Chapter 32: Antenna Arrays
Chapter Learning Objectives: After completing this chapter the student will be able to:

Determine the radiation pattern for an antenna array consisting of two elements.

Determine the radiation pattern for a phased array of arbitrary size.

Determine the radiation pattern for a binomial array of arbitrary size.

Determine the directivity of an antenna array.
You can watch the video associated
with this chapter at the following link:
Historical Perspective: Jill Tarter (1944-present) is an
American astronomer who uses arrays of radio antennas to
search for extraterrestrial life. She is the former director of the
Center for SETI Research and was the inspiration for the
main character in the movie Contact.
Photo credit: https://upload.wikimedia.org/wikipedia/commons/3/3b/Jill_Tarter_at_Starmus_IV_Trondheim_2017.jpg, [CC by
SA 2.0], via Wikimedia Commons.
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32.1
Introduction
We now understand enough about antennas to know that we will want to carefully design our
antenna to have a desirable radiation pattern for our particular application. This can be measured
by the half-power beam width, directivity, gain, and effective area. The problem is that we don’t
actually have very many things to change on a typical antenna. For a finite dipole, it is just
one—the length. But we need more control than that to custom design the radiation pattern for
more advanced applications.
An antenna array is a group of similar antennas that are spaced in a regular manner, typically in a
line or a grid. It turns out that constructive and destructive interference of the radiation from the
individual antennas can create quite desirable radiation patterns for the overall array. In addition,
we now have several things we can adjust:




