UNIT 3

ANTENNA ARRAYS
SYLLABUS- Two-Element Array, Broadside arrays, End fire arrays. N-Element Linear Array: Uniform
Amplitude and Spacing, N-Element Linear Array: Directivity, N-Element Linear Array: Uniform
Spacing, Non uniform Amplitude, Binomial Array, Chebyshev Arrays, Principle of pattern
multiplication, Array pattern Synthesis.
3.1 TWO ELEMENT ARRAY
An antenna, when individually can radiate an amount of energy, in a particular direction, resulting
in better transmission, how it would be if few more elements are added it, to produce more efficient
output. It is exactly this idea, which led to the invention of Antenna arrays.
An antenna array can be better understood by observing the following images. Observe how the antenna
arrays are connected. An antenna array is a radiating system, which consists of individual radiators and
elements. Each of this radiator, while functioning has its own induction field. The elements are placed
so closely that each one lies in the neighbouring one’s induction field. Therefore, the radiation pattern
produced by them, would be the vector sum of the individual ones. The following image shows another
example of an antenna array.
The spacing between the elements and the length of the elements according to the wavelength are also
to be kept in mind while designing these antennas.
The antennas radiate individually and while in array, the radiation of all the elements sum up, to form
the radiation beam, which has high gain, high directivity and better performance, with minimum losses.
Advantages
The following are the advantages of using antenna arrays −
•The signal strength increases
• High directivity is obtained
• Minor lobes are reduced much
• High Signal-to-noise ratio is achieved
• High gain is obtained
• Power wastage is reduced
• Better performance is obtained
Disadvantages
The following are the disadvantages of array antennas −
• Resistive losses are increased
• Mounting and maintenance is difficult
• Huge external space is required
Types of Arrays
The basic types of arrays are −
• Collinear array
• Broad side array
 • End fire array
• Parasitic array
3.1.1 COLLINEAR ARRAY
A Collinear array consists of two or more half-wave dipoles, which are placed end to end. These
antennas are placed on a common line or axis, being parallel or collinear. The maximum radiation in
these arrays is broad side and perpendicular to the line of array. These arrays are also called as broad
cast or Omni-directional arrays.
Frequency range
The frequency range in which the collinear array antennas operate is around 30 MHz to
3GHz which belong to the VHF and UHF bands.
Construction of Array
These collinear arrays are uni-directional antennas having high gain. The main purpose of this
array is to increase the power radiated and to provide high directional beam, by avoiding power loss in
other directions.
Radiation Pattern
The radiation pattern of these collinear arrays is similar to that of a single dipole, but the array
pattern of increasing number of dipoles, makes the difference.

Fig 3.1 Radiation pattern of Collinear Array

The radiation pattern of collinear array when made using two elements, three elements and four elements
respectively are shown in the figure given above.
The broad side array also has the same pattern, in which the direction of maximum radiation is
perpendicular to the line of antenna.
Advantages
The following are the advantages of collinear array antennas −
•Use of array reduces the broad ends and increases the directivity
• Minor lobes are minimised
• Wastage of power is reduced
Disadvantages
The following are the disadvantages of collinear array antennas −
 • Displacement of these antennas is a difficult task
• Used only in outdoor areas
Applications
The following are the applications of collinear array antennas −
• Used for VHF and UHF bands
• Used in two-way communications
• Used also for broadcasting purposes
3.1.2 Broad-side Array
The antenna array in its simplest form, having a number of elements of equal size, equally spaced
along a straight line or axis, forming collinear points, with all dipoles in the same phase, from the same
source together form the broad side array.
Frequency range
The frequency range, in which the collinear array antennas operate is around 30 MHz to
3GHz which belong to the VHF and UHF bands.
Construction & Working of Broad-side Array
According to the standard definition, “An arrangement in which the principal direction of
radiation is perpendicular to the array axis and also to the plane containing the array element” is termed
as the broad side array. Hence, the radiation pattern of the antenna is perpendicular to the axis on which
the array exists.
Radiation Pattern
The radiation pattern of this antenna is bi-directional and right angles to the plane. The beam is
very narrow with high gain.

