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TRANSCRIPT
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MASSACHUSETTS INSTI'rUTE OF TECHNOLOGY
LINCOLN LABORATORY
PATTERN DEGRADATION OF SPACE FED PHASED ARRAYS
J, RUZE
Division 3
PROJECT REPORT SBR-1
5 DECEMBER 1979
Approved for public release; distribution unlimited.
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ABSTRACT
The far field pattern degradation of space fed phased
arrays suitable for a space based radar is examined. The effects
* considered are:
Structural<
' '1) Axial. lens surface distortions)
(2) Uniform radial thermal expansion)
3) Axial and lateral feed displacements~
Electrical
1) Element phase and amplitude excitation errors) ''
2) Failed elements,
An introductory section discusses the size, costand
weight penalties of low sidelobe designs. The final section
presents a method of phase compensation or coherence of large
axial lens distortions.
Acces on ror
DDýC TABUnuiWouncud
tt ci4
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H9OZEQ PAIU BLANK NOT nim
CONTENTS
ABSTRACT iii
1. INTRODUCTION
2. NATURE OF IDEAL RADIATION PATTERNS 3
3. PRACTICAL LIMITATIONS ON ANTENNA SIDELOBES 12
3.1 Antenna Spatial Distortions 21
3.2 Aperture Blockage 12
3.3 Aperture Excitation Errors 13
3.4 Failed Elements 16
4. DISTORTION SENSITIVITY OF DIVERS ANTENNAS 19
5. COMPUTATION OF ANTENNA PATTERNS 23
6. AXIAL DISTORTIONS 26
6.1 Bowl 26
6.2 Linear Fold 31
6.3 Quadratic Fold 38
6.4 Quadratic Astigmatism 52
6.5 Sinusoidal Astigmatism 64
6.6 Eight and Sixteen Gore 75
6.7 Half Linear Fold 105
7. RADIAL DISTORTIONS 113
7.1 Uniform Thermal Expansion 113
8. FEED DISPLACEMENT 117
8.1 Axial 117
8.2 Lateral 118
9. THREE DIMENSIONAL ARRAYS AND PHASE COMPENSATION 119
v
††† ††† ††† †† ††† ††† ††† †† ††† ††† ††† †† ††† ††† ††† †† ††† ††† ††† †† ††† ††† ††† †† ††† ††† ††† †† ††† ††† *.
10. CONCLUSIONS 129
10.1 Electrical and Structural Defects 129
10.1.1 Flectrical 12910.1.2 Structural 129
10.2 Phase Compensation 130
ACKNOWLEDGMENTS 132
REFERENCES 133
APPENDIX 134
A-1 Axial Displacements 134
A-2 Radial Displacements 136
vi
1. INTRODUCTION
Recently interest has been expressed in a large aperture
Space Based Radar for earth surveillance. There is general aaree-
ment that such a structure should cons~sl- of a space fed phased
array. There are significant advantages to this type of antenna
namely:
1) The antenna beam can be electronically scannedover the earth's field of view.
2) The structural tolerances of a space fed lensare at least an order of magnitude greaterthan a corporate fed phased array or a reflect
array.
3) A space feed is a very efficient means of elementexcitation compared to a corporate feed when alarge number of elements are involved. It isfrequently used in large ground base installationsrequiring two dimensional beam scanning.
4) Monopulse operation is readily obtained by useof an appropriate monopulse feed. This isobtained essentially without beam distortionwhereas a corporate feed requires separatesum and difference illumination functions toobtain low sidelobes in the sum and differencemodes.
Radar system studies have indicated that the probable
frequency of operation would lie in the region from L to X band.
Depending on frequency and satellite altitude the antenna diameter
may range from 30 to 100 meters. Present requirements are that
it be space transportable by one space shuttle load and that the
structure be self deployable. Such a structure must be extremely
1
' " • •••"•'-------------------------------- -----------------------------.•" .. •i•'-T
light weight and will be subject to thermal and station keeping
stresses. Such stresses will create distortions of the planar lens
surface.
It is the purpose of this memorandum to calculate the
radiation patterns of a space fed array subject to various array
distortions. As these distortions are not presently known divers
canonical distortions have been assumed. The normal modes of a
circular membrance at first suggested themselves.[I] However,
these are not particularly applicable as our antenna may not be
rigidly clamped at the rim. Instead, various possible simple
distortions were used. Undoubtedly, the motion of the space
antenna will be very complex being a superposition of the simple
types assumed. The computer program employed is flexible so that
when the structurally computed strains are known they can be
inserted.
It is believed that this handbook of antenna patterns will
prove useful to structural and thermal analysts and also for radar
clutter calculations.
2
.1
"2. NATURE OF IDEAL RADIATION PATTERNS
It is planned that on transmission a uniformly
illuminated aperture be used to take advantage of the entire
available aperture and the power available from the element
transmitters in the active array. Neglecting for the present
all losses, aperture errors and blockage such an aperture haslD2
a gain of ( D) a HPBW of 1.02 XD radians; a first sidelobe
of 17.6 dB; and a theoretical sidelobe envelope that has a
power decay of:
p (u) = 8 1 - 2.55 (1)S u3 u3
where
U = sin e (2)
"0" being the angle off bear peak and D0X the antenna diameter
in wavelengths.
Of interest is where the sidelobe envelope attains
isotropic (0 dBi) or the -10 dBi level. For the uniform illumina-
tion considered these levels occur at the angles defined by:
1/3sin 6 0.932 LLW H~ (3)
sin 010 = 2.01 1/3) / 2 3H' (4)
3
So that for typical HPBW's under consideration we have the table
HPBW + 0 + E-o -- 10
10 (0.0175 r) 150 310
0.10 (0.00175 r) 6.90 13.80
We note that depending on orbit, for uniform illumination, a qood
part of the earth's FOV is above isotropic levels.
For the receive mode a low sidelobe illumination taper
is desired to suppress ECM interference. For computation purposes
a truncate.d raussian function of the form
-2r 2f(r) = e (5)
has been used. This illumination has an edge taper of -17.4 dB.
and an aperture efficiency of 76% so that the gain is:
'D 2 2G = 0.7616 ( T (6)
and the
HPBW = 1.25 X/D radians (7)
The first sidelobe is about -34 dB and the sidelobe decay rate is
0.25 (8)P(O) = ý (83
That is a sidelobe envelope 10 dB below the uniform case and we
have the corresponding table for the isotropic and -10 dBi levels:
4
10 0101° (0. 0175 r) 5.4° 11.7°
0.10(0.00175 r) 2.50 5.4
We can summarize this data in the following table
-2r 2
Illumination Uniform e
Gain (.7) 2 7 2
HPBW (radians) 1.02 X/D 1.25 VD
Rim Taper 0 dB -17.4 dB
First Sidelobe -17.6 dB -34 dB
Sidelobe Decay (power) 2.55/u 3 0.25/u 3
Isotropic Level 8°0
HPBW = 10 150 5.40
HPBW = 0.10 6.90 2.50
-10 dBi Below Isotropic Level
HPBW = 10 310 11.70
HPBW - 0.10 13.80 5.40
Relative Antenna Area 1.0 1.50For Same HPBW
5
These theoretical results are of interest as they set the
limits on antenna performance. Although the tapered illumination
has lower sidelobes it requires a 50% increase in antenna area
for the same HPBW or resolution - with a corresponding increase
in the number of elements, modules, etc.
Aperture illuminations can be theoretically designed to
yield any specified sidelobe level. In Fig. 1 we plot the
increase in the number of elements required for a specified HPBW
for the frequently used circular Taylor distribution[2 ] against
the desired near-in sidelobe level. The near-in sidelobe level
is of special importance as it is subject to earth based ECM.
The parameter 1 determines the number of equal height sidelobes,
subsequent sidelobes decay from this design value. It is evident
that low sidelobe design - near isotropic level - requires larg7er
arrays with their corresponding increase in cost and weight.
Thinned arrays are not applicable for this application as they
have relatively high far-out sidelobe levels and low array gain.
Figures 2 and 3 show the theoretical natterns for the
uniform and the assumed Gaussian distribution with their asvrmptctic
fall-off indicated.
The reader may wonder why we did not use an illumination
function for which some optimum properties are claimed, such as
the Taylor Circular Distribution. The truncated Gaussian was
chosen due to analytic and computer convenience. The Taylor
6
rn 2 .5 -StL -3-2,o•7 7
z
Cl)9-12z0"02. 2
LL
w
zZ 1 .5wz3W 5: NUMBER OF EQUAL NEAR IN
> SIDE LOBES
o:I I 1 I20 30 40 50 60 70
DESIGN SIDE LOBE LEVEL (dB)
Fig. 1. Relative increase in area (number of elements, cost,and weight) for specified design sidelobe levels. CircularTaylor taper used.
