# Parabolic antenna

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A parabolic antenna is an antenna that uses a parabolic reflector, a curved surface with the cross-sectional shape of a parabola, to direct the radio waves. The most common form is shaped like a dish and is popularly called a dish antenna or parabolic dish. The main advantage of a parabolic antenna is that it has high directivity. It functions similarly to a searchlight or flashlight reflector to direct the radio waves in a narrow beam, or receive radio waves from one particular direction only. Parabolic antennas have some of the highest gains, meaning that they can produce the narrowest beamwidths, of any antenna type. [1] [2] In order to achieve narrow beamwidths, the parabolic reflector must be much larger than the wavelength of the radio waves used, [2] so parabolic antennas are used in the high frequency part of the radio spectrum, at UHF and microwave (SHF) frequencies, at which the wavelengths are small enough that conveniently-sized reflectors can be used.

In radio engineering, an antenna is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an electric current to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves. In reception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to be amplified. Antennas are essential components of all radio equipment.

A parabolicreflector is a reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis. The parabolic reflector transforms an incoming plane wave traveling along the axis into a spherical wave converging toward the focus. Conversely, a spherical wave generated by a point source placed in the focus is reflected into a plane wave propagating as a collimated beam along the axis.

In mathematics, a parabola is a plane curve that is mirror-symmetrical and is approximately U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define exactly the same curves.

## Contents

Parabolic antennas are used as high-gain antennas for point-to-point communications, in applications such as microwave relay links that carry telephone and television signals between nearby cities, wireless WAN/LAN links for data communications, satellite communications and spacecraft communication antennas. They are also used in radio telescopes.

In telecommunications, a point-to-point connection refers to a communications connection between two communication endpoints or nodes. An example is a telephone call, in which one telephone is connected with one other, and what is said by one caller can only be heard by the other. This is contrasted with a point-to-multipoint or broadcast connection, in which many nodes can receive information transmitted by one node. Other examples of point-to-point communications links are leased lines, microwave radio relay and two-way radio.

Microwave transmission is the transmission of information by microwave radio waves. Although an experimental 40-mile (64 km) microwave telecommunication link across the English Channel was demonstrated in 1931, the development of radar in World War II provided the technology for practical exploitation of microwave communication. In the 1950s, large transcontinental microwave relay networks, consisting of chains of repeater stations linked by line-of-sight beams of microwaves were built in Europe and America to relay long distance telephone traffic and television programs between cities. Communication satellites which transferred data between ground stations by microwaves took over much long distance traffic in the 1960s. In recent years, there has been an explosive increase in use of the microwave spectrum by new telecommunication technologies such as wireless networks, and direct-broadcast satellites which broadcast television and radio directly into consumers' homes.

A wireless network is a computer network that uses wireless data connections between network nodes.

The other large use of parabolic antennas is for radar antennas, in which there is a need to transmit a narrow beam of radio waves to locate objects like ships, airplanes, and guided missiles, and often for weather detection. [2] With the advent of home satellite television receivers, parabolic antennas have become a common feature of the landscapes of modern countries. [2]

Radar is a detection system that uses radio waves to determine the range, angle, or velocity of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. A radar system consists of a transmitter producing electromagnetic waves in the radio or microwaves domain, a transmitting antenna, a receiving antenna and a receiver and processor to determine properties of the object(s). Radio waves from the transmitter reflect off the object and return to the receiver, giving information about the object's location and speed.

An airplane or aeroplane is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine, propeller or rocket engine. Airplanes come in a variety of sizes, shapes, and wing configurations. The broad spectrum of uses for airplanes includes recreation, transportation of goods and people, military, and research. Worldwide, commercial aviation transports more than four billion passengers annually on airliners and transports more than 200 billion tonne-kilometres of cargo annually, which is less than 1% of the world's cargo movement. Most airplanes are flown by a pilot on board the aircraft, but some are designed to be remotely or computer-controlled.

The parabolic antenna was invented by German physicist Heinrich Hertz during his discovery of radio waves in 1887. He used cylindrical parabolic reflectors with spark-excited dipole antennas at their focus for both transmitting and receiving during his historic experiments.

Heinrich Rudolf Hertz was a German physicist who first conclusively proved the existence of the electromagnetic waves theorized by James Clerk Maxwell's electromagnetic theory of light. The unit of frequency, cycle per second, was named the "Hertz" in his honor.

