Phased Array Antennas

Iulian Rosu, YO3DAC / VA3IUL,
http://www.qsl.net/va3iul

pdf version

Introduction

First
Antenna
was invented and built in 1888 by
Heinrich Hertz
in his experiments to
prove the existence of waves predicted by the electromagnetic theory of
James C. Maxwell
.

The name Antenna was coined by
Guglielmo Marconi
in 1895, and is coming from
the Latin name
ANTEMNA
, which is the pole on a mast, from which ship sails are set.

Antennas are essential components of ALL radio equipment

Main Characteristics of Array Antennas

Is a graphical representation (or mathematical function) of the radiation properties of
an antenna as a function of geometric (typically spherical) coordinates.

If that particular direction is not specified, then the direction in which maximum
intensity is observed, can be taken as the directivity of that antenna.

For example, an antenna whose pattern (
First Null Beamwidth
) FNBW / 2 = 2°
has a resolution of 1°, so the antenna may distinguish between two adjacent
geostationary orbit satellites separated by 1°.

Effective area (aperture)
A
e
of an antenna represents the ratio of the available power
at the terminals of the antenna to the power flux density from a plane wave incident
normal to the antenna. The effective area is related to the antenna directivity
D
:

A
e
= (λ
2
*D)/4π

e
a
= A
e
/A
phys
(0 <
e
a
< 1)

In transmitting antenna, these losses involve power fed to the antenna which is not
radiated but heats the antenna structure,

G = k*D

where k (dimensionless) is the
efficiency factor
(0 ≤ k ≤ 1)

Array Antenna Gain = 10*log(N) + G
e
–
Loss
OHMIC
–
Loss
SCAN

Since mismatch will result in additional losses,
realized gain
is smaller than
gain
.

The
gain
in return is smaller than the
directivity
due to the radiation efficiency.

The total amount of energy radiated from a transmitting antenna can be measured in
terms of a
Radiation Resistance
which is the resistance that, when replacing the
antenna, at the feeder will consume the same amount of power that is radiated.

The angular separation between two identical points on opposite sides of the
maximum of the radiation pattern.

Generally, the value definition is the half-power (3dB) point (HPBW).

Indicates the time-varying direction of the
electric field vector
–
vertical, horizontal,
and circular polarization are typical.

The frequencies for which matching is acceptable (e.g. VSWR less than 2), define the
antenna bandwidth.

Antennas are defined as broadband when the factor is equal to or greater than 2.

The ratio of voltage to current at the input terminals of the antenna.

Array Antenna Field Regions

When a high frequency current flows in an antenna, it generates a high frequency
electromagnetic field in the surrounding space. The surrounding space of an antenna is
usually subdivided (classified) into three regions: the
reactive near-field
region, the
radiating near-field
(Fresnel) region and the
far-field
(Fraunhofer) region.

These regions are useful to identify the field structure to know which simplification can be
applied, but there is no precise boundary nor abrupt change in the field configuration.

Even if antenna regions were predicted years before (wave-zones mentioned by
Braun
),
first author who published about regions around the antenna was
Schelkunoff
in the mid
1930’s
, followed later by
Friis
and
Kraus
. Was mentioned that the space around an antenna
may be separated into two regions: one next to the antenna known as the
“
antenna
region
”
and one outside the antenna, known as the
“
outer region
”
. The boundary
between the two regions (which is a sphere whose center is at the middle of the antenna)
may be arbitrary taken to be at a radius
R
.
This distance
R
varies from one type of antenna
to another. For electrically large antennas which have a well defined cophased radiating
aperture, a commonly used criteria is to state that the far-field region starts when:

where
L
is the antenna greatest dimension (length) and
λ
is the wavelength.

Initially, the distinction between fields at a large distance and those nearer to the
antenna was emphasized by subdividing the
“
outer region
”
into two regions: the one near
the antenna called the
near-field
, or Fresnel region, and the one at a large distance called
the
far-field
, or Fraunhofer region.

Later, were coined names as
reactive near-field
for
“
antenna region
”
,
radiating
near-field
for the Fresnel region, and
far-field
for Fraunhofer region.

Antenna field regions

These antenna regions were named after the physicists Fresnel and Fraunhofer due to
the analogy of the antenna fields with their inventions and discoveries in optics.

In the
radiating near-field
Fresnel region, the radial field may be appreciable and the
shape of the field pattern is, in general, a function of the distance to the antenna.

In the
far-field
Fraunhofer region,
E
and
H
field vectors are transverse to the direction of
propagation and orthogonal to each other, and the impedance of the field |
E
|/|
H
| at each
location approaches the free-space wave impedance of 377 Ohms. The shape of the field
pattern is independent of the radius (distance to antenna) at which it is taken.

However,
the distance from an antenna, where far-field conditions are met,
depends on the dimensions of the antenna in respect to the wave length
.

For smaller antennas (e.g. a half-wave dipole) the wave fronts radiated from the antenna
become almost parallel at much closer distance compared to electrically large antennas.

A good approximation for small antennas is that far-field conditions are reached at:

R = 2λ

In the immediate vicinity of the antenna, there is the reactive near field. In this region,
the fields are predominately reactive fields, which means the
Electric-E
and the
Magnetic-H
fields are out of phase by 90° to each other (recall that for propagating or
radiating fields, the fields are orthogonal/perpendicular but are in phase).

For antenna
’s
greatest dimension
L
, the boundary of the
reactive near-field
region
R
is
commonly given as:

The
radiating near-field
or Fresnel region, is the region between the
reactive near-
field
and
far-field
. In this region, the reactive fields are not dominate and the
radiating fields begin to emerge. However, unlike the far-field region, here the shape
of the radiation pattern may vary appreciably with distance
R
from the antenna.

For
antenna dimension
L
,
the boundary of the
radiating near-field
region
R
is commonly
given by:

Note that depending on the values of
R
and the wavelength, this field may or may not exist.

As is defined, the far-field is the region far from the antenna. In this region, the
radiation pattern does not change the shape with distance
R
(although the fields still
die-off as
1/
R
, the power density dies-off as
1/
R
2
). This region is dominated by
radiated fields, with the
Electric-E
and
Magnetic-H
fields orthogonal to each other
and the direction of propagation as with plane waves.

