Geometry of the dual reflector antenna and coordinate description

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Geostationary satellites

Satellites have been in practical use in telecommunications since Echo 1, a 26.5 inch magnesium sphere launched by a Thor Delta rocket on August 12, 1960 bounced a taped message transmitted from Goldstone, California that was received by the Bell telephone laboratory a Holmdel, N.J. Echo I stimulated a great deal of interest in the development of active communication which lead American Telephone and Telegraph Company (AT&T) to build Telstar, launched on July 10,1962. Telstar was an active satellite with a microwave receiver and transmitter. It was the first satellite to transmit live television and conversations across the Atlantic. Geostationary satellites were proposed in 1947 by Arthur C. Clark (1917-), a British physicist and astronomer as a means to relay radio signals from one part of the world to another that is beyond the line of sight. Geostationary orbits are orbits occupied by communications satellites which remain at fixed points in the sky relative to observers on the ground. The are defined by an orbit period of one sidereal day, or about 23 hours 56 minutes 4 seconds. During one sidereal day the earth rotates about its polar axis exactly once. To be geosynchronous, a satellite must orbit the earth in the same period. This period defines the average orbit radius of 42155 km. This value is found from Kepler’s third law. The earth’ s radius (6370 km) subtracted from the orbit radius determines the orbit above the earth to be 35785 km. This definition doesn’t say anything about the shape of the orbit, or the orientation of the orbit plane with respect to the plane of the equator. The orbit can be highly elliptical, and/or it can be inclined with respect to the plane of the equator, and still be synchronous with the earth’s rotation. In this case, a desired class of geosynchronous orbit is the geosationary orbit. A satellite moving in a geostationary orbit remains at a fixed point in the sky at all times. This is desirable for radio communications because it allows the use of stationary antennas on the ground.
To understand the importance of these criteria, consider the result if the orbit fails to  meet them. If the orbit is not geosynchronous, the satellite does not move at the same  rate as the earth’s rotation. Thus, from the point of view of an observer on earth, the  satellite appears to be in continuous motion, and it periodically disappears below the  horizon. If the orbit is not a circle, the satellite does not move at a constant velocity  (Kepler’s second law). Instead, it appears to oscillate east-and-west at a rate of two  cycles per sidereal day. If the orbit does not lie in the equatorial plane, the satellite  does not remain at a fixed point in the sky. Instead, it appears to oscillate north-and south at a rate of one cycle per sidereal day. The terms geosynchronous and  geostationary are not synonymous: geosynchronous specifies only the orbit period, but  geostationary also specifies the shape and orientation of the orbit.

Antenna systems on geostationary satellites

Telecommunication services provided by satellite include television and telephone transponders and direct broadcast television (DBS). In the case of the first two services an operator will provide a service center or hub from where uplinks and downlinks to the satellite are made. This will be made from ground stations with high gain antennas. From the hub, the service is relayed into the terrestrial network. In the case of direct broadcast television a downlink service is provided to many users each using a lower gain antenna. For example, an 18″ aperture parabolic reflector antenna is
used as a DBS receiver. The antenna on the satellite needs to provide coverage over a geographical regIOn called the servIce area as opposed to a single beam to each individual ground station. Shaped or contour beams are used on these satellites to increase antenna efficiency and reduce interference in geographical areas adjacent to the service area. The need for shaped or contour beams was a significant challenge to antenna engineers and several methods of implementing contour beams have been studied and used. These include arrays, array front fed paraboloids and shaped single and dual reflector antenna systems.
From the examples in the previous paragraph, it can be seen that there will be a definite advantage in the ability to reconfigure the contour beam to provide coverage for different geographical service areas and from different satellite geostationary positions. In order to comply with FCC regulations on the level of radiation allowed in areas outside the geographical coverage area, contour beams are subject to much more stringent specifications and this is also likely to be enforced on reconfigurable beams. Reconfigurable contour beams can be implemented in a number of ways, including large aperture arrays, multiple feed reflector antennas and reflector antennas with adjustable main- and/or sub reflector surfaces. The contour beam reflector antenna (CBRA) is widely used because of its versatility and low cost per unit aperture. The disadvantage of using the array fed offset reflector is that the beamforming network is heavy, lossy and expensive. The same disadvantage applies to the phased array antenna. Both type have complex components that need to be space qualified. Space qualification includes thermal, electromagnetic compatibility and mechanical (shock and vibration) tests and account for a significant portion of the total cost of the  antenna subsystem [26]. The relative low cost ofthe shaped reflector antenna made it a popular choice for direct broadcasting satellites. The obvious disadvantage used to be the inability to reconfigure the contour beam. A reconfigurable CBRA has been implemented using an adjustable mesh main reflector and a cluster feed arrangement [4]. Another degree of freedom is added if the subreflector of a dual offset reflector antenna can be made adjustable. Piecewise adjustable subreflectors have been used in the past to correct for gravitational distortion in large axally symmetric dual reflector radio telescope antennas [1,2] and have also more recently been proposed as a way to correct for main reflector distortion in dual offset reflector (DOSR) antennas [3].

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Chapter 1: Introduction
1.1. Geostationary satellites
1.2. Antenna systems on geostationary satellites
1.3. The mechanical finite element diffraction synthesis technique
Chapter 2: Diffraction synthesis and radiation pattern computation for reflector antennas
2.1. Geometry of the dual reflector antenna and coordinate description
2.2. Subreflector analysis
2.3. Calculation of the far-field radiation pattern
2.4. Surface expansion in terms of the Modified Jacobi polynomials
2.5. Verification of the accuracy of the developed codes
2.6. Calculation of the antenna footprint
Chapter 3: Optimization of the contour beam gain cost function
3.1. Cost function
3.2. Optimization methods
3.3. Design of a CONUS beam
Chapter 4: Mechanical design of the reflector
4.1. Mechanical description of the surfaces using shell elements
4.2. DitTraction synthesis results and mechanical design performance
Chapter 5: Contour beam synthesis using the mechanical FEM surface description
5.1. Mechanical FEM diffraction synthesis
5.2. The effect of mechanical surface properties on actuator number and placement
5.3. Synthesis of an adjustable elliptical beam using the mechanical FEM surface description
5.4. Synthesis of an reconfigurable beam using the mechanical FEM surface description
Chapter 6: Conclusion

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