7
-3i-21038j _ ___
0DISTORTION
- _________SCAN ANGLE
F/DAZIMUTH PLANE----------
10
20
30
40
0 5 10 sD2
Fig. 2. Radiation pattern of space fed array uniformillumination f(r) 1.
I-7-21039fl0
DISTORTION ---______SCAN ANGLE----
F/D- - - - - - - - - -
AZIMUTH PLANE--------10
20ENVELOPE SIDELOBE DECAY
() 0.25
so U3
40
5010 6 0152
U Dsin 8
Fig. 3. 1Radiatiog pattern of space fed array Gaussian
taper f(r) =e-
9
circular function yielding the same HPBW and gain has about a 4 dB
lower first sidelobe; however, the first 5 sidelobes are of the
same strength (-38 dB). Whereas with the truncated Gaussian the
fifth sidelobe has already decayed to -43 dB. It was, therfore,
concluded that for this degree of sidelobe suppresssion there was
no significant advantage of either of these forms for these cal-
culations or for the constructed antenna.
10i0i
3. PRACTICAL LIMITATIONS ON ANTENNA SIDELOBES
3.1 Antenna Spatial Distortions
It is expected that antenna surface distortions, whether
we have a reflect array, corporate or space fed phase array, will
have a low spatial frequency or that they may be considered smooth
in wavelength measure. Distortions caused by structural or thermal
strains will then occupy large portions of the antenna aperture
and due to their long spatial period, will degrade the antenna
pattern principally in the vicinity of the main beam and the first
few sidelobes, leaving the far-out sidelobe region essentially
undisturbed.
As the magnitude of the distortions is increased the
pattern degradation further increases and also spreads out in
angle. The latter is due in part to the shorter spatial period
due to the removal of module 27 from the phase front and also
due to higher order terms in the expansion of the phase function.
. ..
3.2 Aperture Blockage
Some of the proposals for a large space fed array included
an axial feed support (unipod). Such a support will not only
block the center portion of the array but also, if of large diameter,
disturb the primary pattern of the feed. It is believed that a
strut feed support to the rim of the array will be desirable.
As the effect on the feed primary pattern is not readily calculable
feed support aperture blockage was not included in these calcula-
tions.
Another form of aperture blockage, not included, is the
presence of seams and hinges, necessary for space deployment.
The effect is similar to that of strut blockage in a parabolic
reflector.
We have also assumed that the array elements are placed
on a uniform square grid. This is not possible with a gore
type of construction as the elements must be fitted into the
available space.
There is no doubt that the above factors will further
degrade the antenna pattern especially in the low sidelobe reqion.
However, our neglect is not without its benefit as it places the
pattern degradation due to spatial distortion in evidence.
12
3.3 Aperture Excitation Errors
The pattern of a constructed array, outside of the main
beam and first few sidelobes, bears little relation to the
theoretically desired pattern. The reason for this is that we
have not achieved our theoretical aperture distribution in our
model. The radiation pattern, in the low sidelobe region, is
determined not by our theoretical aperture distribution but by
the aperture excitation errors.
Nevertheless, if the aperture excitation plus errors are
known the pattern can be computer calculated. This is generally
not the case and recourse is made to estimates of the errors and
statistical methods. Based on some reasonable statistical
assumptions it can be shown that the average sidelobe level
referred to the beam peak is
L-2 + 2 + (l-P) (9)
nPN
where
-= mean square phase error (radians)
= mean square fractional amplitude error
(l-P)= fraction of failed elements
N = number of independent elements.
P = fraction of elements active
n = aperture illumination efficiency
13
As the gain of the array for half wavelength spaced
elements is "nTrPN" the average sidelobe level referred to the
isotropic level is:
-- = +r(1-P) 2 + 2(0)0
Equations (9) and (10) give statistical average levels. The
peak sidelobe level may be as much as 10 dB above the average
value. Whether this peak value occurs for a given array depends
on the number of samples taken - that is pattern cuts, frequencies
and scan angles.
Fig. 4 shows the phase and amplitude tolerance required
for a -10 dBi average and peak sidelobe level. We note that to
achieve an average SLL due to excitation errors we require a 100
rms phase only tolerance or a 1.5 dB rms amplitude only tolerance.
Combinations of phase and amplitude errors may be obtained from
these curves. For a guaranteed -10 dBi peak sidelobe we require
about one third of these values. The excitation error sidelobes
must be added to those due to the theoretical excitation. In
the far sidelobe region the latter are negligible for a tapered
aperture.
It may appear that to achieve an average SLL of -10 dBi
would not be too difficult as we require only about 70 rms phase
error and a 1 dB amplitude error (if phase and amplitude errors
are equally divided). However, a four bit phase shifter has a
14
, ",." . . . . . . . . . * ...i
11.50' 0.2-I0.00.-- -- a2oo
W V
S0.0 I 0.1 FRACTIONI 0.2
o! %
0.5dB 1.5dBAMPLITUDE ERROR (rms)
Fig. 4. Phase and amplitude tolerance required for-10 dBi average and peak sidelobe level.
15
• ,- -L " . . . . .. .. ....................... . . . , , , ., .. . , •. .. .. •' I:
. ,I- V .
bit quantization error of 6.5 degrees rms, a typical manufacturing
error of 5 degrees rms, to which must be added line length errors,
measurement errors, module errors, mismatch errors, mutual coupling
errors etc.
Typical ground based optically fed phase arrays have
average SLL of -5 to -8 dBi in the distant sidelobe region.
3.4 Failed Elements
In any spaced based array the number of failed elements
must be considered. Equations (9) and (10) show this effect on
the average sidelobe level on the assumption that the failed
elements do not radiate and are randomly l~cated.
Fig. 5 shows this effect for elements located on a half
wave spaced grid for an array with no amplitude and phase errors
for the remaining active elements. We note that to achieve a
-10 dBi average sidelobe level only 3% element failure is
allowable and 30% failure yields isotropic levels.
Fig. 5 is also applicable to purposely thinned arrays.
We note that element thinning is not applicable for low sidelobe
design and Fig. 5 indicates that for highly thinned arrays the
sidelobe level approaches the gain of one element which for our
half wave grid spacing is Pi(+5 dBi).
iJ
16
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Wl
rr -1
w
00
LUo
•' r4l
0
u 0)
W H
(OBP) 73A37 380-1 MIS 3M1•3AV
,• 17
." " .. . . .......... ....... ... . . . .. . . .. ... ,,.,, , ,,h i L i." " !' 'J
The pattern degradation discussed in this section is
cumulative - that is the effects of:
a) lens surface distortion
b) aperture blockage
c) element excitation errors
d) failed elements
must be added to the theoretical level as an incoherent power
addition. The element statistical sidelobes ("c" and "d" above)
do decrease with observation angle but for the half wave element
spacing this effect varies only as the element pattern; about
as the cosine square of the angle off boresight.
18
4. DISTORTION SENSITIVITY OF DIVERS ANTENNAS
Fig. 6 shows three common antenna sytems; a parabolic
reflector, a corporate fed phased array, and a space fed phased
array.
If the parabola's surface is distorted from the desired
parabolic shape by an amount "A" the path length error created
in the emerging wavefront approaches "2A".
In contrast a similar distortion of a corporate fed phase
array creates a path length error of only "t'cosQ" where "0" is
the angle of observation or scan angle. A corporate fed phased
array, in addition to its wide angle scanning capabilities, is
less sensitive to surface distortion by a factor better than two
than a reflector.
For a phased array with a large number of elements a
corporate feed becomes inconvenient as each element must be fed
by a transmission line with its inherent ohmic losses. It is
common, in such cases, to employ a "space" or "optical" feed.
Here the array consists of three layers; a backside comprising
receptors (dipoles, waveguides, etc.) illuminated by a horn feed
axially located at a focal distance, "F", a radiating or outside
layer of radiators and a isolating layer or metal ground plane.
The corresponding receptors and radiators are connected by a
network whose function is to transfer power between the two layers
19
A - .4-AZ
E =:Acos9
REFLECTOR CORPORATE FED ARRAY
- [- 2 - 2sin7
Vr 0 f=F/D
F
-3-21042
SPACE FED LENS
Fig. 6. Axial distortion susceptibility of variousantenna systems.
20
......~, .
and to provide the necessary phase shifts to steer the radiated
beam. The networks or "modules" may be passive (phase shifters
only) or active (pre-amplifiers and transmitters additional).
The path length error, "a " caused by an axial displace-
ment, "Aa" of a section of a space fed phase array is (derivation
in Appendix A-i)
C=A oso- 1 ] (11)Ea a IC V/l+ -(r/2f)21
where 0 is the observation angle
f is the f-number = F/D
r is the aperture radial coordinatel normalized to unityat the rim.
