Category Archive Parabolic reflector equation


Parabolic reflector equation

In mathematicsa parabola is a plane curve which is mirror-symmetrical and is approximately U- shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point the focus and a line the directrix. The focus does not lie on the directrix.

The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic sectioncreated from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.

The line perpendicular to the directrix and passing through the focus that is, the line that splits the parabola through the middle is called the " axis of symmetry ". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where the parabola is most sharply curved.

The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The " latus rectum " is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar.

Parabolas have the property that, if they are made of material that reflects lightthen light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel " collimated " beam, leaving the parabola parallel to the axis of symmetry.

The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. They are frequently used in physicsengineeringand many other areas. The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered a way to solve the problem of doubling the cube using parabolas.

The solution, however, does not meet the requirements of compass-and-straightedge construction. The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola.

The name "parabola" is due to Apolloniuswho discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope.

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Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points locus of points in the Euclidean plane:. The horizontal chord through the focus see picture in opening section is called the latus rectum ; one half of it is the semi-latus rectum. The latus rectum is parallel to the directrix. From the picture one obtains. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve.

The implicit equation of a parabola is defined by an irreducible polynomial of degree two:. The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of a function. From the section above one obtains:.Parabolic geometry is the basis for such concentrating solar power CSP technologies as troughs or dishes. Parabolic trough is also considered one of the most mature and most commercially proven technologies in the utility scale CSP facilities Mendelsohn et al.

Geometrically, a parabola is a locus of points that lie on equal distance from a line directrix and a point focus - see Figure 2. The distance VF between the vertex and focus of the parabola is the focal distance f. The line perpendicular to the directrix that passes through the focus is the axis of the parabola; the axis divides the parabola into two parts that are symmetrical.

Parabolic Reflector Antenna Gain

With origin at its vertex, and the axis of the parabola taken as x-axis, a parabola is described by the equation:. By definition of the focal point of the parabola, all incoming rays parallel to the axis of the parabola are reflected through the focus. This provides an opportunity for light concentration by using parabolic surfaces. If we assume that solar light arrives to the surface as essentially parallel rays, and apply the Snell's law the angle of reflection equals the angle of incidencewe can assign the focal point as an ideal location for the receiver Figure 2.

Solar applications deal with a parabola of a finite height Figure 2. The design of the parabolic reflector takes into account the available aperture size afocus location f - i. These parameters are interrelated via the equation Stine and Harrigan, :.

This figure above shows that the flatter the reflecting surface, the longer the focal length. When rim angle increases within the same aperturethe parabola becomes more curved, and the focal distance shortens. Parabolic trough Figure 2. Parabolic trough is one of the most widely implemented technologies for sunlight concentration at the utility scale. This type of collectors relies on sun tracking to ensure that the beam radiation is directed parallel to the parabolic axis.

A parabolic mirror produces an image of the sun on the surface of the receiver, so the receiver size needs to be matched to the image size. Consider Figure 2.

Since the sun is not really a point source, solar beam incident on the reflector is represented as a cone with an angular width 0. Being reflected at a point on the parabolic surface, the beam hits the focal plane, where it produces an image of a certain dimension, centered around the focal point.

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For the linear receiver, the width of the image W produced on the focal plane can be determined as follows:. The equations presented here can be used to estimate the size of the reflected light image on the receiver for different shapes of parabolic reflectors. Note that these are the minimal theoretical dimensions of the reflected image that would be produced by the ideal parabolic mirror that is perfectly aligned.

If there are any flaws in the mirror surface or trueness of the angle, additional spreading of the image may occur.

parabolic reflector equation

If you are interested in more explanation of how these formulas were derived, please refer to Duffie and Beckman, book Section 7. The above-described geometrical concepts apply to the cross-section of a parabolic reflector. In reality, the reflector itself is three-dimensional shape, i.When you kick a soccer ball or shoot an arrow, fire a missile or throw a stone it arcs up into the air and comes down again Get a piece of paper, draw a straight line on it, then make a big dot for the focus not on the line!

