# Parabolic Reflector Antenna Theory & Formulas

### Some simple mathematics provides a significant insight into the operation and theory for the parabolic reflector antenna.

Some of the mathematics and theory behind the parabolic reflector antenna gives a good understanding into its operation and some of the key factors concerning its operation and design.

The parabolic shape of the reflector is key to its operation – in particular its gain and directivity.

## Parabolic reflector theory basics

As the name implies, the parabolic reflector is formed from a shape known as a paraboloid. This shape forms the reflective surface in the antenna that enables waves reflected by the surface to retain their phase relationship, thereby enabling the maximum gain to be obtained.

In other words, RF energy in the form of electromagnetic waves travelling towards the antenna in a plane wavefront will be reflected by the reflector and remain in phase at the focal point. In this way the whole signal remains in phase and there is no cancellation. This means that the maximum signal is maintained. Conversely signals radiated from the focal point will be reflected by the parabolic reflector and form a parallel wavefront (in-phase) travelling outwards from the antenna. The parabolic reflector shape enables the wavefronts to remain in phase
${A}_{1}+{A}_{2}={B}_{1}+{B}_{2}$

In view of the fact that total length A1 + A2 is the same as B1 + B2, etc, this means that the phase integrity of the system is retained. Incoming waves add at the focal point, and outgoing waves produce a single wavefront moving in parallel away from the reflector.

It is this concept that is at the centre of parabolic reflector antenna theory.

## Parabolic reflector shape theory

Parabolic reflector theory relies on the shape of the reflector for its properties.

The reflector uses a parabolic shape to ensure that all the power is reflected in a beam in which the wave traces run parallel to each other. Also all the reflected power is in the same phase, because the path length from the source to the reflector and then outwards is the same wherever it is reflected on the surface of the parabola.

The parabolic curve follows the equation:

The measurements and references for the parabolic reflector antenna formula can be seen on the diagram below:

Parabolic theory shows that the paraboloid curve is the locus of points that are equidistant from a fixed point known as the focus located on the X axis. A fixed line behind the parbolic curve detailed as AB on the diagram is known as the directrix. On this the length FP = PQ wherever it is located on the parabolic curve.

As the surface acts as a reflector, the directix has the same properties when located in front of the reflector. In other words the parabolic reflector theory shows that the emanating wavefront will have the same phase regardless of the point of reflection on the parabolic curve.

The parabolic reflector antenna theory also shows the emanating beam will tend to be parallel.

## Parabolic antenna focal length

One important element of the parabolic reflector antenna theory is its focal length. To ensure that the antenna operates correctly, it is necessary to ensure that the radiating element is placed at the focal point. To determine this it is necessary to know the focal length.

$f=\frac{{D}^{2}}{16c}$

Where:
f is the focal length
D is the diameter of the reflector
c is the depth of the reflector

In addition to this the f/D ratio is important. As the f/D ratio is often specified along with the diameter, the focal length can be obtained very easily by multiplying its f/D ratio by the specified diameter D.