Tae second term of Equation 2 reduces the path length error, in
fact phase array distortions that occur at the radial distance
"ro cause no main beam degradation at a scan angle given by
tan O0 = ro/2f (12)
Axial disiortions at other radial distances will still degrade
the main beam at 00, or distortions at r0 will degrade the beam
at scan angles other than 00.
This form of axial error compensation considerably
decreases the susceptibility of a space fed phased error to
axial distortions and consequently permits lar•.'r distortions.
21
Tho compensating effect is quite complex and cannot be simply
predicted as it depends on the position and magnitude of the
distortion, the illumination excitation, the scan angles involved,
and the "f" number of the system. However, as a round rule of
thumb the space fed array is about one tenth as sensitive to
axial distortion as a corporate fed array.
Further insight into the behavior of the compensating
axial error can be seen from an approximate form of Eq. 22
obtained by expanding the radical:
= [ - 2 sin2 (13)
where we have retained only the dominant term. For zero scan
angle we have
a= Aa[' r)2] (14)
where it is evident that the path length error is reduced from
the corporate fed value by the bracketed term, attaining a
maximum value of one eight at the rim for an F/D=1.0. We note
that the center element of the array produces no path length
length error when displaced for an broadside beam. This can be
seen, physically, from Fig. 6.
A reflect array has the same susceptibility to axial
distortion as a reflector and, therefore, commands no advantage.
22
".......... . ............
5. COMPUTATION OF ANTENNA PATTERNS
Antenna patterns were computed basically by evaluating
the integral
g(Qf) (r) eju[x cos + y sin ] eJia(x,y) dxdy (15)•' A
where (6,f1 are the observation coordinates
f(r) the illumination function, assumed real andcircularly symmetric;
SU=-sin e
x,y are the array coordinates normalized to unit radius
C the path length error caused by the array distortionsa in radians
The integration is over the circular aperture so that points
where x 2 + y 2 >1 are excluded.
For computer calculations the integration must be
"replaced by a summation
ju[xij cos *+ Yij sin fl ja(XijYij)g(u,4) = f(rij) e e
i j (16)
All patterns are normalized by the beam peak with no path length
error.
Patterns are plotted in dB below beam peak. Zero dB
then represents the no error on axis gain
2
G= (T D (17)
23
,• t ,}
This procedure resulted in sets of patterns plotted in
universal coordinates, (u, dB). The "u" coordinate can be
converted to spatial degrees if the antenna diameter and wave-
length are known.
No element pattern is indicated in Eq. 15 as the HPBW's
under consideration are in the tenths of degrees. For the
scanned patterns the dB indicated should be reduced by
20 log (cos 0)
which amounts to 0.54 dB for the 20 degrees scan angle usually
employed.
The summation indicated in Eq. 16 was performed on a
uniformly spaced square grid with 64 intervals on a principal
diameter, a total of about 3200 points.
When the array is scanned to the angle (0, 0 °) the
elements must be phased according to:
=27i D sin 0o [x. cos o + yi sin ýo] (18)
0 1) 0 J
We calculated the pattern in the plan of scan where Eq. 16
becomes:4
S0f( ej(u-uO)[xj cos O + Yij sin •o]G(u,• °) =L• f~rij) ei 0 i00 ij (19)
j
e Ca (xij Yij)
24
. . .
the pattern abscissa is then (u-uo) or distance from the beam peak
in sine space. This procedure of presenting the data in selected
"•o" planes was judged more illuminating than two dimensional
contour plots. Although contour plots are more complete their
generation was just not possible in view of the available computer
time and the large amount of data presented.
Even with this reduction judgement had to be exercised
as to what was to be plotted. The parameters generally used were:2!i e-2r2
taper function, f(r) =e
t . scan angle 0= 0,200
f-number F/D - 1.0, 1.414, 2.0
distortion (axial and radial)
pattern cuts (as deemed significant but in the planeof scan)
I[
25
6. AXIAL DISTORTIONS
Axial distortions, that is displacement of the elements
normal to their planar positions are probably the most common
to be experienced. They would occur due to incomplete deployment,
thermal strains, station keeping torques, etc.
Although some radial displacement will accompany any
axial displacement in a structure with some connection these are
considered as second order effects and are neglected in this
section
6.1 Bowl
Here we consider the lens surface distorted into a bowl,
that is the planar surface becomes:
z = ar 2 (20)where "a" is the rim displacement.
Fig. 7 shows the planar surface and Fig. 8 the bowl
distortion.
Fig. 9 shows the pattern degradation for varioub values
of rim dirplacement for a corporate fed phased array; indicating
that only about a quarter wave rim displacement is allowable.
These curves can also be used for a reflector if the displacements
are halved.
Fig. 10 shows a comparable curve for a space fed lens
with a "f" number of unity and at a scan angle of zero. We note
the marked decrease of distortion susceptibility of the optically
26
S It
Fig. 7. Undistorted planar array.
1• 27
-[ -3-21045-2 ,.
Fig. 8. Bowl distortion z = ar 2 .
28
• .".• . ,,.. • . • 'I,!• •,J i,
0DISTORTION BW
SCAN ANGLEF/D NA-
AZIMUTH PLANE ~i-A-Y-- -- -
10 -
20
3X.
30
40
5010 6 10162
U =113sin e
Fig. 9. Radiation patterns of corporate fed array-2r2
Gaussian taper f(r) =e-
29
U. 0-DISTORTION !iL-- ---
SCAN ANGLE 0
FID
10
20
30
40
50
u vX Dsin 8
Fig. 10. Radiation patterns of space fed array
Gaussian taper f(r) = e
30
fed structure where a 2.5 wavelength rim displacement is tolerable.
Fig. 1i same as Fig. 10 but scanned to 20 degrees. We note
the different shape of the degraded antenna patterns. This is
due to the compensating nature of the path length errors. With
a= ar2
we have from Eq. 13S= al r4 r2 6_
Ea a r 2 ) - 2 sin I (21)aL2 (2f) 21
where we have a quadratic and a fourth order aberration that
tend to compensate.
Fig. 12 and 13 are the same as Fig. 10 and 11 except
that the 'If" number is 1.414. We note that the compensation is
more complete in Fig. 13.
Fig. 14 and 15 are the same as Fig. 12 and 13 except
that the "f" number is 2.0. We have obviously over-compensated
for this scan angle and "f" number. The smaller "f" number is
preferred.
6.2 Linear Fold
This is a distortion where the space fed array is bent
along a diameter. The distortion is portrayed graphically in
Fig. 16 and is represented mathematically by:
z = zlyI (22)
31
0:DISTORTION I3oJL
SCAN ANGLE 200
F/D 1
AZIMUTH PLANE SCAN
K 10
20 ..
30
40
0 5 10 15 20
r'D=U sin 8
Fig. 11. Radiation patterns of space fed array
Gaussian taper f(r) e -2r2
32
S.•
1-3-21049II 0
DISTORTION BO t L_
SCAN ANGLE o
F/D 1,414
AZIMUTH PLANE ANY.10
200
I.5
40
0 610 15 20
u , D sin 8
Fig. 12. Radiation patterns of space fed array Gaussian-2r 2
taper f(r) = e
33
J --3-2n1 o50
0
DISTORTION BowL
SCAN ANGLE 200L2
F/D 1.414
AZIMUTH PLANE SCAN
10
20
40
5010 5 10 15 20
"7'Du Dsin 0
Fig. 13. Radiation patterns of space fed array Gaussian
taper f(r) Ce
34
7T 3-.2 1. 5'-].DISTORTION BOWL
SCAN ANGLE 0-
F/D 2.0
AZIMUTH PLANE ANY
20
10
30 lox
40
500 5 10 15 20
uD sin 8
Fiq. 14. Radiation patterns of space fed array Gaussian-2r2taper f'r) =
354 .I ,-
r0
0 DISTORTION BOWL
SCAN ANGLE Z0
FID 2.0
AZIMUTH PLANE SC.AN
10
20
300
40
501,w0 5 10 15 20
U r -D sin 9
Fig. 15. Radiazion patterns of space fed array Gaussian- 2r2
ta.'er f(r) = r.
4 36
-3-2o53
Fig. 16. Linear fold distortion z = alyl.
37
Li1
As we no longer have azimuthal symmetry the pattern degradation
will be different in various azimuth cuts.
Fig. 17, 18, and 19 show the degradation of the broad-
side beam in the "4" = 0, 450 and 90° azimuth cuts for a unity
"f" number. The large differences between these cuts can be
explained by collapsing the phase error excitation on the line
of the cut taken. The tolerance criteria will then be determined
by the worse cut,namely, Fig. 19. The axial gain reduction is,
obviously the same for all azimuthal cuts.
Fig. 20, 21, and 22 are the same as the previous three
except that the beam is scanned 20 degrees in the direction of
the azimuth cut taken.
Figs. 23-28 inclusive are the same as the previous six
except that the "f" number has been increased to 1.414.
