217 beam solid angle weve already stated that the

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2.1.7 Beam solid angle We’ve already stated that the beam solid angle is always given by = 4 π/D regardless of the beam shape. This can be proven by expressing the total radiated power as an integral and invoking the definition of directivity: P t = integraldisplay P d = integraldisplay D ( θ,φ ) P t 4 π d Here, P t is a constant (the total radiated power) that can be canceled from both sides of the equation, leaving the identity: 4 π = integraldisplay D ( θ,φ ) d 32
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Meanwhile, the formal definition for the beam solid angle is given by Ω = integraltext D ( θ,φ ) d /D max . Substituting above gives the desired result: = 4 π D max where the D max notation is nonstandard but used here to avoid confusion. The beam solid angle of the ideal dipole is 8 π/ 3 Str. 2.2 Reciprocity theorem The proceeding discussion emphasized the performance of antennas used for transmitting radiation. Antennas of course also receive radiation, and their performance as transmitter and receiver are related by the reciprocity theorem. Reciprocity is best illustrated by a kind of thought experiment, where we consider what happens when antennas are used interchangeably for transmission and reception. Tx Rx P 12 A eff 2 G 1 r Rx Tx P 21 A eff 1 G 2 r Z 22 Z l V 2 = I 1 Z 21 Z 11 Z l V 1 = I 2 Z 12 Z g Z 11 V g Z g Z 22 V g I 1 I 2 + - + - Figure 2.4: Communication links illustrating the reciprocity theorem. Two different configurations for a communica- tions link are shown at left and right, where the roles of transmitter and receiver have been reversed. Below each is the Thevenin equivalent circuit, in which V 1 , 2 is the open-circuit voltage of the given receiving antenna. Refer to Figure 2.4. Consider first the diagram on the left which shows a communication link with a transmitter feeding power to a dipole antenna and a receiver collecting power from an aperture antenna. The particular kinds of antennas used are not important. We know that the power received at site 2, the aperture antenna, due to radiation from site 1, the dipole, is given by P 12 = P t D 1 4 πr 2 A eff 2 (2.5) where the subscripts refer to the antennas at the two sites. Consider next the equivalent circuit describing the link. If 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 parenrightbigg parenleftbigg 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
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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 = 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,
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