If the
maximum linear dimension
of an antenna is the length
L
, then the following
three conditions must be all satisfied to be in the
far-field
region:

1)

2)

3)

The equations
1
) and
2
) from above, ensure that the power radiated in a given
direction from distinct parts of the antenna are approximately parallel (see figure below).

This helps ensure the fields in the far-field region behave like plane waves.

In the Far-Field the Rays from any point on the antenna are approximately parallel

Thus, we can see how greater could be the far-field starting range of an Array
Antenna compared to the far-field range of a
single antenna element
.

The far-field equation number 3) come from the statement that near a radiating antenna,
there are reactive fields (see reactive near field region, above), that typically have the
Electric-E
fields and
Magnetic-H
fields die-off with distance as
1/
R
2
and
1/
R
3
.

The equation number 3) ensures that these near fields are gone, and we are left with the
radiating fields, which fall-off with distance as
1/
R
.

Techniques how to increase the Antenna Gain and change the Radiation Pattern

For some applications, single element antennas are unable to meet the
gain
or the
radiation pattern
requirements.

To create a
high gain antenna
, which radiates radio waves in a narrow beam pointed to
a desired direction, few techniques can be used:

  1. 1.
    One technique is to use
    large metal surfaces
    such as parabolic reflectors, horns or
    dielectric lenses which
    change the direction of the radio waves by reflection
    or
    refraction
    , to focus the radio waves from a single low gain antenna into a beam.

This type of antenna is called an
Aperture Antenna
. Parabolic dish and horns are
examples of aperture antennas. Their gain increases with increased dimension.

Parabolic dish and horn antennas

  1. 2.
    Increasing the
    size of the antenna
    (Antenna Aperture), the larger antenna becomes
    more directive due to the periodic current distribution
    across the antenna.

Although this method does not require external circuitry for control, the direction of
the beam is fixed and the number of sidelobes increases.

Examples include electrically long dipoles, horns, and waveguides.

  1. 3.
    An antenna that consists of a single driven element connected to the feed-line, and
    other elements which are not connected (called parasitic elements) is named
    Parasitic Array
    . Yagi-Uda antenna (invented in 1926 by
    Shintaro Uda
    and published
    by
    Hidetsugu Yagi
    ) is an example of Parasitic Array.
  2. -
    Yagi-Uda antenna use a driven element (which is a dipole or a folded dipole),
    and one or more parasitic elements, as reflectors and directors. This antenna
    can provide
    high gain in a particular direction
    (from the driven element to the
    directors).

Yagi-Uda antenna
Antenna Pattern

Optical equivalent of Yagi-Uda antenna

This results in a significantly more complex feeding network with higher losses
than the other methods.

-- In a Phased Array Antenna, the radio waves radiated by each individual
antenna element combine and superpose, adding together (interfering
constructively) to enhance the power radiated in desired directions, and
cancelling (interfering destructively) to reduce the power radiated in other
directions --

Phased Array Antennas

The
Phased Array Antenna
was invented in 1905 by
Karl F. Braun
.

Karl F. Braun also discovered the point-contact semiconductor (1874) and also invented
and built the first cathode-ray tube CRT, and the first CRT oscilloscope (1897).

He shared a Nobel Prize in Physics (1907) together with Guglielmo Marconi.

Karl F. Braun
mentioned in his Nobel Prize lecture the following experiments he did:

“
I found in 1902 that an antenna, inclined at somewhat less than 10° to the horizon, formed
a kind of directional receiver. The receptivity showed a clearly defined maximum for waves
passing through the vertical plane in which the antenna was situated. The results were
published in March of 1903. A directional transmitter is made up in the following way Fig12.

It is assumed that the antennae A and B, located at corners of an equilateral triangle, are
equal in phase, but are delayed by a quarter of a cycle of oscillation relative to antenna C,
which is in the third corner. The height CD of the triangle is to be a quarter wavelength.
The radiation will then prefer the direction CD. The wave emanating from C will reach AB at
the moment that A and B start to oscillate.

In Fig.13 is shown, schematically, the layout used.

The field was measured at a fair distance away, that is to say, in the so-called wave-zone.

There was satisfactory agreement between theory and observation, and the results were
checked in various ways. It was further shown that the experimental layout functioned in
the desired sense. By suitable distribution of the amplitudes in the three transmitters, a field
as in Fig. 14 was calculated (the
singly dotted curve is the measured field). The radial
vectors represent the range. If the roles of the three transmitters are exchanged - by simply
tripping a changeover switch - the preferred direction can be rotated through 120° or 60°.
”

The basic property of a
Phased Array Antenna
is that the relative position of the antenna
elements with respect to each other introduces relative phase shifts in the radiation vectors,
which can then add
constructively
in some directions or
destructively
in others.

This is a direct consequence of the translational phase-shift property of Fourier transforms:

A translation in Space or in Time becomes a Phase Shift.

Phased Array Antennas
are used to radiate power towards a desired angular
sector.

Once a Phased Array Antenna has been designed to focus the beam towards a particular
direction, the beam it can be
steered
towards some other direction by changing the relative
phases of the array elements (the physical antenna structure can be stationary).

This process is called
Steering
or
Scanning
.

By changing the relative amplitude and phase across the beam, we can steer the
beam and reduce the sidelobes of the resulting beam pattern.

Each antenna element has an excitation parameter, which is: current for a dipole,
voltage for a slot, and mode voltage for a multiple-mode element.

Array Antenna Elements

one needs to understand the radiated field intensity of an antenna element in the far-
field region.

element as a function of two far-field coordinates, while the radial distance remains

constant.

There are many categories of antennas: wire antennas (e.g., dipole, monopole, loop);
aperture antennas (e.g., horn); reflector antennas (e.g., parabolic, corner); lens antennas;
microstrip or printed antennas (e.g. patch, slot, PIFA).

