Phased Arrays - MATLAB Simulations for Radar Systems Design - page 321

z

#N

( N – 1 ) d

#2

d

y

ψ

#1

d sin ψ

x

Figure 8.4. Linear array of equally spaced elements.

Substituting Eq. (8.30) into Eq. (8.31) and collecting terms yield

) 2 ) 2

( 1 – cos Nkd sin ψ + ( sin Nkd sin ψ

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

(8.32)

) 2 ) 2 ( 1 – cos kd sin ψ + ( sin kd sin ψ

1 –

1

cos Nkd sin ψ

= ----------------------------------------

– cos kd sin ψ

) 2

and using the trigonometric identity 1 – cos θ = 2 ( sin θ 2 ⁄ yields

sin ( Nkd sin ψ ⁄ 2 )

E ( sin ψ ) =

------------------------------------------

(8.33)

sin ( kd sin ψ ⁄ 2 )

which is a periodic function of kd sin ψ , with a period equal to 2π .

The maximum value of E ( sin ψ ) , which occurs at ψ = 0 , is equal to N . It

follows that the normalized intensity pattern is equal to

1
----

sin ( ( Nkd sin ψ ) ⁄ 2 )

E n ( sin ψ ) =

-----------------------------------------------

(8.34)

sin ( ( kd sin ψ ) ⁄ 2 )

The normalized two-way array pattern (radiation pattern) is given by