A method for evaluating the performance of a four-reflection or quadreflex antenna is reported. Geometrical optics was used initially to determine the ideal feed pattern required to produce uniform illumination on the aperture of the conical reflector and the reverse problem of quickly finding the aperture illumination given an arbitrary feed pattern. The knowledge of the aperture illumination makes it possible to compute the antenna efficiency, which is useful for comparing antenna performance during tradeoff studies. Scattering calculations, using physical optics techniques, were then used to more accurately determine the performance of a specific design

Cramer, P. W., Jr.

English

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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Technical Report 32-1591
Conical Quadreflex Antenna Analytical Study
Paul W. Cramer, Jr.
Unclas
G3/09 2k863
JET PROPULSION LABORATORY
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA
December 15, 1973
https://ntrs.nasa.gov/search.jsp?R=19740004807 2020-03-23T11:57:08+00:00Z
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Technical Report 32-1591
Conical Quadreflex Antenna Analytical Study
Paul W. Cramer, Jr.
JET PROPULSION LABORATORY
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA
December 15, 1973
I
Prepared Under Contract No. NAS 7-100
National Aeronautics and Space Administration
Preface
The work described in this report was performed by the Telecommunications
Division of the Jet Propulsion Laboratory.
JPL TECHNICAL REPORT 32-1591 PRECEDING PAGE BLANK NOT FILMED
Contents
I. Introduction . . . . . . . . . . . . . . . .. . 1
II. Geometrical Optics . . . . . .. .. . . . . . . . . . . 2
III. Efficiency Determined by Scattering Calculations . . . . . . . . 5
References .................... . 5
Table
1. Antenna efficiency ........ . . . . . . . 5
Figures
1. Quadreflex geometry . ....... . . . . . . 1
2. Equivalent parabola configuration ............ 3
3. Horn pattern required to produce uniform aperture
illumination by ray tracing . . . . .. ... . . . . . . . 3
4. Exponential feed patterns for quadreflex antenna . . . . . . . 3
5. Aperture fields for quadreflex antenna .. . . . . . ... 3
6. Spillover, illumination, and overall efficiencies of
quadreflex antenna ................ . 4
7. Flow diagram of computer programs for calculating
efficiency .................. . 4
8. Subreflector and feed parameters . . . . . . . . 4
JPL TECHNICAL REPORT 32-1591
PRECEDING PAGE BLANK NOT FILMED
Abstract
Conical antennas have recently been shown to effectively meet the requirements
for large erectable spacecraft antennas. One configuration investigated consists of
an antenna with three scattering surfaces requiring four reflections for an electro-
magnetic wave. This article presents a method for evaluating the performance of a
four-reflection or "quadreflex" antenna. Geometrical optics was used initially to
determine the ideal feed pattern required to produce uniform illumination on the
aperture of the conical reflector and the reverse problem of quickly finding the
aperture illumination given an arbitrary feed pattern. The knowledge of the
aperture illumination makes it possible to compute the antenna efficiency, which is
useful for comparing antenna performance during tradeoff studies. Scattering cal-
culations, using physical optics techniques, were then used to more accurately
determine the performance of a specific design.
vi JPL TECHNICAL REPORT 32-1591
Conical Quadreflex Antenna Analytical Study
I. Introduction- - - -
The conical quadreflex antenna is a four-reflection an-
tenna consisting of a feed horn, ellipse of revolution sub- EQUIVALENT/
reflector, an "urn" with a parabola of revolution shape, PARABOLA / \3
and a conical reflector (Fig. 1) (Refs. 1 and 2). / RE ALOR91.44 cm IREFLECT
(36.0 in.) R FEED
In its simplest form, the antenna can be considered to GENERATING HORN
consist of an equivalent parabola which generates a sur- I PARABOLA FOCUS
face by being rotated about an axis parallel to and offset 9 RING 8 10
from the axis of parabola. The focus of the surface thus (5.242736cm URN (PARABOLA) 4LLIPSE
generated is a ring formed by the focus of the parabola E FOCUS 2
as it is rotated about the offset axis. The ellipse serves POINT -14.1867 cm
the purpose of transforming a point focus on the axis of 17.4318 cm
rotation to the ring focus required by the equivalent 98.6912 m
parabola of revolution. (38.8548 in.)
