NASA
Technical
Memorandum
30164
PRELIMINARY DESIGN OF LARGE REFLECTORS
WITH
FLAT
FACETS
Pradeep
K.
Agrawal
;
Melvin S. Anderson;
and Michael
F.
Card
January
1980
(IASA-TH-80164) PRELIRINABP
CESIGI
OF
LABGE
N80-18301
REFLECTORS
YITH
PLAT
FACETS
(NASA)
35
p
EC
~031~~
a01
CSCL
09C
Unclas
63/33
47298
Nat~onal Aeronaut~cs and
Space
Adrnlnlstrat~on
bngley
Research
Center
Hamplon, V~rg~nla
23665
PRELIMINARY DESIGN
OF
LARGE
REFLgCTORS
iJITH
FLAT
FACETS
Pradeep
K.
Agrawal,* Melvin
S.
Anderson and Michael
F.
Sard
ABSTRACT
A
concept for approximating curved antenna surfaces using flat facets
is
discussed.
A
preliminary design technique for determining the size of
the reflector surface facets necessary to meet antenna surface accuracy
requirements
is
presented.
A
proposed large Microwave Radiometer Satellite
is
selected
as
an application,
and
the far-field electromagnetic response
of a faceted reflector surface
is
compared
with
that
from
a spherical
reflector surface.
INTRODUCTION
Recent research studies [I], [2] indicate increased interest in the
feasibility of assembling and operating large-aperture antennas in space.
A
challenging potential application of
a
large space antenna system
is
the
Microwave Radiometer Satellite
(MRS)
[3]
,
[b]
.
When orbiting at
an
altitude
of
650
km
and operating
at
a
frequency of
1
GHz,
this antenna system
is
required to produce simultaneously
200
contiguous beams each having
a
foot-
print diameter of
1
km
on earth.
This resolution requirement plus the
requirement for
90 percent beam efficiency yields a reflector with
a
1150m
spherical radius and an aperture diameter of
660111.
A
typical structural concept for such a system
is
shown in Fig.
1.
The
concept consists of a large
str.
.:ural
truss network with a mesh or
membrane reflector
surface.
The tr,.ngular arrangement of nodes shown in
Fig.
1
is
typical of
many
proposed concepts for large space structures
because of desirable structural characteristics. Of interest in the present
paper
is
how the structural nodal pattern might be used to support flat
*P.
Agrawal was vith
NASA
Langley Research Center, Hampton,
VA
23665.
He
is
now with
RCA
Missile and Surface Radar Div., Moorestown,
NJ
08057.
M.
Anderson and
b:.
Card are
with
NASA
Langley Research Center, Hampton,
VA
23665.
membrane facets that would satisfactorily approximate the desired reflector
surface.
One attractive approach for erecting
a
large reflecting antenna
in
space
is
to assemble
an
array of deployable truss modules each of which
supports a reflecting facet
as
depicted in Fig.
2.*
Near-optical flatness
of these facets in this reflector concept can be achieved by tensioning
a
membrane with
as
few
connections to the truss
as
possible
151.
A
nearly
triangular membrane facet
is
shown in Fig.
3
in which mirror-quality flatness
has been achieved using
only
three tensioned-cable attachments.
Antenna accuracy requirements are often specified by the root-mean-
square deviation
(dm)
of the actual surface from the desired surface.
The purpose of the present paper
is
to present
a
preliminary design
technique for determining the size of the reflector surface facets required
to meet specified antenna surface accuracy requirements. Only the
regulw
but known deviation of the
flat
facet
from the desired surface
is
considered
and
errors
from
other sources
are
not included. Equations and generali-ed
design curves
are
presented in the paper for both triangular and hexagonal
surface facets which approximate spherical or paraboloidal surfaces.
The
proposed Microwave Radiometer Satellite reflector
is
chosen
as
an illustrative
example, and the far-field electromagnetic response of reflector surfaces
with either
triangular or hexagonal facets
is
compared with that from a
spherical reflector surface.