6.3 Quadratic Fold
This distortion is similar to the linear fold except
that the displacement from the diameter varies in a quadratic
manner. it is shown graphically in Fig. 29 and mathematically
represented by2
z = ay (23)
38
-3-21o54
DISTORTION LINEAR FOLD
____-"-___SCAN ANGLE 0F/D L ----AZIMUTH PLANE 00
10
20
30
40
5010 5 10 15 20
s in e
Fig. 17. Radiati?n patterns of space fed array Gaussian-2r•
taper f(r) - e
39
.... . .. . . . . .
F-3-21•55p 0
DISTORTION LINEAR FOLD -
SCAN ANGLE 0-F/D L--- -.-----
AZIMUTH PLANE 0-.-,-10
20
2.5X 5X30
40
0 5 10 15 20
u =-'R"sin 8
Fig. 18. Radiation patterns of space fed array Gaussian-2r 2
taper f(r) = e
40
.....I'.1.1 -
-3-21o I0
DISTORTION LINEAR FOLD~00SCAN ANGLE 0
•,F/D L --...- -
AZIMUTH PLANE 90 -10
20
30 om0
40
S50 v I-AY
0 5 10 15 20
U=. D sin 8
Fig. 19. Radiation patterns of space fed array Gaussian-2r 2
taper f(r) = e -
41
A. ..-
0 --3-l bo77
DISTORTION LINEAR FOLD
SCAN ANGLE 2.°
F/D -..0
AZIMUTH PLANE O - - - - - -
20
llo
302.5
40 -101x
0 5 10 15 20
u = • sin 8
Fig. 20. Radiation patterns of space fed array Gaussian-2rZtaper f(r) e
42
-. .
0DISTORTION IME.. P-n.fl,-..
$CAN ANGLE 20°F/DAZIMUTH PLANE i.,.° .
1 0 -.. . . . .. ,
20lox
30
40
500 5 10 15 20
7rDu = sin 8
Fig. 21. Radiation patterns of space fed array Gaussian
taper f(r) =e-
43
- -------- -
] -3-21o5, 1
DISTORTION LIEAR FOLD
SCAN ANGLE 2o
F/D 1
AZIMUTH PLANE 900 SCAN10
20
30
40
0 5 10 15 20
Ssin e
Fig. 22. Radiatipn patterns of space fed array Gaussiantaper f(r) = e
44
isI*I I I
"'--3-21060j
0 DISTORTION LINEAR FOLD
S_ SCAN ANGLE 0f
F/D 1.414
AZIMUTH PLANE 0o0--10
20 -,olox30
401
0 5 10 15 20
u sin 8
Fig. 23. Radiation patterns of space fed array Gaussian
taper f(r) -2re
45
..... ...... . . ......
1-3-2 1061
0
DISTORTION L --- -U
SCAN ANGLE f -'F/D 1.414
AZIMUTH PLANE 450
10
' •"45°
20
302.5X
40
5011 vV L-0 5 10 15 20
Ssin 8
Fig. 24. Radiation patterns of space fed array Gaussian-2r2taper f(r) = e-
46
I -3-21062 1K 0
DISTORTION L NL ,.R _F.o L . . . . .
-SCAN ANGLE 0°
I It F/D0 AZIMUTH PLANE U2---
20
00
30 •x
Ii 40
5010 5 10 15 20
U sin G
Fig. 25. Radiation patterns of space fed array Gaussian
taper f(r) = e"2r 2 .
47
"L-3-21063 10
F DISTORTION LINEAR FOLD
_ SCAN ANGLE 202F/D 1.414.
AZIMUTH PLANE SCA -
20 - -
30
0 5 10 10 20
Ssin 8
Fig. 26. Radiation patterns of space fed array Gaussian-2r2
taper f(r) = e
48
- -3-21064
0DISTORTION LIANEAR FOLD
_____ _____SCAN ANGLE
F/D .1.414 - - - - - - -AZIMUTH PLANE -5cm------
10~ - _ _ _ _ _ _ _ _
20 -
30-
40
501 1 \0 5 10162
U= vx Dsin 8
Piq. 27. Radliation patterns of space fed array Oaussi.nntaper f (r) v r
49
I_____
-T -- 3-21065-.
0DISTORTION LINEAR FOLD
$ BCAN ANGLE ?0°
F/D 1.414
AZIMUTH PLANE 90° SCAN
10
20
30
40
500 5 10 15 20
u~ r in 8
Fig. 28. Radiation patterns of space fed array Gaussiantaper f(r) = e-2r 2 .
50
Fig. 29. Quadratic fold z =ay4
51
Fig. 30-35 inclusive show the pattern degradation in the
three azimuth cuts (4 = 0,45,90 ) for a broadside beam and one
"scanned to 8= 200. This distortion is more benign than the
linear fold as regions of high illumination have smaller displace-
ments for the same rim displacement. Only the case for unity
"f" number was calculated.
6.4 Quadratic Astigmatism
This is a distortion called astigmatism in optics,
characterized by having different focal lengths in the two
principal planes. It is shown graphicelly in Fig. 36. We see
that the lens is bent upward in one principal, plane and downward
in the orthogonal plane. This distortion is represented
mathematically by:
z - ar 2 cos 24 (24)
There are two azimuthal cuts of interest, namely 4= 0 and 4 = 450.
Succeeding 45 degree cuts will have identical patterns.
Fig. 37 and 38 show these two cuts for a broadside beam.
Fig. 39 and 40 for the beam scanned to 200 in the direction of
the cut taken. All with unity "f" number.
Comparison of Fig. 37 with Fig. 10 (bowl distortion)
indicates that the astigmatic degraded patterns have higher
sidelobes but smaller gain loss for the same rim distortion.
52
I* * . K
r .......... iJ'
0DISTORTION .UADMA. Z.O.LP__ - -
SCAN ANGLEFID 1'•
AZIMUTH PLANE 0°10
20
30
30
40 -
5010 5 10 15 20
u= -D sin e
Fig. 30. Radiation patterns of space fed array Gaussian52r2taper f(r) =e
53
0ITRTO QUADRATIC FOLD
SCAN ANGLE ----------
F/D----AZIMUTH PLANE
10--
20
30
40 ______
50
0 5 10 15 20
U-_Dsinl 9
Fig. 31. Radiation patterns of space fed array Gaussian-2r2taper f(r) =e
54
-T -3-210-69 10DISTORTION .U.ADRATIC FOLD
________SCAN ANGLE 00-
F/DAZIMUTH PLANE 900 ---
10-
20
__0______
30
40 _ _ _ _ _ _
50.0 5 10 15 20
U sin 8
Fig. 32. Radiation patterns of space fed array Gaussian
taper fir) -er.
L -3-21070 10
DISTORTION QUADRATIC POLD
_.... SCAN ANGLE 200
SFtD 1
AZIMUTH PLANE o%...-.-;; -
20
30
Iv
150 5
so \
0 5 10 15 20
Ssin 8
Fig. 33. Radiation patterns of space fed array Gaussian-2r2
taper f(r) e-
56
-3-21071 7
DISTORTION QUADRATIC FOLD
__________SCAN ANGLE---------
I- ~F/D1AZIMUTH PLANE 45 SCAN
10
20
lox'
30
S2.5)'
50
0 5 10 20
Ul = sin 8
Fig. 34. Radiation patterns of space fed array Gaussian
taper f(r) e-2r2
57
0 -3-21072
DISTORTION QUADRATIC FOLD
SCAN ANGLE 20•tFID L-----
AZIMUTH PLANE 9o .- -
20
10)
30 5
40
0 5 10 15 20
Ssin e
Fig. 35. Radiatign patterns of space fed array Gaussiantaper f(r) = e-
5B
' 1~..VaA2.<.i.AJL~ - r-Vmt~~L~ALJL. . . . .
N
U,0U
N
IIN
��5�
N
59
� � U
I-3-210740
DISTORTION QUADP~LbQ AS=Mism
__________SCAN ANGLE 0
F/D 3
AZIMUTH PLANE 0-------
10
lox.
20
40 -J
600
0 5 10 15 20
Fig. 37. Radiation patterns of space fed array Gaussian-2r
2taper f(r) e
60
1 A--
L -3-21075
0DISTORTION Q.WADTS ArA zT.a.SM
SCAN ANGLE
F/D -
AZIMUTH PLANE 45-.........
20
co
302.5 X
40__ _ _ _ _ _
I V
6010 5 10 15 20
U= TR sin e
Fig. 38. Radiation patterns of space fed array Gaussian-2r
2
taper f(r) = e
61
L-3-21Of0
N ~DISTORTION Qu&V.Txc AsZX~auIsm...--_ BOSCAN ANGLE 20-
F/D L ------ ---AZIMUTH PLANE 9lfSqAN - - - - - -
10 - _ _ _ _ _ _ _ _ _ _ _
20
30
501 •..0 5 10 15 20
u sin 8
Fig. 39. Radiation patterns of space fed array Gaussiantaper f(r) e -2r 2
62
"'"''.........................................•' . ....... •...... ... ........ ;....'"-'','•?''',.