Fig 3.2 Radiation Pattern of Broadside Array

The above figure shows the radiation pattern of the broad side array. The beam is a bit wider and minor
lobes are much reduced in this.
3.1.3 End-fire Array
The physical arrangement of end-fire array is same as that of the broad side array. The
magnitude of currents in each element is same, but there is a phase difference between these currents.
This induction of energy differs in each element, which can be understood by the following diagram.
 Fig 3.3 End Fire Array
The above fig3.3 shows the end-fire array in top and side views respectively.
There is no radiation in the right angles to the plane of the array because of cancellation. The first and
third elements are fed out of phase and therefore cancel each other’s radiation. Similarly, second and
fourth are fed out of phase, to get cancelled.
The usual dipole spacing will be λ/4 or 3λ/4. This arrangement not only helps to avoid the
radiation perpendicular to the antenna plane, but also helps the radiated energy get diverted to the
direction of radiation of the whole array. Hence, the minor lobes are avoided and the directivity is
increased. The beam becomes narrower with the increased elements.
Radiation Pattern
The Radiation pattern of end-fire array is uni-directional. A major lobe occurs at one end, where
maximum radiation is present, while the minor lobes represent the losses.

Fig 3.4 Radiation Pattern of End Fire Array

The fig 3.4 explains the radiation pattern of an end-fire array. Figure 1 is the radiation pattern for
a single array, while figures 2, 3, and 4 represent the radiation pattern for multiple arrays.
End-fire Array Vs Broad Side Array
To compare the end-fire and broad side arrays, along with their characteristics.

Fig 3.5 Comparison of End-fire Array Vs Broad Side Array

The fig 3.5 illustrates the radiation pattern of end-fire array and broad side array.
•Both, the end fire array and broad side array, are linear and are resonant, as they consist
of resonant elements.
• Due to resonance, both the arrays display narrower beam and high directivity.
• Both of these arrays are used in transmission purposes.
• Neither of them is used for reception, because the necessity of covering a range of
frequencies is needed for any kind of reception.
3.1.4 Parasitic Array
The antenna arrays as seen above, are used for the improvement of gain and directivity.
A parasitic element is an element, which depends on other’s feed. It does not have its own feed. Hence,
in this type of arrays we employ such elements, which help in increasing the radiation indirectly. These
parasitic elements are not directly connected to the feed.The arrays are used at frequencies ranging
from 2MHz to several GHz. These are especially used to get high directivity, and better forward gain
with a uni-directional. The most common example of this type of array is the Yagi-Uda antenna. Quad
antenna may also be quoted as another example.
3.2 ARRAY OF TWO POINT SOURCES
The simplest array configuration is array of two point sources of same polarization and
separated by a finite distance. The concept of this array can also be extended to more number
of elements and finally an array of isotropic point sources can be formed.

Based on amplitude and phase conditions of isotropic point sources, there are three types of
arrays:
(a) Array with equal amplitude and phases
(b) Array with equal amplitude and opposite phases
(c) Array with unequal amplitude and opposite phases
3.2.1 Two Point Sources with Currents Equal in Magnitude and Phase

...(1)

Fig 3.6 Two element array

Consider two point sources A1 and A2, separated by distance d as shown in the Fig.3.6 .
Consider that both the point sources are supplied with currents equal in magnitude and phase.
Consider point P far away from the array. Let the distance between point P and point sources
A1 and A2 .be r1 and r2 respectively. As these radial distances are extremely large as compared
with the distance of separation between two point sources i.e. d, we can assume,

r1= r2 = r

The radiation from the point source A2 will reach earlier at point P than that from point source
A1 because of the path difference. The extra distance is travelled by the radiatedwave from
point source A1 than that by the wave radiated from point source A2.

