E j ωt kr e θ idl sin θ jz 4 πk bracketleftbigg k

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e j ( ωt kr ) E θ = Idl sin θ jZ 4 πk bracketleftbigg k 2 r jk r 2 1 r 3 bracketrightbigg e j ( ωt kr ) where Z is called the “impedance” of free space and has a numerical value of radicalbig μ or about 120 π = 377 Ohms in MKS units. The terms that vary as r 1 are known as the radiation or far field terms. These terms alone carry average power away from the source. Terms that decay faster with radial distance are called the near field terms and are associated with energy storage in the vicinity of the dipole. The radiation fields dominate at distances much greater than a wavelength ( r λ ). Here, the electric and magnetic fields are normal to one another and to the (radial) direction of wave propagation, E and H are in phase, and their ratio is E θ /H φ = Z . Power flow is represented by the Poynting vector P E × H This is the instantaneous definition, not yet written in phasor notation. The Poynting vector has units of power per unit area and represents the power density carried by electromagnetic radiation along with the direction. In phasor notation, the time-averaged Poynting vector is given by ( P ) = 1 2 ( E × H ) 29
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The average power density propagating away from an antenna centered at the origin is given by the dot product of the Poynting vector with the unit radial direction, ˆ r . For the elemental dipole, this is ( P ) · ˆ r = P r = | I | 2 ( dl ) 2 Z k 2 sin 2 θ 1 32 π 2 r 2 ( W/m 2 ) (2.3) Only the far fields contribute to this result, although the derivation is completely general. The near fields are associated with reactive power flow and do not contribute to the time average flow of power away from the antenna. This must be the case; power density must decay as r 2 (inverse square law) since the area of the spherically expanding wavefronts that carry it increases as r 2 . Only the far field terms combine to give an expression that obeys the inverse square law. We next evaluate the characteristics of the elemental dipole antenna. We need only consider the far zone fields in the calculations, since the near field terms do not contribute to the radiation. 2.1.3 Radiation pattern z x y θ φ Figure 2.2: Elemental dipole radiation pattern. (left) E-plane. (right) H-plane. The overall pattern is a torus. We should now have a better appreciation for the radiation pattern discussed earlier in the chapter. The average power density (2.3) for the ideal electric dipole can be expressed as P r ( θ,φ ) = Z 8 parenleftbigg Idl sin θ parenrightbigg 2 ( W/m 2 ) (2.4) sin 2 θ The radiation pattern expresses the distribution of radiated power versus bearing ( θ,φ ) and is conventionally normal- ized to a maximum value of unity. Radiation patterns are typically plotted in polar coordinates. The E and H planes are the planes in which electric and magnetic field lines reside, respectively.
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