I -3-21077.1
DISTORTION QLIAI)IRAP1C ASTTIONATIS'M
SCAN ANGLEF/D
10AZIMUTH PLANE 4 - - -
20
30
40'
50V0 510 is 20
wD sin e
Il'itT. 40. Piidiat iocn pat-teril-I otf :spzwc ted arraty (iwllsiall
t a''or f*Cr) V
This can be explained by the fact that the astigmatic distortion
has a larger peak to peak distortion creating higher sidelobes in
certain cuts, whereas the gain loss depends on the mean square
path length error which is smaller for astigmatism as the 450
planes (seam planes) are not displaced.
6.5 Sinusoidal Astigmatism
For want of a better name we call the distortion
represented by
z = a[sin krrr] cos 2 4 (25)
sinusoidal astigmatism, as it is astigmatic as far as the azimuth
coordinate is concerned but wihh a radial sinusoidal variation.
We would expect the pattern degradation to be greater for this
type of distortion than for those considered previously as now,
depending on the parameter "k", regions of high distortion move
toward the center of the array.
Fig. 41 shows graphically Eq. 25 for "k = 1/2" where
greatest distortion occurs at the rim.
Figs. 42-45 inclusive show the radiation patterns for the
* two planes of interest (0= 0, 450) for the broadside beam and one
scanned to 20 degrees. All for unity "f" number.
Fig. 46 shows graphically Eq. 25 for "k = 1.0". Here
the rim is undistorted. Figs.47-50 inclusive show the corre-
*: sponding patterns.
64
................ .......
Fig. 41. Sinusoidal astigmatism (k = 1/2) z - a(sin r) cos 20.
65
•~~~~t J.. IiI
- -3-21079
0-.. (k-1/2)DISTORTION _I N..OLALA.TI QM,&T ISM
8CAN ANGLE
F/D1AZIMUTH PLANE 45 SCAN
10
20
30
40
50 z I0 5 10 15 20
Ssin 9
Pig. 42. Radiation patterns of space fed array Gaussian-2r2taper f(r) = e
66Ii
[-3-21080
0 (k-1/2)DISTORTION SINUSOIDAL ASTIGMATISM
__........__ SCAN ANGLE 0
AZIMUTH PLANE i5.10
20 _ _ _ _ _ _ _ _ _ _ _ _ _
30
40
0 5 10 15 20
s in 8
Fig. 43. Radiation patterns of space fed array Gaussian-2r2taper f(r) = e-
67
0 (k-1/2)
DISTORTION SINUSOIDAL ASTIGMATIsm
_______SCAN ANGLE 200F/D 1
10 AZIMUTH PLANE Q...i L - - - - - -
20
30
400
501 ..... \0 5 10 15 20
Ssin 8
Fic:. 44. Radiation patterns of space fed array Gaussiantaper f(r) e-2r 2
68
. ....... ... '
1 -3-21082
0 • (k-1/2)
DISTORTION lSINUSOIDAL ASTIGMATISM
SCAN ANGLE 20°
F/D 1AZIMUTH PLANE -. - --,-
10 ....
20
30,, •
50so I, AY/ l0 5 10 15 20
u v-D sin 8
Fig. 45. Radiation patterns of space fed array Gaussian* -2r2taper f(r) er
69
0U
"-4
rU
Ise
M
Cl 4
700
-!Tr
-3-21084
r ISTORTION §.IjUSO ID.AWLI ASIGAT IS M
_____________ AN ANGLE
F/D1AZIMUTH PLANE 0----
10
20
V
30
44
0 5 10 16 20
L) r Dsin 8
Fig. 47. Radiation patterns of space fed array Gaussiantaper f (r) =e-2r 2
K-3-2 1085
0DISTORTION SINUSOIDAL ASTrIGMATISM
___________SCAN ANGLE
F/D1
106AZIMUTH PLANE
10
30 __ _ _ _ _ __ _ _
40
0 5 10 1s 20
irDsin e
Fig. 48. Radiation patterns of space fed array Gaussian
taper f(r) e-r
72
I-3-2_1086 1
DISTOTION SINEISOIDAL ASTIGMATISM
-SCAN ANGLE V20
F/D -1
AZIMUTH PLANE LaciScm - - - - - -
10
20 _ _ _ _ __ _ _ _ _ _
V
30 _ _ _ _ __ _ _
40
So0ý0 5 10 15 20
iD
Fig. 49. Radiation patterns of space fed array Gaussian
taper fir) = e -r
73 .
0 (k-1)
DISTORTION SiNUSOI1DAL ASTIGMATISM
SCAN ANGLE 20
F/D1AZIMUTH PLANE 40SA
10 -
20
40
50 1_ _ _ __ _ _
0 5 10 15 20
u=-! Dsin *
Fig. 50. Radiation patterns of space fed array Gaussian
taper f(r) e r
74
Fig. 51 shows graphically Eq. 25 for "k=l.5." Fig. 52
to 55 inclusive show the patterns. We note that due to the high
distortion near the array center that a 2.5 wavelength distortion
is no longer acceptable.
Examination of the pattern degradation in the two
astigmatic cases shows that it is more benign in the 450 or seam
planes. This is due to the fact that about these planes the
phase error has conjugated symmetry. Collapsing the complex
illumination function onto the 450 plane results in a modified
real illumination function.
6.6 Eight and Sixteen Gore
As the circular space fed lens may be of gore construction
pattern calculations were made with increased azimuthal frequency
variations. That is axial distortions of the type:
8 Gore z = a sin rTkr (cos 40) (26)
16 Gore z = a sin Pkr (cos 8f) (27)
k = 1/2, 1, 3/2.
Graphics and degraded patterns are shown in Fig. 56-79. Few
general comments can be made about these and the "four gore"
case of Section 5 above; namely:
75
ILi
-L-3-21088 ]_ _
Fig. 51. Sinusoidal astigmatism (k = 1.5) z = a(sin 1.5rr) cos 2•.
76..
-3-21089
0(k- I SDISTORTION tNtJOIDA, ASTIGMA
SCAN ANGLE
F/D 1
10 AZIMUTH PLANE- - -
20 .. .. 0 x
30 -_°__ _ __ _
30 5 10 15
Ssin 8
FIiq. 52. Radiation |l,.terz.; of •pace f.d array •,1U5,ssij-2r2
l t (r) = 7
77
DISTORTION SINUSOIDAL ASTIGMATISM
SCAN ANGLE 0-F/D1AZIMUTH PLANE10
'
20
5)'
30
2.5
40 __ _ _ _ - _ _ _ _ _ _
50-0 5 10 is 20
Ssin e
Fig. 53. Radiatign patterns of space fed array Gaussiantaper f(r) e
78
.1I
7-3-21091 ]
0 (k-1. 5)DISTORTION SINUSOIDAL ASTIGMATISM
SCAN ANGLE 200
F/D1"AZIMUTH PLANE 0 SCAN
20 -
0 5 10 15 20
Ssin e
Fig. 54. Radiation patterns of space fed array Gaussian-2r 2
taper f(r) = e
79
00
F/D1AZIMUTH PLANE !A- 0 b --
10
20,,
30
40 5 0 10l
U sin 8
Fig. 55. Radiation patterns of space fed array Gaussian
taper f(r) e= 2r
80
Fig. 56. Eight gore distortion z a(sin - r) cos 4p.2
81
r
0-DISTORTION EIGHT G.ORE (k-1/2)
SCOAN ANGLE 0-
F/D 1
AZIMUTH PLANE --
10
20
30
0
40~~~ aA--"--ý
50 ~0 5 10 15 20
U=-M2 D sin 8
Fig. 57. Radiation patterns of space fed array Gaussian-2r2
taper f(r) =e
82
..... . .. .
,1)
"L-3-21095 1,
DISTORTION EIGHT GORE (k-I/2)
.SCAN ANGLE 0-
F/D 1
AZIMUTH PLANE 22.5010
20 20 ' .. . .. :5.0 ...x
301\ . .
40
0 510 Is 20
vD sin 6
Fig. 58. Radiation patterns of space fed array Gaussian
taper f(r) e2r 2 .
83
0
DI8TORTION I9HT •, Q J•,..(/_.
\S $CAN ANGLE 20F /D -
AZIMUTH PLANE °A - - - - - -
10 ,
20
30
40 __ _ _ _ _ _
I f
50
0 5 10 15 20
U in 8
Fig. 59. Radiation patterns of space fed array Gaussian-2r2
taper f(r) = e
84
1 i -'1lF • /' m i-.'•''i ''•: , / P Ii'•l I '1-I• -tll t;,1I ' :'I I I • 1'
--3-21-09-70
EIGHT GORE (k-1/2)
SCAN ANGLE 20
F/D 1
AZIMUTH PLANE V2.50 SCAN10 . . . .