Small individual antennas, such as quarter-wave monopoles and half-wave dipoles (or
derivates of them), don't have much directivity (or gain); they are omnidirectional antennas
which radiate radio waves over a wide angle. However, these antennas can be used as
antenna elements in a Phased Array Antenna system.

normalized with respect to the square root of the average radiated power per unit

solid angle.

with respect to that of an isotropic radiator that radiates an equal amount of power.

Due to their multiple advantages and easier implementation, one of the most used
antenna elements
are the
Printed Microstrip Antennas
.

Patch Antenna Elements in a Phased Array

A microstrip patch antenna consists of a very thin metallic patch placed a small fraction
of a wavelength above a conducting ground-plane.

The patch and ground-plane are separated by a dielectric. The patch conductor is
normally copper and can assume any shape, but simple geometries generally are used,
and this simplifies the analysis and performance prediction.

The substrate is usually non-magnetic. The relative permittivity of the substrate is
normally in the region between 1 and 4, which enhances the fringing fields that account for
radiation, but higher values may be used in special circumstances.

Patch Antenna with direct feed connection

The width
W
of the radiating edge, which is not critical, is chosen first.

If a square geometry is chosen it can be arranged to produce circularly polarized
waves. The length
L
is slightly less than a half-wavelength in the dielectric
(λ
g
/2).

The calculation of the precise value of the dimension
L
of the square patch is carried
out by an iteration procedure.

L
=
c / (2f
r
√ε
r
)

The current it is maximum and minimum at the center and beginning of the patch,
respectively.

The voltage is at a maximum of +V at the end, while it is at a minimum of −V at the
beginning.

Patch Antenna Impedance Matching and Return Loss

line printed on the same substrate, to transform the high input impedance of the patch
to a lower system impedance of 50
Ω or 75
Ω

However, the microstrip matching section is itself a radiating element due to the
discontinuity in line width, and the radiation from it may add to that of the antenna in ways
that are difficult to determine.

A match to 50
Ω
may be achieved by suitably locating the feed point.

The variation of
input resistance
R
(X)
as a function of feed-point position is approximated:

where
R
0
is the resistance at the edge of the patch and
x
is the distance from the edge.

Patch Antenna Cross-Polarization

For example, if an antenna is designed to be horizontally polarized, its cross-
polarization will be vertically polarized. This unwanted signal can cause interference
and reduce the overall performance of the antenna, especially in systems like Phased
Array Antennas where it can become significant when scanning (steering) the beam.

Feeding the Patch Antenna element

Different methods are available to feed the microstrip patch antennas. These
methods can be contacting and non-contacting methods.

There are many patch antenna feed methods but the four most used and popular feeding
techniques are: microstrip line feed, coaxial probe feed (both contacting schemes),
aperture coupling and proximity coupling (both non-contacting schemes), inset feed.

Line Feed has the advantage that the feed can be etched on the same substrate to
provide a planar structure. There are three main methods for
microstrip line feed
:

The inset feed distorts the equivalent slot radiation due to the change in geometry.

The λ/4 feed minimizes the equivalent slot field distortion due to the narrower,
high-impedance line required for impedance matching.

Inset feed
λ/4 transformer feed
Loosely gap coupled
Bottom layer feed

A coaxial connector is used to connect to the antenna (central pin connected to the
patch, and the outer conductor to the ground).

The major advantage of this is that the feed can be placed at any location inside the
patch in order to match with its input impedance.

The disadvantage is that it provides narrow bandwidth (5%) and is complex to model.

In this technique, the radiating patch and the microstrip feed line are separated by the
ground plane. The patch and the feed line are coupled through a slot in the ground
plane. The coupling slot is centered below the patch, leading to low cross polarization
due to symmetry of the configuration. Since the ground plane separates the patch
and the feed line, spurious radiation is minimized.

The main disadvantage of this feed technique is that it is difficult to fabricate due to
multiple layers, which also increases the antenna thickness.

This type of feed is also called as the electromagnetic coupling scheme.

Two dielectric substrates are used and the feed line is between the two substrates.

The radiating patch is on top of the upper substrate.

The main advantage of this feed technique is that it eliminates spurious feed radiation
and provides very high bandwidth (as high as 13%).

The major disadvantage of this feed scheme is that it is difficult to fabricate because
of the two dielectric layers which need proper alignment.

There is an equation to calculate the position of inset feed point for a desire
impedance, usually where the input impedance is 50 ohms:

where,
Y
o
is the distance from the feeding point to the edge of the patch,
L
is the
patch length, and
Z
in
and
R
in
are the resonant input impedance and resonant input
resistance respectively.

Patch Antenna Inset Feeding

Slot Antenna Elements in a Phased Array

A
slot antenna
consists of a metal surface (ground plane), usually a flat plate, with one or
more holes or slots cut out.

Thus, the radiation pattern of a slot antenna can be the same as of a dipole antenna.

Microstrip slot antenna fed by a center oriented microstrip line

Where
W
is the width of the slot,
L
is the length of the slot,
f
r
is the resonant frequency,
ε
o
is the permittivity in free space,
μ
o
is the permeability in free space,
ε
r
is the dielectric
substrate,
ε
reff
is the effective dielectric constant of the substrate, and
d
is the height of the
substrate. In the equation could be also a
ΔL
(mainly related to the height of the substrate),
which is the extended increment length, that should be added to the length of the slot, but
usually this is not a significative length.

In a conventional
microstrip-fed slot antenna
(figure above), a narrow rectangular slot
is cut in the ground plane, and the slot is excited by a microstrip feedline with a short or an
open circuit termination.

Four elements rectangular slotted array antenna

For round-ended slots, the modified round-ended slot length values, differed from the
typical rectangular slot lengths by 1% to 3% only.

When the radiowaves are conducted by a waveguide, and the antenna consists of multiple
slots in the waveguide, this is called a
slotted waveguide array antenna
.

Longitudinal Slotted Waveguide Array Antenna

There are two widely used types of slotted waveguide array antennas:

1.
Longitudinal slotted waveguide array antenna
, where the slots' axis is parallel
to the axis of the waveguide, and the antenna pattern is similar to a collinear antenna.

Slotted waveguide antenna (offset from the center line)

There are a number of ways for suppressing unwanted off-axis lobes. Each of them tries to
have a uniform E-field (a collinear array) on the aperture:

different waveguide heights on the sides of the ridge.

or posts placed in the waveguide.

region and the non-linear placement effect of the slots will be eliminated.

