# 9.9 Antenna Arrays

AM broadcast services operate in the 535 to 1605 kHz band. The antennas they use are vertical dipoles mounted along tall towers. The antennas range in height from λ/6 to 5 λ/8, depending on the operating characteristics desired and other considerations. Their physical heights vary from 46 m (150 ft) to 274 m (900 ft); the wavelength at 1 MHz, approximately in the middle of the AM band, is 300 m. Because the field radiated by a single dipole is uniform in the horizontal plane (as discussed in Sections 9-1 and 9-3), it is not possible to direct the horizontal pattern along specific directions of interest, unless two or more antenna towers are used simultaneously. Directions of interest may include cities serviced by the AM station, and directions to avoid may include areas serviced by another station operating at the same frequency (thereby avoiding interference effects). When two or more antennas are used together, the combination is called an antenna array.

The AM broadcast antenna array is only one example of the many antenna arrays used in communication systems and radar applications. Antenna arrays provide the antenna designer the flexibility to obtain high directivity, narrow beams, low side lobes, steerable beams, and shaped antenna patterns starting from very simple antenna elements. Figure 9-25 shows a very large radar system consisting of a transmitter array composed of 5,184 individual dipole antenna elements and a receiver array composed of 4,660 elements. The radar system, part of the Space Surveillance Network operated by the U.S. Air Force, operates at 442 MHz and transmits a combined peak power of 30 MW!

# Figure 9-25 The AN/FPS-85 Phased Array Radar Facility in the Florida panhandle is near the city of Freeport. A several-mile no-fly zone surrounds the radar installation as a safety concern for electroexplosive devices, such as ejection seats and munitions that are carried on military aircraft.

NASA

Although an array need not consist of similar radiating elements, most arrays actually use identical elements, such as dipoles, slots, horn antennas, or parabolic dishes. The antenna elements comprising an array may be arranged in various configurations, but the most common are the linear one-dimensional configuration—wherein the elements are arranged along a straight line—and the two-dimensional lattice configuration in which the elements sit on a planar grid. The desired shape of the far-field radiation pattern of the array can be synthesized by controlling the relative amplitudes of the array elements’ excitations.

This flexibility of the array antenna has led to numerous applications, including electronic steering and multiple-beam generation.

The purpose of this and the next two sections is to introduce the reader to the basic principles of array theory and design techniques used in shaping the antenna pattern and steering the main lobe. This presentation is confined to the one-dimensional linear array with equal spacing between adjacent elements.

A linear array of N identical radiators is arranged along the z axis, as shown in Fig. 9-26. The radiators are fed by a common oscillator through a branching network. In each branch, an attenuator (or amplifier) and phase shifter are inserted in series to control the amplitude and phase of the signal feeding the antenna element in that branch.

In the far-field region of any radiating element, the element electric-field intensity E˜e(R,θ,ϕ) may be expressed as a product of two functions: the spherical propagation factor e−j k R/R, which accounts for the dependence on the range R, and f˜e(θ,ϕ), which accounts for the directional dependence of the element’s electric field. Thus, for an isolated element, the radiated field is

E˜e(R,θ,ϕ)=e−j k R R f˜e(θ,ϕ),(9.93)

and the corresponding power density S e is

S e(R,θ,ϕ)=1 2 η 0|E˜e(R,θ,ϕ)|2=1 2 η 0 R 2|f˜e(θ,ϕ)|2.(9.94)

Hence, for the array shown in Fig. 9-26(b), the far-zone field due to element i at range R i from observation point Q is

E˜i(R i,θ,ϕ)=A i e−j k R i R i f˜e(θ,ϕ),(9.95)

where A i=a i e j ψ i is a complex feeding coefficient representing the amplitude a i and phase ψ i of the excitation giving rise to E˜i, which is relative to a reference excitation. In practice, the excitation of one of the elements is used as reference. Note that R i and A i may be different for different elements in the array, but f˜e(θ,ϕ) is the same for all of them because they are all identical; hence, they exhibit identical directional patterns.

