# The power delivered to the receiver under matched

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is the only thing driving a current. The power delivered to the receiver under matched load conditions, Z l = Z 22 , is then given by P 12 = | I 1 Z 21 | 2 8 R 2 where R 2 = Z 22 = Z l . This gives the ratio P 12 P t = | Z 21 | 2 4 R 1 R 2 (2.6) which must be consistent with the result from (2.5). Now, reverse the roles of the sites so that transmission occurs from site 2 and reception at site 1, as shown on the right side of Figure 2.4. This time, the power received at site 1 due to transmission at site 2 under matched load conditions is given by P 21 = P t D 2 4 πr 2 A eff 1 (2.7) where the transmit power is kept the same as before. In terms of equivalent circuit theory, we find that P 21 P t = | Z 12 | 2 4 R 1 R 2 (2.8) which must be consistent with P 21 /P t from (2.7). In the appendix, it is shown that for linear, isotropic (but not necessarily homogeneous) intervening media, Z 12 = Z 21 . This behavior is sometimes referred to as strong reciprocity . In view of equations (2.6) and (2.8), one consequence of strong reciprocity is that P 12 = P 21 , which is a somewhat less general condition referred to as weak reciprocity . We can restate this condition using (2.5) and (2.7) as P t D 1 4 πr 2 A eff 2 = P t D 2 4 πr 2 A eff 1 which demands that D 1 D 2 = A eff 1 A eff 2 This fundamental result shows that the ratio of the directivity for any antenna to its effective area is a universal constant (that may depend on λ , which is held fixed in the swap above). That constant can be found by performing a detailed calculation for some specific antenna configuration. The results can be stated as: D ( θ,φ ) = 4 π λ 2 A eff ( θ,φ ) (2.9) The factor of λ 2 is to be expected; directivity is a dimensionless quantity and can only be equated with another dimensionless quantity, such as the effective area measured in square wavelengths. Note also that the antenna effective area has been generalized and is now function of bearing, just like the directivity. When quoting a single figure for the effective area, the maximum value is given. Two important points should be recognized: 34

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1. For aperture antennas, we will see in chapter 4 that the effective area is related to the physical area but that A eff A phys . In practice, the former is always less than the latter. 2. All antennas have an effective area even if they have no obvious physical area. For the elemental dipole antenna, we found that D =1.5. Consequently, A eff = 1 . 5 λ 2 / 4 π independent of the length of the antenna! For the AM car radio example, A eff = 1 × 10 4 m 2 ! As stated above, the reciprocity theorem assumes that the antennas are matched electrically to the transmission lines, which we take to be matched to the transmitters and receivers. If not, then reflections will occur at the antenna terminals, and power will be lost to the communication budgets. The reflection coefficient at the terminals is given by ρ = Z l Z t Z l + Z t where the l and t subscripts denote the antenna (load) and transmission line impedances, respectively. The fraction of power reflected is then given by | ρ | 2 . In the AM car radio example, | ρ | 2 0.9992.
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