This solution gives restrictions to the array size.

2. Transverse slotted waveguide antenna
, where the slots are almost
perpendicular to the axis of the waveguide but skewed at a small angle, with alternate
slots skewed at opposite angles.

Transverse slotted waveguide antenna

Due to this sharp radiated beam, this array antenna is used in microwave marine
radars, mounted horizontally on a mechanical drive that rotates the antenna scanning
the arrays fan-shaped beam by 360°.

A familiarization with the modal fields within a waveguide is necessary to understand
where to place slots, so that they are properly excited.

Slots cut in the walls of a rectangular waveguide (Volakis)

edge. Therefore, by offsetting longitudinal slots such as slot
a
from the center of the
waveguide, the power coupled to the slots can be adjusted.

Feeding the Patch Antenna Elements in a Phased Array

There are few possibilities how to feed and how to arrange patch elements to form a
Phased Array Antenna.

Each feeding option have advantages and disadvantages on their implementation.

Series Feeding Linear Arrays
Corporate (Parallel) Feeding Linear Arrays
Combined Parallel/Series feeding

Offer the possibility of frequency scanning by changing frequency which changes
the electrical line length between elements, and thus the phase.

A series feed introduces dispersion that distorts short pulses and limits bandwidth.

The equal line lengths in corporate feed makes the network frequency
independent, and thus wide bandwidth.

Such an antenna does not have the obvious identical performance for every element
that the infinite active array has. Furthermore, reflections from the elements do not
necessarily return to the generator, but may be absorbed in terminations or reradiated
in other directions depending on the particular network used. Thus, the passive array
is actually more complicated to analyze than the active one. The active array not only
is relatively simple, but also it provides the concepts and fundamental limitations for
any array antenna.

Hertzian Dipole Array Antennas

The infinitesimal
Hertzian Dipoles
(dipole with total length <<
λ
) are the simplest
antenna elements outside of point sources (isotropic).

Placing several of these dipole antennas in the vicinity of each other causes the dipole
elements to interact. In other words, the dipoles all radiate and receive time-varying fields
from each other. This interaction is called
mutual coupling
, which will be discussed later.

Examples:

Ex.1)
. An 8-element linear array lying along the
x-axis
with spacing
d
=
λ
/2 has the array
antenna pattern shown in figure below.

(8 isotropic elements)
(8 Hertzian dipole elements)

Antenna pattern of an 8-element linear array (dipole elements placed on
x
-axis and oriented in
z
-direction)

The peak occurs at
Ф
= 90°
for all
ϴ
angles.

The directivity of this dipole array is 11.9dB and can be calculated using numerical
integration of the
array factor
times the
element pattern.

Ex.2)
. An 8-element linear array lying along the
z
-axis with spacing
d
= λ/2
has the array
antenna pattern shown in figure below:

(8 isotropic elements)
(8 Hertzian dipole elements)

Antenna pattern of an 8-element linear array (dipole elements placed on
z
-axis and oriented in
z
-direction)

Array Antenna Radiation Patterns

Antenna pattern in polar 2D coordinates
Antenna pattern in rectangular coordinates in dB

Antenna fields pattern in 3D coordinates

For
Phased Antenna Arrays
using identical radiating antenna elements, there are at least
five types of controls that can be used to shape the overall pattern of the antenna system:

1.
The
geometrical configuration
of the overall Array Antenna (linear, circular,
rectangular, spherical, etc.).

2.
The relative
spacing
between the radiating elements.

3.
The excitation
amplitude
of the individual radiating elements.

4.
The excitation
phase
of the individual radiating elements.

5.
The relative
pattern
of the individual radiating elements.

Array Factor (AF)

where
N
is the number of antenna elements,
a
i
is the excitation amplitude,
d
i
is the inter-
element spacing,
ϕ
i
is the excitation phase, and
k
is the propagation constant for the
i
th
element.

In the figure below is plotted the Array Factor (AF) of an antenna array by varying the
number of the antenna elements while keeping the spacing
d
between two consecutive
elements as λ
/2
and ϕ
i
= 90°. The results are normalized with respect to the maximum
value of the main lobe.

Array Factor (AF) varied with number of antenna elements (N from 2 to 12)

In the plot below could be observed the effect of spacing
d
on the radiation characteristics
of an array antenna. A simulation has been carried out with spacing
d
changing from λ/2 to
a relatively small separation, up to
d
= λ/10
.

It has been demonstrated that the distance
d
between the elements should be close to λ
/2.

Array Factor (AF) varied with distance between elements
, d decreasing from λ/2 to λ/10

In the plot below, a comparative analysis of weighted and un-weighted array antenna
design is presented. Could be observed that for equal amplitude excitation, the
Half Power
Beam Width
(HPBW) is decreased and the
Side Lobe Level
(SLL) is increased, while for
unequal amplitude excitation, the SLL is reduced and the HPBW is increased.

Array Factor (AF) varied for equal and unequal excitation amplitude

Array Factor (AF) varied with N and optimized excitation amplitude

Array Factor (AF) when inter-element spacing was optimized

The
Normalized Array Factor
𝐟(𝚿)
for an
N
element, uniformly excited, equally spaced
linear array (UE, ESLA) that is centered about the coordinate origin is:
f(Ψ)
=
sin (
NΨ/2)
N sin(Ψ/2)

where the
wave number
(array phase function):
Ψ
= k
∙
d
∙
sin
θ + δ

δ
is the phase difference between the two sources, and
θ
is the angle of incidence or the
angle between the wave vector
k
and the array axis.

Conclusions after analyzing the Array Factor plots for various number of elements
N
:

The number of full lobes (one main lobe and the side lobes) in one period of
f(Ψ)
equals N - 1.

There will be N - 2 side lobes, and one main lobe in each period.