The number of antennas
The placement of the antennas
The amplitude of the signal applied to each antenna
The phase of the signal applied to each antenna
We will consider two main types of antenna arrays: phased arrays and binomial arrays (which
might be better called amplitude arrays). But first we will study the simplest case: an array of
two identical antennas.
32.2
Two Antennas
The radiation coming from an antenna has both an amplitude and a phase angle. When two such
signals interact with each other, they can constructively interfere if the signals are “in phase,” or
they can destructively interfere if the signals are “out of phase.” Figure 32.1 illustrates a
situation with two antennas whose signals are interacting at an observation point P.
P
r1
r2
r
q
d
Figure 32.1. The Two-Element Array
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It is clear from this figure that the path length r1 will be greater than the path length r2 for the
configuration shown. In general, the path lengths will be different unless q=90°. Our task is to
figure out the difference in the path lengths and determine whether the waves will interfere
constructively or destructively. If we assume that the observation point P is in the far field (very
far away from the antenna array), then the dashed lines in Figure 32.1 will be approximately
parallel. The two paths r1 and r2 will come to resemble Figure 32.2.
r1
r2
q
q
d
Figure 32.2. The Two-Element Array with Observation Point in the Far Field
Studying the bottom left corner of this figure, we can see that the right triangle that is formed has
a hypotenuse of d, meaning that the side adjacent to q is dcosq as shown. Geometrically, this is
the difference in path length between r1 and r2. If we then multiply this path length difference by
the wave number, we will find the phase difference that is created by the path length difference:
(Equation 32.1)
Of course, it is also possible that we could intentionally introduce a phase difference between the
signal applied to the two antennas. We will call such a phase difference . Combining this
intentional phase difference with Equation 32.2 gives the total phase difference between the
signals from the two antennas when they reach the observation point in the far field:
(Equation 32.2)
We will arbitrarily say that the signal from one antenna has a phase angle of +/2 while the other
antenna has a phase angle of –/2. (This decision has no impact on the final answer other than
an overall phase factor, but it makes the math much easier.) We can then write the sum of the
electric fields from these antennas as:
(Equation 32.3)
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If we assume that the two antennas are identical, this equation can be rewritten as:
(Equation 32.4)
Factoring a 2 out of the parentheses gives:
(Equation 32.5)
The term in parentheses is now in the form of a cosine, so we can write:
(Equation 32.6)
Substituting Equation 32.2 into this expression gives:
(Equation 32.7)
Notice that what has essentially happened is that we have multiplied the electric field of a single
antenna (E1(r)) by a new function to obtain the electric field of a two-element array. This is a
particular example of the pattern multiplication rule, which says that the radiation pattern of an
antenna array will be the product of the radiation pattern of a single antenna, the “element factor”
F1(q,), multiplied by an “array factor” Fa(q,) that corresponds to the pattern that constitutes the
array:
(Equation 32.8)
From inspection of Equation 32.7, it is apparent that the array factor for a two-element array
spaced a distance d apart with an intentional phase difference of  is:
(Equation 32.8)
Since we typically normalize the radiation pattern to remove any constants multiplying it, we
will omit the “2” from this function, giving the final array factor:
(Equation 32.9)
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Example 32.1: What is the radiation pattern for a pair of isotropic antennas with no applied
phase difference?
The radiation pattern for a two-element array of isotropic antennas, derived in Example 32.1,
gives very interesting behavior as the distance d between the antennas increases. The Figure
32.2 shows the radiation pattern for d=10cm with a 1.5GHz wave. Clicking on this figure will
also show an animation of this pattern from d=0 to d=50cm.
Figure 32.3. Radiation Pattern of the Two-Element Array with f=1.5GHz and d=10cm.
Already, we can see that antenna arrays will give us interesting opportunities, even if they only
contain two antennas. Let’s look at what happens if we include more than two.
32.3
Phased Arrays
A “phased array” is simply a linear array of antennas, all equally spaced at a distance d, with the
same amplitude applied to each antenna. Sometimes, the phase is also the same, but sometimes
we will introduce an intentional phase difference to each antenna. Essentially, we are
generalizing the two-element array to have more than two elements.
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P
r6
q6
q5
r5
q4
r4
q3
r3
q2
r2
r1
q1
d
d
d
d
d
Figure 32.4. A Phased Array with Six Elements (N=6)
Again, if the observation point P is in the far field, then all of the angles q1-q6 become
approximately equal as the lines become approximately parallel:
q
d
q
d
q
d
q
d
q
q
d
Figure 32.5. A Phased Array with Six Elements Observed in the Far Field
Notice, then, that we can apply our “dcosq” trick from the two-element array to each of the six
antennas. This means that each antenna will have a path length that is dcosq longer than the
element to its right. There is also the possibility once again that we will intentionally introduce a
phase shift  to each antenna signal, meaning that the phase difference at the observation point of
each antenna compared to the one to its right is the same as it was for the two-element array:
(Equation 32.10)
Remember that each element of the array will have an additional phase shift, so they add up as
you move from antenna #1 (on the far right) to antenna #6 (on the far left). If each antenna has
an element factor of E1(q,), we can write the total electric field at the observation point as:
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(Equation 32.11)
If we generalize to the case of an array with N elements, this becomes:
(Equation 32.12)
Mathematicians have proven that:
(Equation 32.13)
Applying this to Equation 32.12 where q=ej, we can find a closed form of this series:
(Equation 32.14)
If we consider only the magnitude of the electric field, this becomes:
(Equation 32.15)
By inspection, we can see that the array factor for a phased array is:
(Equation 32.16)
Bringing in Equation 32.10, this can in turn be written as:
(Equation 32.17)
We can also determine (using l’Hopital’s rule) that the maximum value of this function will
occur when =0. This also makes intuitive sense, because if =0, all of the signals will arrive in
phase, and they will add constructively. To achieve this goal, we must introduce a negative
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phase shift to each antenna that cancels out the positive phase shift that comes from the path