Hence path difference is given

by, Path difference = d cos ʋ
The path difference can be expressed in terms of wavelength as,

Path difference = (d cos ʋ) / λ ...(2)

Hence the phase angle ʋis given by,

Phase angle ʋ = 2π (Path difference)

But phase shift β= 2π/λ,thus equation (3) becomes,

Let E1 be the far field at a distant point P due to point source Al. Similarly let E2 be the far
field at point P due to point source A2. Then the total field at point P be the
addition of the two field components due to the point sources A1 and A2. If thephase
angle between the two fields is ʋ = βdcosʋ then the far field component at point P due to point
source A1 is given by,

Similarly the far field component at point P due to the point source A2 is given by,

Note that the amplitude of both the field components is E0 as currents are same andthe point
sources are identical.
The total field at point P is given by,
Rearranging the terms on R.H.S., we get,

By trigonometric identity,

Hence equation (7) can be written as,

Substituting value of Ψfrom equation (4), we get,.

Above equation represents total field in intensity at point P. due to two point sourceshaving
currents of same amplitude and phase. The total amplitude of the field at
point P is 2E0 while the phase shift is βdcosʋ/2
The array factor is the ratio of the magnitude of the resultant field to the magnitudeof the
maximum field.

But maximum field is Ernax =2E0

The array factor represents the relative value of the field as a function of ʋdefinesthe
radiation pattern in a plane containing the line of the array.
Maxima direction

From equation (9), the total field is maximum when is maximum. As we know,the
variation of cosine of a angle is ± 1. Hence the condition for maxima is given by,
Let spacing between the two point sources be λ/2. Then we can write,

If n = 0, then

Minima direction

Again from equation (9), total field strength is minimum when is minimum
i.e.0 as cosine of angle has minimum value 0. Hence the condition for minima is given by,

Again assuming d = λ/2and β=2π/λ, we can write

Half power point direction:

When the power is half, the voltage or current is 1/√2 times the maximum value.
Hence the condition for half power point is given by,

Let d=λ/2 and β=2π/λ, then we can write,

The field pattern drawn with ET against ʋ for d=λ/2, then the pattern is bidirectional as shown
in Fig 3.7. The field pattern obtained is bidirectional and it is a figure of eight.
0
If this pattern is rotated by 360 about axis, it will represent three dimensional doughnut
shaped space pattern. This is the simplest type of broadside array of two point sources and it
is called Broadside couplet as two radiations of point sources are in phase.

Fig. 3.7 Field pattern for two point source with spacing d=λ/2 and fed with currents
equal inmagnitude and phase.
3.2.2Two Point Sources with Currents Equal in Magnitudes but Opposite in Phase

Consider two point sources separated by distance d and supplied with currentsequal in
magnitude but opposite in phase. Consider Fig. 5 all the conditions areexactly same
except the phase of the currents is opposite i.e. 180°. With thiscondition, the total field
at far point P is given by,

Assuming equal magnitudes of currents, the fields at point P due to the pointsources A1

and A2 can be written as,

Substituting values of E1 and E2 in equation (1), we get

Rearranging the terms in above equation, we get,

By trigonometry identity,

Equation (4) can be written as,

Now as the condition for two point sources with currents in phase and out of phaseis
exactly same, the phase angle can be written as previous case.
Phase angle = βdcosʋ...(6) Substituting value of phase angle inequation (5),
we get,
Maxima direction

From equation (7), the total field is maximum when is maximum i.e. ±1 asthe
maximum value of sine of angle is ±1. Hence condition for maxima is given by,

Let the spacing between two isotropic point sources be equal to d=λ/2Substituting d=λ/2
and β=2π/λ,in equation (8), we get,

If n = 0. then

Minima direction

Again from equation (7), total field strength is minimum when is

minimum i.e. 0.
Hence the condition for minima is given by,

Assuming d=λ/2 and β=2π/λin equation (10), we get,

If n = 0, thenHalf Power Point Direction (HPPD)

When the power is half of maximum value, the voltage or current equals to 1/√2 times the
respective maximum value. Hence the condition for the half power point can be obtained from
equation (7) as,

Let d=λ/2 and β=2π/λ, we can write,

Thus from the conditions of maxima, minima and half power points, the field pattern can be
drawn as shown in the Fig. 7.

