure. It is shown that
finite arrays can be scanned up to 90 degrees from broadside, still retaining
substantial radiation. The endfire radiation can be enhanced through an
appropriate design of the element feed network. With minor modifications
the method can be simply applied to the approximate investigation of the
radiation of an array on a cylinder (having a large radius in terms of wave-
lengths) in the axial direction.
xii
UNCLASSIFIED
TABLE OF CONTENTS
Page
ABSTRACT ii
ENDFIRE RADIATION FROM PLANAR AND LARGE CYLINDRICAL ARRAYS 1
1. Background 1
2. Edge Effects in Planar Periodic Arrays Z
3. Endfire Radiation in a Planar Array 5
4. Finite Arrays on Cylinders - An Approximate Analysis of the Axial Radiation 6
APPENDDC 8
REFERENCES 19
-. ■ ■■■ ■■ ■
UNCLASSIFIED
LIST OF ILLUSTRATIONS
Figure Page
1 Array Lattice 10
2 Amplitude of Element Voltage 11
3 Amplitude of Element Reflection Coefficient 12
4 Array Pattern, Broadside Match, Broadside Scan 13
5 Array Pattern, Broadside M-ich, Endfire Scan 13
6 Amplitude of Element Voltage 14
7 Amplitude of Element Reflection Coefficient 15
8 Array Pattern, Match at 80 Degrees from Broadside, Broadside Scan 16
9 Array Pattern, Match at 80 Degrees from Broadside, Endfire Scan 16
10 Gain versus Scan, Planar Array, Broadside Match 17
11 Gain versus Scan, Planar Array, Match at 80 Degrees from Broadside 17
12 Gain versus Scan, Cylindrical Array R = 50X 18
13 Gain versus Scan, Cylindrical Array R = 50X 18
Vll
iiii.iijIWI.JIJ iiii|Wiii|i. i.i iw^iiw iiiinmipi. WIBHW»JI,.IU.IIII.IIUIII.II»IIIIL A.m.» m. .HipiW. ■ «."W—"«JJWIWUi. .-<~
UNCLASSIFIED
ENDFIRE RADIATION FROM PLANAR AND LARGE CYLINDRICAL ARRAYS
1. Background
The analysis of planar and cylindrical periodic arrays is usually based
on the infinite array model [ 1-6 ] . In this way all the analytical problems
related to "edge effects" are avoided by simply postulating that the environ-
ment is the same for each element. It is well known that according to this
model a planar array cannot have any endfire gain. Also for cylindrical
infinite periodic arrays the gain can be shown to go necessarily to zero in
the direction of the cylinder axis. It is consequently clear that the infinite
periodic array model becomes useless for those directions for which end-
effect plays a fundamental role, and a formalism which better reflects the
physics of the phenomena must be sought.
Considering the finiteness of the array makes the analysis very involved,
and leads invariably to facing the problem of the inversion of large matrices
with complex elements. The problem can however be circumvented through
the use of a perturbation technique, based on the recognition that for a large
array the "eigenexcitations" of the structure (eigenvectors of the element
scattering matrix) [7] are "not too different" from those of an infinite
structure (with the same element and spacing).
A technique based on this idea, which does not require any large matrix
inversion, has been recently introduced and applied to the analysis of an array
of uniform slits on a conducting ground plane, considering the radiators as
"one mode" elements [7] . This report extends the results of [7] in three
respects:
• The practical case of circular apertures in a triangular
arrangement is considered.
• A technique for enhancing radiation in directions close to
endfire is developed, based on the idea of modifying mutual
coupling among elements, through the us * of matching networks
in the waveguides feeding the elements. 1
~~ : «"^V-BIHWSW"
UNCLASSIFIED • An approximate simple method for the analysis of the radia-
tion of a finite cylindrical structure for axial scan has been
introduced.
In Section 2 the formalism for planar arrays is very briefly discussed.
Computed examples are presented in Section 3, focusing the attention on the
influence of element matching networks on array endfire or quasi endfire
radiation. In Section 4 it is shown how the planar array results can be useful
in cylindrical array performance evaluation.
The development in this report is based on the algebraic technique
developed in Reference [7] , which for reasons of brevity, will be assumed
familiar to the reader.
2. Edge Effects in Planar Periodic Arrays
The structure investigated consists of a periodic array having as
elements circular waveguides terminated on a ground plane. The elements
are arranged in a triangular grid and the array is infinite in y direction and
has a finite length in x direction. The lattice geometry is indicated in
Figure 1. The elements in each infinite column are assumed to be in phase:
thus only scan in a plane orthogonal to the array edge will be considered.
Free array excitation is considered, i.e., the elements are excited by a
set of incident waves in the element waveguides.
