Page 560 - Engineering Electromagnetics, 8th Edition

542 ENGINEERING ELECTROMAGNETICS

resistances, provided ohmic losses in all conductors and all losses to surrounding
objects are reduced to zero. We will assume this, in addition to far-zone operation, to
be true here. The trans-impedances, Z 12 and Z 21 , depend on the spacing and relative
orientation between the antennas, as well as on the characteristics of the surrounding
medium. A critical property of the transimpedances in a linear medium is that they
are equal. This property is the embodiment of the reciprocity theorem. Stated simply,

Z 12 = Z 21 (85)

Further insights can be found by inverting (84a) and (84b), and invoking the admit-
tance parameters, Y ij :

I 1 = Y 11 V 1 + Y 12 V 2 (86a)
I 2 = Y 21 V 1 + Y 22 V 2 (86b)
where, again, the reciprocity theorem tells us that Y 12 = Y 21 .
Now, suppose that the terminals of antenna 2 are shorted, so that V 2 = 0. In this
case, Eq. (86b)gives I = Y 21 V , where the single prime denotes the condition of a

1
2
shorted antenna 2. Instead, we could short antenna 1, resulting in I = Y 12 V (with

2
1
the double prime indicating conditions with antenna 1 shorted). Because reciprocity
holds, it follows that
V 2 V 1 (87)
I 1 = I 2
Equation (87) applies regardless of the relative positioning and orientation of the
two antennas. We know that in a given direction, each antenna will transmit a power
density whose value is determined by the antenna radiation pattern. Furthermore, we
would expect to see the current that is set up on the receiving antenna depend on that
antenna’s orientation; that is, there is a reception pattern that the receiving antenna
presents to the incoming signal. Now, for a fixed relative orientation between the two
antennas, with antenna 1 as the transmitter, and antenna 2 shorted, a certain ratio
V /I will be obtained. This ratio will depend on the relative orientation, which in

2
1
turn will depend on the radiation pattern of antenna 1 and on the reception pattern of
antenna 2. If roles are reversed such that the transmitter now becomes the receiver,
and with antenna 1 shorted, a ratio V /I will be obtained that by Eq. (87) is the same

2
1
as before. The conclusion we must come to is that the extent to which the receiving
antenna accepts power will be determined by its radiation pattern. This means, for
example, that the main beam direction in the radiation pattern of the receiving antenna
corresponds to the direction from which it is most sensitive to incoming signals. The
radiation and receiving patterns of any antenna are the same.
Wenextconsideramoregeneraltransmissioncase,inwhichthereceivingantenna
is to deliver power to a load. Antenna 1 (Figure 14.16) serves as the transmitter, while
antenna 2 is the receiver, at which the load is attached. A primary assumption is that the
antennas are far enough away from each other so that only forward coupling (through
Z 21 ) will be appreciable. The large separation distance means that the induced current