Page 554 - Engineering Electromagnetics, 8th Edition

536 ENGINEERING ELECTROMAGNETICS

Eqs. (75) and (76) give the field dependence on φ as
E θ P (r,π/2,φ) ∝ A(π/2,φ) = cos 1 (ξ + kd cos φ) (77)
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The H-plane pattern depends on the choices of the relative current phase, ξ, and the
element spacing, d.

EXAMPLE 14.3
Investigate the H-plane pattern when the currents are in phase (ξ = 0).
Solution. With ξ = 0, Eq. (77) becomes
kd πd

A(π/2,φ) = cos cos φ = cos cos φ
2 λ
This reaches a maximum at φ = π/2 and 3π/2, or along the direction that is normal
to the plane of the antennas (the y axis). This occurs regardless of the choice of d,
and the array is thus referred to as a broadside array.Now,by choosing d = λ/2,
we obtain A = cos[(π/2) cos φ], which becomes zero at φ = 0 and π (along the x
axis), and we have single main beams along the positive and negative y axis. When
d is increased beyond λ/2, additional maxima (sidelobes) appear as φ is varied, but
zeros still occur along the x axis if d is set to odd multiples of λ/2.

The broadside array of the previous example can be regarded as the simplest case.
More interesting behavior occurs when a nonzero phase difference exists between the
two currents, and adjustments can be performed in the phase and element spacing.

EXAMPLE 14.4
Determine the necessary conditions to establish an endfire array, in which the maxi-
mum radiation is directed along the x axis.
Solution. Setting φ = 0or π in Eq. (77) and requiring the equation to achieve a
maximum results in the condition:
ξ
πd
A = cos ± =±1
2 λ
or
ξ πd
± = mπ
2 λ
where m is an integer that includes 0, and where the plus sign in the bracket applies
for φ = 0, and the minus sign for φ = π. One case of practical interest occurs when
m = 0, d = λ/4, and ξ =−π/2, which satisfies the above condition when the
positive sign is chosen. Equation (77) now becomes
π
A(π/2,φ) = cos (cos φ − 1)
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This function maximizes at φ = 0, and reaches zero at φ = π.Wehave thus created
an array that radiates a single main lobe along the positive x axis. The way this