In this case the radiation resistance is predicted to

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In this case, the radiation resistance is predicted to be R rad 0.01 , a virtual short circuit. Such electrically short antennas pose practical problems for impedance matching and are likely to be inefficient. Radio engineering at very low frequencies (VLF and ULF) where antennas are necessarily short compared to the wavelength is consequently very challenging, and exotic solutions have been proposed, including the use of superconducting wires, long wires trailed from aircraft, and even modifying the conductivity of the ionosphere in such a way that it can function as an antenna. 31
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Returning to the AM car radio example, despite the poor efficiency and impedance matching problems, an electri- cally short antenna provides acceptable performance. What ultimately matters in a communications (or radar) link is the signal-to-noise ratio rather than the absolute signal power, since the latter can always be boosted with amplifica- tion. Antenna inefficiency and mismatches attenuate the signal and atmospheric or sky noise equally, and so long as the final noise delivered by the antenna to the receiver is large compared to the noise generated within the receiver, the link is not compromised. Background sky noise at AM radio frequencies is very intense, so antenna performance is not a critical issue. Note that this reasoning does not apply to antennas used for transmission, however! Transmitting antennas must be efficient and readily matched and are generally physically large ( λ/ 4 or longer). We’ll return to these ideas later. 2.1.6 Directivity We previously defined the directivity D as the ratio of the radiated power density in the direction of the main beam to the average radiated power density. One can also generalize the concept and consider the directivity in any direction, D ( θ,φ ) . It is once again expedient to work with radiated power per unit solid angle rather than per unit area. Therefore, define D ( θ,φ ) power radiated per unit solid angle in ( θ,φ ) direction avg . power radiated per unit solid angle = 4 π power radiated per unit solid angle in ( θ,φ ) direction total power radiated A single number is sometimes quoted for the directivity, referring to the directivity in the direction of the main beam ( D ( θ,φ ) max ). For the elemental dipole, we have D ( θ,φ ) = 4 π ( Z / 8)( Idl sin θ/λ ) 2 Z ( π/ 3)( Idl/λ ) 2 = 1 . 5 sin 2 θ so that D =1.5 for this antenna. Note once again that directivity is not a function of range. It may be a function of wavelength, although not in this case (so long as dl λ .) Recall that the elemental dipole antenna is an idealization. All practical antennas (i.e. that you can build) have D 1.5, and there is no such thing as an isotropic radiator with D =1. In practice, the gain of an antenna, which reflects ohmic losses, will be smaller than the directivity, although most often not by very much.
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