## Design

The operating principle of a parabolic antenna is that a point source of radio waves at the focal point in front of a paraboloidal reflector of conductive material will be reflected into a collimated plane wave beam along the axis of the reflector. Conversely, an incoming plane wave parallel to the axis will be focused to a point at the focal point.

In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by aberrations of the imaging optics. In the absence of significant aberrations, the smallest possible blur circle is the Airy disc, which is caused by diffraction from the optical system's aperture. Aberrations tend to get worse as the aperture diameter increases, while the Airy circle is smallest for large apertures.

In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space.

A typical parabolic antenna consists of a metal parabolic reflector with a small feed antenna suspended in front of the reflector at its focus, [2] pointed back toward the reflector. The reflector is a metallic surface formed into a paraboloid of revolution and usually truncated in a circular rim that forms the diameter of the antenna. [2] In a transmitting antenna, radio frequency current from a transmitter is supplied through a transmission line cable to the feed antenna, which converts it into radio waves. The radio waves are emitted back toward the dish by the feed antenna and reflect off the dish into a parallel beam. In a receiving antenna the incoming radio waves bounce off the dish and are focused to a point at the feed antenna, which converts them to electric currents which travel through a transmission line to the radio receiver.

In parabolic antennas such as satellite dishes, a feed horn is a small horn antenna used to convey radio waves between the transmitter and/or receiver and the parabolic reflector. In transmitting antennas, it is connected to the transmitter and converts the radio frequency alternating current from the transmitter to radio waves and feeds them to the rest of the antenna, which focuses them into a beam. In receiving antennas, incoming radio waves are gathered and focused by the antenna's reflector on the feed horn, which converts them to a tiny radio frequency voltage which is amplified by the receiver. Feed horns are used mainly at microwave (SHF) and higher frequencies.

In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.

Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around twenty thousand times per second to around three hundred billion times per second. This is roughly between the upper limit of audio frequencies and the lower limit of infrared frequencies; these are the frequencies at which energy from an oscillating current can radiate off a conductor into space as radio waves. Different sources specify different upper and lower bounds for the frequency range.

### Parabolic reflector

The reflector can be of sheet metal, metal screen, or wire grill construction, and it can be either a circular "dish" or various other shapes to create different beam shapes. A metal screen reflects radio waves as well as a solid metal surface as long as the holes are smaller than one-tenth of a wavelength, so screen reflectors are often used to reduce weight and wind loads on the dish. To achieve the maximum gain, it is necessary that the shape of the dish be accurate within a small fraction of a wavelength, to ensure the waves from different parts of the antenna arrive at the focus in phase. Large dishes often require a supporting truss structure behind them to provide the required stiffness.

A reflector made of a grill of parallel wires or bars oriented in one direction acts as a polarizing filter as well as a reflector. It only reflects linearly polarized radio waves, with the electric field parallel to the grill elements. This type is often used in radar antennas. Combined with a linearly polarized feed horn, it helps filter out noise in the receiver and reduces false returns.

Since a shiny metal parabolic reflector can also focus the sun's rays, and most dishes could concentrate enough solar energy on the feed structure to severely overheat it if they happened to be pointed at the sun, solid reflectors are always given a coat of flat paint.

### Feed antenna

The feed antenna at the reflector's focus is typically a low-gain type such as a half-wave dipole or more often a small horn antenna called a feed horn. In more complex designs, such as the Cassegrain and Gregorian, a secondary reflector is used to direct the energy into the parabolic reflector from a feed antenna located away from the primary focal point. The feed antenna is connected to the associated radio-frequency (RF) transmitting or receiving equipment by means of a coaxial cable transmission line or waveguide.

At the microwave frequencies used in many parabolic antennas, waveguide is required to conduct the microwaves between the feed antenna and transmitter or receiver. Because of the high cost of waveguide runs, in many parabolic antennas the RF front end electronics of the receiver is located at the feed antenna, and the received signal is converted to a lower intermediate frequency (IF) so it can be conducted to the receiver through cheaper coaxial cable. This is called a low-noise block downconverter. Similarly, in transmitting dishes, the microwave transmitter may be located at the feed point.