Fig. 1. Quadreflex geometry
The antenna has one unique feature which complicates
the design procedure. The pattern illuminating the urn- the feed becomes rays 8-9, 9-9, 9-9, and 9-10. Thus, the
conical surface structure is inverted from the original feed outer ray from the feed becomes the inner ray after final
pattern. This can be visualized by referring to Fig. 1. scattering from the antenna.
Beginning with the feed at the focus point, an axial ray 1-2
is scattered from the ellipse as ray 2-3, which becomes This phenomenon causes a number of design problems
rays 3-4, 4-5, and 5-6 in sequence. An off-axis ray 1-8 from since the traditional edge taper problem is now affecting
JPL TECHNICAL REPORT 32-1591 1
the performance of the center of the equivalent parabola II. Geometrical Optics
and the center of the feed pattern, which contains little
of the pattern power due to the sin 0 effect, is now illumi- Two objectives were to be met:
nating the outer portions of the equivalent parabola, (1) Given the quadreflex antenna configuration, deter-
which accounts for a large part of the scattering surface mine the feed pattern required to provide a uniform
area. The outer edge of the surface, therefore, experiences amplitude aperture distribution. The feed phase
an unusually high level of edge taper. Therefore, it is pattern is constant since the antenna geometry of
necessary to determine the effect of this property on the the quadreflex antenna transforms a spherical phase
antenna performance. front from a feed into a uniform phase pattern on
the aperture. Thus, the phase pattern transforma-
In the following discussion an equivalent parabolic tion is defined and need not be determined.
surface will be used to replace the urn and conical re-
flector to simplify this analysis. (2) Conversely, given practical feed patterns, determine
what aperture plane distributions will result and
To facilitate the analysis of such an antenna, two tech- what aperture efficiencies can be obtained. Again,
niques were used: due to the antenna geometry, for each ray originat-
ing from the feed, the phase where these rays
(1) Geometrical optics was used as a fast and efficient intercept the aperture will differ from the feed
tool for trade-off studies. phase by a constant. Thus, the phase transformation
is simply defined.
(2) Scattering calculations were made on the 1.83-m
(6-ft) version at 16.33 and 8.448 GHz to obtain a To meet these two objectives, by ray optics, an expres-
more accurate determination of the antenna per- sion was derived which related the aperture amplitude
formance. fields with the feed pattern:
sn( + p) _ _ _ { in(j 3)A sin( ) + 1 - 2a sin/3
sin ( - 0) sin ( - 9)
EA(r) = EF() (1)
sin (0 + /3) sec() 2aF{A + 2F tan
where
EA(r) = aperture fields
E,(0) = feed pattern
and
sin 3 (cos q - M cos 0) + sin 4 cos 0 - cos 0 sin 0
S= cos - M sin 0 - sin (2)
Equation (2) must be solved for 3 = f(9), where the dental Equation Solving Program" has been written to
aperture fields are desired, and 0 = g(p), where the feed evaluate Eq. (2). It should be pointed out in Eq. (1) that
pattern is to be obtained. The parameters of these equa- the independent variable is either E,(9) and its associated
tions are defined in Fig. 2. A program entitled "Conical polar angle 0 or, by simple rearrangement, EA(r) and itsQuadreflex Antenna Efficiency Program" has been written associated radius r. 0, A, a, and F are fixed parameters
to evaluate Eq. (1) and a program entitled "Transcen- defining the antenna geometry. r, 8, and p are functions
2 JPL TECHNICAL REPORT 32-1591
0 i I I
1.83-m (6-ft) DIAM CONICAL
REFLECTOR
EQUIVALENT
PARABOLA -10
F -15
S-25
FOCAL -30
.. ING -35
SU ELLIPSE -40 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30
FOCAL POLAR ANGLE 0, deg
POINT 8 AXIS OF
SYMMETRY Fig. 3. Horn pattern required to produce uniform aperture
L F illumination by ray tracing
M = a/c
2c = DISTANCE BETWEEN ELLIPSE FOCII 50 I I I I I I I
1.83-m (6-ft) DIAM CONICAL REFLECTOR 22a = u + v EXPONENTIAL FEED PATTERN E = exp (-K2
THUS a AND c ARE PARAMETERS OF AN ELLIPSE 40 -- 8-dB TAPER
--- 11-dB TAPER
Fig. 2. Equivalent parabola configuration 
- 15-dB TAPER AT POLAR ANGLE =22 deg30130 - -*- 18-dB TAPER
0 -- 24-dB TAPER
of each other as shown by Eq. (2) and by the following 24-dB TAPER
expression: I
r =A + 2Ftan (3) "
The computed optimum feed pattern for a uniform
aperture distribution is shown in Fig. 3. The infinite peak -o
required on the axis is the result of the pattern inversion
property of the quadreflex antenna and can be visualized -50 -40 -30 -20 -10 0 10 20 30 40 50
as the zero area of the feed pattern center being required POLAR ANGLE e, deg
to illuminate the finite area of the outer edge of the Fig. 4. Exponential feed patterns for quadreflex antenna
conical reflector.