𝐒𝐋𝐋
=
|𝐦𝐚𝐱𝐢𝐦𝐮𝐦 𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 𝐥𝐚𝐫𝐠𝐞𝐬𝐭 𝐬𝐢𝐝𝐞 𝐥𝐨𝐛𝐞|
|𝐦𝐚𝐱𝐢𝐦𝐮𝐦 𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 𝐦𝐚𝐢𝐧 𝐥𝐨𝐛𝐞|
(often expressed in dB)

HPBW vs SLL in a 30 element Linear Array Antenna

The
Directivity
of the array antenna can be defined as the ratio of radiation intensity in a
given direction from the array antenna to the radiation intensity averaged over all directions.

Therefore,
inter-
element spacing equal to λ
/2 is favored to achieve higher
directivity in planar array antenna
.

2D plot of directivity for optimized excitation amplitude and inter-element spacing

Array antennas can form a
highly directive beam (high directivity) by controlling
the phase and amplitude of each element
. This allows for constructive interference
in a desired direction and destructive interference elsewhere, focusing the radiated
power.

A smaller aperture is needed for a larger EOC coverage area.

However, the amplitude distribution should not be uniform for dissimilar elements;
thus the maximum-gain theorem has very important significance in designing a pencil
beam array with dissimilar elements.

Another important application of linear array antenna is the
Null Control
:

Ideally, the power is decreased in the direction of interferers and the main beam is
increased in the same direction. Generally, it is hard to accomplish this, and it is
needed to do a tradeoff.

Array
Factor (AF) when nulls are imposed at θ=50°, 55°, 125°, and 130°

Array Antenna Patterns
–
Broadside (Boresight) and End-fire

Ψ
=
β
+
kd cosθ
|
θ = 90°
= 0

=> β
= 0 (phase angle)

For a broadside array, in order the above equation to be satisfied with θ = 90°,
the
phase
angle β
must be zero, which means, all elements of the Array Antenna must be driven with
the same phase.

Ψ
=
(
2Π
λ
d cosθ)
=
Π
cosθ

The maximum of the end-fire array factor occurs when the array phase function
Ψ
= 0

Ψ
= β + kd cosθ |
θ=0° or θ=180°
= 0

β =
-
kd for θ = 0°

β = kd for θ = 180°

HPBW = 0.886
λ
Nd
csc θ
0
near broadside

and

HPBW =2
√0.886
λ
Nd
end-fire

(θ
o
= main beam pointing angle)

Example
: A broadside beam (with maximum at
θ
=90°), if the pattern goes first to
zero at 60° and also at 120°, the
First Null Beamwidth (FNBW)
is: 120° - 60° = 60°

The main beam nulls are where the
Array Factor
𝐟(𝚿)
first goes to zero in a plane
containing the linear array.

For long array
antennas (length L = Nd >> λ), we can approximate the
First Null
Beamwidth FNBW
(or Null-to-Null Beamwidth) as follows:

FNBW
=
2λ
Nd
near broadside

FNBW
= 2
√
2λ
Nd
end-fire

Antenna Array directivity is determined entirely from the radiation pattern.

The directivity
D
of a broadside array of isotropic elements is given by:

D = 2
Nd
λ
where: N=nr. of elements, d=spacing between elements

Instead, they have
directionality
that is proportional to their size.

The antenna elements also have
frequency, impedance
, and
polarization
properties which are not associated with isotropic point sources.

As such, the
element pattern
is too small to have sidelobes.

A typical element pattern for an antenna array in the X-Y plane can be reasonably
approximated by
cosϴ
or
sinФ
or the change in the projected area of the element.

Array pattern = Element pattern
x
Array factor

There are several important differences between the
array pattern
and the
array factor:

However, it is possible to orient the elements in a way that causes the array antenna
pattern to have a different polarization from the element pattern.

The element pattern enhances the array factor in the direction of the element pattern
peak and suppresses the array factor in direction of element pattern minima.

element pattern, as was stated above. The important subtlety in pattern multiplication
is that while the array factor scans with phase change, the antenna element pattern
remains fixed. As a phased array is steered, the peak of the total array pattern follows
the element pattern shape.

Thus, the
array element patterns
remain fixed in space.

Thus, the product of the array factor and main beam changes as the main beam
is steered.

Thus, a correction to the steering phase for the array factor is necessary to
make sure that the peak of the antenna pattern points to the desired angle.

Null-free Array Antenna Pattern

Some antenna applications require patterns without nulls, as the airport beacon
antennas which must radiate uniformly, to be able to communicate with aircrafts arriving
from all directions.

To obtain a null-free array antenna pattern, typically need to optimize the array's element
excitations and their arrangements to minimize or eliminate nulls in the radiation pattern.

Here are some common approaches:

Phased Array Antenna Beamforming

A sharper beam means a narrower 3dB beamwidth (HPBW).

Linear antenna arrays with 8 and 16 elements and their antenna pattern

Two-dimensional array antennas and their antenna pattern

Principle of Beamforming

There are three main approaches to get Beamforming in an Array Antenna system:

The RF signal is phase adjusted per antenna element in the RF domain, for a single
signal after DAC in transmit mode. Analog beamforming can only apply a spatial filter
on a single signal. Analog beamforming is relatively cheap to integrate and allows
better coverage of a system.

Analog Beamforming

Advantages of Analog Beamforming:

Disadvantages of Analog Beamforming:

Digital Beamforming

It also allows frequency-selective beamforming.

Advantages of Digital Beamforming:

Disadvantages of Digital Beamforming:

In Hybrid Beamforming, there is a formation of analog (sub-array) beams from a portion of
the full array.

Hybrid Beamforming
Formation of analog (sub-array) beams

If the calibration of these errors is implemented, the resulting array sidelobe level
decreases further.

Array Antenna Scanned Beam (Beam Steering)

But in some applications the goal is to direct the main lobe of the radiation pattern at an
angle other than the
broadside
or
end-fire
directions.

The scan angle of the pattern dictates this
steered angular location
.

This is the basic principle of
electronic scanning
for phased array antennas.

The antenna elements phases are adjusted to form a phase front that is planar and
oriented to steer the main beam in a direction normal to the formed planar wavefront.

At +/- 60º from broadside a phased array antenna exhibits half the gain at
broadside, and zero at end-fire conditions (+/- 90º from broadside).