An advantage of parabolic antennas is that most of the structure of the antenna (all of it except the feed antenna) is nonresonant, so it can function over a wide range of frequencies, that is a wide bandwidth. All that is necessary to change the frequency of operation is to replace the feed antenna with one that works at the new frequency. Some parabolic antennas transmit or receive at multiple frequencies by having several feed antennas mounted at the focal point, close together.

Dish parabolic antennas
Shrouded microwave relay dishes on a communications tower in Australia.
A satellite television dish, an example of an offset fed dish.
Cassegrain satellite communication antenna in Sweden.
Offset Gregorian antenna used in the Allen Telescope Array, a radio telescope at the University of California Berkeley, US.
Shaped-beam parabolic antennas
Vertical "orange peel" antenna for military height finder radar, Germany.
Early cylindrical parabolic antenna, 1931, Nauen, Germany.
Air traffic control radar antenna, near Hannover, Germany.
ASR-9 Airport surveillance radar antenna.
"Orange peel" antenna for air search radar, Finland.

## Types

Parabolic antennas are distinguished by their shapes:

• Paraboloidal or dish – The reflector is shaped like a paraboloid truncated in a circular rim. This is the most common type. It radiates a narrow pencil-shaped beam along the axis of the dish.
• Shrouded dish – Sometimes a cylindrical metal shield is attached to the rim of the dish. [3] The shroud shields the antenna from radiation from angles outside the main beam axis, reducing the sidelobes. It is sometimes used to prevent interference in terrestrial microwave links, where several antennas using the same frequency are located close together. The shroud is coated inside with microwave absorbent material. Shrouds can reduce back lobe radiation by 10 dB. [3]
• Cylindrical – The reflector is curved in only one direction and flat in the other. The radio waves come to a focus not at a point but along a line. The feed is sometimes a dipole antenna located along the focal line. Cylindrical parabolic antennas radiate a fan-shaped beam, narrow in the curved dimension, and wide in the uncurved dimension. The curved ends of the reflector are sometimes capped by flat plates, to prevent radiation out the ends, and this is called a pillbox antenna.
• Shaped-beam antennas – Modern reflector antennas can be designed to produce a beam or beams of a particular shape, rather than just the narrow "pencil" or "fan" beams of the simple dish and cylindrical antennas above. [4] Two techniques are used, often in combination, to control the shape of the beam:
• Shaped reflectors – The parabolic reflector can be given a noncircular shape, and/or different curvatures in the horizontal and vertical directions, to alter the shape of the beam. This is often used in radar antennas. As a general principle, the wider the antenna is in a given transverse direction, the narrower the radiation pattern will be in that direction.
• "Orange peel" antenna – Used in search radars, this is a long narrow antenna shaped like the letter "C". It radiates a narrow vertical fan shaped beam.
• Arrays of feeds – In order to produce an arbitrary shaped beam, instead of one feed horn, an array of feed horns clustered around the focal point can be used. Array-fed antennas are often used on communication satellites, particularly direct broadcast satellites, to create a downlink radiation pattern to cover a particular continent or coverage area. They are often used with secondary reflector antennas such as the Cassegrain.

Parabolic antennas are also classified by the type of feed , that is, how the radio waves are supplied to the antenna: [3]

• Axial, prime focus, or front feed – This is the most common type of feed, with the feed antenna located in front of the dish at the focus, on the beam axis, pointed back toward the dish. A disadvantage of this type is that the feed and its supports block some of the beam, which limits the aperture efficiency to only 55–60%. [3]
• Off-axis or offset feed  – The reflector is an asymmetrical segment of a paraboloid, so the focus, and the feed antenna, are located to one side of the dish. The purpose of this design is to move the feed structure out of the beam path, so it does not block the beam. It is widely used in home satellite television dishes, which are small enough that the feed structure would otherwise block a significant percentage of the signal. Offset feed can also be used in multiple reflector designs such as the Cassegrain and Gregorian, below.
• Cassegrain – In a Cassegrain antenna, the feed is located on or behind the dish, and radiates forward, illuminating a convex hyperboloidal secondary reflector at the focus of the dish. The radio waves from the feed reflect back off the secondary reflector to the dish, which reflects them forward again, forming the outgoing beam. An advantage of this configuration is that the feed, with its waveguides and "front end" electronics does not have to be suspended in front of the dish, so it is used for antennas with complicated or bulky feeds, such as large satellite communication antennas and radio telescopes. Aperture efficiency is on the order of 65–70% [3]
• Gregorian – Similar to the Cassegrain design except that the secondary reflector is concave, (ellipsoidal) in shape. Aperture efficiency over 70% can be achieved. [3]

