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Unformatted text preview: the intervening material between the antennas is linear, the link can be regarded as a two-port linear network, with parenleftbigg Z 11 Z 12 Z 21 Z 22 parenrightbiggparenleftbigg I 1 I 2 parenrightbigg = parenleftbigg V 1 V 2 parenrightbigg The lower portion of Figure 2.4 shows the corresponding Thevenin equivalent circuit. For this discussion, we sup- pose that the transmitting and receiving antennas are sufficiently distant to neglect the coupling term Z 12 that would 33 otherwise appear as an additional source in the transmitter circuit. The total power delivered to the antenna by the transmitter to the transmitting antenna is then P t = 1 2 | I 1 | 2 R 1 where R 1 = ℜ Z 11 is the radiation resistance. Coupling cannot be neglected in the receiver circuit, of course, since that 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 = ℜ Z 22 = ℜ 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....
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- Spring '13
- The Land, power density, Solid angle