Furthermore we can approximate de θ jz k 4 πr i z

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The current distribution is approximately triangular on any sufficiently short wire. Furthermore, we can approximate dE θ = jZ k 4 πr I ( z ) dz sin θe jkr e jkz cos θ bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright 1 again, since all the distances measured along the wire antenna are small compared to a wavelength. The integration over dz now only involves I ( z ) and is tantamount to computing the average current on the wire, i.e. L ( I ( z ) ) = I L/ 2 . Finally, we have E θ = jZ k 4 πr I L 2 sin θe jkr for the electric field due to a short dipole antenna, precisely the same result as for the elemental dipole, only with dl replaced by L/ 2 . This means that the effective length of a short dipole is L/ 2 (times the factor of sin θ ), half its visible physical length. The reduction is associated with the incomplete utilization of the wire length due to the tapered current distribution. By comparison to an elemental dipole with the same physical length, the short dipole generates electric fields half as strong and radiates one quarter the Poynting flux and total power. Consequently, its radiation resistance is also reduced by a factor of 4 (from an already small figure). Its radiation pattern is identical, however, meaning that its directivity, HPBW, solid angle, etc., are the same. 2.3.7 Small current loop Another important antenna is a small current loop with radius a λ , as shown in Figure 2.9. Here, the observer is at location ( r,θ,φ ) in spherical coordinates. The sources of radiation are differential line segments Idl = Iadφ at locations ( a,π/ 2 ) in the x-y plane. (As elsewhere, primed and unprimed coordinates are used to represent source and observation points here.) The direction of the source current anywhere on the loop is given by ˆ φ = ˆ x sin φ + ˆ y cos φ . The vector potential at the observation point can then be expressed in terms of a line integral of the source current elements on the loop: A = μ 4 π Ia integraldisplay 2 π 0 e jkR R ( ˆ x sin φ + ˆ y cos φ ) where R is the distance between the source and observation points. In the limit r a , R is approximately equal to r minus the projection of the vector from the origin to the source on the vector from the origin to the observer: R r a sin θ cos( φ φ ) = r a sin θ (cos φ cos φ + sin φ sin φ ) 41
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x y z R θ φ r a φ r θ φ Figure 2.9: Radiation from a small current loop of radius a carrying current I . Since we are interested in calculating the far field, we may replace R with r in the denominator of the vector potential expression. More care is required in the exponent in the numerator, however, where we instead note that ka 1 and so use the small-argument approximation: e jkR = e jkr e jka... e jkr (1 + jka sin θ (cos φ cos φ + sin φ sin φ )) The vector potential therefore becomes A = μ Ia 4 πr e jkr integraldisplay 2 π 0 (1 + jka sin θ (cos φ cos φ + sin φ sin φ )) ( ˆ x sin φ + ˆ y cos φ ) The only terms that survive the φ integration are those involving sin 2 φ and cos 2 φ .
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