a Matlab package for phased array beam shape inspection

42 A FIGURES TO CHAPTERS 2–7
I A
E T ∝ G T (û)I A
û
V (B)
A
(a) Transmission from A.
V (A)
B
û
E R ∝ I B
I B
(b) Transmission from B.
Figure 8: Antenna directional gain in reception. A small test antenna, e.g. a short
dipole, is at a fixed distance from the antenna of interest, always oriented
perpendicularly to the direction unit vector û. We envision û but not the
distance to be varied by moving the test antenna, while always keeping it
perpendicular to û, and inquiry the dependency on û of the voltage V (A)
B
at
A caused by the current at B, the situation shown in panel (b). We first
consider the transmission from A as in panel (a). Total transmitted power is
proportional to IA 2 , so by definition of the directional amplitude gain G T (û), the
far field E T in direction û is proportional to G T (û)I A . On the other hand, the
induced voltage V (B)
A
to I A and G T . We write this as V (B)
depends linearly on E T , so that also V (B)
A
A
is proportional
= Z AB I A , where Z AB = a G T (û) and
the factor a that does not depend on û. By reciprocity, the impedance Z AB
gives also the coupling from B to A in panel (b): V (A)
B
= a G T (û) I B . The
(linear) relation between ‖E R ‖ and I B cannot depend on û, so we must also
have V (A)
B
= b G T (û) ‖E R ‖, with some b that is independent of û. That is,
all directional dependency in V (A)
B
comes via G T (û). This shows that the
gain pattern in reception is equal to the gain pattern in transmission. To
find the constants of proportionality a and b requires knowledge of the actual
radiation fields associated with a short dipole. With standard assumptions of
polarization and impedance matching in the receiving system, this leads to
the general result A eff = (G T (û)) 2 λ 2 /4π for the effective area of an receiving
antenna.