oots for an N-element array, there can be up to N - 1 independent nulls. Beam Steering Consider a general array excitation an (n = 0,1,... N - 1). The corresponding radiation pattern is as in equation 9.29, where Z = exp(jkd cc 0). Fa(2) (9.29) Now apply a linear phase gradient to the array; that is, let the array excitations become av, a1 exp(jp), a2 exp(j20), and SO on. Let bn be given by equation 9.30; then the new radiation pattern is given by equation 9.31, where z' is given by equation 9.32. bn=anexp(jno) (9.30) (9.31) Chapter 9 Antennas z'=exp(j(kdcos0++p)) (9.32) Then, if equations 9.33 and 9.34 are true, the radiation pattern keeps the same form but with an angular shift related to the incremental phase, Q. (9.33) (9.34) From these equations, equation 9.35 can be obtained. The right-hand side is proportional to the phase gradient (dq/dx) along the array. (9.35) If the main beam peak is initially at 0=90° broadside to the array, then COS Oa = 0, and COS Ob = sin a, where a is the beam deflection angle. Equation 9.36 is thus obtained. (9.36) If d = 2/2 and a are small, SO that sin a = a, equations 9.37 and 9.38 are obtained. a(radians)=-(P(radians) = (9.37) a°===" (9.38) Thus for an incremental phase value of 30°, the deflection angle is 10°. Using equation 9.36, an incremental phase of 90° for 2/2 spaced elements gives a deflection angle of 30°. The use of electronically controlled phase shifters on each element of a "phased" array allows the beam-pointing direction of an array antenna to be varied rapidly and without any mechanical antenna movement. This is in contrast to a fixed-beam antenna, such as a simple horn-feed parabolic reflector, where the antenna is rotated mechanically to "scan" the beam. However, the latter is in general much cheaper! The question of grating lobes arose in the Nulls section, where d < 2 was required to avoid them for a beam with a broadside peak. The radiation pattern of a uniformly excited N-element array is given by equation 9.39. sin(1/2Nkdcos0a) (9.39) Now consider this