length difference:
(Equation 32.18)
(Equation 32.19)
In this case, all the signals will arrive at the observation point perfectly in phase, meaning they
simply add magnitudes:
(Equation 32.20)
Example 32.2: Determine the radiation pattern for a phased array with two halfwavelength electric dipoles separated by a distance d. Assume the antennas are transmitting the
signals with no phase difference.
The result of this calculation can be seen in Figure 32.6 for d=. Clicking on this figure also
provides an animation for d=0 to 3.
Figure 32.6. A Phased Array of Two Half-Wave Dipoles Spaced  Apart
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Figure 32.7 shows the radiation pattern for an array of four half-wavelength antennas (N=4)
spaced one-half-wavelength apart from each other. Clicking on the image shows an animation
for d=0 to d=3.
Figure 32.7. A Phased Array of Four Half-Wave Dipoles Spaced 2 Apart
Finally, Figure 32.8 shows a phased array of four half-wavelength antennas each spaced on-halfwavelength apart and with an intentional phase difference of /2 included. Clicking on this
image shows an animation for =- to =+.
Figure 32.8. A Phased Array of Four Half-Wave Dipoles Spaced 2 Apart with =/2.
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This last animation is especially important, because it shows how the radiation pattern can be
adjusted simply by adjusting the phase angle of the signals sent to each antenna. This means that
no physical adjustment of the antennas is needed. The movement of the pattern is obtained
simply by adjusting the electrical signals.
32.4
Binomial Arrays
The second major type of antenna array we will study is called a binomial array. You’ll
understand why it’s called that shortly, but for now, it is probably better to think of it is an
“amplitude array,” because each of the antennas in the array will receive the same signal at the
same phase but with different amplitudes.
Consider Figure 32.9, which has three uniformly spaced antennas, but the middle antenna
receives twice as much voltage as the other two.
V0
2V0
/2
V0
/2
Figure 32.9. A Binomial Array with Three Antennas (N=2)
We can think of this array is being a “pair of a pair” of antennas, as illustrated in Figure 32.10. If
we conceptually split the middle antenna into two identical antennas, each transmitting with an
intensity of V0, then this array of three antennas becomes identical to an array of two arrays, each
of which is composed of two antennas.
V0
V0
V0
V0
/2
Figure 32.10. A Binomial Array of Three Antennas is a Pair of Antenna Pairs
We already know the array factor for a pair of antennas:
(Equation 32.21)
Since we are not introducing a phase difference to the antennas, this can be written as:
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(Equation 32.22)
Each antenna is a half wavelength from its neighbors, so we can substitute d=/2 and k=2/:
(Equation 32.22)
Simplifying this expression gives:
(Equation 32.23)
Since we are working with a “pair of pairs,” both the array factor and the element factor are the
same:
(Equation 32.24)
Multiplying the element factor by the array factor gives the total radiation pattern:
(Equation 32.25)
This element factor, array factor, and radiation pattern are shown in Figure 32.11. Notice that
the radiation pattern is narrower than the element factor and the array factor..
Element Factor
Array Factor
Radiation Pattern
Figure 32.11. Radiation Patterns for a Binomial Array of Three Antennas (N=2)
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This same strategy can be expanded beyond three antennas. Figure 32.12 shows four antennas in
which this middle two are each transmitting three times as strongly as the outer two:
V0
3V0
/2
3V0
/2
V0
/2
Figure 32.12. Binomial Array of Four Antennas (N=3)
This can be thought of as a pair of N=2 binomial arrays (the 1-2-1 array we just studied), as
shown in Figure 32.13.
Figure 32.13. Binomial Array of Four Antennas (N=3) as a Pair of Binomial Arrays with N=2
Since this is a pair of arrays, the array factor will be the same as in Equation 32.23, and the
element factor will be the radiation pattern of a 1-2-1 antenna, as shown in Equation 32.25.
Multiplying these together gives:
(Equation 32.26)
The element factor, array factor, and radiation pattern for a binomial array with four antennas
(N=3) is shown in Figure 32.14.
Element Factor
Array Factor
Radiation Pattern
Figure 32.11. Radiation Patterns for a Binomial Array of Four Antennas (N=3)
We can generalize the binomial array to any number of antenna elements. Notice that we have
been consistently defining N to be one less than the number of antennas. There will always be
N+1 antennas in a binomial array of order N.
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The intensity of each of the N+1 elements can be calculated using a binomial distribution (thus
the name, “binomial array.”)
(Equation 32.27)
Here, there are N+1 antennas, and k=0, 1, 2, …, N.
The radiation pattern for an order N binomial array (with N+1 elements) is:
(Equation 32.28)
As N increases, the power of the cosine will increase, making the lobes narrower and increasing
the directivity of the radiation pattern.
Example 32.3: What is the intensity for each element of a binomial array with N=4? What
is the radiation pattern for this array?
32.5
Directivity of Antenna Arrays
We can calculate the directivity of a given antenna array that contains N identical elements, each
separated by a distance d=/2:
(Equation 32.29)
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Example 32.4: What is the directivity of a phased array with three antennas?
Example 32.5: What is the directivity of a binomial array with three antennas?
Figure 32.12 shows the radiation patterns of a three-element phased array and a three-element
binomial array. As you can see, the phased array is slightly narrower, but the binomial array has
no side-lobes. Depending on the application, one or the other will be preferable.
Phased Array
Binomial Array
Figure 32.12. Radiation Patterns for Three-Element Phased and Binomial Arrays
32.6

Summary

Arrays of antennas give us a great deal more flexibility in designing the radiation pattern
because there are so many more variables that can be adjusted.
The overall radiation pattern of an antenna array is the product of the element factor (the
radiation pattern of a single antenna) multiplied by an array factor, which depends on how
the array is arranged.

A two-element array has an array factor of:
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
A phased array is composed of a number of identical antennas, spaced uniformly, and
receiving the same amplitude of signal. The phase can vary. The array factor for a phased
array is:

A binomial (amplitude) array consists of a number of identical antennas, spaced uniformly,
and receiving a signal with the same phase. The amplitude of each element can vary,
according to:

The radiation pattern of such an array is:

You can calculate the directivity of an array of identical antennas spaced l/2 apart from each
other by the following formula:
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