a Matlab package for phased array beam shape inspection

3.2 Reception 13
equations. But first we need to be careful about what we, actually, want to proof. What
are the “standard situations” in RX and TX?
Conceivable, we could imagine really running the transmission backwards, with energy
flows coming in from all over the sky. But that would be a very special case, and, would
anyone really want to do that, could be handled as a consequence of linear superposition,
once we have handled the standard case of a point source first. So, in standard reception
situation, we have a small radiation source in direction û as viewed from the receiving
antenna.
• The standard concept of “gain in reception” seeks to quantify the variation of the
received power, or the size of the induced voltage on the antenna terminals, when
the point source’s direction but not its distance is varied. Thus, a driving plane
wave field’s magnitude but not direction in front of the receiving antenna is kept
constant, and we want to know how much of the wave’s power actually gets in, as
function of û.
• On the other hand, the standard concept of “gain in transmission” quantifies how
much a transmitting antenna’s far field intensity, or amplitude, varies as function
of direction when the distance to the point of observation is fixed.
There are two different entities involved: (1) the gain-in-transmission and (2) the gainin-reception.
But, fortunately, there are also two different situation to plug these entities
in: the TX and the RX situations. Therefore, it is possible, though still miraculous, that
the two different gain-quantities can be numerically represented by a single function.
This function is then simply called the directional power gain G(û) (or the directional
amplitude gain G(û) = √ G(û), if we want to deal with field amplitudes rather than
field intensities). Another point of view is to maintain that there exists only a single
gain quantity, namely, the gain in transmission, and the reception directional sensitivity
is expressed in terms of the transmission gain.
To proceed this, we need the principle of reciprocity. This is a statement that (under
certain conditions) there is a symmetry, reciprocity, in the mutual coupling between
subsystems of an electromagnetic system. For a simple electronic two-port circuit, consisting
only of linear elements such as shown in Fig. 7, the statement of reciprocity can
be phrased in the following way: If a generator current I A = I 0 in port A causes a
voltage U (B)
A
= U 0 in port B, then, if we interchange the current generator and the
voltmeter, and again extract I 0 from the generator, the voltmeter again will show the
value U 0 . For ordinary circuit elements, this result is already built-in into the Kirchoff
rules used to compute the internal currents and voltages in the circuit, so that for any
given configuration, such as shown in Fig. 7, one may prove the reciprocity by directly
computing the voltage U 0 resulting from the driving current I 0 .
Although it cannot any more proved from Kirchoff rules, the same reciprocity statement
holds also for the two-port system shown in Fig. 8, where subsystem A is the
radar antenna of interest, and subsystem B is a small far-away test antenna. From the
reciprocity and the linearity properties of the system, we argue in the caption text of
Fig. 8 that the directional dependancy of the voltage V REC observed at the terminals of
the receiving antenna when the antenna is illuminated with transmission from the test
antenna in direction û, is solely via G T (û), where G T (û) has originally been defined as
the gain-in-transmission quantity:
V REC = b × G T (û) × ‖E REC ‖ . (33)