Page 176 - Maxwell House

156 Chapter 3

consider the case when the bounding surface A is extended to infinity and rewrite the left-side
term in (3.103) in scalar form similar to (3.95)

∯ ( x − x ) ∘ = ∯ � x − x � ∘ (3.104)
2 1 1 2 ,2 ,1 ,1 ,2

Here { ,1 , ,1 } and { ,2 , ,2 } are the tangential component of electric and magnetic
fields on the bounding surface A. Evidently, for such far-far away fields any limited in size
source looks like the point source alike any star for us in the night sky. The field magnitudes
depending on the source power can be different by the same way as star brightness but these
far away field structure must be identical. It means that far-off from sources the relationship
between tangential components is independent of the source type. If so, ,1 = ,1
⁄
⁄
and ,2 = ,2 . We met this value = � = 376.6� [Ohms] and in
0

0

Chapter 1 and called it the free space impedance. Putting these relationships into (3.104) we
will discover that the surface integral in (3.104) is equal to zero and (3.103) comes to the
following simplification
∫ ∘ = ∫ ∘ (3.105)
2 1 2 1 2 1
Note that is evaluated in the domain and is evaluated in the domain . Eventually,
1
1
2
2
nothing will change in (3.105) if we make the transform 1 ↔ 2 putting the 1-source in the
position of the 2-source and vice versa. That means that the interchange of sources does not
change the power balance (3.105). Note that due to the “dot” product in (3.105) the maximum
of power could be transmitted or received when || and || . Such mutual vector
2
1
1
2
orientation is customary calls the polarization match and fundamental requirement in the
antenna applications and measurements.
The next paragraphs are devoted to the application of Lorentz’s reciprocity theorem in antenna
theory, measurement, and development. If so, we need some definition of the “antenna” term.
Widely speaking, anything that can radiate or absorb electromagnetic fields is an antenna: our
sun and all stars, our earth, and earth atmosphere lighting, our brain and body, towers for
broadcast and communication antennas on towers, radio telescopes and satellite antennas,
northern lights, and lasers, etc. This list can be infinite long because everything in our universe
consists of atoms carrying moving charged particles like electrons that are equivalent to small
currents, therefore, capable of radiation. The most difficult engineering problem is to prevent
or shield EM radiation not to create it. But technically speaking, we will define an antenna as
an electrical device which converts electric power into electromagnetic radiation and vice
versa.

3.4.4 Receive-Transmit Antenna Reciprocity

In the case of polarized matched sources, we can present (3.105) in a scalar form. Assuming
that both volume and are small cylinders of length and cross section ( = )
2
1
oriented in parallel to the field and current vectors, we obtain integrating around the sources
(like (3.11))