**Parabolic Reflector Antenna**

Image Credit – “On board friendship” (CC BY-NC-ND 2.0) by Elf-8

**Points for discussion**

**Introduction to parabolic reflector antenna****Overview of parabolic reflector antenna****Applications of parabolic reflector antenna****Characteristics****Geometry Analysis****Directivity of parabolic reflector antenna****Aperture efficiency of parabolic reflector antenna****Mathematical problems**

**Introduction to Parabolic Reflector Antenna**

Antenna or radiator is a means for radiating and receiving electromagnetic information. Parabolic reflector antenna is one of the widely used antennae. It is a particular type of reflector antennas. The use of reflector antennas started with the start of second world war with the advancement of communication technologies.

The most straight-forward reflector and more comfortable to implement the reflector antenna is ‘Plane Reflector’ antenna. There are some other types of reflectors also, like – corner reflector, parabolic reflector, Cassegrain reflectors, spherical reflectors. Parabolic reflectors have another type known as ‘Front fed parabolic reflector antenna’.

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**Overview of Parabolic reflector antenna**

The parameters of radiation of a reflector antenna are upgradeable by improving the structural pattern of the ground. The optical science comes into play in this field for this parabolic reflector. The optical mathematics proves that incoming parallel rays can be converged to a specific spot (known as the focal point), upon reflection on a parabola shaped structure.

The reflected waveforms will be outgoing as a parallel shaft of ray. This is a mathematical phenomenon and known as ‘rule of reciprocity’. The proportioned point is termed as the vertex. The outgoing, reflected rays are termed as collimated (as they are parallel). Though the practical observations have revealed that the emerging rays can not be called a parallel beam of rays, they are slightly different from the proper form.

The transmitter of this antenna is generally placed at the focal points of the dish or reflector. This type of set up is termed as ‘front-fed’. We will discuss an analysis of this type of parabolic reflectors in the next part of this article.

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**Applications of Parabolic Reflector Antenna**

One of the biggest reflector antenna set up in Germany for satellite communication, Image Credit – Richard Bartz, Munich aka Makro Freak, Erdfunkstelle Raisting 2, CC BY-SA 2.5

Parabolic reflectors are one of the widely used, highly efficient antennae whose demand is going up day-by-day. From receiving the signal for our TV to transmitting the signal for the space stations, this type of antenna has applications in almost every communication technology field. Some of the notable ones are – in airports, in satellites, in space stations, in telescopes etc.

**Characteristics**

Some significant properties of the parabolic reflector are given below. The properties are about aperture amplitude, polarization properties, phase angles, etc.

- The magnitude part has a dependency on the distance of the feed to the reflector’s surface. The proportionality varies from structure to structure. Like for parabola shaped, it is inversely proportional to the square of the radius of the parabola, and for a cylindrical structure, the relation is inversely proportional to ρ.
- The focal point of the reflector acts differently for different types of geometrical configurations. The cylindrical structure has a line source, and parabolic structures have point source.
- If there are linear polarizations from the feed that is parallel to the cylinder’s axis, then there is no chance of cross-polarizations. Parabolic structures don’t have the same property.

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**Geometrical analysis**

If a geometrically perfect parabola is rotated on its axis, there will be another structure. That structure is known as a parabolic reflector. That is how a parabolic shaped reflector is formed. There is a specific reason behind the shape of this reflector. The parabolic shape helps to generate simple and plane waveform from the emerging rays.

We can observe from the image that the geometrical length OP+PQ gives a constant value for designing.

We can write, **OP + PQ = 2f**; **2f is the constant term.**

Let us assume that** OP = r** and thus PQ comes as **PQ = r * cosϴ.**

Now the value of OP + PQ, after substituting the values,

**OP + PQ = r + r * cosϴ = 2f**

**Or, r (1 + cosϴ) = 2f**

**Or, r = 2f / (1 + cosϴ) = f * sec ^{2}(ϴ/2)**

Now, in antenna theory, we have to keep in kind the co-ordinate system basics. The above equation can be written in rectangular co-ordinate systems using x`, y`, z`. That turns into the following form.

**r + r * cosϴ = √ [ (x`)2 + (y`)2 + (z`)2] + z` = 2f**

Let us find out the unit vector, which is the perpendicular to the tangent of the reflection point.

**f – r * cos ^{2}(ϴ/2) = 0 = S**

by doing some calculus operations, we find the unit vector. It is described below.

**n = N / | N | = – (a)` _{r} cos(ϴ/2) + – (a)`_{ϴ} sin(ϴ/2)**

Now, using the geometrical analysis, we can find out an expression for the subtended angle. It is described below.

**tan(ϴ _{0}) = (d/2) Z_{0}**

The Z_{0} is the measurement of the distance from the axis to the focal point. Mathematical expressions can also represent it.

**Z _{0} = f – [(x_{0}^{2} + y_{0}^{2}) / 4f]**

**Or, Z _{0} = f – [ (d/2)^{2}/ 4f]**

**Or, Z _{0}= f – d^{2} / 16f**

Let us check the value of tan(ϴ_{0}) after substituting the value of Z0.

**tan(ϴ _{0}) = [(f/2d) / {(f/d)^{2} – (1/16)}]**

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**Directivity of parabolic reflector antenna**

Before we jump into finding out the directivity of a parabolic antenna, let us know about the directivity of an antenna.

