# D 4 π r 2 power radiated per unit area at center of

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• cornell2000
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D 4 π R 2 (power radiated per unit area at center of main beam) total power transmitted = 4 π power radiated per unit solid angle at center of main beam total power transmitted = 4 π P P t We can also characterize the main beam shape in terms of the total solid angle it occupies. Imagine for example that the main beam is conical, with uniform power radiated within and no power radiated outside of that. Then P = P t / , and we find that = 4 π/D . This is in fact a general result for any beam shape (i.e. not just conical). We will prove this later. Since can be no more than 4 π , D can be no less than unity. In fact, it is never less than 1.5. Roughly speaking, the area illuminated by the antenna can be written in terms of the product of the half-power beamwidths in the E and H planes: R 2 ( R · HPBW E )( R · HPBW H ) D 4 π HPBW E HPBW H where the latter term follows from the definition of . Note that the half-power beamwidths must be evaluated in radians here. This approximation permits approximations of the directivity knowing only something about the beamwidth of the main beam. It is not likely to give a very good approximation in most cases, however, since antenna 12

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patterns normally have tapered main beams and sidelobes. A more useful approximation is given by the rule of thumb: D 26 , 000 HPBW E HPBW H where the beamwidths are in degrees this time. Exact means of calculating directivity and gain will follow, but the approximations given here are sufficient for many purposes. Antenna effective area We’ve defined the effective area of an antenna as the ratio of the power it delivers to a matched (no reflected power) receiver to the incident power density, i.e. P rx ( W ) = P inc ( W/m 2 ) A eff ( m 2 ) Whereas antenna directivity and gain are more naturally discussed in terms of transmission, effective area is more of a reception concept. However, directivity and effective area are intimately related, and high gain antennas have large effective areas. Effective area has a natural, intuitive interpretation for aperture antennas (reflectors, horns, etc.), and we will see that the effective area is always less than or equal to (less than in practice) the physical area of the aperture. The effective area of a wire antenna is unintuitive. Nevertheless, all antennas have an affective area, which is governed by the reciprocity theorem . We will see that D = 4 π λ 2 A eff where λ is the wavelength of the radar signal. High gain implies a large effective area, a small HPBW, and a small solid angle. For wire antennas with lengths L large compared to a wavelength, we will see that HPBW λ/L generally. Just as all antennas have an effective area, they also have an effective length. This is related to the physical length in the case of wire antennas, but there is no intuitive relationship between effective length and antenna size for aperture antennas. High gain antennas have large effective lengths.
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• Spring '13
• HYSELL
• The Land, power density, Solid angle

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