## Feed pattern

The radiation pattern of the feed antenna has to be tailored to the shape of the dish, because it has a strong influence on the aperture efficiency, which determines the antenna gain (see Gain section below). Radiation from the feed that falls outside the edge of the dish is called "spillover" and is wasted, reducing the gain and increasing the backlobes, possibly causing interference or (in receiving antennas) increasing susceptibility to ground noise. However, maximum gain is only achieved when the dish is uniformly "illuminated" with a constant field strength to its edges. So the ideal radiation pattern of a feed antenna would be a constant field strength throughout the solid angle of the dish, dropping abruptly to zero at the edges. However, practical feed antennas have radiation patterns that drop off gradually at the edges, so the feed antenna is a compromise between acceptably low spillover and adequate illumination. For most front feed horns, optimum illumination is achieved when the power radiated by the feed horn is 10 dB less at the dish edge than its maximum value at the center of the dish. [5]

### Polarization

The pattern of electric and magnetic fields at the mouth of a parabolic antenna is simply a scaled up image of the fields radiated by the feed antenna, so the polarization is determined by the feed antenna. In order to achieve maximum gain, the feed antenna in the transmitting and receiving antenna must have the same polarization. [6] For example, a vertical dipole feed antenna will radiate a beam of radio waves with their electric field vertical, called vertical polarization. The receiving feed antenna must also have vertical polarization to receive them; if the feed is horizontal (horizontal polarization) the antenna will suffer a severe loss of gain.

To increase the data rate, some parabolic antennas transmit two separate radio channels on the same frequency with orthogonal polarizations, using separate feed antennas; this is called a dual polarization antenna. For example, satellite television signals are transmitted from the satellite on two separate channels at the same frequency using right and left circular polarization. In a home satellite dish, these are received by two small monopole antennas in the feed horn, oriented at right angles. Each antenna is connected to a separate receiver.

If the signal from one polarization channel is received by the oppositely polarized antenna, it will cause crosstalk that degrades the signal-to-noise ratio. The ability of an antenna to keep these orthogonal channels separate is measured by a parameter called cross polarization discrimination (XPD). In a transmitting antenna, XPD is the fraction of power from an antenna of one polarization radiated in the other polarization. For example, due to minor imperfections a dish with a vertically polarized feed antenna will radiate a small amount of its power in horizontal polarization; this fraction is the XPD. In a receiving antenna, the XPD is the ratio of signal power received of the opposite polarization to power received in the same antenna of the correct polarization, when the antenna is illuminated by two orthogonally polarized radio waves of equal power. If the antenna system has inadequate XPD, cross polarization interference cancelling (XPIC) digital signal processing algorithms can often be used to decrease crosstalk.

### Dual reflector shaping

In the Cassegrain and Gregorian antennas, the presence of two reflecting surfaces in the signal path offers additional possibilities for improving performance. When the highest performance is required, a technique called "dual reflector shaping" may be used. This involves changing the shape of the sub-reflector to direct more signal power to outer areas of the dish, to map the known pattern of the feed into a uniform illumination of the primary, to maximize the gain. However, this results in a secondary that is no longer precisely hyperbolic (though it is still very close), so the constant phase property is lost. This phase error, however, can be compensated for by slightly tweaking the shape of the primary mirror. The result is a higher gain, or gain/spillover ratio, at the cost of surfaces that are trickier to fabricate and test. [7] [8] Other dish illumination patterns can also be synthesized, such as patterns with high taper at the dish edge for ultra-low spillover sidelobes, and patterns with a central "hole" to reduce feed shadowing.