The polarization is in the plane of scan and the waveguides are filled
with a dielectric material having a dielectric constant e = 2» 5. A matching
network, located far enough from the aperture not to interfere with the
evanescent waveguide mode, completes the idealized element.
In order to simplify the analysis, the functional form of the element
electrir: transverse field distribution is assumed independent of scan condi-
tion and equal to that of the fundamental waveguide mode polarized in the x
direction. Thus, the element interactions are assumed to effect only the
element relative complex voltage levels. For small elements and polariza-
tion in the plane of scan this approximation has proved to be very good in
infinite array analysis, and it can be safely conjectured that this will be true
in the finite case also. The relative complex levels of the element voltage
qpr^SW« A W Rf JPI^t !«!P. ■'I'"» •■-■A M a.*^*"-^ -■■
^ jr-rv. *%,
1
UNCLASSIFIED
ai* oH*i»*d by a.*ti* - ■&£■ Vrciavlq«« of transforming the field problem into a
n*lw<»yfe pyx>tl",m. 1 bs> x&p&Hifäi consists of considering each (infinite)
column ae a «sistfl« e*e>t;c.ts£ d an equivalent linear array, and determining
tht> relative- net oi seif »-^ rautua? admittances via the Fourier Transform
method £8]. Denote by ?-. Jie excitation of an element of the ith column
^be excitatioB A aU the ether elements of the same column being the same).
The set oi ihs quantities a. can be concisely indicated as a vector column
a w fi = I. . . N-l) (1)
in the input space of the array having a dimensionality N equal to the number
of array columns. The element voltages similarly are denoted by the vector
V_related to a by:
V = -»/T(YTU + Yf1 Y ~1'2 a —"" -MZ2 3E JL» "•
(2)
where standard normalization has been used (see for example [8]).
In Equation (2) U is the unit matrix of order N and Y. is the admittance
looking into the element waveguides, assumed equal for each element. Thus,
Y. depends upon the internal admittance of the equivalent generator feeding
the element and upon the element matching network. The elements of the
Nth order matrix Y are the admittances between columns (as discussed in
some detail in the appendix) given by
+O0
it k» h ZLrf (-1) (i-t) f +00
m=-oo .00
i2/ 2 , ke„ +w -JL.
w
(3)
eju(i~t)ddu
mir "IT
where 6 and 6 , are the polar components of the Fourier Transforms of P «I»
the fundamental waveguide mode (nominally polarized in the x direction)
whose expression is given in the appendix, w is the wavenumber in the z
direction related to the two coordinates u, v of the wavenumber plane by:
w = JJ u (4)
UNCLASSIFIED
(the appropriate branch of Equation (4) being chosen in order to satisfy
radiation condition), k and rj are free space propagation constant and charac-
teristic impedance. The various integrals in Equation (3) must be calculated
for v = rmr/h, as the notation indicates.
To avoid the direct matrix inversion Equation (2), through a series of
manipulations (described in detail in [7]) the following Neumann series is
obtained:
v*1 21+(i)ä -ll" -l v = 2^ ™w y +Y(i) + M {= V= iSYg+S i=o g ' L
N-l +.., r ^s
L i=0
where m(i) are the set of vectors
M+a (5)
m(i) = j N"1/2 exp (-j2TTik/N)[ (i, k=0, 1 ... N-l) (6)
M is a matrix whose columns are the vector m(i). Y(i) is the active admit-
tance for a uniform excitation and a phase progression equal to 2iri./N for an
infinite reference array with the same elements and spacing. D is a diagonal
matrix whose elements different from zero are given by Y(i), (suitably
ordered) and:
Q = M+YM-D (7)
Once the aperture voltages are found, the element active admittances and
array patterns are determined through standard procedures (see for example
[7]). Since this investigation aims at determining the scan limitations of a
finite array, the attention will be mainly focussed on endfire or quasi endfire
scan. It is apparent that the realized gain pattern of the array in the direc-
tion <>i scan (gain referred to the power of the "free excitation" :.e.,
including mismatch loss) depends essentially upon the nature of the network
located in the element waveguide. In an infinite array it is possible to match
the structure for radiation in any assigned direction (different from endfire).
In a finite array instead the input admittance is different for each element,
'■"-r~■■■" -■ ./^wanajwB
UNCLASSIFIED and a perfect match of the array can be obtained only by using different
matching networks for different elements. However, it is physically clear
that placing in each element transmission line equal networks which would
yield perfect match at a direction close to grazing angle for an infinite array
will improve endfire or quasi endiire radiation. This is obtained, of course,
at the expense of broadside gain.
