Project Cyclops, A Design... - Department of Earth and Planetary ...

APPENDIX
L
TUNNEL AND CABLE LENGTHS
Exactly n-I lines are needed to connect n points, so
n-I tunnels between adjacent antennas suffice to connect
an array of n antennas. One more tunnel is then
needed to tie the array to the central control headquarters.
However, certain main tunnels may be larger
than the branch tunnels and, depending on the array
configuration, not all tunnels may be of the same length,
so some analysis of this problem is in order.
To minimize the tunnel and cable lengths required,
the array should be made as compact as possible and the
control and processing center should indeed be at the
physical center of the array. Also the outline of the
array should be circular, although small departures from
circularity cause only a second-order change in the total
tunnel and cable length. Two obvious configurations
come under consideration: a square lattice in which the
antennas are placed in equally placed rows and columns,
and a hexagonal lattice in which the antennas are at the
centers of the cells of a honeycomb. These configurations
correspond to tessellating the array area with
squares and hexagons, respectively.
The ratio of the area of a circle to the area of a
circumscribed square is 1r/4 while that of a circle and the
circumscribed hexagon is 7r/2x/_. However, the spacing s
between antennas must be greater than their diameter d
to prevent shadowing of the antennas by their neighbors
.at low elevation angles. If e is the minimum elevation
angle and 0m = 0r/2) -e is the maximum angle form the
zenith for unobstructed reception then
d d
s - - (L1)
cos 0m sin e
The filling factor for a square lattice array is therefore
= = - sin 2e (L2)
fs 4 4
while that for a hexagonal array is
lh=2v " = sin e (L3)
Equations (L2) and (L3) apply to circular dishes: If
elliptical dishes are used, only the vertical dimension
(minor axis) must be reduced to d = s sin e; the
horizontal dimension (major axis) need be only slightly
less than s. Thus, for elliptical dishes the filling factors
involve sin e rather than sin2e. Table L-1 shows the
realizable filling factors without obstruction.
TABLE
L-I
Circular
Elliptical
e 0m s/d fs fh fs fh
45 ° 45 ° 1.1414 .393 .453 .455 .641
30° 60° 2.000 .196 .227 .393 .453
20 ° 70 ° 2.924 .092 .106 .269 .310
10° 80 ° 5.760 .024 .027 .I 36 .I 57
The filling factor for a hexagonal lattice array is always
2/V_ = 1.155 times as great as for a square lattice array.
If we wish unobstructed operation down to a 20 °
elevation angle, the array diameter must be roughly
three times as great with circular dishes, or x/_ times as
great with elliptical dishes, as would be required for
zenith operation only. Clearly, the use of elliptical dishes
would permit significant reduction in the array size and
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