# 9.10 N-Element Array with Uniform Phase Distribution

We now consider an array of N elements with equal spacing d and equal-phase excitations; that is, ψ i=ψ 0 for i=1,2,…,(N−1). Such an array of in-phase elements is sometimes referred to as a broadside array because the main beam of the radiation pattern of its array factor is always in the direction broadside to the array axis. From Eq. (9.104), the array factor is given by

F a(θ)=e j ψ 0∑i=0 N−1 a i e j i k d cos θ 2=|e j ψ 0|2∑i=0 N−1 a i e j i k d cos θ 2=∑i=0 N−1 a i e j i k d cos θ 2.(9.105)

The phase difference between the fields radiated by adjacent elements is

γ=k d cos θ=2 π d λ cos θ.(9.106)

In terms of γ, Eq. (9.105) takes the compact form

For a uniform amplitude distribution with a i=1 for i=0,1,…,(N−1), Eq. (9.107) becomes

F a(γ)=|1+e j γ+e j 2 γ+⋯+e j(N−1)γ|2.(9.108)

This geometric series can be rewritten in a more compact form by applying the following recipe. First, we define

F a(γ)=|f a(γ)|2,(9.109)

with

f a(γ)=[1+e j γ+e j 2 γ+⋯+e j(N−1)γ].(9.110)

Next, we multiply f a(γ) by e j γ to obtain

f a(γ)e j γ=(e j γ+e j 2 γ+⋯+e j N γ).(9.111)

Subtracting Eq. (9.111) from Eq. (9.110) gives

f a(γ)(1−e j γ)=1−e j N γ,(9.112)

which in turn gives

f a(γ)=1−e j N γ 1−e j γ=e j N γ/2 e j γ/2(e−j N γ/2−e j N γ/2)(e−j γ/2−e j γ/2)=e j(N−1)γ/2 sin(N γ/2)sin(γ/2).(9.113)

After multiplying f a(γ) by its complex conjugate, we obtain the result:

From Eq. (9.108), F a(γ) is maximum when all terms are 1, which occurs when γ=0 (or equivalently, θ=π/2). Moreover, F a(0)=N 2. Hence, the normalized array factor is given by

F an(γ)=F a(γ)F a,max=sin 2 (N γ/2)N 2 sin 2 (γ/2)=sin 2 N π d λ cos θ N 2 sin 2 π d λ cos θ. (9.115)

A polar plot of F an(θ) is shown in Fig. 9-30 for N=6 and d=λ/2. The reader is reminded that this is a plot of the radiation pattern of the array factor alone; the pattern for the antenna array is equal to the product of this pattern and that of a single element, as discussed earlier in connection with the pattern multiplication principle.

# Figure 9-30 Normalized array pattern of a uniformly excited six-element array with interelement spacing d=λ/2.

Figure 9 30 Full Alternative Text

# Example 9-7: Multiple-Beam Array

Obtain an expression for the array factor of a two-element array with equal excitation and a separation d=7 λ/2, and then plot the array pattern.

Solution: The array factor of a two-element array (N=2) with equal excitation (a 0=a 1=1) is given by

F a(γ)=∑i=0 1 a i e j i γ 2=|1+e j γ|2,=|e j γ/2(e−j γ/2+e j γ/2)|2=|e j γ/2|2|e−j γ/2+e j γ/2|2=4 cos 2 (γ/2),

where γ=(2 π d/λ) cos θ. The normalized array pattern shown in Fig. 9-31 consists of seven beams; all have the same peak value but not the same angular width. The number of beams in the angular range between θ=0 and θ=π is equal to the separation between the array elements, d, measured in units of λ/2.

# Figure 9-31 Normalized array pattern of a two-element array with spacing d=7 λ/2.

Figure 9 31 Full Alternative Text