Fig. 3.8 Field pattern for two point sources with spacing d = d=λ/2 and fed with currents
equal inmagnitude but out of phase by 1800.

As compared with the field pattern for two point sources with inphase currents, the maxima
have shifted by 90° along X-axis in case of out-phase currents in two point source array. Thus
the maxima are along the axis of the array or along the line joining two point sources. In first
case, we have obtained vertical figure of eight. Now in above case, we have obtained horizontal
figure of eight. As the maximum field is along the line joining the two point sources, this is
the simple type of the end fire array.
3.2.3 Two point sources with currents unequal in magnitude and with any phase
Let us consider Fig. 5. Assume that the two point sources are separated by distance d and
supplied with currents which are different in magnitudes and with any phase difference say α.
Consider that source 1 is assumed to be reference for phase and amplitude of the fields

E1 and E2, which are due to source 1 and source 2 respectively at the distant point P. Let us

assume that E1 is greater than E2 in magnitude as shown in the vector diagram in Fig. 8.

Fig. 3.9 Vector diagram of fields El and E2

Now the total phase difference between the radiations by the two point sources at any far point
P is given by,

where α is the phase angle with which current I2 leads current Il. Now if α = 0, then the
condition is similar to the two point sources with currents equal in magnitude and phase.
Similarly if α = 180", then the condition is similar to the two point source with currents equal
in magnitude but opposite in phase. Assume value of phase
0
difference as 0 <α < 180 . Then the resultant field at point P is given by,

Note that E1> E2, the value of k is less than unity. Moreover the value of k is given by, 0 ≤k

≤1

The magnitude of the resultant field at point P is given by,

The phase angle between two fields at the far point P is given by,

3.3 n Element Uniform Linear Arrays

At higher frequencies, for point to point communications it is necessary to have a
pattern with single beam radiation. Such highly directive single beam pattern can be obtained
by increasing the point sources in the arrow from 2 to n say. An array of n elements is said to
be linear array if all the individual elements are spaced equally alonga line. An array is said to
be uniform array if the elements in the array are fed with currents with equal magnitudes and
with uniform progressive phase shift along the line. Consider a general n element linear and
uniform array with all the individual elements spaced equally at distance d from each other
and all elements are fed with currents equal in magnitude and uniform progressive phase shift
along line as shown in the Fig. 9.

Fig.3.10 Uniform, linear array of n elements

The total resultant field at the distant point P is obtained by adding the fields due to n
individual sources vectorically. Hence we can write,

Note that ʋ= (βdcosʋ + α) indicates the total phase difference of the fields from adjacent
sources calculated at point P. Similarly α is the progressive phase shift between two adjacent

Multiplying equation (1) by e , we

ʋ get,

Subtracting equation (2) from (1), we get,

.
Simply mathematically, we get

According to trigonometric identity,

The resultant field is given by,

This equation (4) indicates the resultant field due to n element array atdistant
point P. The magnitude of the resultant field is given by,

The phase angle θ of the resultant field at point P is given by,

3.3.1 Array of n elements with Equal Spacing and Currents Equal in Magnitude and Phase •
Broadside Array
Consider 'n' number of identical radiators carries currents which are equal in magnitude

and in phase. The identical radiators are equispaced. Hence the maximum radiation occurs in

the directions normal to the line of array. Hence such an array is known as Uniform broadside

array.Consider a broadside array with n identical radiators as shown in the Fig. 10.

Fig 3.11 Array of n elements with Equal Spacing

The electric field produced at point P due to an element A0 is given by,

As the distance of separation d between any two array elements is very small as compared to
the radial distances of point P from A0, A1, ...An-1, we can assume r0, r1, ...rn-1 are
approximately same.

