# 9.11 Electronic Scanning of Arrays

The discussion in the preceding section was concerned with uniform-phase arrays, in which the phases of the feeding coefficients, ψ 0 to ψ N−1, are all equal. In this section, we examine the use of phase delay between adjacent elements as a tool to electronically steer the direction of the array-antenna beam from broadside at θ=90° to any desired angle θ 0. In addition to eliminating the need to mechanically steer an antenna to change its beam’s direction, electronic steering allows beam scanning at very fast rates.

As shown in Fig. 9-32, the phase of the ith element relative to that of the zeroth element is

ψ i=−i δ,(9.116)

where δ is the incremental phase delay between adjacent elements. Use of Eq. (9.116) in Eq. (9.104) leads to

F a(θ)=∑i=0 N−1 a i e−j i δ e j i k d cos θ 2=∑i=0 N−1 a i e j i(k d cos θ−δ)2=∑i=0 N−1 a i e j i γ′2=F a(γ′),(9.117)

where we introduced a new variable given by

γ′=k d cos θ−δ. (9.118)

# Figure 9-32 The application of linear phase.

Figure 9 32 Full Alternative Text

For reasons that become clear later, we define the phase shift δ in terms of an angle θ 0, which we call the scan angle, as follows:

δ=k d cos θ 0.(9.119)

Hence, γ′ becomes

The array factor given by Eq. (9.117) has the same functional form as the array factor developed earlier for the uniform-phase array [see Eq. (9.107)], except that γ is replaced with γ′. Hence,

If the amplitude distribution is symmetrical with respect to the array center, the array factor F a(γ′) is maximum when its argument γ′=0. When the phase is uniform (δ=0), this condition corresponds to the direction θ=90°, which is why the uniform-phase arrangement is called a broadside array. According to Eq. (9.120), in a linearly phased array, γ′=0 when θ=θ 0. Thus, by applying linear phase across the array, the array pattern is shifted along the cos θ axis by an amount cos θ 0, and the direction of maximum radiation is steered from the broadside direction (θ=90°) to the direction θ=θ 0. To steer the beam all the way to the end-fire direction (θ=0), the incremental phase shift δ should be equal to k d radians.

# 9.11.1 Uniform-Amplitude Excitation

To illustrate the process with an example, consider the case of the N-element array excited by a uniform-amplitude distribution. Its normalized array factor is given by Eq. (9.115). Upon replacing γ with γ′, we have

F an(γ′)=sin 2 (N γ′/2)N 2 sin 2 (γ′/2)(9.121)

with γ′ as defined by Eq. (9.120). For an array with N=10 and d=λ/2, plots of the main lobe of F an(θ) are shown in Fig. 9-33 for θ 0=0°, 45°, and 90°. We note that the half-power beamwidth increases as the array beam is steered from broadside to end-fire directions.

# Figure 9-33 Normalized array pattern of a 10-element array with λ/2 spacing between adjacent elements. All elements are excited with equal amplitude. Through the application of linear phase across the array, the main beam can be steered from the broadside direction (θ 0=90°) to any scan angle θ 0. Equiphase excitation corresponds to θ 0=90°.

Figure 9 33 Full Alternative Text

# 9.11.2 Array Feeding

According to the foregoing discussion, to steer the antenna beam to an angle θ 0, two conditions must be met: (1) the phase distribution must be linear across the array and (2) the magnitude of the incremental phase delay δ must satisfy Eq. (9.119). The combination of these two conditions provides the necessary tilting of the beam from θ=90° (broadside) to θ=θ 0. This can be accomplished by controlling the excitation of each radiating element individually through the use of electronically controlled phase shifters. Alternatively, a technique known as frequency scanning can be used to provide control of the phases of all the elements simultaneously. Figure 9-34 shows an example of a simple feeding arrangement employed in frequency-scanning arrays. A common feed point is connected to the radiating elements through transmission lines of varying lengths. Relative to the zeroth element, the path between the common feed point and a radiating element is longer by l for the first element, by 2 l for the second, and by 3 l for the third.

