Phased Arrays - MATLAB Simulations for Radar Systems Design - page 320

In general, an array can be fully characterized by its array factor. This is true
since knowing the array factor provides the designer with knowledge of the
arrayÓs (1) 3-dB beamwidth; (2) null-to-null beamwidth; (3) distance from the
main peak to the first sidelobe; (4) height of the first sidelobe as compared to
the main beam; (5) location of the nulls; (6) rate of decrease of the sidelobes;
and (7) grating lobesÓ locations.

8.4. Linear Arrays

Fig. 8.4 shows a linear array antenna consisting of identical elements. The
element spacing is (normally measured in wavelength units). Let element #1
serve as a phase reference for the array. From the geometry, it is clear that an
outgoing wave at the

N

d

nth element leads the phase at the ( n + 1 ) th element by

kd sin ψ , where k = 2πλ ⁄ . The combined phase at the far field observation

point P is independent of φ and is computed from Eq. (8.24) as

Ψψφ ( , ) = kr i ( • r 0 ) = ( n – 1 ) kd sin ψ (8.26)

Thus, from Eq. (8.25), the electric field at a far field observation point with
direction-sine equal to sin ψ (assuming isotropic elements) is

N
∑

e jn 1

( – ) kd ( sin ψ )

E ( sin ψ ) = (8.27)

n = 1

Expanding the summation in Eq. (8.27) yields

e jkd

sin ψ

… e jN 1

( – ) kd ( sin ψ )

E ( sin ψ ) = 1 + + +

(8.28)

The right-hand side of Eq. (8.29) is a geometric series, which can be expressed
in the form

a N

1 –

aa 2 a 3

( N – 1 )

1 +++++ … a = ---------------

(8.29)

1 – a

ae jkd

sin ψ

Replacing by yields

e jNkd

sin ψ

1 – 1 –

1

( cos Nkd sin ψ ) – j ( sin kd ψ sin )

E ( sin ψ ) = ----------------------------- = -----------------------------------------------------------------------------------------

(8.30)

e jkd

sin ψ

– ( cos kd sin ψ ) – j ( sin kd ψ sin )

1 –

The far field array intensity pattern is then given by

) E ∗ E ( sin ψ ) = E ( sin ψ ( sin ψ )

(8.31)