*An antenna’s directivity is defined as the ratio of radiation intensity of an antenna in a particular direction to the averaged radiation intensity over all the directions. *

Directivity is considered as a parameter for calculating the figure of merit of the antenna. The following mathematical expression describes the directivity.

**D = U / U _{0} = 4πU / P_{rad}**

When the direction is not given, the default direction is the direction of maximum radiation intensity.

**D _{max} = D_{0} = U_{max} / U_{0} = 4πU_{max} / P_{rad}**

Here, ‘D’ is the directivity, and it has no direction as it is a ratio. U is the radiation intensity. U_{max} is the maximum radiation intensity. U_{0} is the radiation intensity of the isotropic source. P_{rad} is the total radiated power. Its unit is Watt (W).

**U = ½ r ^{2} * | E (r, ϴ = π) |^{2} * √(ε/μ)**

Now for U (ϴ = π) and substituting the value of energy E the previous turns into –

**U (ϴ = π) = [16 π ^{2} f^{2 }* Pt * | ∫_{0} ^{ϴ} tan (ϴ /2) * √ (G_{f} (ϴ)) dϴ |^{2}] / 4πλ^{2}**

The directivity comes as – **D = U / U _{0} = 4πU / P_{rad}**

**Or, D = [16 π ^{2} f^{2 }* | ∫_{0} ^{ϴ} tan (ϴ /2) * √ (G_{f} (ϴ)) dϴ |^{2}] / λ^{2}**

**Aperture efficiency of parabolic reflector antenna**

Microwave relay dishes, a type of reflector antenna, Image Credit- Bidgee, Parabolic antennas on a telecommunications tower on Willans Hill, CC BY-SA 2.5 AU

The mathematical expression for the parabolic reflector antenna is given below.

** ε _{ap} = ε_{s} * ε_{t} * ε_{p} * ε_{x} * ε_{b} * ε_{r}**

Here,

*ε _{ap} represents aperture efficiency.*

*ε _{s} is spillover efficiency. It can be defined as the part of the power which is transmitted by the feed and paralleled by the surface of the reflection.*

*ε _{t} represents the efficiency of the taper. It can be described as the singularity of spreading the magnitude for feed design over the reflector’s surface.*

*ε _{p} gives us the efficiency of phase. It can be described as the uniformity of the practical field phase over the plane of the aperture.*

*ε _{x} represents the efficiency of polarization.*

*ε _{b} is the efficiency of the backlog.*

*And, ε _{r} represents the efficiency of error, calculated over the whole reflector area.*

**Mathematical problem**

**1. A parabolic reflector antenna has a diameter of 10 meters. The f/d ratio is given as 0.5. The frequency of the operation is set to 3 GHz. The antenna which is fed with the reflector is symmetrical in design. It is also given that –**

**G**_{f} (ϴ) = 6 cos^{2}ϴ; where ϴ^{o} ≤ ϴ ≤ 90^{o} and null at any other point.

_{f}(ϴ) = 6 cos

^{2}ϴ; where ϴ

^{o}≤ ϴ ≤ 90

^{o}and null at any other point.

**Now calculate i) the aperture efficiency (ε**_{ap}). ii) Directivity of the antenna. iii) taper efficiency and efficiency of spillover. iv) Find out the directivity of the antenna if the aperture phase deviation set to π / 4 radian.

_{ap}). ii) Directivity of the antenna. iii) taper efficiency and efficiency of spillover. iv) Find out the directivity of the antenna if the aperture phase deviation set to π / 4 radian.

**Solution:**

We know that, subtended angle is given by the following expression.

tan(ϴ_{0}) = [(f/2d) / {(f/d)^{2} – (1/16)}]

Or, tan(ϴ_{0}) = [ (0.5 *0.5) / {(0.5 * 0.5) – (1/16)}]

Or, tan(ϴ_{0}) = 0.25 / 0.0625

Or, ϴ_{0} = 53.13^{o}

The aperture efficiency is given as –

ε_{ap} = 24 [(sin^{2} (26.57^{o}) + ln {cos (26.57^{o})}]^{2} * cot^{2}(26.57^{o})

or, ε_{ap} = 0.75

*So, the aperture efficiency comes as 75 %.*

Now, let us find out the directivity of the antenna.

It can be calculated as below.

D = 0.75 * [ π * (100)]^{2}

Or, D = 74022.03

**Or, D = 48.69 dB.**

The spillover frequency will be ε_{s}.

ε_{s} = 2 cos^{3} ϴ |_{0} ^{53.13} / 2 cos^{3} ϴ |_{0} ^{90}

or, ε_{s} = 0.784

**So, the spillover efficiency of the antenna comes as 78.4%.**

Now time for calculation of efficiency of the tapper. The tapper efficiency is represented as ε_{t}.

ε_{t} = (2 * 0.75)/ 1.568

or, ε_{t} = 0.9566

**So, the tapper efficiency is 95.66 % for the parabolic reflector antenna.**

Now aperture phase deviation is set to π / 4 radian.

That means m = π / 4 = 0.7854

We know that D/ D_{0} ≥ [ 1 – m^{2}/2]^{2}

Or, D/ D_{0} ≥ [ 1 – (0.7854 * 0.7854) /2]^{2}

Or, D/ D_{0} ≥ 0.4782737

Or, D ≥ 0.4782737 * D_{0.}

Or, D = 0.4782737 * 74022.03

Or, D = 35402.8

Or, D = 45.5 dB.

*The directivity at the given condition comes to be 45.5 dB.*

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