REFLECTOR
SURFACE
ERRORS
Faceted Surface Geometry
To subdivide a surface of revolution the geometric scheme proposed in
[6]
was a3opted. The mapping procedure
is
illustrated in Fig.
4.
A
surface
of revolution
is
divided at the aperture circle into
N
equal segments.
A
line connecting these points to the center of the reflector forms the
N
sided pyramid shown in the upper part of
Fig.
4.
Each side of the pyramid
is
then subdivided into
M
equal parts (bays) to form subelements.
Finally, the points of intersection of these triangles are projected or
mapped on the surface using
a
suitable origin of coordinates to obtain
the final nodal coordinates
of
the members.
It
was
found
[6]
that for
shallcw reflectors six aperture divisions
(~=6)
and
a center of projection
at
the center of curvature for the point
at
the center
of
the reflector
resulted in nearly uniform triangular facets that were close
.to having
equal sides. This method of subdivision was used
in
all the studies of
this report.
*A
patent disclosure
on
this "Deployable Module
for
Constructing Large
Surfaces" has been filed at the Langley Research Center by
H.
Bush,
M.
Mikulas,
Jr.,
and
R.
Wallsom.
2
Root-Mean-Square Surface Deviation
To
determine the acceptable size of a facet for preliminary antenna
reflector designs, the root-mean-square deviation
(62ms) of the flat triangular
surfaces from the ideal reflector surface
is
employed. Equations for
determining
6- for
a
general triangle with vertices located on
a
sphere
or paraboloid
are
given in the Appendix.
An
illustrative preliminary design
result
for the Microwave Radiometer
Satellite reflector
is
shown in Fig.
5.
In this example, a spherical reflector
was investigated with six circumferential subdivisions
(~=6) to obtain
nearLy equal-sided facets.
The surface deviation
is
shown in Fig.
5
for the number of beys
M
ranging from 13 to
25.
The vertical bar for
each
M
indicates the range of discrete member lengths and
is
plotted at
the
6-
for that configuration. The geometry of the surface with ~=16
is
shown in Fig. 6.
Simplified Facet Sizing Equations
If the reflector under consideration
is
shallow, the facets tend toward
equilateral triangles and
the pincipal curvatures are nearly equal for any
surface of revolution. Thus the
6ms
calculation for an equilatrial
triangle on a spherical surface should be a good approximation for the
actual
geometry.
In this case the equations of the Appendix are greatly
simplified. The maximum deviation
H
of a sphere of radius
R
from the
plane of an equilateral triangle with vertices on the sphere
is
where
L
is
the length of one side of the triangle. The effective sphere
is
defined as that which minimizes the
6,,
with respect to the triangular
facet.
As
shown in the appendix the distance between the sphere containing
the vertices and the effective sphere
is
314
H.
The corresponding
drms
is
1
L~
6
=-
-
rms
86
R
Thus, the member length required to satisfy a certain
6rms requirement
is
then
where
F
is
the focal length of the reflector taken
as
one-half the radius
>f
curvature at the center
of
the reflector
and
D
is
the aperture
diameter.
The general design curve contained in
Eq.
(3)
is
plotted in
Fig.
7(a).
Assuming the reflector
is
shallow
(F/D
>
0.5)
the number of
bays
M
(assuming ~=6)
is
given approximately by
The total number of members
Nm
in the reflector
is
given in
161
by
and the total number of triangular facets
Nf
is
Figure 7(b) presents generalized curves for the number of members and facets
based on Eqs. (5)
and
(6).
For the Microwave Radiometer Satellite, the accuracy of the design
formula
(~q. (3))
is
indicated in Fig. 5, where results from
Eq.
(3)
are
seen to
be
in the middle of the range of actual member lengths for each
value of number of bays
M.
Thus, the accuracy of estimated facet size
presented in
Eq.
(3)
is
adequate for preliminary design.