-
Phase scanning
. The beam of antenna points in a direction that is normal to the
radiated phase front. In phased array antennas, this phase front is adjusted to steer
the beam by individual control of the phase excitation of each radiating element.

The phase shifters are electronically actuated to permit rapid scanning and are
adjusted in phase to a value between 0 and 2
π
rad.

-
Time-delay scanning
. Phase scanning are frequency-sensitive. However, time-
delay scanning is independent of frequency. Delay lines are used instead of phase
shifters, providing an incremental delay from element to element. Individual time-
delay circuits are naturally too cumbersome to be added to each radiating element,
and a reasonable compromise may be reached by adding one time-delay network to
a group of elements (subarray) where each element has its own phase shifter.

Time-delay overcomes instantaneous bandwidth limitation of phase shifters.

-
Frequency scanning
. Frequency rather than phase may be used as the active
parameter to exploit the frequency-sensitive characteristics of phase scanning. At one
particular frequency, all antenna radiators are in phase. As the frequency is changed,
the phase across aperture tilts linearly, and the beam is scanned. Frequency-
scanning array antennas are relatively simple and inexpensive to implement.

-
Beam switching
. Avoids use of variable phase shifters. With properly designed
antenna lenses or reflectors, a number of independent beams may be formed by
feeds at the focal surface. Each beam has substantially the gain and beamwidth of
the whole antenna. All the beams lie in one plane, and greater antenna complexity is
required for switching beams in both planes.

-
Digital beamforming
. For receiving, the output from each antenna element may be
amplified and digitized. The signal is then processed by a computer, which can
include the formation of multiple simultaneous beams (formed with appropriate
aperture illumination weighting) and adaptively derived nulls in the beam patterns to
avoid spatial interference or jamming.

-
Analog or digital phase shifters
(ferrite or semiconductor diodes) are also used
for beam scanning.

- Phase scanned in only a plane containing line of elements.

- Beamwidth in a plane perpendicular to the line of element centers is determined by
the element beamwidth in that plane (limitation of realizable gain).

When the scanning is required to be continuous, the feeding system must be capable of
continuously varying the progressive phase
θ
between the elements.

Assuming that the maximum radiation of the array antenna is required to be oriented at
angle
θ
0
, or
in other words, to “electronically” rotate, or steer, the array pattern towards
some other direction, without physically rotating the antenna.

To accomplish this, the progressive phase excitation
β
between the elements must be
adjusted so that:

Ψ
= β +kd cosθ
0
= 0

β =
-
kd cosθ
0
which is named
steering phase

Basically, a
Phased Array Antenna
system is a combination of
N
antennas made to get:

The fundamental configuration for elements in an array is the
Linear Antenna Array
shown in the picture below.

Scanned beam array block diagrams

Array Antenna Beam Distortion (Scan Loss)

For example, for
patch antenna elements
, the HPBW is limited to about 90°-100°.

For rectangular arrays, the 3dB beam contour is approximately elliptical.

Amplitude weighting of each antenna element for reducing grating lobes during steering

On the other hand, if the distribution has an inverse taper with maximum amplitude at
the edges and none at the center of the array (edge distribution), the minor lobes are
accentuated, being in fact equal to the main lobe.

Array Radiation Pattern = Primary Pattern x Array Factor

Frequency Scanning - Beam Squint

W
hen a wavefront approaches an array of elements, there’s a time delay between
elements based on the wavefront angle
θ
relative to broadside radiation.

Such a situation can be intuitively explained if is acknowledged that a time delay is a linear
phase shift vs. frequency.

Previously, was shown that the phase shift
ΔФ
applied to the elements across the array
antenna as a function of beam angle is:

where:
d
is the distance between antennas,
ϴ
is the beam angle, and
λ
is the wavelength.

Inversely, can compute the beam angle as a function of the phase shift:

The phase in the last equation is periodic and repeats every
2
π
.

Using equations above, the beam direction deviation (beam squint
Δϴ
) can be calculated:

where:
f
o
is the center frequency,
f
o
/f
is the frequency deviation, and
ϴ
o
is the angle at
f
o
.

A few observations on beam squint:

Implementing
beam steering
with
true time-delay
units doesn’t have this problem
.

Grating Lobes of Phased Array Antenna

direction of the main beam is consequently reduced.

Array Antennas with inter-element spacing
d
greater than the wavelength
λ
,
always have Grating Lobes (multiple main beams or fringes).

Usually, a side lobe has much lower peak intensity than that of a grating lobe.

Grating lobes of two-element antenna array

where
β
max
is the largest phase difference, and
ϴ
max
is the scan angle that maximize the
phase difference
β
where the first grating lobe occur.

lobe. The level of a grating lobe is equal to that of the main beam, since a grating

lobe is a repeated main beam.

the grating lobe itself, copies of the radiation pattern around broadside.

The Ordinary End-fire Array

Another way to achieve a single pencil beam is by the proper design of an end-fire array.

Ordinary end-fire conditions to produce a single pencil beam are:

d <
λ
2
(1
−
1
2N
)
;
α = +/
-
βd

An array satisfying these conditions produces a single end-fire beam and no grating lobe
with a peak in the direction θ = 0° for α =
-
βd and in the direction θ = 180° for α = βd.

Thinned Phased Array Antenna

Regular
thinning
produces grating lobes, but these can be partially suppressed by
randomizing the element spacings.

A special kind of thinned array uses variable element spacing to produce an
equivalent amplitude taper. The goal is to produce a sidelobe that tapers down.

a periodic lattice in order to obtain a
spatial taper
that results in low sidelobes.

The normalized desired amplitude taper serves as a probability density function for a
uniform array that is to be thinned.

Elements with an amplitude of one are connected to the feed network, while elements
with an amplitude of zero are connected to a matched load and do not contribute to a
signal to the array output.

Nonuniformly Spaced Array Antennas

Sidelobe synthesis of array antennas could be done also by variation of antenna
elements
spacings
, designing a
nonuniformly spaced array antenna
.

Initial attempts at nonuniformly spaced arrays were based upon trial and error.

Mutual Coupling between Antenna Elements

The array element radiation pattern is determined as a pattern taken with a feed at a
single element in the array, and all other elements are terminated by the matched
loads.

