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A more useful approximation is given by the rule of

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Unformatted text preview: 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. We can consider the Arecibo S-band radar system as an extreme example. This radar operates at a frequency of 2.38 GHz or a wavelength of 12 cm. The diameter of the Arecibo reflector is 300 m. Consequently, we can estimate the directivity to be about 6 × 10 7 or about 78 dB, equating the effective area with the physical area of the reflector. In fact, this gives an overestimate, and the actual gain of the system is only about 73 dB (still enormous). We will see how to calculate the effective area of an antenna more accurately later in the semester. 1.3.2 Phasor notation Phasor notation is a convenient way of describing linear systems which are sinusoidally forced. If we excite a linear system with a single frequency, all components of the system will respond at that frequency (once the transients have decayed), and we need only be concerned with the amplitude and phase of the response. The results can be extended to multiple frequencies and generalized forcing trivially using superposition. Let us represent signals of the form A cos( ωt + φ ) as the real part of the complex quantity A exp( j ( ωt + φ )) . Here, the frequency ω is presumed known and the amplitude A and phase φ are real quantities, perhaps to be determined....
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A more useful approximation is given by the rule of thumb D...

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