Utilizing Eq. (1) the aperture distribution was com- 30 I I I I I I
puted for an E = exp(- K0) feed pattern. The exponen- EXPONENTIAL FEED PATTERN E = exp (-K 2
tial form was selected as it quite closely models the fields 20 -- 8-dB TAPER
of the corrugated feed horn utilized in the quadreflex -- II-dB TAPER
antenna. The parameter K is determined by the sub- 10 - - - 15-dB TAPER AT POLAR ANGLE 22 deg
reflector edge illumination desired. Figure 4 illustrates -- 24-dB TAPER
typical computed feed patterns selected to give edge D .
tapers of 8 to 24 dB on the subreflector edge. The corre- .< Z
sponding aperture distributions are shown in Fig. 5. It < - _
should be pointed out that in Fig. 5 the aperture fields
were plotted as a function of the polar angle p (Fig. 2). 20
Since p = 0 is the center of the plot and corresponds to a
point on the aperture a distance A from the axis of the -30
antenna, the hole or null that appears in the aperture
distribution within the radius A does not show in Fig. 5. -40
If the fields had been plotted as a function of the radius -50 -40 -30 -20 -10 0 10 20 30 40 50
r, the hole would not be suppressed, since the region POLAR ANGLE 9, deg
within A would appear on the plot. Fig. 5. Aperture fields for quadreflex antenna
JPL TECHNICAL REPORT 32-1591 3
100
SPILLOVER -
> ILLUMINATION
Z 90 OVERALL
0 17.4318 cm 14.1867 cm
(6.8629 in.) (5.5853 in.)
ELLIPSE PARABOLA
DIAMETER =1.83 m (6 ft)FOCII OCUS
701 FEED 13.3165 cm 13.3223 cm
10 12 14 16 18 20 in) (5.245 in.)HORN (5.2427 in.)
EDGE TAPER, dB
Fig. 6. Spillover, illumination, and overall efficiencies ELLIPSE 13.3165 cm
of quadreflex antenna PHASE R (5.2427 in.)P PHASE (5.2427 in.)
CENTER
31.6184 cm
(12.4482 in.)
HORN PATTERNS 32.958425 cm
(12.975758 in.)
SCATTERED
FIELDS 32.959563 cm
ELLIPSE ELLIPTICAL FROM SPHERICAL (12.976206 in.)
PARAMETERS ELLIPSE SURFACE SCATTERING ELLIPSE WAVE
lSURFACE ,PROGRAM EXPANSION
PROGRAM FREQUENCY POSITION P, cm(in.) S, cm (in.) R, cm (in.)
KU-BAND OPTICAL 2.125218 0.0 33.588249 23.368031
(0.8367) (13.223720)
COEFFICIENTS
KU-BAND MAX GAIN 5.704225 0.0 30.336035 26.050143
(2.245758) (11 .943321)
OFFSET PARABOLIC X-BAND MAX GAIN 12.943225 0.2032 24.043541 33.648099
PARABOLA PARAMETERS PARABOLIC SURFACE FIELDS (5.095758) (0.08) (9.465961)
SURFACE PROGRAM
PROGRAM OTHER PARAMETERS (SEE FIG. 2)
A = 13.3165 cm (5.2427 in.) 2a = 51.075803 cm (20.108584 in.)
