Page 558 - Engineering Electromagnetics, 8th Edition

540 ENGINEERING ELECTROMAGNETICS

An endfire array requires a principal maximum to occur along the x axis. In the
H plane, we may therefore write

ψ = 0 = ξ + kd cos(0) = ξ + kd
or ξ =−kd to obtain endfire operation with a maximum occurring along the positive
x axis. This may or may not result in a main beam occurring along the negative x
axis as well.

EXAMPLE 14.5
For arrays of 4 and 8 elements, select the current phase and element spacing that will
give unidirectional endfire operation, in which the main beam exists in the φ = 0
direction, whereas no radiation occurs in the direction of φ = π, nor in the broadside
directions (φ =±π/2).
Solution. We want ψ = 0 when φ = 0. Therefore, from ψ = ξ + kd cos φ,we
would require that 0 = ξ +kd,or that ξ =−kd. Using 4 or 8 elements, we find either
from Eq. (81) or from Figure 14.14 that zeros will occur when ψ =±π/2 and ±π.
Therefore, if we choose ξ =−π/2 and d = λ/4, we obtain ψ =−π/2at φ = π/2,
3π/2, and ψ =−π at φ = π.We thus have ψ =−(π/2)(1 − cos φ). Polar plots of
the resulting array functions are shown in Figure 14.15a and b.Again, the move from
4to8 elements has the effect of decreasing the main beamwidth, while increasing the
sidelobe count from 1 to 3, in this case. If an odd number of elements is used with the
above choices in phasing and spacing, a small sidelobe will be present in the φ = π
direction.

In general, we may choose current phasing and element spacing to establish the
main beam in any direction. Choosing the ψ = 0 principal maximum, we may write
ξ
ψ = 0 = ξ + kd cos φ max ⇒ cos φ max =−
kd
so that the main beam direction can be changed by varying the current phasing.
y y

x x

(a) (b)

Figure 14.15 H-plane plots of (a) 4-element and (b)
8-element arrays having element spacing of d = λ/4, and
current phasing ξ =−π/2.