20
30 5.0 x
40
500 5 10 18 20
U = sin 8
Fig. 60. Radiation patterns of space fed array Gaussian-2r 2
taper f(r) = e
S-3-21098
Flg. 61.. Eight gore distortion z = a (sinir) oos 4ý.
86
0SDISTORTION EIGHT GORE (k-1)
SCAN ANGLE
AZIMUTH PLANE 0-0 --
10 _
20 _ _ _ _ _ _ _
40
50 II0 5 10 16 20
U=_D $ine9
Fig. 62. Radiation patterns of spaoe fed arrayGaussian taper f(r) =er.
87
T3-21100
0DISTORTION EIGHT GORE (k-1)
SCAN ANGLE O0
F/D 1
AZIMUTH PLANE10
20
500
0 5 10 15 20
U5 X
U sin 9
Fig. 63. Radiation patterns of space fed array
Gaussian taper f(r) = e-2r 2
88
-21
0
"DISTORTION FQ .I T. o 1 ) -L
,,, SCAN ANGLE 20I
AZIMUTH PLANE o..•_, - --
10
20 2 X
30
40
0 5 10 15 20Sx sin 8
Fig. 64. Radiation patterns of space fed array
Gaussian Taper f(r) = e-2r 2
89
II
• " ' " . .. ... .. • ..... , ........ . .... ... .. . , ,, - .... ,... . .. ..: , .. ., . - .. ,,• [- •: - ' , ' ' - i
-3-21102
0DISTORTION ;p.JL•Q2. (.&-(I. L-
_SCAN ANGLE 2--
AZIMUTH PLANE 22.50 SCAN
10
20
5.0 x
30 A2.5.
40
501
0 5 10 15 2
u ! sin 8
F'ig. 65. Radiation patters of spaoe fed array
Gaussian taper f (r) e 2r
Fig. 66. Eight gore distortion z = a (sin 3/2 irr) cos 40.
91
0DISTORTION EIGHT GORE (k-1.5)
SCAN ANGLE 0(iFID 1
AZIMUTH PLANE---10 -
290
1.25%
30
40 I V
500 5 10 15 20
u -!D si e
Fig. 67. Radiation pat~terns of space fed array
Gaussian taper f (r) e-r 2
92
0DISTORTION EIGhHTGORE (k-1.5)
SCAN ANGLE Q%.
"F/D 1
AZIMUTH PLANE 2?.5_O
10
20 S~5.oh,
30
40
50
0 6 10 16 20
sin 8
Fig. 68. Radiation patterns of space fed array
Gaussian taper f(r) e-2r 2 .
i,
93
i iZE
S-3-21106__
0DISTORTION -P LG W_90.RL Lk a1. 1)SCAN ANGLE 200
FID -1AZIMUTH PLANE _Q SCAN
2 0 A_ • -"
30
40
0 5 10 15 20
u =-! Dsin 8
Fig. 69. Radiation patterns of space fed array
Gaussian taper f(r) = '2r 2 .
94
I ,U - ~ •,
-3-21107
0DISTORTION LIQIL MM-ULk-1. .1
_ ..-.- ,_-- SCAN ANGLE 10°
F/D 1IAZIMUTH PLANE L2 .L . .
10 ..
20
30
40
50
0 5 10 15 20
r D s in e
Pig. 70. Radiation patterns of space fed array-2r 2
Gaussian taper f(r) e .
95
-321108
Fig. 71. Sixteen gore distortion z = a (sin r) oos 8ý.
96
-3-211
DISTORTION SIXTEEN GORE (k•1/2)
SCAN ANGLE 00
F/D 1AZIMUTH PLANE 00
10
20
30 //z
40
5010 5 10 15 20
U 'D sin 8
Fig. 72. Radiation patterns of space fed array
Caussian taper f(r) e2r2 .
97
LL~~ ~~ '7-"-- 7
DISTORTION 8.IXTERM GOREL..Lk%3/2)
_______SCAN ANGLE 20'
FIDAZIMUTH PLANE 0 SCA-N
10
20
30ISx
40
501 v
Fig. 73. Radiation patrs of space fed arrayGaussian taper f(r) :%=~ r2
98
4.l
Fig. 74. Sixteen gore distortion z a (sinrnr) cos 84'.
99
S-3-2,If22]
DISTORTION SIXTEEN GORE (k-i)
SCAN ANGLE o-
F/DAZIMUTH PLANE 02_
1 0 .... . . -
20
30
40
0 5 10 16 20
U--v Dsin e
Fig. 75. Radiation patterns of space fed arrayGaussian taper f(r) = e-2r 2
100
L -3-21113 I
DISTORTION 5IXTU .
SCAN ANGLEF/D1AZIMUTH PLANE OSCAN
10
20
30 IA
505011 oi 1 / \ 10 5 10 15 20
Us Dsin 8
Fig. 76. Radiation patterns of space fed array
Gaussian taper f(r) = e .
101
A t . .. .. & ... .. ....... ..h.... ..U
-3-21114I
Fig. 77. Sixteen gore distortion z = a (sin 3 ir) oCs 8ý.
102
- •
DISTORTION SIXjTrEN CORE (ký1.5)
-...... SCAN ANGLE 00°
F/D I
AZIMUTH PLANE o-10
20
\V
40
50-0 5 10 1s 20
U"-D sin G
Fiq. 78. Radiation patterns of space fed array
claussian taper f(r) =
103
.... .....
-.3-21116]}
DISTORTION ,T&T• .. • (-•l..5)
\_ SCAN ANGLE 200F/D L............
AZIMUTH PLANE 2o° CAN
10
20
30
40 /O
50 IV 1 z0 5 10 15 20
u rD sin 8
Fig. 79. Radiation patterns of spaoe fed array-2r2Gaussian taper f(r) = e
104
.iL-;& ... st.o.s. aLA4L.~g -
a) For moderate loss of main beam gain the loss isindependent of the number of gores.
b) The pattern degradation is more benign in the seamplane cut and was not calculated in all cases.
c) The pattern degradation was markedly dependenton radial axial distortion, that is on the valueof Ilk".
d) The pattern degradation was more spread out withthe larger number of gores, probably as thisdistortion represents a higher spatial frequency.
The last point mentioned was further investigated, by
calculating the patterns over a larger angular interval. Fig. 80
to 83 show the principal and seam plane pattern of a sixteen
gore distortion with a five and ten wavelength maximum distor-
tion.
6.7 Half Linear Fold
Next we consider an unsymmetric case where one half of
the array experiences a linear distortion, Fig. 84. Fig. 85
and 86 show the radiation patterns across the fold for zero and
twenty degree scan angle. The zero scan case clearly shows the
expected main beam shift and a coma type degradation.
105
r
-3-21117
DISTORTION IXTEIEN GORE (k.1.
SCAN ANGLE 0
F/D-10 AZIMUTH PLANE cO0
-20
~-30 -__
-40 - - - - _ _ _ _
-60
-60 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
0 10 20 30 40 50 60
U=-' sin a
Fig. 80. Radiation pattern of space fed array
Gaussian taper f (r) = e-2r 2
106
O -3110
DISTORTION SIXTEEN GORE (k,1.5)
SCAN ANGLE 2 -
F/D 1
-10 AZIMUTH PLANE .25 - - - - - -
-20
S-30 - ____ __
5x'
-40
-50
-60 -
0 10 20 30 40 50 60'D
SU sin 8
"Fig. 81. Radiation pattern of space fed array
Gaussian taper f(r) e 2 r.
107
a I I
I -3-21119 10
DI8TORTION SIXTEEN GORE (k-1. 5)
SCAN ANGLE o°
F/D 1
-10 AZIMUTH PLANE 0o------- - -
-20
*u-30 - _ _
-40 - _ _
-60
-600 10 20 30 40 50 so
Ssin 8
Fig. 82. Radiation pattern of spaoe fed array
Gaussian taper f(r) - e-2r 2 .
108
"" • ' • • • ... .... .. •' . , : ,, ".:' ."• I". ''• • '':- ...... .. .. a•', ' 2.•2 .. :•. .. . . ....."., ,..' .....•
0DISTORTION jIJT~EEE CSffq_(jt-1 5)
SCAN ANGLE ~F/D1
-10 AZIMUTH PLANE I!-1.250-
-20
'a-30
-40
-60
0 10 20 30 40 60 80
U r Dsin 9
Fig. 83. R~adiation pattern of space fed array*Gaussian taper f(r) e 2r2
109
Fig. 84. Half linear fols z = ay, y>O;z = 0 , y<O.