3. Endfire Radiation in a Planar Array
Consider an array with a number of columns N = 26. The element size
and lattice are those of Figure 1. The "free" array excitation a consists
of terms having equal magnitude for each column with a linear phase taper
from one column to the other. Different choices of the element tuning net-
work (equivalent to a shunt susceptance and a perfect transformer) are
considered, yielding perfect impedance matches for different scan angles
in an infinite reference array (i. e., with the same element and lattice ).
In Figure 2 the magnitudes of the aperture voltages of the elements versus
element positions have been shown for broadside match. Three different
scan directions have been considered. A long "spatial transient" is present
for 80-and 90-degree scan conditions. The aperture voltages for endfire
scan vary along the array tending to the short-circuit condition typical of an
infinite array. The phase of the voltages is essentially linear with only little
difference from that of the free excitation, and thus has not been indicated.
Figure 3 shows the magnitude of the reflection coefficients for the same
match condition and three scan directions. For endfire radiation the moduli
of the reflection coefficients increase going toward the edge from which the
radiation occurs. All the elements in this scan condition are badly mis-
matched as expected. Figures 4 and 5 show two array patterns (for this
match condition) for broadside and endfire scan. Figures 6 through 9 show
the effect of a different matching network. The reference infinite array is
matched for scan at 80 degrees from broadside. It is apparent that now the
The evaluation of the active admittance in the finite reference array necessary for the calculation of the parameters of the tuning network is done through well known methods recently developed [5-6],
UNCLASSIFIED array is matched in a much better way for extreme scan angle. The element
active reflection coefficient now decreases when approaching tl e radiating
edge (Figure 7). Figures 8 and 9 show the radiation patterns for broadside
and endfire scan. For this match condition the difference of gain scanning
from broadside to endfire is about 3. 7 dB.
The difference of behavior between the two element match conditions is
clearly illustrated by the curves of gain envelope versus scan angle in
Figures 10 and 11.
4. Finite Arrays on Cylinders - An Approximate Analysis of the Axial Radiation ' ' " '
The "infinite array" method of analysis consists of considering a
cylindrical array as the excited section of an infinite structure periodic in
both the longitudinal and azimuthal directions [10], This model neglects
endfire effects and therefore is of no use in predicting the radiation pattern
for directions close to the cylinder axis. Here below a very simple method
is presented which is believed to yield reasonably accurate performance
predictions for radiation patterns in directions belonging to the axial
symmetry plane of the array.
Consider a structure consisting of N rings of elements on a cylindrical
surface. The elements are the same and are arranged with the same lattice
as in the planar array previously considered for the planar case. Suppose
that only a region azimuthally confined between the two angles ~4> and <j> is
excited. The analysis of the radiative properties of this structure can be
performed through a procedure requiring the inversion of matrices of Nth
order only. As discussed in detail elsewhere [9], this is obtained by
decomposing the original problem into a set of simpler reduced problems
concerning the analysis of linear arrays having as "elements" entire rings
of radiators excited with constant amplitude and linear phase taper. These
reduced problems can be in turn approached by using the technique described
in Section 2, avoiding in this way even the direct inversion of the Nth order
matrices. The procedure will not be described in detail because in most
cases it can be replaced by a useful simple approximate technique, based on
the recognition that for a cylinder with a large radius both the realized gain
UNCLASSIFIED pattern and the pattern of the isolated elements are, in the axial plane,
practically identical to their counterparts for a planar array. Consequently,
in the axial plane the cylindrical array pattern can be obtained from that of
a finite planar array simply by multiplication by a factor a depending upon
the azimuthal angular extension 2$ of the array. This factor takes into
account the polarization effects arising from the fact that axially polarized
elements located in different azimuthal positions give differently polarized
contributions to the radiation field on the cylinder axis.
Suppose that the array aperture has a rectangular shape, limited by the
angular abscissa <j> and -<J> , symmetric with respect to the axial scan
plane. Consider a planar array having the same elements and lattice, and
a number of columns equal to the number of array rings. When all the
elements are phased to contribute in phase in the scan direction the pattern
of the conformal array in the axial plane is approximately obtained by
multiplying the planar array pattern by the factor
-, ? sin <}> a (9) = cos^ 6 + sin^ 9 ° (8) p s s s <j>o
9 being the angle from broadside in the plane of scan. Expression (8) is
determined on the basis of simple geometrical considerations. In many
cases for large cylinder radii and relatively small 4> 's, the expression
(Equation (8)) is very close to unity.
Figures 12 and 13 show the gain envelope versus scan angle for a 273-
element cylindrical array with a rectangular aperture and an angular exten-
sion 2(j> = 49 degrees. The element size and lattice are those of Figure 1.