Now play around with some measurements until you have another dot that is exactly the same distance from the focus and the straight line. Keep going until you have lots of little dots, then join the little dots and you will have a parabola!

Any ray parallel to the axis of symmetry gets reflected off the surface straight to the focus. We also get a parabola when we slice through a cone the slice must be parallel to the side of the cone. If you want to build a parabolic dish where the focus is mm above the surface, what measurements do you need?

Try to build one yourself, it could be fun! Just be careful, a reflective surface can concentrate a lot of heat at the focus. Hide Ads About Ads. Parabola When you kick a soccer ball or shoot an arrow, fire a missile or throw a stone it arcs up into the air and comes down again Except for how the air affects it.

Conic Sections Geometry Index. So the parabola is a conic section a section of a cone.To make your own custom parabolic reflector in any size or depth out of tissue, you need to calculate the shape of the tissue segments and where to sew it. I did the math for you… here is how. Few people know it although it is quite powerful. Those of you who use a Mac can download my grapher file below and calculate your own reflectors within minutes. For everybody else I have included all the formulas so you can use a program of your choice or even pen and paper.

Any parabolic reflector can be defined by two numbers: the diameter and the depth. Since the tissue is flat and the reflector is composed of multiple segments, we also need the number of rods:. My sample reflector has a diameter of millimeters, but you can actually use any unit you like with these formulas centimeters, inches, … and obtain correct results.

The radius is simply half the diameter. Since the rods face away from the center in a circular pattern, we can calculate their points using trigonometric functions sin, cos.

The projected length of a rod i. The actual Z value finally is our parabolic function, since the rod is bent according to a parabola. To get a more realistic plot, we want to draw the outer rim by connecting the endpoints of the rods.

This can be done by the following equation:. The above equation tells Grapher to draw the focal point as a little ball in the diagram. For our example reflector, the focal point is roughly mm inside the reflector.

But how long are the rods really, and which shape does the tissue need to have? To answer these questions, we first have to calculate a flat unbent rod. Mathematically speaking, we need to determine the arc length of the parabola from the vertex to the outer rim. Basically, rodRunLength is a function of prjLen which tells us for any projected point on the ground plane how long the bent rod actually is from the vertex to the point above this projected point.

Now we know how long a segment of tissue needs to be, but not yet how wide it needs to be. But that is not difficult. We can use Pythagoras and simply determine the distance between two points on adjacent rods:. For a tissue segment, segWid gives the width for any projected point, while rodRunLength gives the length at which the rod has that width. In order to graph this, I made a new Grapher document with only 2 dimensions like the tissue.

The following equation plots the side walls of a tissue segment:. This equation draws the top wall of the tissue segment i.

And this is what it looks like:. Now you can generate a multitude of different reflectors. All you have to do is modify the three key parameters to your liking:. If you are super- lazy efficient, you can even calculate the tissue right here on this page, because I put the formulas into JavaScript for you. Just modify the 3 constants below to your liking and the page will generate two series of points: one for the rod i.

Assemble all the segments to get the parabolic reflector. By the way: you can easily generate the drawing in AutoCAD by typing plinepressing enter and copying the series of points from this page into AutoCAD.

Update Mar Reader Mike S. Check out Mike's work here. Constants Any parabolic reflector can be defined by two numbers: the diameter and the depth. Since the tissue is flat and the reflector is composed of multiple segments, we also need the number of rods: My sample reflector has a diameter of millimeters, but you can actually use any unit you like with these formulas centimeters, inches, … and obtain correct results.

From these constants we derive two more constants: The radius is simply half the diameter.Antennas List. Antenna Theory Home. Antenna Basics. Radiation Patterns. The most well-known reflector antenna is the parabolic reflector antennacommonly known as a satellite dish antenna. Examples of this dish antenna are shown in the following Figures. Figure 1. The "big dish" antenna of Stanford University. Figure 2. A random direcTV dish antenna on a roof.