PARABOLIC
SURFACE 20 
= 
43.187755' FOR 8 = 0.0 6= 37.3768990
F = 98.6912 cm (38.8548 in.) c = 10.968073 cm (4.318139 in.)
PROGRAM F/D = 0.631635 D = 365.76 cm (144.0 in.)
PHASE FOR
CORRECTIONS PROJECTING Fig. 8. Subreflector and feed parameters
SURFACE
FIELDS
ONTO
APERTURE
APERTURE
FIELDS
FIELDS ON
SPHERE FOR
POWER
CALCULATIONS APERTURE
EFFICIENCY
PROGRAM
Fig. 7. Flow diagram of computer programs for
calculating efficiency
4 JPL TECHNICAL REPORT 32-1591
A series of aperture distributions were calculated for solution was to use the asymmetrical scattering program
edge tapers between 9 and 20 dB, and the aperture to obtain the scattered fields from the elliptical sub-
efficiencies were in turn calculated from these distribu- reflector when the subreflector is illuminated by a corru-
tions. Since ideal feed patterns with spherical phase fronts gated horn. Then, by taking a spherical wave expansion
were considered, the phase was constant across the of the scattered fields, the fields on the surface of the
aperture and, therefore, did not contribute to any effi- equivalent parabola were calculated and then projected
ciency degradation. The spillover, illumination, and over- onto the antenna aperture. Last, the antenna efficiency
all efficency is plotted in Fig. 6. As can be seen, the was then calculated based on the projected fields in the
overall efficiency increases rapidly from 9-dB edge tapers aperture plane. Figure 7 shows a flow chart describing
to about 14 dB and then levels out. The broad efficiency the sequence of programs used to obtain the final effi-
peak results from the fact that the spillover efficiency is ciency value. Table 1 shows the efficiencies obtained for
improving at the same rate the illumination efficiency was X-band and Ku-band antennas, each with a two-section
decreasing. In measured efficiencies versus edge taper, the corrugated horn feed. In each case the feed was located
overall efficiency begins to drop off at about 14 to 15 dB. at the position that was experimentally found to give the
The difference is attributed to the fact that feed patterns highest efficiency. As a final case, the Ku-band configura-
in reality do not have ideal phase patterns. For most tion was adjusted to place the phase center at the focus
patterns the phase patterns begin to change rapidly, as the of the ellipse. The efficiency that resulted is also pre-
pattern amplitude drops. For cases where increased edge sented in Table 1. Figure 8 shows the subreflector dimen-
tapers are used, it would then be expected that over a sions for each of the three cases.
greater portion of the aperture there would be widely
varying phase values. The varying phase would then Table 1. Antenna efficiency (1.83-m (6-ft) diameter)
detract from the phase efficiency. Thus, if the efficiency
computed from calculated aperture fields could be ad- Frequency Ku-band Ku-band X-band
justed for phase, it would be expected to show the same Maximum Maximum
dropoff observed from measured efficiencies. Feed phase center location Optical gain gain
Efficiency
III. Efficiency Determined by Scattering Subreflector blockage 0.979 0.979 0.979
Calculations Spillover, main reflector 0.884 0.906 0.825
Because of the relatively small sizes of antennas under Urn blockage 0.972 0.987 0.989
consideration, the various scattering surfaces were in the Illumination 0.886 0.905 0.904
near field of the fields incident upon them. Therefore, Phase 0.984 0.933 0.967
using ray optics techniques does not provide sufficient Overall 0.734 0.740 0.698
accuracy. The approach taken to obtain a more rigorous
References
1. Oliver, R. E., and Wilson, A. H., Furlable Spacecraft Antenna Development:
An Interim Report, Technical Memorandum 33-537, pp. 6-7, Jet Propulsion
Laboratory, Pasadena, Calif., Apr. 15, 1972.
2. Ludwig, A., Woo, K., Gustincic, J., and Hardy, J., "Recent Development of
Conical Reflector Antennas", IEEE G-AP 1973 International Symposium Record,
pp. 314-317, Aug. 1973.
JPL TECHNICAL REPORT 32-1591 5
NASA - JPL - Coml., L.A., Calif.


Conical quadreflex antenna analytical study

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