nts is given by equation 9.17, which results in equation 9.18. <)=1+z+z2+...+z"- (9.17) 1-exp(jkd/Ncos6) -exp(jkdcos0) |F(z)|==F(z)=F*(z) 2-2cos(Nkdcos0) -2cos(kdcos0) (9.18) sin2(1/2Nkdcos0) sin2(1/2kdcos0) sin(1/2Nkdcos0) (9.19) The result of equation 9.19 can be compared with the radiation pattern of a uniform aperture of length a, as in equation 9.20. sin(1/kkdsina) (9.20) The numerator in equation 9.20 is exactly the same if a = Nd, noting that sin a = cos O. The denominators differ but are similar if sin (1/2kd cos 0) = 1/2kd cos 0-that is, if 1/2kd cos O << /2 and if dcos O << A/2. (They also differ by a factor of N, but that is simply a constant of proportionality.) If d = A/2, this condition requires COS O<<1, SO that the patterns are similar near broad- side to the aperture or array. The zero-to-zero beamwidth is the same for the two cases. In general, the pattern is similar to that of a diffraction grating in optics, and the denominator can cause "grating lobes." The condition for these is given by equations 9.21 and 9.22. 1/2kdcos0 (9.21) (9.22) Chapter 9 Antennas At such angles, the denominator in equation 9.19 becomes 0 simultaneously with the numerator, and a repeated main lobe occurs in the array factor radiation pattern. The first 0 of the pattern occurs where x = /N, that is COS O (= sin a) = + al (Nd), while the first grating lobes occur where x = t-that is, cos O = + A/d. The main beamwidth is thus governed by the complete array length, while the grating lobes are governed by the spacing of the array elements. The relation of equation 9.22 implies that, if grating lobes are to be avoided, d < a is required. When beam steering is considered (see the Beam Steering section), further restrictions on d occur. Now consider nonisotropic elements. The radiation pattern becomes as shown in equation 9.23. (9.23) Here fn(0, Q) is the radiation pattern of the nth array element. If the array elements have identical patterns and similar alignment, the element pattern can be factored out SO that equation 9.2