Page 234 - Maxwell House

214 ANTENNA BASICS

3
Here the binomial expansion was truncated since the terms proportional to � ⁄ � and all the

0
followings quickly converge toward zero. Therefore, a good approximation for ∆ can be

written as
2
2
∆ = ( − ) ≅ cos − cos (5.26)
0

1
2 0
The presence of the quadratic term in (5.26) incredibly complicates the far field theoretical
analysis as well the interpretation of far field graphs and measurements. So a compromise was
made to start counting the far field zone or Fraunhofer’s region from the point where
2 2

≫ =
0 � (5.27)
2
∆ ≅ cos

Note that in this zone
1 1 1 1 1 1

≅ ≅ �1 − � ⁄ �cos� = − cos ≅ (5.28)
0

⁄
1 0 1−� 0 �cos 0 0 2 0
0
Here in Taylor’s expansion (1 + ) −1 ≅ 1 − we kept the zeroth- and first-order terms only.
On the other hand, the emitting elements can be arranged not only along y-axis but be shifted
also in the direction of x- or z-axis that creates additional phase shifts between the fields in far
field zone. Using the same approach we obtain
2

≫ ⎫
0
2 ⎪
∆ ≅ cos = sin ∙ cos (5.29)

∆ ≅ cos = sin ∙ sin⎬

⎪
∆ ≅ cos = cos ⎭

2
2
2
In (5.29) = � + + while the total phase shift can be found as ∆ = ∆ + ∆ +

∆ . For example, one of the biggest steering radio telescope in the world built in Spain and

has a parabolic dish of a 30-meter diameter. According to (5.29) the radiation pattern of such
an antenna is fully formed at a distance beyond 27 km (!) at = 10.5 cm.
5.2.6 Radiation Pattern. Main Beam, Beamwidth, and Sidelobes
An antenna can be isotropic or directional as shown in 5.2.6a and 5.2.6b, respectively. The
isotropic antenna as it follows from its name radiates or receives EM waves of the same
intensity in or from any direction of 3D-space. A good example of the isotropic radiator is the
sun shining almost uniformly in all direction (horizontally and vertically). An ideal isotropic
antenna is an imaginative fiction useful only as a reference radiation source in antenna
parameter definitions such as directivity and gain (see later). As it radiates uniformly in all
directions, it is also called omnidirectional radiator and has a directivity of 1 (see definition of
directivity later).