Now the electric field produced at point P due to an element A1 will differ in phaseas r0
and r1 are not actually same. Hence the electric field due to A1 is given by,

Exactly on the similar lines we can write the electric field produced at point P due toan
element A2 as,
But the term inside the bracket represent E1

From equation (2), substituting the value of E1, we get,

Similarly, the electric field produced at point P due to element An-1 is given by,

The total electric field at point P is given by,

Let βdcosʋ= ʋ, then rewriting above equation,

Consider a series given

2
by s = 1 + r + r + +
n-1 jʋ
r where r = e ... (i)
Multiplying both the sides of the equation (i)
2 n
by r, s . r = r + r + +r
... (ii)
Subtracting equation (ii) from (i), we
n
get. s(1-r) = 1- r
Using equation (iii), equation (5) can be modified as,

From the trigonometric identities,

Equation (6) can be written as,

The exponential term in equation (7) represents the phase shift. Now consideringmagnitudes of the
electric fields, we can write,

Properties of Broadside Array

1. Major lobe
In case of broadside array, the field is maximum in the direction normal to theaxis
of the array. Thus the condition for the maximum field at point P is given by,

0 0
Thus ʋ = 90 and 270 are called directions of principle maxima.
2. Magnitude of major lobe
The maximum radiation occurs when ʋ=0. Hence we can write,

where, n is the number of elements in the array.

Thus from equation (10) and (11) it is clear that, all the field components addup
together to give total field which is ‘n’ times the individual field when ʋ =
0 0
90 and 270 .
3. Nulls
The ratio of total electric field to an individual electric field is given by,

Equating ratio of magnitudes of the fields to zero,

The condition of minima is given by,

Hence we can write,

 where, n= number of elements in the
array d=spacing between elements
in meter

λ= wavelength in meter
m= constant= 1, 2 , 3....
Thus equation (13) gives direction of nulls
4. Side Lobes Maxima
The directions of the subsidary maxima or side lobes maxima can be obtainedif in
equation (8),

Hence sin(nʋ/2), is not considered. Because if nʋ/2=π/2 thensin nʋ/2 =1which is the
direction of principle maxima.
Hence we can skip sin nʋ/2 = ±π/2value Thus, we get

Now equation for ʋcan be written as,

The equation (15) represents directions of subsidary maxima or side lobes maxima.
5. Beamwidth of Major Lobe
Beamwidth is defined as the angle between first nulls. Alternativelybeamwidth is the
angle equal to twice the angle between first null and the major lobe maximum direction.
Hence beamwidth between first nulls is given by,

Also

Hence
Taking cosine of angle on both sides, we get
If γ is very small, then sin γ ≈ γ. Substituting n above equation we get,

For first null i.e. m=1,

But nd≈(n-1)d if n is very large. This L= (nd) indicates total length of thearray.
BWFN in degree is written as,

Now HPBW is given by,

HPBW in degree is written as,

6. Directivity
The directivity in case of broadside array is defined as,

where, U0 is average radiation intensity which is given by,

From the expression of ratio of magnitudes we can write,

or
For the normalized condition let us assume E0 = 1, then

Thus field from array is maximum in any direction θ when ʋ = 0. Hencenormalized

field pattern is given by,

Hence the field is given by,

where ʋ = βdcosʋ
Equation (23) indicated array factor, hence we can write electric field due to narray
as

Assuming d is very small as compared to length of an array,

Then,

Substituting value of E in equation (24) we get

Let
Rewritting above equation we get,

For large array, n is large hence nβdis also very large (assuming tending toinfinity).
Hence rewriting above equation.

Interchanging limits of integration, we get

By integration formula,

Using above property in above equation we can write,

From equation (23), the directivity is given by,

But Umax = 1 at ʋ =90° and substituting value of U0 from equation (28), we get,
 But β= 2π/λ
Hence
The total length of the array is given by, L = (n - 1) d ≈nd, if n is very large.
Hencethe directivity can be expressed in terms of the total length of the array as,

3.4 Array of n Elements with Equal Spacing and Currents Equal in Magnitude but with
Progressive Phase Shift - End Fire Array
Consider n number of identical radiators supplied with equal current which are not in phase as
shown in the Fig. 11. Assume that there is progressive phase lag of βd radians in each radiator.