Parabolic reflectors typically have a very high gain dB is common and low cross polarization. The smaller dish antennas typically operate somewhere between 2 and 28 GHz.

The large dishes can operate in the VHF region MHzbut typically need to be extremely large at this operating band. The basic structure of a parabolic dish antenna is shown in Figure 3. It consists of a feed antenna pointed towards a parabolic reflector.

The feed antenna is often a horn antenna with a circular aperture. Figure 3. Components of a dish antenna. Unlike resonant antennas like the dipole antenna which are typically approximately a half-wavelength long at the frequency of operation, the reflecting dish must be much larger than a wavelength in size. The distance between the feed antenna and the reflector is typically several wavelenghts as well.

This is in contrast to the corner reflectorwhere the antenna is roughly a half-wavelength from the reflector.In fact the parabolic reflector antenna gain can be as high as 30 to 40 dB. These figures of gain are not easy to achieve using other forms of antenna.

At microwave frequencies where these antennas are normally used, they are able to produce very high levels of gain, and they offer a very convenient and robust structure that is able to withstand the rigours of external use. By contrast, many other types of antenna design are not practicable at these frequencies. The one common feature of all these examples is the parabolic antenna gain, or parabolic dish gain.

While the larger antennas have greater levels of parabolic antenna gain, the performance of all these antennas is of prime importance. There are a number of factors that affect the parabolic antenna gain. These factors include the following:. The parabolic antenna gain can easily be calculated from a knowledge of the diameter of the reflecting surface, the wavelength of the signal, and a knowledge or estimate of the efficiency of the antenna. The parabolic reflector antenna gain is calculated as the gain over an isotropic source, i.

This is a theoretical source that is used as the benchmark against which most antennas are compared. The gain is quoted in this manner is denoted as dBi. From this it can be seen that very large gains can be achieved if sufficiently large reflectors are used.

Parabolic Reflector Antenna Theory & Formulas

However when the antenna has a very large gain, the beamwidth is also very small and the antenna requires very careful control over its position. In professional systems electrical servo systems are used to provide very precise positioning. It can be seen that the parabolic reflector gain can be of the order of 50dB for antennas that have a reflector diameter of a hundred wavelengths or more. Whilst antennas of this size would not be practicable for many antennas designs such as the Yagi, and many others, the parabolic reflector can be made very large in comparison to the wavelength and therefore it can achieve these enormous gain levels.

More normal sizes for these antennas are a few wavelengths, but these are still able to provide very high levels of gain. In the overall gain formula for the antenna, an efficiency factor is included.

Parabolic reflector

The parabolic reflector antenna gain efficiency is dependent upon a variety of factors. These are all multiplied together to give the overall efficiency.

The term km is used to denote the various miscellaneous efficiency elements that are often more difficult to determine. These include those due to surface effort, cross polarisation, aperture blockage, and the non-single point feed. Normally the beamwidth is defined as the points where the power falls to half of the maximum, i.

It is possible to estimate the beamwidth reasonably accurately from the following formula. All dimensions must be in the same units for the calculation to be correct, e. To provide the optimum illumination of the reflecting surface, the level of illumination should be greater in the centre than at the sides.

It can be shown that the optimum situation occurs when the centre is around 10 to 11 dB greater than the illumination at the edge. Lower levels of edge illumination result in lower levels of side lobes. The reflecting surface antenna forms a major part of the whole system.

In many respects it is not as critical as may be thought at first. Often a wire mesh may be used. Provided that the pitch of the mesh is small compared to a wavelength it will be seen as a continuous surface by the radio signals. If a mesh is used then the wind resistance will be reduced, and this provides significant mechanical advantages.

Parabolic Reflector Part 1 Drawing and Measuring the Parabola

The parabolic reflector antenna is able to provide a significant level of gain which can be put to good use, especially for microwave frequencies where the size of the antenna for a given level of gain becomes very managemable. The high level of gain is one of the main reasons why parabolic reflector antennas are used.