## History

The idea of using parabolic reflectors for radio antennas was taken from optics, where the power of a parabolic mirror to focus light into a beam has been known since classical antiquity. The designs of some specific types of parabolic antenna, such as the Cassegrain and Gregorian, come from similarly named analogous types of reflecting telescope, which were invented by astronomers during the 15th century. [9] [2]

German physicist Heinrich Hertz constructed the world's first parabolic reflector antenna in 1888. [2] The antenna was a cylindrical parabolic reflector made of zinc sheet metal supported by a wooden frame, and had a spark-gap excited 26 cm dipole as a feed antenna along the focal line. Its aperture was 2 meters high by 1.2 meters wide, with a focal length of 0.12 meters, and was used at an operating frequency of about 450 MHz. With two such antennas, one used for transmitting and the other for receiving, Hertz demonstrated the existence of radio waves which had been predicted by James Clerk Maxwell some 22 years earlier. [10] However, the early development of radio was limited to lower frequencies at which parabolic antennas were unsuitable, and they were not widely used until after World War 2, when microwave frequencies began to be exploited.

Italian radio pioneer Guglielmo Marconi used a parabolic reflector during the 1930s in investigations of UHF transmission from his boat in the Mediterranean. [9] In 1931 a 1.7 GHz microwave relay telephone link across the English Channel using 10 ft. (3 meter) diameter dishes was demonstrated. [9] The first large parabolic antenna, a 9 m dish, was built in 1937 by pioneering radio astronomer Grote Reber in his backyard, [2] and the sky survey he did with it was one of the events that founded the field of radio astronomy. [9]

The development of radar during World War II provided a great impetus to parabolic antenna research, and saw the evolution of shaped-beam antennas, in which the curve of the reflector is different in the vertical and horizontal directions, tailored to produce a beam with a particular shape. [9] After the war very large parabolic dishes were built as radio telescopes. The 100 meter Green Bank Radio Telescope at Green Bank, West Virginia, the first version of which was completed in 1962, is currently the world's largest fully steerable parabolic dish.

During the 1960s dish antennas became widely used in terrestrial microwave relay communication networks, which carried telephone calls and television programs across continents. [9] The first parabolic antenna used for satellite communications was constructed in 1962 at Goonhilly in Cornwall, England to communicate with the Telstar satellite. The Cassegrain antenna was developed in Japan in 1963 by NTT, KDDI and Mitsubishi Electric. [11] The advent in the 1970s of computer design tools such as NEC capable of calculating the radiation pattern of parabolic antennas has led to the development of sophisticated asymmetric, multireflector and multifeed designs in recent years.

## Gain

The directive qualities of an antenna are measured by a dimensionless parameter called its gain, which is the ratio of the power received by the antenna from a source along its beam axis to the power received by a hypothetical isotropic antenna. This is

${\displaystyle G={A_{\text{antenna}} \over A_{\text{isotropic}}}}$

The aperture ${\displaystyle A_{\text{antenna}}}$ of the antenna is equal to the area of the physical aperture ${\displaystyle A}$ multiplied by a factor ${\displaystyle e_{\text{A}}}$ between 0 and 1 called the aperture efficiency: ${\displaystyle A_{\text{antenna}}=e_{\text{A}}A}$. The aperture of an isotropic antenna (see the article Antenna aperture) is

${\displaystyle A_{\text{isotropic}}={\lambda ^{2} \over 4\pi }}$

Thus the gain of a parabolic antenna is: [12]

${\displaystyle G={\frac {4\pi A}{\lambda ^{2}}}e_{A}=\left({\frac {\pi d}{\lambda }}\right)^{2}e_{A}}$

where:

• ${\displaystyle A}$ is the area of the antenna aperture, that is, the mouth of the parabolic reflector. For a circular dish antenna, ${\displaystyle A=\pi d^{2}/4}$, giving the second formula above.
• ${\displaystyle d}$ is the diameter of the parabolic reflector, if it is circular
• ${\displaystyle \lambda }$ is the wavelength of the radio waves.
• ${\displaystyle e_{A}}$ is a dimensionless parameter between 0 and 1 called the aperture efficiency . The aperture efficiency of typical parabolic antennas is 0.55 to 0.70.