Example
: if we place two antenna elements (
A
and
B
) nearby, the resulting antenna
coupling will affect both antenna radiations and their terminal properties as is mentioned

If the antenna
A
is driven with a signal, the radiated field by antenna
A
and
intercepted by antenna
B
, will cause also a radiation from antenna
B
.

Hence, the total radiation is a combination of
A
and
B
antenna elements, so the
effective radiation when we drive antenna
A
is thus changed.

Power radiated by antenna element
B
is intercepted again by antenna
A
, which
results in changes of current flowing to antenna
A
, so the input impedance of
antenna element
A
is modified.

Further, some power will be received by antenna
A
, thus the current on antenna
A
is modified (so the input impedance of the antenna element
A
is modified).

- the radiation patterns of the antenna elements.

- the distribution of the antenna elements near-fields.

- the spacing between antenna elements.

- antenna elements loading.

This effect can lead to a
poor impedance match
and often, depending on the feed
mechanism, to a
poor aperture illumination
and increase of sidelobes.

The
impedance match
is very dependent on the
phase slope
.

Impedance matrix of the antenna array contains all the inter-element mutual
impedances.

Mutual impedance is computed by placing a generator on the input of one antenna,

finding the voltage appearing at the input terminals of the second antenna, and then
taking the ratio of the voltage to the current. The mutual impedance values decrease
with increasing separation distance between antenna elements.

A second way to quantify coupled arrays in addition to the mutual impedance
approach is to use the scattering parameter method that is commonly used in RF
circuit analysis.

The returned power is caused by departure from conjugate match between the
antenna element impedance and the generator impedance.

The element active impedance is the impedance measured while all the other
elements are excited with the appropriate phase.

Also, the element pattern, measured in the proper environment of surrounding
elements, deviates from the ideal pattern in proportion to this effect.

Examples of impedance measurements for coupling between antenna elements

No. rows

No.
columns

Total number
of elements

Reactive portion of
impedance (ohms)

3

3

9

+j5.9

5

5

25

+j3.2

7

7

49

+j2.4

9

9

81

+j2.1

11

11

121

+j2.0

25

25

625

+j1.8

Reactive impedance of a central antenna element vs array size

Conclusions:

Element position

Reactive portion of
impedance (ohms)

Center element (0,0)

+j2

Edge element of the center row

+j7

Corner element

+j12

Isolated element (Reference)

+j21

Reactive impedance of an antenna element as a function of its location in a 11x11 array

Conclusions:

The distance between the adjacent elements i
s about half of the wavelength λ/2.

Element #

1

2

3

4

5

S
0n
(dB)

-10

-22

-32

-38

-43

Coupling coefficient (in dB) between central array element and the element number n

Conclusions:

EM simulation software
is commonly used to simulate antennas with multiple feeds,
including phased arrays, stacked radiators with different polarizations, and single
apertures with multiple feed points.

An extreme example of this is
scan blindness
, where the interaction between the
elements causes no radiation to occur at certain scan angles.

It is critical to simulate the feed network itself since resonances can build up due to
the loading at the antenna ports.

Frequency Bandwidth of Array Antenna

In both effects, it is the path-length differences that contribute to the frequency
bandwidth sensitivity of a phased array.

For a parallel-fed array (equal line length), the feed network does not contribute to a
change in phase with frequency, and so only the aperture effect remains.

for wideband operation, constant lengths rather than constant phases are required.

The approximate limit is given by:
Bandwidth (%)
≈
Beamwidth (°)

Antenna Element Failure Analysis

For example, a rectangular 16x4 array antenna
with λ/2 spaced elements, if some of
the antenna elements fails working, the antenna array pattern will change, as is shown in
the plots below.

Antenna pattern of a 16x4 array antenna with failing elements

Array Element failure, always results in side lobe response degradation.

Scan Blindness

Scan Blindness Phenomenon

In the transmitting case, when the feed network is configured to steer the beam to a
blind scan angle there is power reflected back into the transmitter, which can cause
damage.

In a microstrip array antenna, the severe surface waves increase the mutual coupling
between array elements, which cause impedance and pattern anomalies associated
with
scan blindness.

The interaction between antenna elements degrades S-parameters, and as a result,
the scan blindness of the radiation pattern for the adjacent coupled elements in the
array increases.

The mutual coupling can result in severe degradation to
the antenna’s radiation
characteristics. While surface waves are weakly excited in very thin grounded
dielectric substrates, space-waves dominate and show strong coupling when
antennas are in close proximity.

It can usually be avoided by using smaller element spacings, and very careful use of
dielectric substrates or superstrates, or any coupling structures like transmission lines
and baluns at the array face.

However, the maximum scan angle limits how far apart elements can be in order to

avoid grating lobes as given by and to avoid scan blindness.

Realization problem of cavity-backed structures require a two-step fabrication
process; one is the conventional PCB process to print the planar radiating elements
and the other is a CNC machining or metal casting processing in order to fabricate
the waveguide metalized cavities.

Array Factor plots for Phased Array Antennas

In 1937
George Brown
working for RCA published for the first time the
array factor
plots
for Phased Array Antennas with two isotropic elements with equal amplitude
excitations for various combinations of excitation phase
α
, and element spacing,
d
.

The plots also show a unit circle (dashed) representing the radiation from a hypothetical
isotropic point source with the same input current.

Patterns of Phased Array Antennas with two isotropic elements (G.Borrow)

Sparse Array Antennas

The arrangement of the antenna elements on the array aperture is chosen such that
the synthesized continuous aperture current distribution can be sampled optimally in a
discrete manner.

This can be done either by placing the unequally fed elements at equal intervals
separated by a half-
wavelength (λ/2) or smaller, or by placing equally fed elements at
nonuniform separations. The former category is called the
Dense Antenna Arrays (DAA)
and the latter one is called the
aperiodic or nonuniform antenna arrays
.

The configuration of the sparse array design is widely needed in various practical
communication systems that have limited antenna size and weight, as well as cost
efficiencies, such as radar equipment, satellite communications, and other space
communications.

LDS arrays could use about 80% less elements than a fully populated array on a
square grid.