Fig.3.12 End fire array

Consider that the current supplied to first element A0 be I0. Then the current supplied to A1
is given by,

Similarly the current supplied to A2 is given by,

Thus the current supplied to last element is

The electric field produced at point P, due to A0 is given by,

The electric field produced at point P, due to A1 is given by,

But r1 = r0 – dcosʋ
Let ʋ = βd(cosʋ-1)

The electric field produced at point P, due to A2 is given by,

Similarly electric field produced at point P, due to An-1 is given by,

The resultant field at point p is given by,

Considering only magnitude we get,

Properties of End Fire Array

1. Major lobe
For the end fire array where currents supplied to the antennas are equal in amplitude
but the phase changes progressively through array, the phase angle is given by,
ʋ = βd(cosʋ -1) ...(9)

In case of the end fire array, the condition of principle maxima is given by,
ʋ = = 0 i.e.

i.e. cosʋ= 1
0
i.e. ʋ = 0 ...(11)
 0
Thus ʋ = 0 indicates the direction of principle maxima.

2. Magnitude of the major lobe

The maximum radiation occurs when ʋ= 0. Thus we can write,

where, n is the number of elements in the array.

3. Nulls
The ratio of total electric field to an individual electric field is given by,

Equating ratio of magnitudes of the fields to zero,

The condition of minima is given by,

Henc
e we can write,

Substituting value of ʋfrom equation (9), we get,

But β= 2π/λ

Note that value of (cosʋ-1) is always less than 1. Hence it is always negative.
Hence only considering -ve values, R.H.S., we get
 where, n= number of elements in the array
d=spacing between elements in
meter

λ= wavelength in meter
m= constant= 1, 2 , 3....
Thus equation (15) gives direction of nulls
Consider equation(14),

Expressing term on L.H.S. in terms of halfangles, we get,

4. Side Lobes Maxima

The directions of the subsidary maxima or side lobes maxima can be obtainedif in
equation (8),

Hence sin(nʋ/2), is not considered. Because if nʋ/2=±π/2 then sin nʋ/2

=1 which is the direction of principle maxima.
Hence we can skip sin nʋ/2= ±π/2value Thus, we get

Putting value of ʋ from equation (9) we get

 Now equation for ʋ can be
written as,But β = 2π/λ

Note that value of (cosʋ-1) is always less than 1. Hence it is always negative. Hence
only considering -ve values, R.H.S., we get

5. Beamwidth of Major Lobe

Beamwidth is defined as the angle between first nulls. Alternativelybeamwidth is the
angle equal to twice the angle between first null and the major lobe maximum direction.
From equation (16) we get,

ʋminis very low

Hence sin ʋmin/2 ≈ ʋmin/2

But nd≈ (n-1)d if n is very large. This L= (nd) indicates total length ofthe
array. So equation (20) becomes,
 BWFN is given by,

BWFN in degree is expressed as

For m=1,

6. Directivity
The directivity in case of endfire array is defined as,

where, U0 is average radiation intensity which isgiven by,

For endfire array, Umax = 1and

The total length of the array is given by, L = (n - 1) d ≈nd, if n is very large.
Hencethe directivity can be expressed in terms of the total length of the array as,

3.5 Multiplication of patterns

In the previous sections we have discussed the arrays of two isotropic point sources radiating
field of constant magnitude. In this section the concept of array is extended to non-isotropic
sources. The sources identical to point source and having field patterns of definite shape and
orientation. However, it is not necessary that amplitude of individual sources is equal. The
simplest case of non-isotropic sources is when two short dipoles are superimposed over the two
isotopic point sources separated by a finite distance. If the field pattern of each source is given
by

Then the total far-field pattern at point P becomes

...(1)

where

Equation (1) shows that the field pattern of two non-isotropic point sources (short dipoles) is
equal to product of patterns of individual sources and of array of point sources. The pattern of
array of two isotropic point sources, i.e., cos ʋ/2 is widely referred as an array factor. That is

ET= E (Due to reference source) x Array factor

This leads to the principle of pattern multiplication for the array of identical elements. In
general, the principle of pattern multiplication can he stated as follows:

The resultant field of an array of non-isotropic hut similar sources is the product of the fields
of individual source and the field of an array of isotropic point sources, each located at the
phase centre of individual source and hating the relative amplitude and phase. The total phase
is addition of the phases of the individual source and that of isotropic point sources. The same
is true for their respective patterns also.