UNCLASSIFIED
APPENDIX
Denote by E^ (x, y) the transverse electric field distribution on the
reference element, whose center is at the origin of the x, y coordinate
system. Thus, the electric transverse field of the ith column can be
written
E(l) (x, y) = V* E^ [x - id, y - (2m+i)h] m=-oo
where d and h are the distances between columns and between rows,
respectively. Introduce the Fourier Transform £ (u,v) of E (x, y):
+00 +00
£o(X« y) = 27 / / £ (U' v)e"j(UX+Vy) du dv
(Al)
(A2)
.00 -00
The transverse electric field of the ith column can be expressed as follows:
+oo +00 +00
E(i>(x,y)^ f f iKv^-^e^^^ «<v-E*)dudv (AJ)
_oo -oo m=-oo
Use has been made in Equation (A3) of the well known Fourier representation
of a periodic delta function.
The mutual admittance Y.. between two columns i and t is defined here
as the short circuit aperture current of an element in the column i when all
the elements of column t are excited with equal unitary voltages. By using
the results of [5] it is easy to establish the expression (3) for Y...
The polar components of the fundamental circular waveguide mode
polarized in x direction are given by:
8
/
t
i
UNCLASSIFIED
/Z Veil»"1
where a is the radius of the element aperture, x*. is the first root of the
equation JWx) = 0 and
4. / 2 J 2 t = vu + v
u COS [I
777 2 V
V sin |x =
vu i v
If in Equation (A3) |i-t| >> 1 (practically > 3) the following asymptotic
expression can be conveniently used:
2 +°° v * Y™* I. 2r2 2 2 „ (2) . |. .... Yit = k^h L, k*p
+rf+w | Ho '(um|x-t|d) m=-oo m
with:
u m i^)
UNCLASSIFIED
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12 NOT REPRODUCIBLE
UNCLASSIFIED
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Figure 4 - Array Pattern, Broadside Match, Broadside Scan
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13 N0T REPRODUCIBLE
UNCLASSIFIED
Figure 6 - Amplitude of Element Voltage
14
HOT REPRODUCIBLE
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Figure 7 - Amplitude of Element Reflection Coefficient
15 NOT REPRODUCIBLE
UNCLASSIFIED
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NOT REPRODUCIBLE
16
UNCLASSIFIED
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Figure 11 - Gain versus Scan, Planar Array, Match at 80 Degrees from Broadside
17
NOT REPRODUCIBLE
UNCLASSIFIED
Figure 12 - Gain versus Scan, Cylindrical Array R = 50X
Figure 13 - Gain versus Scan, Cylindrical Array R = 50X
18 N0T REPRODUCIBLE
UNCLASSIFIED
REFERENCES
[1] S. Edelberg and A. A. Oliner, "Mutual Coupling Effects in Large Antenna Arrays: Part 1 - Slot Arrays", IRE Trans, on Antennas and Propagation, Volume AP-8, pp. 286-2 97, May I960.
[2] L. Stark, "Radiation Impedance of a Dipole in an Infinite Planar Phased Array", Radio Science, Volume 1, pp. 361-377, March 1966.
[3] H.A. Wheeler, "The Grating-Lobe Series for the Impedance Variation in a Planar Phased Array Antenna", IEEE Trans, on Antennas and Propagation, Volume AP-14, pp. 707-714, November 1966.
[4] G. F. Farrell and D. H. Kuhn, "Mutual Coupling in Infinite Planar Arrays of Rectangular Waveguide Horns", IEEE Trans, on Antennas and Propagation, Volume AP-16, pp. 405-415, May 1968.
[5] V. Galindo, and C. P. Wu, "Numerical Solutions for an Infinite Phased Array of Rectangular Waveguides with Thick Walls", IEEE Trans, on Antennas and Propagation, Volume AP-14, pp. 149-158, March 1966.
[6] G. V. Borgiotti, "Modal Analysis of Periodic Planar Arrays of Apertures", IEEE Proceeding, Volume 5o, No. 11, pp. 1881-1892, November 1968.
[7] G. V. Borgiotti, "Edge Effects in Finite Arrays of Uniform Slits on a Ground Plane", IEEE Transactions on Antennas and Propagation, September 1971 (to be published).
[8] G. V. Borgiotti, "A Novel Expression for the Mutual Admittance of Planar Radiating Elements", IEEE Trans, on Antennas and Propagation, Volume AP-16, in 3, pp. 329-333, May 1968.
[9] G. V. Borgiotti and Q. Balzano, "Conformal Arrays on Surfaces with Rotational Symmetry" to be published in the Proceedings of the Symposium on Phased Arrays, Polytechnic Institute of Brooklyn, Farming dale, N. Y., June 1970.
[10]G. V. Borgiotti and Q. Balzano, "Analysis and Element Pattern Design of Periodic Arrays of Circular Apertures on Conducting Cylinders", submitted for publication to IEEE Transactions on Antennas and Propagation.
19