It can be seen that, as with any aperture antenna, the larger the aperture is, compared to the wavelength, the higher the gain. The gain increases with the square of the ratio of aperture width to wavelength, so large parabolic antennas, such as those used for spacecraft communication and radio telescopes, can have extremely high gain. Applying the above formula to the 25-meter-diameter antennas often used in radio telescope arrays and satellite ground antennas at a wavelength of 21 cm (1.42 GHz, a common radio astronomy frequency), yields an approximate maximum gain of 140,000 times or about 50 dBi (decibels above the isotropic level). The largest parabolic dish antennas in the world are the Five-hundred-meter Aperture Spherical radio Telescope in southwest China, and the Arecibo radio telescope in Arecibo, Puerto Rico, US, which both have effective apertures of about 300 meters. The gain of these dishes at 3 GHz is roughly 90 million, or 80 dBi.

Aperture efficiency eA is a catchall variable which accounts for various losses that reduce the gain of the antenna from the maximum that could be achieved with the given aperture. The major factors reducing the aperture efficiency in parabolic antennas are:. [13]

• Feed spillover - Some of the radiation from the feed antenna falls outside the edge of the dish and so doesn't contribute to the main beam.
• Feed illumination taper - The maximum gain for any aperture antenna is only achieved when the intensity of the radiated beam is constant across the entire aperture area. However the radiation pattern from the feed antenna usually tapers off toward the outer part of the dish, so the outer parts of the dish are "illuminated" with a lower intensity of radiation. Even if the feed provided constant illumination across the angle subtended by the dish, the outer parts of the dish are farther away from the feed antenna than the inner parts, so the intensity would drop off with distance from the center. So the intensity of the beam radiated by a parabolic antenna is maximum at the center of the dish and falls off with distance from the axis, reducing the efficiency.
• Aperture blockage - In front-fed parabolic dishes where the feed antenna is located in front of the dish in the beam path (and in Cassegrain and Gregorian designs as well), the feed structure and its supports block some of the beam. In small dishes such as home satellite dishes, where the size of the feed structure is comparable with the size of the dish, this can seriously reduce the antenna gain. To prevent this problem these types of antennas often use an offset feed, where the feed antenna is located to one side, outside the beam area. The aperture efficiency for these types of antennas can reach 0.7 to 0.8.
• Shape errors - random surface errors in the shape of the reflector reduce efficiency. The loss is approximated by Ruze's Equation.

For theoretical considerations of mutual interference (at frequencies between 2 and c. 30 GHz - typically in the Fixed Satellite Service) where specific antenna performance has not been defined, a reference antenna based on Recommendation ITU-R S.465 is used to calculate the interference, which will include the likely sidelobes for off-axis effects.

In parabolic antennas, virtually all the power radiated is concentrated in a narrow main lobe along the antenna's axis. The residual power is radiated in sidelobes, usually much smaller, in other directions. Because in parabolic antennas the reflector aperture is much larger than the wavelength, due to diffraction there are usually many narrow sidelobes, so the sidelobe pattern is complex. There is also usually a backlobe, in the opposite direction to the main lobe, due to the spillover radiation from the feed antenna that misses the reflector.

### Beamwidth

The angular width of the beam radiated by high-gain antennas is measured by the half-power beam width (HPBW), which is the angular separation between the points on the antenna radiation pattern at which the power drops to one-half (-3 dB) its maximum value. For parabolic antennas, the HPBW θ is given by: [5] [14]

${\displaystyle \theta =k\lambda /d\,}$

where k is a factor which varies slightly depending on the shape of the reflector and the feed illumination pattern. For an ideal uniformly illuminated parabolic reflector and θ in degrees, k would be 57.3 (the number of degrees in a radian). For a "typical" parabolic antenna k is approximately 70. [14]

For a typical 2 meter satellite dish operating on C band (4 GHz), this formula gives a beamwidth of about 2.6°. For the Arecibo antenna at 2.4 GHz the beamwidth is 0.028°. It can be seen that parabolic antennas can produce very narrow beams, and aiming them can be a problem. Some parabolic dishes are equipped with a boresight so they can be aimed accurately at the other antenna.