# Figure 9-26 Linear-array configuration and geometry.

(a) Array elements with individual amplitude and phase control

Figure 9 26-a Full Alternative Text

(b) Array geometry relative to observation point

Figure 9 26-b Full Alternative Text

The total field at the observation point Q(R 0,θ,ϕ) is the sum of the fields due to the N elements:

E˜(R 0,θ,ϕ)=∑i=0 N−1 E˜i(R i,θ,ϕ)=∑i=0 N−1 A i e−j k R i R i f˜e(θ,ϕ),(9.96)

where R 0 denotes the range of Q from the center of the coordinate system, which is chosen to be at the location of the zeroth element. To satisfy the far-field condition given by Eq. (9.73) for an array of length l=(N−1)d, where d is the interelement spacing, the range R 0 should be sufficiently large to satisfy

R 0≥2 l 2 λ=2(N−1)2 d 2 λ.(9.97)

This condition allows us to ignore differences in the distances from Q to the individual elements as far as the magnitudes of the radiated fields are concerned. Thus, we can set R i=R 0 in the denominator in Eq. (9.96) for all i. With regard to the phase part of the propagation factor, we can use the parallel-ray approximation given by

R i≈R 0−z i cos θ=R 0−i d cos θ,(9.98)

where z i=i d is the distance between the i th element and the zeroth element (Fig. 9-27). Employing these two approximations in Eq. (9.96) leads to

E˜(R 0,θ,ϕ)=f˜e(θ,ϕ)e−j k R 0 R 0 ∑i=0 N−1 A i e j i k d cos θ,(9.99)

and the corresponding array-antenna power density is given by

S(R 0,θ,ϕ)=1 2 η 0|E˜(R 0,θ,ϕ)|2=1 2 η 0 R 0 2|f˜e(θ,ϕ)|2∑i=0 N−1 A i e j i k d cos θ 2=S e(R 0,θ,ϕ)∑i=0 N−1 A i e j i k d cos θ 2,(9.100)

where use was made of Eq. (9.94). This expression is a product of two factors. The first factor, S e(R 0,θ,ϕ), is the power density of the energy radiated by an individual element, and the second, called the array factor, is a function of the positions of the individual elements and their feeding coefficients but not a function of the specific type of radiators used.

# Figure 9-27 The rays between the elements and a faraway observation point are approximately parallel lines. Hence, the distance R i≈R 0−i d cos θ.

Figure 9 27 Full Alternative Text

Denoting the array factor by

F a(θ)=∑i=0 N−1 A i e j i k d cos θ 2,(9.101)

the power density of the antenna array is then written as

This equation demonstrates the pattern multiplication principle. It allows us to find the far-field power density of the antenna array by first computing the far-field power pattern with the array elements replaced with isotropic radiators, which yields the array factor F a(θ), and then multiplying the result by S e(R 0,θ,ϕ), which is the power density for a single element (and is the same for all elements).

The feeding coefficient A i is, in general, a complex amplitude consisting of an amplitude factor a i and a phase factor ψ i:

A i=a i e j ψ i.(9.103)

Insertion of Eq. (9.103) into Eq. (9.101) leads to

The array factor is governed by two input functions: the array amplitude distribution given by the a i’s and the array phase distribution given by the ψ i’s.

# Example 9-5: Array of Two Vertical Dipoles

An AM radio station uses two vertically oriented half-wave dipoles separated by a distance of λ/2, as shown in Fig. 9-28(a). The vector from the location of the first dipole to the location of the second dipole points toward the east. The two dipoles are fed with equal-amplitude excitations, and the dipole farther east is excited with a phase shift of −π/2 relative to the other one. Find and plot the antenna pattern of the antenna array in the horizontal plane.

Solution: The array factor given by Eq. (9.104) was derived for radiators arranged along the z axis. To keep the coordinate system the same, we choose the easterly direction to be the z axis, as shown in Fig. 9-28(b), and we place the first dipole at z=−λ/4 and the second at z=λ/4. A dipole radiates uniformly in the plane perpendicular to its axis, which in this case is the horizontal plane. Hence, S e=S 0 for all angles θ in Fig. 9-28(b), where S 0 is the maximum value of the power density radiated by each dipole individually. Consequently, the power density radiated by the two-dipole array is

S(R,θ)=S 0 F a(θ).

# Figure 9-28 Two half-wave dipole array of Example 9-5.

(a) Dipole array

Figure 9 28-a Full Alternative Text

(b) Observation plane

Figure 9 28-b Full Alternative Text

(c) Horizontal-plane pattern

For two elements separated by d=λ/2 and excited with equal amplitudes (a 0=a 1=1) and with phase angles ψ 0=0 and ψ 1=−π/2, Eq. (9.104) becomes

F a(θ)=∑i=0 1 a i e j ψ i e j i k d cos θ 2=1+e−j π/2 e j(2 π/λ)(λ/2) cos θ 2=1+e j(π cos θ−π/2)2.