The Goldstone parabolic reflector antenna has a very high level of gain Image courtesy NASA At microwave frequencies where these antennas are normally used, they are able to produce very high levels of gain, and they offer a very convenient and robust structure that is able to withstand the rigours of external use. Factors affecting parabolic reflector antenna gain There are a number of factors that affect the parabolic antenna gain.A parabolic or paraboloid or paraboloidal reflector or dish or mirror is a reflective surface used to collect or project energy such as lightsoundor radio waves.

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Its shape is part of a circular paraboloidthat 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. Parabolic reflectors are used to collect energy from a distant source for example sound waves or incoming star light. Since the principles of reflection are reversible, parabolic reflectors can also be used to focus radiation from an isotropic source into a narrow beam.

In radio parabolic antennas are used to radiate a narrow beam of radio waves for point-to-point communications in satellite dishes and microwave relay stations, and to locate aircraft, ships, and vehicles in radar sets.

In acousticsparabolic microphones are used to record faraway sounds such as bird callsin sports reporting, and to eavesdrop on private conversations in espionage and law enforcement. Strictly, the three-dimensional shape of the reflector is called a paraboloid. A parabola is the two-dimensional figure. The distinction is like that between a sphere and a circle.

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However, in informal language, the word parabola and its associated adjective parabolic are often used in place of paraboloid and paraboloidal. See " Parabola In a cartesian coordinate system ".

All units must be the same. If two of these three quantities are known, this equation can be used to calculate the third. A more complex calculation is needed to find the diameter of the dish measured along its surface. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. The parabolic reflector functions due to the geometric properties of the paraboloidal shape: any incoming ray that is parallel to the axis of the dish will be reflected to a central point, or " focus ".

For a geometrical proof, click here. Because many types of energy can be reflected in this way, parabolic reflectors can be used to collect and concentrate energy entering the reflector at a particular angle. Similarly, energy radiating from the focus to the dish can be transmitted outward in a beam that is parallel to the axis of the dish.

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In contrast with spherical reflectorswhich suffer from a spherical aberration that becomes stronger as the ratio of the beam diameter to the focal distance becomes larger, parabolic reflectors can be made to accommodate beams of any width.

However, if the incoming beam makes a non-zero angle with the axis or if the emitting point source is not placed in the focusparabolic reflectors suffer from an aberration called coma. This is primarily of interest in telescopes because most other applications do not require sharp resolution off the axis of the parabola.

The precision to which a parabolic dish must be made in order to focus energy well depends on the wavelength of the energy. If the dish is wrong by a quarter of a wavelength, then the reflected energy will be wrong by a half wavelength, which means that it will interfere destructively with energy that has been reflected properly from another part of the dish.

parabolic reflector equation

Microwaves, such as are used for satellite-TV signals, have wavelengths of the order of ten millimetres, so dishes to focus these waves can be wrong by half a millimetre or so and still perform well. It is sometimes useful if the centre of mass of a reflector dish coincides with its focus.

This allows it to be easily turned so it can be aimed at a moving source of light, such as the Sun in the sky, while its focus, where the target is located, is stationary. The dish is rotated around axes that pass through the focus and around which it is balanced. If the dish is symmetrical and made of uniform material of constant thickness, and if F represents the focal length of the paraboloid, this "focus-balanced" condition occurs if the depth of the dish, measured along the axis of the paraboloid from the vertex to the plane of the rim of the dish, is 1.

The radius of the rim is 2. The focus-balanced configuration see above requires the depth of the reflector dish to be greater than its focal length, so the focus is within the dish. This can lead to the focus being difficult to access. An alternative approach is exemplified by the Scheffler Reflectornamed after its inventor, Wolfgang Scheffler. This is a paraboloidal mirror which is rotated about axes that pass through its centre of mass, but this does not coincide with the focus, which is outside the dish.

If the reflector were a rigid paraboloid, the focus would move as the dish turns. To avoid this, the reflector is flexible, and is bent as it rotates so as to keep the focus stationary.

parabolic reflector equation

Ideally, the reflector would be exactly paraboloidal at all times.

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