It can be seen there is an inverse relation between gain and beam width. By combining the beamwidth equation with the gain equation, the relation is: [14]

${\displaystyle G=\left({\frac {\pi k}{\theta }}\right)^{2}\ e_{A}}$

### Radiation pattern formula

The radiation from a large paraboloid with uniform illuminated aperture is essentially equivalent to that from a circular aperture of the same diameter D in an infinite metal plate with a uniform plane wave incident on the plate. [15]

The radiation-field pattern can be calculated by applying Huygens' principle in a similar way to a rectangular aperture. The electric field pattern can be found by evaluating the Fraunhofer diffraction integral over the circular aperture. It can also be determined through Fresnel zone equations. [16]

${\displaystyle E=\int \int {\frac {A}{r_{1}}}e^{j(\omega t-\beta r_{1})}dS=\int \int e^{2\pi i(lx+my)/\lambda }dS}$

where ${\displaystyle \beta =\omega /c=2\pi /\lambda }$. Using polar coordinates ${\displaystyle x=\rho \cdot \cos \theta ,\quad y=\rho \cdot \sin \theta }$. Taking account of symmetry,

${\displaystyle E=\int \limits _{0}^{2\pi }d\theta \int \limits _{0}^{\rho _{0}}e^{2\pi i\rho \cos \theta l/\lambda }\rho d\rho }$

and using first-order Bessel function gives the electric field pattern ${\displaystyle E(\theta )}$,

${\displaystyle E(\theta )={\frac {2\lambda }{\pi D}}{\frac {J_{1}[(\pi D/\lambda )\sin \theta ]}{\sin \theta }}}$

where ${\displaystyle D}$ is the diameter of the antenna's aperture in meters, ${\displaystyle \lambda }$ is the wavelength in meters, ${\displaystyle \theta }$ is the angle in radians from the antenna's symmetry axis as shown in the figure, and ${\displaystyle J_{1}}$is the first-order Bessel function. Determining the first nulls of the radiation pattern gives the beamwidth ${\displaystyle \theta _{0}}$. The term ${\displaystyle J_{1}(x)=0}$ whenever ${\displaystyle x=3.83}$. Thus,

${\displaystyle \theta _{0}=\arcsin {\frac {3.83\lambda }{\pi D}}=\arcsin {\frac {1.22\lambda }{D}}}$.

When the aperture is large the angle ${\displaystyle \theta _{0}}$ is very small, so ${\displaystyle \arcsin(x)}$ is approximataly equal to ${\displaystyle x}$. This gives the common beamwidth formulas, [15]

${\displaystyle \theta _{0}\approx {\frac {1.22\lambda }{D}}\,{\text{(in radians)}}={\frac {70\lambda }{D}}\,{\text{(in degrees)}}}$

## Related Research Articles

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In antenna theory, a phased array usually means an electronically scanned array, a computer-controlled array of antennas which creates a beam of radio waves that can be electronically steered to point in different directions without moving the antennas. In an array antenna, the radio frequency current from the transmitter is fed to the individual antennas with the correct phase relationship so that the radio waves from the separate antennas add together to increase the radiation in a desired direction, while cancelling to suppress radiation in undesired directions. In a phased array, the power from the transmitter is fed to the antennas through devices called phase shifters, controlled by a computer system, which can alter the phase electronically, thus steering the beam of radio waves to a different direction. Since the array must consist of many small antennas to achieve high gain, phased arrays are mainly practical at the high frequency end of the radio spectrum, in the UHF and microwave bands, in which the antenna elements are conveniently small.

In the field of antenna design the term radiation pattern refers to the directional (angular) dependence of the strength of the radio waves from the antenna or other source.

A directional antenna or beam antenna is an antenna which radiates or receives greater power in specific directions allowing increased performance and reduced interference from unwanted sources. Directional antennas provide increased performance over dipole antennas—or omnidirectional antennas in general—when greater concentration of radiation in a certain direction is desired.

A helical antenna is an antenna consisting of one or more conducting wires wound in the form of a helix. In most cases, directional helical antennas are mounted over a ground plane, while omnidirectional designs may not be. The feed line is connected between the bottom of the helix and the ground plane. Helical antennas can operate in one of two principal modes — normal mode or axial mode.

In radio and telecommunications a dipole antenna or doublet is the simplest and most widely used class of antenna. The dipole is any one of a class of antennas producing a radiation pattern approximating that of an elementary electric dipole with a radiating structure supporting a line current so energized that the current has only one node at each end. A dipole antenna commonly consists of two identical conductive elements such as metal wires or rods. The driving current from the transmitter is applied, or for receiving antennas the output signal to the receiver is taken, between the two halves of the antenna. Each side of the feedline to the transmitter or receiver is connected to one of the conductors. This contrasts with a monopole antenna, which consists of a single rod or conductor with one side of the feedline connected to it, and the other side connected to some type of ground. A common example of a dipole is the "rabbit ears" television antenna found on broadcast television sets.

