# Since the radiation is uniform in the h plane here

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. Since the radiation is uniform in the H-plane here, the HPBW is not defined in that plane. Another measure is the beamwidth between first nulls (BWFN), which usually works out to be about twice the HPFB. In this case, it is 180 (exactly twice). Directivity A more objective measure of the degree of confinement is the directivity. The definition of directivity, D , can be expressed as D power density at center of main beam at range R avg . power density at range R = power density at center of main beam at range R total power transmitted / 4 πR 2 where the average means an average over all directions. This definition is slightly clumsy. If we interpret power density as power per unit area, then we have to specify some range R where this can be evaluated. It’s more natural to calculate power densities per unit solid angle, since that quantity does not vary with range. The latter definition will have to wait until we have discussed solid angles, however. Gain Gain G (sometimes called directive gain) and directivity differ only for antennas with losses. Whereas the directivity is defined in terms of the total power transmitted, the gain is defined in terms of the power delivered to the antenna by the transmitter. Not all the power delivered is transmitted if the antenna has significant ohmic losses. G power density at center of main beam at range R power delivered to antenna / 4 πR 2 Note that G D . In many applications, antenna losses are negligible, and the gain and directivity are used inter- changeably. Both directivity and gain are dimensionless quantities, and it is often useful to represent them in decibels (dB). D G = 10 log 10 P max P avg 11

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Note also that a high gain implies a small HPBW. Beam solid angle We can characterize the main beam of an antenna pattern by the solid angle it occupies. Solid angle is the generalization of an angle to a radial shape in three dimensions. Just as the arc length swept out on a circle by a differential angle is Rdθ , the differential area swept out on a sphere by a differential solid angle is dA = R 2 d . Thus, we define d = dA/R 2 . In spherical coordinates, we can readily see that d = sin θdθdφ . The units of solid angle are steradians (Str.) The total solid angle enclosed by any closed surface is 4 π Str. Tx R d θ φ dA x y z Figure 1.4: Antenna radiation pattern characterized by its solid angle d . As mentioned earlier, the definitions of directivity and gain given above are somewhat clumsy since they are stated in terms of power densities which vary with range even though the directivity and gain do not. We can now redefine the directivity more naturally in terms of solid angle with the knowledge that a per-unit-area quantity multiplied by R 2 becomes a per-unit-solid-angle quantity.
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