# Figure 9-34 An example of a feeding arrangement for frequency-scanned arrays.

Figure 9 34 Full Alternative Text

Thus, the path length for the ith element is

l i=i l+l 0,(9.122)

where l 0 is the path length of the zeroth element. Waves of frequency f propagating through a transmission line of length l i are characterized by a phase factor e−j β l i, where β=2 π f/u p is the phase constant of the line and u p is its phase velocity. Hence, the incremental phase delay of the ith element relative to the phase of the zeroth element is

ψ i(f)=−β(l i−l 0)=−2 π u p f(l i−l 0)=−2 π i u p f l.(9.123)

Suppose that at a given reference frequency f 0 we choose the incremental length l so that

l=n 0 u p f 0,(9.124)

where n 0 is a specific positive integer. In this case, the phase delay ψ 1(f 0) becomes

ψ 1(f 0)=−2 π f 0 l u p.=−2 n 0 π(9.125)

Similarly, ψ 2(f 0)=−4 n 0 π and ψ 3(f 0)=−6 n 0 π. That is, at f 0 all the elements have equal phase (within multiples of 2 π) and the array radiates in the broadside direction. If f is changed to f 0+Δ f, the new phase shift of the first element relative to the zeroth element is

ψ 1(f 0+Δ f)=−2 π u p(f 0+Δ f)l=−2 π f 0 l u p−2 π l u p Δ f=−2 n 0 π−2 n 0 π Δ f f 0=−2 n 0 π−δ,(9.126)

where use was made of Eq. (9.124) and δ is defined as

Similarly, ψ 2(f 0+Δ f)=2 ψ 1 and ψ 3(f 0+Δ f)=3 ψ 1. Ignoring the factor of 2 π and its multiples (since they exercise no influence on the relative phases of the radiated fields), we see that the incremental phase shifts are directly proportional to the fractional frequency deviation (Δ f/f 0). Thus, in an array with N elements, controlling Δ f provides a direct control of δ, which in turn controls the scan angle θ 0 according to Eq. (9.119). Equating Eq. (9.119) to Eq. (9.127) and then solving for cos θ 0 leads to

As f is changed from f 0 to f 0+Δ f, k=2 π/λ=2 π f/c also changes with frequency. However, if Δ f/f 0 is small, we may treat k as a constant equal to 2 π f 0/c; the error in cos θ 0 resulting from the use of this approximation in Eq. (9.128) is on the order of Δ f/f 0.

# Example 9-8: Electronic Steering

Design a steerable six-element array with the following specifications:

1. All elements are excited with equal amplitudes.

2. At f 0=10 GHz, the array radiates in the broadside direction, and the interelement spacing d=λ 0/2, where λ 0=c/f 0=3 cm.

3. The array pattern is to be electronically steerable in the elevation plane over the angular range extending between θ 0=30° and θ 0=150°.

4. The antenna array is fed by a voltage-controlled oscillator whose frequency can be varied over the range from 9.5 to 10.5 GHz.

5. The array uses a feeding arrangement of the type shown in Fig. 9-34, and the transmission lines have a phase velocity u p=0.8 c.

Solution: The array is to be steerable from θ 0=30° to θ 0=150° (Fig. 9-35). For θ 0=30° and

k d=2 π λ 0 λ 0 2=π,

# Figure 9-35 Steerable six-element array (Example 9-8).

Figure 9 35 Full Alternative Text

Eq. (9.128) gives

0.87=2 n 0 Δ f f 0.(9.129)

We are given that f 0=10 GHz and the oscillator frequency can be varied between (f 0−0.5 GHz) and (f 0+0.5 GHz). Thus, Δ f max=0.5 GHz. To satisfy Eq. (9.129), we need to choose n 0 so that Δ f is as close as possible to, but not larger than, Δ f max. Solving Eq. (9.129) for n 0 with Δ f=Δ f max gives

n 0=0.87 2 f 0 Δ f max=8.7.

Since n 0 is not an integer, we need to modify its value by rounding it upward to the next whole-integer value. Hence, we set n 0=9.

Application of Eq. (9.124) specifies the magnitude of the incremental length l:

l=n 0 u p f 0=9×0.8×3×10 8 10 10=21.6 cm.

In summary, with N=6 and k d=π, Eq. (9.121) becomes

F an(γ′)=sin 2 (3 γ′)36 sin 2 (γ′/2)

with

γ′=k d(cos θ−cos θ 0)=π(cos θ−cos θ 0)

and

cos θ 0=2 n 0 π k d Δ f f 0=18 f−10 GHz 10 GHz.(9.130)

The shape of the array pattern is similar to that shown in Fig. 9-30, and its main-beam direction is along θ=θ 0. For f=f 0=10 GHz, θ 0=90° (broadside direction); for f=10.48 GHz, θ 0=30°; and for f=9.52 GHz, θ 0=150°. For any other value of θ 0 between 30° and 150°, Eq. (9.130) provides the means for calculating the required value of the oscillator frequency f.

Concept Question 9-11: Why are antenna arrays useful? Give examples of typical applications.

Concept Question 9-12: Explain how the pattern multiplication principle is used to compute the radiation pattern of an antenna array.

Concept Question 9-13: For a linear array, what roles do the array amplitudes and phases play?

Concept Question 9-14: Explain how electronic beam steering is accomplished.

Concept Question 9-15: Why is frequency scanning an attractive technique for steering the beam of an antenna array?