A horn antenna or microwave horn is an antenna that consists of a flaring metal waveguide shaped like a horn to direct radio waves in a beam. Horns are widely used as antennas at UHF and microwave frequencies, above 300 MHz. They are used as feed antennas for larger antenna structures such as parabolic antennas, as standard calibration antennas to measure the gain of other antennas, and as directive antennas for such devices as radar guns, automatic door openers, and microwave radiometers. Their advantages are moderate directivity, low standing wave ratio (SWR), broad bandwidth, and simple construction and adjustment.

A Luneburg lens is a spherically symmetric gradient-index lens. A typical Luneburg lens's refractive index n decreases radially from the center to the outer surface. They can be made for use with electromagnetic radiation from visible light to radio waves.

In electromagnetics and antenna theory, antenna aperture, effective area, or receiving cross section, is a measure of how effective an antenna is at receiving the power of electromagnetic radiation. The aperture is defined as the area, oriented perpendicular to the direction of an incoming electromagnetic wave, which would intercept the same amount of power from that wave as is produced by the antenna receiving it. At any point , a beam of electromagnetic radiation has an irradiance or power flux density which is the amount of power passing through a unit area of one square meter. If an antenna delivers watts to the load connected to its output terminals when irradiated by a uniform field of power density watts per square meter, the antenna's aperture in square meters is given by:

An antenna reflector is a device that reflects electromagnetic waves. Antenna reflectors can exist as a standalone device for redirecting radio frequency (RF) energy, or can be integrated as part of an antenna assembly.

An isotropic radiator is a theoretical point source of electromagnetic or sound waves which radiates the same intensity of radiation in all directions. It has no preferred direction of radiation. It radiates uniformly in all directions over a sphere centred on the source. Isotropic radiators are used as reference radiators with which other sources are compared, for example in determining the gain of antennas. A coherent isotropic radiator of electromagnetic waves is theoretically impossible, but incoherent radiators can be built. An isotropic sound radiator is possible because sound is a longitudinal wave.

An offset dish antenna or off-axis dish antenna is a type of parabolic antenna. It is so called because the antenna feed is offset to the side of the reflector, in contrast to the common "front-feed" parabolic antenna where the feed antenna is suspended in front of the dish, on its axis. As in a front-fed parabolic dish, the feed is located at the focal point of the reflector, but the reflector is an asymmetric segment of a paraboloid, so the focus is located to the side.

Antenna measurement techniques refers to the testing of antennas to ensure that the antenna meets specifications or simply to characterize it. Typical parameters of antennas are gain, radiation pattern, beamwidth, polarization, and impedance.

In electromagnetics, directivity is a parameter of an antenna or optical system which measures the degree to which the radiation emitted is concentrated in a single direction. It measures the power density the antenna radiates in the direction of its strongest emission, versus the power density radiated by an ideal isotropic radiator radiating the same total power.

A corner reflector antenna is a type of directional antenna used at VHF and UHF frequencies. It was invented by John D. Kraus in 1938. It consists of a dipole driven element mounted in front of two flat rectangular reflecting screens joined at an angle, usually 90°. Corner reflectors have moderate gain of 10-15 dB, high front-to-back ratio of 20-30 dB, and wide bandwidth.

Radar engineering details are technical details pertaining to the components of a radar and their ability to detect the return energy from moving scatterers — determining an object's position or obstruction in the environment. This includes field of view in terms of solid angle and maximum unambiguous range and velocity, as well as angular, range and velocity resolution. Radar sensors are classified by application, architecture, radar mode, platform, and propagation window.

Dual-Band Blade Antenna, a type of commercial RF "Blade antenna" that uses a "plane and slot design" to get efficient omni-directional coverage at two distinctly different RF bands. It has additive properties similar to a monopole antenna.

Leaky-Wave Antenna (LWA) belong to the more general class of Traveling wave antenna, that use a traveling wave on a guiding structure as the main radiating mechanism. Traveling-wave antenna fall into two general categories, slow-wave antennas and fast-wave antennas, which are usually referred to as leaky-wave antennas.

## References

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