110
.......
I-3-2112210
DISTORTION_ -HALF LI NEAR FOLD
SCAN ANGLE q0F/D
-10 - AZIMUTH LN E10 -
-20
~-30
-40
-60
-20 -15 -10 -6 0 5 10 15 20
U sin 8
Fig. 85. Radiation patterns of space fed arrayGaussian taper~f(r) =e-2r 2 .
... .. .. .. .. .. ...... ...11.1
S-3-21123
DISTORTION UL•iF •.•WA .OLD
SCAN ANGLE 200.....
-10 r AZIMUTH PLANE 9O° SCAN
-20 -
2.5>X
-40 --
-50 - -
-60 - - - -
-20 -15 -10 -5 0 5 10 15 20
Ssin 8
Fig. 86. Radiation patterns of space fed array
Gaussian Taper f(r) e2r2 .
112
7. RADIAL DISTORTIONS
In Appendix A-2 we derive that the total path length
advance due to an element radial displacement Ar is r:
E r A Fmine cos(-') -r/2f (28)IS L 1- +(r/2f)2]
where (r, 4) are aperture coordinates
and (9,4) are observational coordinates.
7.1 Uniform Thermal Expansion
We consider the case of uniform thermal expansion where
a module element at a radial distance "Ro0 " has expanded to a
radial distance "R" given by
R - R0 (1 + eAT] (29)
where R is the original radius used for the phase shift
comiands
e is the thermal coefficieint of expansion
AT is the temperature change
The radial displacement is
Ar = RoeAT (30)
The radiation pattern for the circular aperture scanned
to (6 may be written as
113
Rm 2rg f/ R jkRsinS° COS(4 °-ý 1) -jkRsine 0cOs( -' X
g(O,•) = f ff(R) e 0 o(c~' j~i~o(-'
0 0
eJkReAT sinO cos(ý-0') x
e RdRdo' (31)
where k - 2w/X
= -ReAT R/FV1 + (R/F) T
f(R) = aperture taper function
in the plane of scan 0 = and we haveRm 2T
g(6 4ý =f f f(R) e JkR[v°-v(l'eAt)] cos(½°-V) + JkE" RdRd4'
O o (32)
where vO sin 0
v sin 6
since "e" is a function only of R the "• integration can be
performed with the result :Rm
g(eo 0 ) 0 f f(R) Jo[kR(vo-v(l-eAT))] ejkel' RdR (33)0
where JC(x) is the zero order Bessel FunctionA
The beam maximum occurs at " 8"
A sin 0sin 1 eAT (34)
114
* Ui
A
or at a slightly larger angle than commanded. The pointing error
"66" is then
A e AT tan 6 radians (35)0
a result independent of the HPBW and the "f" number.
Typical space craft materials have expansion coefficients
given as:
Material e
Kapton 20 x 10 6 /°C
Aluminum 28 x 10 6 /°C
Taking a + iOO 0C temperature change we have a beam pointing
error of + 2.5 milliradians (0.140) at a scan angle of 45 deg.
A 100 meter aperture at L-band has a HPBW of about 0.2 so that
the effect is not insignificant. The beam pointing error
caused by uniform thermal expansion therefore merits considera-
tion for narrow beamwidth arrays scanned to large angles.
We still have to consider the term "kc" in Eq. (33).
This term
ke,'= 27 (eAT) R2 F (36)/1 + (R/F)236
is a parabolic type radial phase error considered in Fig. 9 where
a quarter wave rim displacement caused a tolerable gain degradation
(about 1 dB). Setting this as a criteria we have the relation:
115
F)() e AT (37)
V-1 + (D/2F) 2
where "D" is the array diameter.
For a 400 wavelength aperture (100 m at L-Band) we have an
allowable temperture change of +1120C for a unity "f" number
system. We note larger "f" numbers permit larger antenna
diameters measured in wavelengths.
It is evident that thermal expansion will set a
limit on the antenna size in wavelengths unless some means of
phase correction is employed.
116
* -. .... .. . . . • ' .. .• • |
8. FEED DISPLACEMENT
Due to faulty deployment or thermal expansion of the
feed supports the feed may not be in its correct axial position.
Considering an axial displacement, "A" from the focal point, we
have for the path length error (PLE)
D/2
- ~ R
A F
PLE -L_~R' - (F+A)] l-'R2P (38)
where the second bracketed term is the phase shifter correction
programmed for the correct focal position. For small displace-
ments:
-Il-'T ~ 7 -- 239PLE - AL 1+ (R/F) (39)
which is a quadratic type error for which we can choose the
quarter wave criteria at the rim for a 1 dB loss and we have:
2A 2 (40)
117
The axial tolerance, therefore, increases as the square of the "f"
number and is about two wavelengths for F=D.
8.2 Lateral
On the assumption that the unipod feed support (required
for deployment) remains straight but is depressed by an angle
"a" an aperture phase error is created given by:
4/2 ~ 1/2,27 LAD +(r)2 ..rsin acos 1/2 +/r\ 2( 1/2
(41)
The beam shift and pattern degradation can be computer calculated
using this expression. However, a tolerance can be set on "a"
as it can be shown by expansion of Eq. 41 that the beam pointing
error (BPE) is " al" essentially independent of the f number.
The allowable BPE, therefore, sets the tolerance on the permitted
lateral feed displacement.
118
9. THREE DIMENSIONAL ARRAYS AND PHASE COMPENSATION
A space fed array with axial surface distortions forms
a three dimensional array of elements. This is a form of conformal
array where the element location conform to the distorted lens
surface. Other forms of conformal arrays such as those mounted
on a sphere or cylinder frequently exhibit high sidelobes in the
farout angular region even though the near in sidelobes show the
expected sidelobe behavior. This condition, belately realized,
led us to question the assumption, originally made in this report,
that the pattern degradation will be confined to the vicinity of
the main beam if the distortion spatial period is large. This
behavior can be placed in evidence by returning to Eq. 13.
Ca = Aa(r'•') (.) - 2 sin2 (13)
For small observation angles, 8, the effective path
length error is considerably smaller than the lens distortion
and the lens structure has its alleged distortion insensitivity.
However at large observation angles, the effective error ap-
proaches the lens uistortion and the farout sidelobe level will
rise above the low theoretical value. It is, therefore,
necessary to examine the complete pattern in the forward half
space. To do this a specific array must be examined. To limit
computation time we have chosen a 32 wavelength square grid
(3217 elements, 38.85 dB gain and 2.240 HPBW) and with our
119
rr
Gaussian taper.
Another consideration that arises due to our three
dimensional array is the problem of array directivity (loss-less
gain). The directive gain should be calculated from
G O4r p(e (42)p(e,l) sinededý
where p(e,O) is the power pattern in all space
p( o, o) is the power pattern at beam peak. The gain reduc-
tion shown in all of the previous graphs is the reduction of all
the element contributions at beam peak taken as a vector sum. For
a discussion of the two methods of gain computation see reference [4].
Conformal arrays are normally phased cohered to radiate
a plane wave in the desired scan direction. This technique can
also be applied to a distorted lens array if the element locations
are known. Their position can be determined by a laser radar
located at the focal point or by other means. As the distortions
are expected to be smooth and of long correlation length only a
reasonable number of survey points are required to determine the
distorted surface. To dtermine the phase compensation required
to cohere the distorted array we return to Eq. 11 for the effective
phase error.
2_ a __ Aa(r,, cos e( 11X X [ 1 1 + (r/2f)2
120
i
. . .. . . . . . . .
To cohere the array at an observation angle 8o, we apply a phase
F correction (function of aperture coordinates and desired scan
angle e only).0
Phase Correction = - -r- a o (ros 000 %Il + (r/2Z)
(43)
The residual phase error becomes:
S~~~~~~~2 Tr r$)[csO os0] (4Residual Phase = 7-A (rW) cos 8- cos 0a (44)
which vanishes at the cohered angle but which remains significant
for angles far from the compensated angle.
To avoid the double integration of the radiation pattern
required by Eq. 42, we have chosen an azimuthally symmetric
distortion (Radial ripple) specified by
Aa(r) = a sin 1.5lr (45)
We show in Figs. 87, 88, and 89 the radiation patterns
for a= 0,2, and 5 wavelengths. The two wavelength case shows
the degradation of the near-in sidelobes, the five wavelength
case we see further degradation and a farout sidelobe near
isotropic levels.
The array was then phase cohered for boresight radiation.
Fig. 90 (5 O distortion) shows that the undistorted main beam
121
* i-..-'. . . . . . . . . . . . . . . . . . . . . . ..
K-3-21124j
DISTORTION V9W.P1AJ BE.EPLE.._
SCAN ANGLE 0F/D ._
-10 - - AZIMUTH PLANE .l-
"-20
V -30
-40- - - - --
-50-- -
-601
-
0 10 20 30 40 50 60 70 80 goU =IDW8
Fig. 87. Radiation pattern of space fed array
Gaussian taper f(r) e-2r2,
122
L .L i.