The normalized total field (i.e., ETn), given in Eq. (1), can re-written as

where E1(θ) = sin θ= Primary pattern of array

= Secondary pattern of array.

Thus the principle of pattern multiplication is a speedy method of sketching the field pattern
of complicated array. It also plays an important role in designing an array. There is no restriction
on the number of elements in an array; the method is valid to any number of identical elements
which need not have identical magnitudes, phase and spacing between
 then). However, the array factor varies with the number of elements and theirarrangement,
relative magnitudes, relative phases and element spacing. The array of elements having
identical amplitudes, phases and spacing provides a simple array

factor. The array factor does not depend on the directional characteristic of the array elements;
hence it can be formulated by using pattern multiplication techniques. The proper selection of
the individual radiating element and their excitation are also important for the performance of
array. Once the array factor is derived using the point-source array, the total field of the actual
array can be obtained using Eq. (2).

3.5.1 Binomial Array

In order to increase the directivity of an array its total length need to be increased. In this
approach, number of minor lobes appears which are undesired for narrow beam applications.
In has been found that number of minor lobes in the resultant pattern increases whenever
spacing between elements is greater than λ/2. As per the demand of modern communication
where narrow beam (no minor lobes) is preferred, it is the greatest need to design an array of
only mainlobes. The ratio of power density of main lobe to power density of the longest minor
lobe is termed side lobe ratio. A particular technique used to reduce side lobe level is called
tapering. Since currents/amplitude in the sources of a linear array is non-uniform, it is found
that minor lobes can be eliminated if the centre element radiates more strongly than the other
sources. Therefore tapering need to be done from centre to end radiators of same specifications.
The principle of tapering are primarily intended to broadside array but it is also applicable to
end-fire array. Binomial array is a common example of tapering scheme and it is an array of n-
isotropic sources of non-equal amplitudes. Using principle of pattern multiplication, John
Stone first proposed the binomial array in 1929, where amplitude of the radiating sources arc
arranged according to the binomial expansion. That is. if minor lobes

appearing in the array need to be eliminated, the radiating sources must have current
amplitudes proportional to the coefficient of binomial series, i.e. proportional to the coefficient
of binomial series, i.e.

...(1)
where n is the number of radiating sources in the array.
For an array of total length nλ/2, the relative current in the nth element from the one
end is given by

where r = 0, 1, 2, 3, and the above relation is equivalent to what is known as Pascal's

triangle.
For example, the relative amplitudes for the array of 1 to 10 radiating sources are as
follows:

Since in binomial array the elements spacing is less than or equal to the half-
wavelength, the HPBW of the array is given by

and directivity

Using principle of multiplication, the resultant radiation pattern of an n-

sourcebinomial array is given by
In particular, if identical array of two point sources is superimposed one above other,
then three effective sources with amplitude ratio 1:2:1 results. Similarly, in casethree
such elements are superimposed in same fashion, then an array of four sources is
obtained whose current amplitudes are in the ratio of 1:3:3:1.
 The far-field pattern can be found by substituting n = 3 and 4 in the above expression
and they take shape as shown in Fig. 14(a) and (b).

Fig. 3.13(a) Radiation pattern of 2-element array with amplitude

ratio 1:2:1.
Fig 3.13(b) Radiation pattern of 3-element array with amplitude ratio 1:3:3:1.

It has also been noticed that binomial array offers single beam radiation at the cost
of directivity, the directivity of binomial array is greater than that of uniform array
for the same length of the array. In other words, in uniform array secondary lobes
appear, but principle lobes are narrower than that of the binomial array.
Disadvantages of Binomial Array
(a) The side lobes are eliminated but the directivity of array reduced.
As the length of array increases, larger current amplitude ratios are required