# 64 only slightly more than 15 for an elemental dipole

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The overall directivity is therefore 1.64, only slightly more than 1.5 for an elemental dipole, since the quotient in (2.10) is unity when θ = 0. The most important difference between the half-wave dipole and the elemental dipole is its radiation resistance. Equating P t = 1 2 | I | 2 R rad R rad 73 . 2 Ω 37

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Figure 2.6: Radiation pattern of a half-wave dipole. The vertical axis here is the polar ( ˆ z ) axis. The higher impedance of the half-wave dipole permits a practical match using commercially available 75 transmis- sion line. In fact, the impedance of the half-wave dipole is slightly reactive. Shortening the antenna slightly below λ/ 2 makes it purely resistive and reduces the radiation resistance to a figure close to 50 , permitting a practical match using standard 50 transmission line. Furthermore, since this radiation resistance is likely to be much greater than the ohmic resistance of the antenna, efficiency is very high, and we are justified in neglecting ohmic losses and equating directivity with gain. These qualities make the half-wave dipole practical. The gain of the antenna is, however, still quite low. 2.3.2 Antenna effective length As an aside, we can introduce the idea of antenna effective length. The effective length is that quantity which, when multiplied by the electric field in which the antenna is immersed, gives the open circuit voltage appearing across the antenna terminals. There is a simple, intuitive interpretation of effective length that applies to wire antennas, which is described below. All antennas, including aperture antennas, have an effective length, however. The concept of effective length is useful for calculating the power received by an antenna without invoking the reciprocity theorem — see problem ????. We’ve seen that the electric fields for a half-wave dipole and an elemental dipole are given respectively by: E θ = jZ I 2 cos(( π/ 2) cos θ ) sin θ e jkr 4 πr E θ = jZ Idlk sin θ e jkr 4 πr Let us rewrite the second of these as E = jZ ke jkr 4 πr I h (2.11) h = dl sin θ ˆ θ where h represents the effective length of the elemental dipole. It obviously has the apparent physical length of the elemental antenna as viewed by an observer. We assert that the effective and physical lengths of the elemental dipole are identical since the current is uniform, meaning that the entire length dl is fully utilized. 38
By extension, if we can cast the radiation field of any antenna in the form of (2.11), the effective length of that antenna is whatever becomes of h . In the case of the half-wave dipole, we have h = λ π cos(( π/ 2) cos θ ) sin θ ˆ θ Note that | h | ≤ λ/π λ/ 2 . The effective length of the half-wave dipole is less than its physical length, reflecting the fact that the current distribution on the wire is tapered and that the wire is therefore not fully utilized for transmission and reception. Similar results hold for all practical wire antennas, since it is impossible to drive spatially uniform currents in practice.

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