L-3-21125 I
0
DISTORTION 2.X RADIAL RIPPLE
SCAN ANGLEF/D
1 -0 AZIMUTH PLANE -- -
-20 - - -- - -- -- --
• -30
-40 -
-60-- --
0 10 20 30 40 50 60 70 80 90
' U -1-D sin 9
Fig. 88. Radiation pattern of space fed array-2r2Gaussian taper f(r) - e .
123
"7-a---1126
0 -
DISTORTION 5X RADIAL RIPPLE
SCAN ANGLE o
F/D 1-10 -AZIMUTH PLANE"-L-
-20
S-30 -
ISOTROPIC LEVEL
-40.
-50 -
-60 "0 10 20 30 40 50 60 70 80 g0
U sin 9
Fig. 89. Radiation pattern of space fed array
Gaussian taper f(r) e-2r2.
124
.-3-211270
5X RADIAL RIPPLE-DISTORTION WAsL W&BUI,. _ _ _
SCAN ANGLE o--F/D .
-10 AZIMUTH PLANE ALL
-20 - --
S-30
ISOTROPIC LEVEL
-40 - -f-- __
-50 - --
-60 -..0 10 20 30 40 50 60 70 - o go
U = -ZDsiSsin U
Fig. 90. Radiation pattern of space fed array-2r2 iGaussian taper f (r) e .
125
gain and beam shape was recovered but the farout sidelobe
degradation remained. A cosine square element power pattern is
included in these calculations.
From the computer printouts we summarize some data of
interest%
Uncompensated
a Gain Loss Gain
0 0.00 dB 38.97 dB
ix 0.20 38.77
2X 0.77 38.20
5x 3.85 34.95
Phase Compensated (cohered)
1% 0.00 dB 38.97 dB
2X 0.00 38.97
5x 0.00 38.68
From summation of elements in phase and magnitude**
Prom pattern integration
126
We note that the gain loss obtained by summing the element
contributions and that by pattern integration agree rather well,
expecially for reasonable distortions justifying the previous
work in the body of this report. When the array is boresight
phase cohered the gain loss is only 0.3 dB with the five wave-
length distortion.
Another item of interest is the distribution of the
radiated energy shown in Fig. 91. For the undistorted array,
due to the high illumination taper, essentially 99% of the
energy is in the main beam. With a five wavelength uncompensated
distortion this is reduced to about 80%; phase compensation
increases this to about 93%.
127
t. I :A..
i coL! 0
0 W~
0
~0O
•eco
o w_0-L ;5 (.)
U w- < 0 zQLV
°W C
- ,N0<--0
CV ccO"
0 l 00
"128
NI
10. CONCLUSIONS
10.1 Electrical and Structural Defects
Various electrical and structural defects that
cause antenna pattern degradation have been examined. The
results may be summarized as:
10.1.1 Electrical
1. Element Excitation Errors
Well known antenna tolerance theoryindicated that to achieve a -10 dBiaverage sidelobe level requires a10 electrical degree rms or a 1.5 dBamplitude rms tolerance. Combinationof errors is indicated in Fig. 4.This desired SLL level will bedifficult to achieve.
2. Element Failure
A 3% element failure will cause anaverage SLL of -10 dBi with no otherelectrical defects. This degree ofreliability may be difficult toachieve over the projected life ofthe space craft.
10.1.2 Structural
1. Axidl Array Distortions
It was shown that a space fed arrayis comparatively insensitive to axialsurface distortions compared to otherantenna types. Many canonical distor-tions were examined. Although thepattern degradations differ, a generaltolerance on flatness can be stated asplus or minus one wavelength for unityf numbers. Larger f numbers reduce themain beam and near-in sidelobe patterndegradation.
129
-
2. Radial Array Distortions
Uniform thermal expansion causes aa beam pointing error and a loss ofmain beam gain. The beam pointingerror is proportional to the tangentof the array scan angle and is indepen-Sdent of the array IIPBW and the f number.The gain loss is independent of thescan angle but increases with arraydiameter in wavelengths and decreaseswith increasing f number. For typicalspacecraft material having a thermal 6coefficient of expansion of 25 x 10" /Ortemperature change causes a + 25 mr(0.140) beam pointing error it a 450scan and a 1 dB gain loss for a 400wavelength diameter array with unityf number.
3. Axial Feed Displacement
The permitted axial feed displacementincreases as the square of the fnumber. For a one dB gain loss andunity f number the axial feed toleranceis two wavelengths.
4. Lateral Feed Displacement
A lateral feed displacement producesa beam squint or a beam pointing errorequal to the angular feed displacementindependent of the f number.
10.2 Phase Compensation
Element excitation errors or failed elements cannot in
general be corrected. Uniform lens thermal expansion or axial
feed displacement due to thermal expansion can be corrected with
thermal sensors and quadratic phase corrections stored in the
module microprocessor memory. Axial lens distortions are more
130
difficult to compensate as the distorted lens surface must be
measured and the element-modules independently addressed with
the required phase correction. This phase coherence is a
function of the scan angle. It essentially restores the main
beam gain and the near-in low sidelobes. It does not affect
the farout sidelobe degradation caused by large axial lens
distortion.
131
ACKNOWLEDGMENTS
I am indebted to Joseph Zeytoonian for computer
programming, to William Steinway for the use of his graphics
program, and to David Bernella and Edward Barile for informative
discussions. Finally, my thanks go to Linda Theobald for
preparing this manuscript.
132
REFERENCES
1. P. M. Mort%, Vibration a -nd'Sound (McGraw-Hill, New York,1936) , 18 edition, pp.147-160."
2. R, C. Hansen, microwave scanning Antennas Vol. I (AcadamicPress, New York# 1964).
3. R. C. Collini and F. J. Zucker, Antenna TheoryPart I,Chapter 6, "Non Uniform Arrays"o (McGraw-Hi1, New York,1969) pp. 227-233.
4. R. C. Rudduck, and D. C. F. Wup "Directive Gain of CircularTaylor Patterns," Radio Sci. 6, pp. 1117-1121 (1971).
1331
APPENDIX
A-I Axial DisplacementsRrn• D/2
LENS
FOCUS F
Element module at M provides a phase to:
1) Correct the spherical wavefront from the focus;that is provide a phase correction of%
2 Yr •F 2 ]J-- [7 - F2 (Al)
2) Scan the beam to (8, ,o); that is provide a phase2 T7- R sin eo cos (0-0o) (A2)
When the element module suffers an axial displacement, Aa thea
above module functions remain unchanged. However, the module
is excited by a path length delay (error) of:
'r AF F/ i + R2 (A3)
where the expansion is valid for A <<F.
134
Eq. A3 may be written as:
Aa (A3)14 f + (r/2)
where r - normalized radial coordinate
f = F/D, the system f number
There is also an error term in the direction of observation
LENS D
Y
UV is the observation vector sin 6 cos $ x + sin sin y + cos 0
D-V is the displacement vector- aA-i
The additional path length advance of
-V. D-= A a Cos 8 (A4)
The total path length advance is then
Ca " Aa [Cos e -(i +5(r/2f) .
This result was first derived in the Grumman-Raytheon study.
135
A-2 Radial Displacements
q. R
:1 LENS
when a module suffers a radial displacement, Ar a path length
delay is incurred of:
4F+ (R + &r) - = (M6)
Which may be written, for A r «<F as:
a + (r/2fy)2
Similarly to the axial displacement a component in the direction
of observation (0,0~) is incurred.
136
OV (6,#)
.DV
Observation vector. V - sine coo 00 + sin e sin C + coo 6 Z
Displacement vector DV - Cor ýo * + A sin ý yr r
path length advance - Ar sin 6 COB (cs-')
The total path length advance is then
Er Ar [sine cos ((A- '
I..
iI
137
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IS. 3UPPLEMENTARY NOTES6
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19. KEY WORDS (Colitinme di reve.rse side if necesaryaa~nd identify by block quinb.,)
space-based radar antenna radiation patteinsfar field pattern degradation axial lens distortionsspace fed phased arrays radial lens~ distort-ions
4 large aperture
XI. AISS TRýC T (Comilt i rves jig if block number)T efa fe pttrn egi%,atesjinyoropiacnalielaphased arrays, suitable for a space based radar
its examined, T1e effects considered are:Structulai0Ax~ial ens surface distortions2) Uniform radial thermal expansion3) Axial and lateral feed displacemonts
W'1) Eeent phase and amplitude excitation errors2) Failed elements
Ali introductory section discusses the size, Cost, and weight penalties of low sidelobe designs. Thefinal section presents a method of phase compensation or coherence of large axial lens distortions.
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