Page 555 - Engineering Electromagnetics, 8th Edition

CHAPTER 14 ELECTROMAGNETIC RADIATION AND ANTENNAS 537

works can be understood by realizing that the phase lag in current in the element
at x = d just compensates for the phase lag that arises from the propagation delay
between the element at the origin and the one at x = d. The second element radiation
is therefore precisely in phase with the radiation from the first element. The two fields,
therefore, constructively interfere and propagate together in the forward x direction.
In the reverse direction, radiation from the antenna at x = d arrives at the origin to
find itself π radians out of phase with the radiation from the x = 0 element. The
two fields therefore destructively interfere, and no radiation occurs in the negative x
direction.

D14.6. In the broadside configuration of Example 14.3, the element spacing
is changed to d = λ. Determine (a) the ratio of the emitted intensities in the
φ = 0 and φ = 90 directions in the H plane, (b) the directions (values of φ)
◦
of the main beams in the H-plane pattern, and (c) the locations (values of φ)of
the zeros in the H-plane pattern.
Ans. 1; (0, ±90 , 180 ); (±45 , ±135 )
◦
◦
◦
◦
D14.7. In the endfire configuration of Example 14.4, determine the directions
(values of φ) for the main beams in the H plane if the wavelength is shortened
from λ = 4d to (a) λ = 3d,(b) λ = 2d, and (c) λ = d.
Ans. ±41.4 ; ±45.0 ; ±75.5 ◦
◦
◦
14.6 UNIFORM LINEAR ARRAYS

We next expand our treatment to arrays of more than two elements. By doing this,
more options are given to the designer that enable improvement of the directivity
and possibly an increase in the bandwidth of the antenna, for example, As might
be imagined, a full treatment of this subject would require an entire book. Here, we
consider only the case of the uniform linear array to exemplify the analysis methods
and to present some of the key results.
The uniform linear array configuration is shown in Figure 14.13. The array is
linear because the elements are arranged along a straight line (the x axis in this
case). The array is uniform because all elements are identical, have equal spacing, d,
and carry the same current amplitude, I 0 , and the phase progression in current from
element to element is given by a constant value, ξ. The normalized array factor for
the two-element array can be expressed using (71) as:

1
|A(θ, φ)| =|A 2 (θ, φ)|=| cos (ψ/2) |= 1 + e jψ (78)
2
where the subscript 2 is applied to A to indicate that the function applies to two
elements. The array factor for a linear array of n elements as depicted in Figure 14.13