A function of the form |1+e j x|2 can be evaluated by factoring out e j x/2 from both terms:

|1+e j x|2=|e j x/2(e−j x/2+e j x/2)|2=|e j x/2|2|e−j x/2+e j x/2|2=|e j x/2|2 2[e−j x/2+e j x/2]2 2.

The absolute value of e j x/2 is 1, and we recognize the function inside the square bracket as cos(x/2). Hence,

|1+e j x|2=4 cos 2 x 2.

Applying this result to the expression for F a(θ), we have

F a(θ)=4 cos 2 π 2 cos θ−π 4.

The power density radiated by the array is then

S(R,θ)=S 0 F a(θ)=4 S 0 cos 2 π 2 cos θ−π 4.

This function has a maximum value S max=4 S 0, and it occurs when the argument of the cosine function is equal to zero. Thus,

π 2 cos θ−π 4=0,

which leads to the solution θ=60°. Upon normalizing S(R,θ) by its maximum value, we obtain the normalized radiation intensity given by

F(θ)=S(R,θ)S max=cos 2 π 2 cos θ−π 4.

The pattern of F(θ) is shown in Fig. 9-28(c).

# Example 9-6: Pattern Synthesis

In Example 9-5, we were given the array parameters a 0, a 1, ψ 0, ψ 1, and d, and we were then asked to determine the pattern of the two-element dipole array. We now consider the reverse process; given specifications on the desired pattern, we specify the array parameters to meet those specifications.

Given two vertical dipoles, as depicted in Fig. 9-28(b), specify the array parameters so that the array exhibits maximum radiation toward the east and no radiation toward the north or south.

Solution: From Example 9-5, we established that, because each dipole radiates equally along all directions in the y–z plane, the radiation pattern of the two-dipole array in that plane is governed solely by the array factor F a(θ). The shape of the pattern of the array factor depends on three parameters: the amplitude ratio a 1/a 0, the phase difference ψ 1−ψ 0, and the spacing d (Fig. 9-29(a)). For convenience, we choose a 0=1 and ψ 0=0. Accordingly, Eq. (9.101) becomes

F a(θ)=∑i=0 1 a i e j ψ i e j i k d cos θ 2=|1+a 1 e j ψ 1 e j(2 π d/λ) cos θ|2.

Next, we consider the specification that F a be equal to zero when θ=90° (north and south directions in Fig. 9-29(a)). For any observation point on the y axis, the ranges R 0 and R 1 shown in Fig. 9-29(a) are equal, which means that the propagation phases associated with the time travel of the waves radiated by the two dipoles to that point are identical. Hence, to satisfy the stated condition, we need to choose a 1=a 0 and ψ 1=±π. With these choices, the signals radiated by the two dipoles have equal amplitudes and opposite phases—thereby interfering destructively. This conclusion can be ascertained by evaluating the array factor at θ=90° with a 0=a 1=1 and ψ 1=±π:

F a(θ=90°)=|1+1 e±j π|2=|1−1|=0.

# Figure 9-29 (a) Two vertical dipoles separated by a distance d along the z axis; (b) normalized array pattern in the y–z plane for a 0=a 1=1, ψ 1=ψ 0=−π, and d=λ/2.

(a) Array arrangement

Figure 9 29-a Full Alternative Text

(b) Array pattern

Figure 9 29-b Full Alternative Text

The two values of ψ 1, namely π and −π, lead to the same solution for the value of the spacing d to meet the specification that the array radiation pattern is maximum toward the east, corresponding to θ=0°. Let us choose ψ 1=−π and examine the array factor at θ=0°:

F a(θ=0)=|1+e−j π e j 2 π d/λ|2=|1+e j(−π+2 π d/λ)|2.

For F a(θ=0) to be a maximum, we require the phase angle of the second term to be zero or a multiple of 2 π. That is,

−π+2 π d λ=2 n π,

or

d=(2 n+1)λ 2,n=0,1,2,…

In summary, the two-dipole array will meet the given specifications if a 0=a 1, ψ 1−ψ 0=−π, and d=(2 n+1)λ/2.

For d=λ/2, the array factor is

F a(θ)=|1+e−j π e j π cos θ|2=|1−e j π cos θ|2=2 j e−j(π/2) cos θ e j(π/2) cos θ−e−j(π/2) cos θ 2 j 2=4 sin 2 π 2 cos θ.

The array factor has a maximum value of 4, which is the maximum level attainable from a two-element array with unit amplitudes. The directions along which F a(θ) is a maximum are those corresponding to θ=0 (east) and θ=180° (west), as shown in Fig. 9-29(b).