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and containing the directrix (see App. B). Although the characteristics of the reflector antenna are more readily described in terms of radiation, it should be kept in mind that the reciprocity theorem makes these applicable to the receiving mode as well. Now although there are near- and far-field components present in the reflector region, the radio link is made through the far-field component, and only this need be considered. For this, the reflected wave is a plane wave, while the wave originating from the isotropic source and striking the reflector has a spherical wavefront. The power density in the plane wave is independent of distance. However, for the spherical wave reaching the reflector from the source, the power density of the far-field component decreases in inverse proportion to the distance squared, and therefore, the illumination at the edge of the reflector will be less than that at the vertex. This gives rise to a nonuniform amplitude distribution across the aperture plane, which in effect means that the illumination efficiency is reduced. Denoting the focal distance by and the focal length by f as in Fig. 6.17b, then, as shown in App. B, f sec2 2 (6.27)
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The space attenuation function (SAF) is the ratio of the power reaching point P to that reaching point A, and since the power density is inversely proportional to the square of the distance, the ratio is given by SAF f 2 a b (6.28) cos
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For satellite applications, a high illumination efficiency is desirable. This requires that the radiation pattern of the primary antenna, which is situated at the focus and which illuminates the reflector, should approximate as closely as practical the inverse of the space attenuation factor. An important ratio is that of aperture diameter to focal length. Denoting the diameter by D, then, as shown in App. B, f D 0.25 cot
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(6.29)
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The position of the focus in relation to the reflector for various values of f/D is shown in Fig. 6.18. For f/D 0.25, the primary antenna lies in the space between the reflector and the aperture plane, and the illumination tapers away toward the edge of the reflector. For f/D 0.25, the primary antenna lies outside the aperture plane, which results in more
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f < 0.25 D
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f = 0.25 D Position of the focus for various f/D values.
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nearly uniform illumination, but spillover increases. In the transmitting mode, spillover is the radiation from the primary antenna which is directed toward the reflector but which lies outside the angle 2 0 . In satellite applications, the primary antenna is usually a horn (or an array of horns, as will be shown later) pointed toward the reflector. In order to compensate for the space attenuation described earlier, higher-order modes can be added to the horn feed so that the horn-radiation pattern approximates the inverse of the space attenuation function (Chang, 1989). The radiation from the horn will be a spherical wave, and the phase center will be the center of curvature of the wavefront. When used as the primary antenna for a parabolic reflector, the horn is positioned so that the phase center lies on the focus. The focal length can be given in terms of the depth of the reflector and its diameter. It is sometimes useful to know the focal length for setting up a receiving system. The depth d is the perpendicular distance from the aperture plane to the vertex. This relationship is shown in App. B [Eq. (B.37)] to be f D 16d
(6.30)
The gain and beamwidths of the paraboloidal antenna are as follows. The physical area of the aperture plane is Area D2 4 (6.31)
From the relationships given by Eqs. (6.14) and (6.15), the gain is G 4 l2
I area 2
D b Ia l
(6.32)
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The radiation pattern for the paraboloidal reflector is similar to that developed in Example 6.1 for the rectangular aperture, in that there is a main lobe and a number of sidelobes, although there will be differences in detail. In practice, the sidelobes are accounted for by an envelope function as described in Sec. 13.2.4. Useful approximate formulas for the half-power beamwidth and the beamwidth between the first nulls (BWFN) are HPBW > 70 l D (6.33) (6.34)
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