A
random distribution of elements in the sparse array lowers the grating lobes
to a level of the surrounding sidelobes
.

An antenna element is said to be essential if its failure/absence alters or introduces
holes into the antenna pattern.

Sparse array antenna design techniques can be classified as:

- Mathematical calculation methods and structured algorithms,

- Probability statistics and stochastic processes,

- Polynomial factorization method,

- a combinatory mathematical method by using cyclic different sets,

- Low Discrepancy Sequence method,

- Mutual coupling effects method,

- Fourier transform-based techniques,

- Deterministic methodologies,

- Hybridization of the above techniques.

Sparse Array antenna optimization methods:

- Gradient-based optimizers,

- Array thinning,

- Convex optimization,

- Random sampling,

- Utilize special array structures.

Advantages of Sparse Array Antennas:

Disadvantages of Sparse Array Antennas:

The sparse array antenna configuration can be synthesized based on the desired radiation
pattern. This pattern is generated based on the phase excitation parameters of each
element, the amplitude, and the relative phase difference between the array elements.

Example of Sparse Array Antenna design:

The configuration of a sparse array antenna is designed by developing a 9-element sparse
array, as shown in figure below:

9-element sparse array antenna

Determination of the distance between the elements of the sparse array could be carried
out using equation:

d
n
= 1/I
n

where:
d
n
is the coefficient of distance between array elements, and
I
n
is the excitation
coefficient of each element amplitude.

Furthermore, the amplitude current excitation coefficient will be transformed to the element
spacing coefficient for each element and multiplied by the ideal distance between array
elements λ/2.

This parameter is the coefficient for determining the distance of each element in the sparse
array configuration and determining its location in the sparse array arrangement.

The process of this method starts by determining the aperture length of the sparse array
antenna configuration and calculating the distance coefficient between elements
d
n
for each
array element. Then, we calculate the number of elements that match the aperture length,
then multiply the distance coefficient
d
n
by the distance between standard elements λ/2,
and finally set up a linear configuration of the sparse antenna array from the result of
multiplying the distance coefficient by λ/2.

The radiation pattern produced by this 9-element sparse array antenna configuration has a
main lobe magnitude of 25.1dB with a half power bandwidth (HPBW) of 0.5° and a peak
SLL performance of −33.4dB. These results show a significant improvement when
compared to the conventional uniform spacing array configuration with equal aperture,
which has the main lobe magnitude of 19.6dB with HPBW 3.8° and a peak SLL of −13.3dB.

Phase Shifters used in Electronically Controlled Phased Array Antennas

The
scanning accuracy
of a given volume is dependent upon the size of the
incremental step or the
phase shift per step
of the phase shifters.

This can be effectively accomplished by reducing the magnitude of the incremental
steps, more steps are needed to provide the proper amount of total phase shift from
the phase shifter.

Many phase shifters are analog devices, wherein the differential phase between
states is a function of voltage or pulse length or some other analog parameter.

Each phase shifter has a special electrical control circuit that can change
progressively the phase of the received (or transmitted) signal.

Digital Phase Shifters
realize phase shifts with a discrete difference equal to

Δ
= 2
Π
/ 2
q
, where
q
is the number of bits, and
2
q
is the number of states of the
digital phase shifter.

Digital Phase Shifters
–
Number of phase steps vs Number of bits

A two-bit phase shifter (q=2) provides four phase states within the phase range:
0°-360° (0°, 90°, 180°, 270°), with discrete steps of
90°
(360° / 4).

The array factor value does not change for scan angles from 21° to 30° and from
30.5° to 40.5°.

A three-bit phase shifter (q=3) gives eight possible phase states starting from 0° to
360°, with discrete steps of
45°
(360° / 8). (0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°)

The beam pointing remains constant for the angular ranges: 43°-49° and 49°-55.5°.

Phase error due to digital phase shift steps

It is seen that the error between the ideal curve and its approximation is a periodic
function of the X coordinate.

Quantization lobe values depend only on the scanning angle position and do not
depend on the array amplitude distribution.

In this case the phase errors assume a random nature.

case is approximated by a continuous case.

Phase Shifters topologies

From the phase
ψ
equation, we can see the relation of phase with other parameters:

where
f
is frequency,
l
is length,
ε
is permittivity (dielectric constant), and
μ
is permeability.

The phase equation above reveals all phase-shifting possibilities at a glance.

The possibilities are:

These phase shifters of a rod of ferrimagnetic material, centrally positioned inside a
waveguide, where a solenoid is wound around the waveguide.

By changing the current through the solenoid, the magnetic field is changed and
thereby the
permeability
of the ferrimagnetic rod and thus the phase of a wave going
through the waveguide is changed. The phase can be changed continuously, function
of current, making this phase shifter an analog phase shifter.

switched-line

Phase shifters can use switched transmission lines and PIN diodes to change the phase in
discrete phase steps.

Switched-line Phase Shifters
(d)

The standard switched-line Phase Shifter is using switched transmission line segments,
getting different path length and determining in this way the amount of phase shift.

𝚫
=
𝟐𝛑 (𝐋𝟐
−
𝐋𝟏)
𝛌

where
Δ
is the phase shift, (L2 - L1) is the difference between the physical lengths
of the delay line L2 and the reference line L1, and
𝛌
g
is the
guide wavelength
.

𝛌𝐠
=
𝛌
√𝛆𝐫
𝛌𝐠
=
𝐜
𝑓√𝛆𝐫

𝛌
is the wavelength in vacuum,
ε
r
is the dielectric constant of the substrate,
f
is
the frequency, and
c
is the speed of light.

In this circuit, for the 180° phase shift, both PIN diodes are in the OFF position, and the RF
signal passes through line L2 with λ/2 length, providing a 180° phase shift.

The shunt diode is placed at the middle of L2, at λ/4 wavelength
from its ends.

Three-Bit switched-line phase shifter and Signal Path

The term “phase
shifter” often includes any of
these means for changing the phase:

delay and phase shift are combined. This will reduce the spreading of a pulse when

the beam is scanned compared to a phase-only feed.

References:

http://www.qsl.net/va3iul