Elsewhere on the wire we may expect a standing wave

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current to nearly vanish at the ends of the wire in view of the demands of current continuity. Elsewhere on the wire, we may expect a standing wave pattern. A distribution that satisfies these conditions is: I ( z ) = I sin ( k [ L/ 2 − | z | ]) This turns out to be a reasonably good approximation for the actual current distribution on a long wire antenna, except that the actual distribution never exactly vanishes at the feedpoint, as this one can. Substituting I ( z ) and integrating yields the following expression for the observed electric field: E θ = jkZ sin θI 4 πr e jkr bracketleftBigg integraldisplay 0 L/ 2 sin k ( L/ 2 + z ) e jkz cos θ dz + integraldisplay L/ 2 0 sin k ( L/ 2 z ) e jkz cos θ dz bracketrightBigg = jkZ sin θI 2 πr e jkr bracketleftBigg cos( k L 2 cos θ ) cos( k L 2 ) k sin 2 θ bracketrightBigg = jZ I 2 πr e jkr bracketleftBigg cos( k L 2 cos θ ) cos( k L 2 ) sin θ bracketrightBigg = Z H φ 36
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The Poynting flux can then be calculated. P r = 1 2 ( E × H ) · ˆ r = 1 2 ( E θ H φ ) = Z 2 | I | 2 (2 πr ) 2 bracketleftBigg cos( k L 2 cos θ ) cos( k L 2 ) sin θ bracketrightBigg 2 Like the elemental dipole, this antenna does not radiate power in the direction of the wire (despite the sin 2 θ term in the denominator) and has a maximum in the equatorial plane. 2.3.1 Half-wave dipole An illustrative and practical case is that of a half-wave dipole antenna with L = λ/ 2 . Now, kL/ 2 = π/ 2 and the second cosine term in the numerator vanishes. This leaves: P r = | I | 2 Z 8 π 2 r 2 bracketleftbigg cos( π 2 cos θ ) sin θ bracketrightbigg 2 We will see that trigonometric functions of trigonometric functions are rather common expressions in antenna theory. While the shape of this particular expression is not immediately obvious, it is actually not very different from sin 2 θ (see Figure 2.6). Note that the term in square brackets is already normalized to unity. The total power radiated by the half-wave dipole antenna is found by integrating the Poynting flux over the surface of a sphere of any radius. P total = | I | 2 Z 8 π 2 integraldisplay 2 π 0 integraldisplay π 0 bracketleftbigg cos( π 2 cos θ ) sin θ bracketrightbigg 2 sin θdθ bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright 1 . 219 36 . 6 I 2 ( W ) where the last result was obtained using Z = 120 π and with the help of numerical integration. (Without the aid of numerical integration, the theta integral can be expressed in terms of the cosine integral function C i ( x ) = integraltext x cos u/udu , which is a tabulated function.) As noted above, the radiation pattern for this antenna is not very different from that of an elemental dipole. The HPBW in the E-plane is 78 and therefore only slightly narrower than the elemental dipole. The BWFN is still 180 . A more meaningful statistic is the directivity: D ( θ,φ ) = P P total / 4 π = Z | I | 2 8 π 2 bracketleftBig cos( π 2 cos θ ) sin θ bracketrightBig 2 36 . 6 | I | 2 / 4 π 1 . 64 bracketleftbigg cos( π 2 cos θ ) sin θ bracketrightbigg 2 (2.10) The overall directivity is therefore 1.64, only slightly more than 1.5 for an elemental dipole, since the quotient in (2.10)
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