Page 553 - Engineering Electromagnetics, 8th Edition

CHAPTER 14 ELECTROMAGNETIC RADIATION AND ANTENNAS 535

which simplifies to

E 0 F(θ)
E θ P (r,θ,φ) = e − jkr 1 + e jψ (71)
r
where

ψ = ξ + kd sin θ cos φ (72)
ψ is the net phase difference between the two antenna fields that is observed at
P(r,θ,φ). Equation (71) can be further simplified by factoring out the term e jψ/2 to
obtain

2E 0 F(θ)
e
E θ P (r,θ,φ) = e − jkr jψ/2 cos(ψ/2) (73)
r
from which we may determine the field amplitude through

2E 0
∗
|E θ P (r,θ,φ)|= E θ P E θ P = |F(θ)|| cos(ψ/2)| (74)
r
Equation (74) demonstrates the important principle of pattern multiplication that
applies to arrays of identical antennas. Specifically, the total field magnitude consists
of the product of the pattern function magnitude, or element factor for the individual
antennas, |F(θ)|, and the normalized array factor magnitude, given by | cos(ψ/2)|.
The array factor is often denoted by

A(θ, φ) = cos(ψ/2) = cos 1 (ξ + kd sin θ cos φ) (75)
2
Equation (74) then becomes

2E 0
|E θ P (r,θ,φ)|= |F(θ) ||A(θ, φ)| (76)
r
This principle can be extended to arrays of multiple elements by appropriately modi-
fying the array factor, as we will find. The underlying assumption is that the individual
array elements are essentially uncoupled; that is, they induce negligible currents in
each other. With appreciable coupling, the problem is far more complicated, and
pattern multiplication cannot be used.
In the field pattern expressed in (76), the E plane (or θ dependence) is primarily
determined by the individual elements, or by |F(θ)|.Itisinthe H plane where the
effect of the array is the strongest. In fact, the main reason for using an array of this
configuration is to enable control of the H-plane pattern. In the H plane (θ = π/2),