Microwave Horn Antenna Theory & Equations

- some of the key details about the theory of the horn antenna along with some of the key equations / formulas.

Horn Antenna Tutorial Includes

As with any antenna the theory can become complicated, although there are several formulas that can be used to calculate some key elements and also explain the theory of operation.

For a given frequency and horn length, there is some flare angle that gives minimum reflection and maximum gain.

Horn antenna theory basics

The horn antenna is essentially a section of waveguide where the open end is flared to provide a transition to the areas of free space.

Waveguides are generally one of two shapes: rectangular or circular. By far the most widely used of these two is the rectangular form.

Waveguide theory indicates that there are several modes of propagation that can occur within a waveguide. The most widely used is the TE10 mode and this is indicated in the diagram below.

The E field for a TE10 mode signal in a rectangular waveguide such as that which may be used in a horn antenna
Field in TE10 waveguide

As the waveguide is rectangular it will have different dimensions for each side. For the horn antenna theory and calculations these will be taken that it has a width b and height a, with b>a.

The horn antenna is a simple development of the waveguide transmission line. Using some simple theory, it is quite possible to see how the horn antenna works.

It is quite possible to leave a waveguide open and let signal radiate from this. However this is not particularly efficient. Signals passing along the waveguide see a sudden transition from the waveguide to free space which has an impedance of around 377Ω.

The result of this sudden transition is to cause signals to be reflected back long the waveguide as standing waves - theory shows that this is exactly the same as for poor matches at the end of coaxial or other forms of wire based transmission lines.

To overcome this issue, the waveguide can be tapered out or flared. This has the effect of providing a gradual transition from the impedance of the waveguide to that of free space. In effect it acts like a progressive matching transformer. The flare functions similarly to a tapered transmission line, or an optical medium with a smoothly varying refractive index. In addition, the wide aperture of the horn projects the waves in a narrow beam.

The horn type that provides the most effective match is the exponential horn. However pyramid or conical horns give a sufficiently good match for most applications, and they are far easier and much cheaper to fabricate.

Horn antenna angle of flare

One of the key properties of the horn antenna is the angle at which the horn flares out. This affects many areas of the performance including the gain and directivity as described below.The angle of flare is defined in the diagram below and there can be a different angle for both the E-plane (E field) and the H-plane (H field. These are referred to as θE and θH.

The flare angle for both fields is sued in areas of horn antenna theory.
Horn antenna flare angle

Horn antenna theory for radiation

In order to understand how a horn antenna radiates, some simple explanations and theory can be used.

The waves of the signal will propagate down the horn antenna towards the aperture. As they travel along the flared opening, the waves travel as spherical wavefronts, having their apex at the apex of the horn - a point referred to as the phase centre of the horn antenna.

As the phase front progressing along the horn antenna are spherical, the phase increases smoothly from the edges of the aperture plane to the centre.

The difference in phase between the centre point and the edges is called the phase error. This increases with the flare angle reducing the gain, but increasing the beamwidth. As a result horn antennas have wider beamwidths when compared to similar-sized plane-wave antennas like parabolic reflectors.

The theory also shows that as the size of a horn antenna increases in terms of its electrical size, i.e. the number of wavelengths for the various dimensions, so the phase error increases. This has the effect of giving the horn antenna a wider beamwidth. In order to provide a narrow beamwidth a longer horn is required, i.e. having a smaller angle of flare. This enables the phase angle to be kept more constant. However the phase error issues mean that horn sizes are practically limited to around 15 wavelengths otherwise larger sizes would require a much longer antenna.

Horn antenna gain

Theory dictates that as the frequency used by a horn antenna increases, so does the gain and directivity (beamwidth decreases). The reason for this is that the aperture of the horn remains constant in terms of physical dimensions (obviously), but increases in terms of the number of wavelengths, i.e. it is electrically larger.

As antennas tend to have higher gain levels as they become larger, so it can intuitively be seen that the gain and directivity of the horn antenna will increase with frequency.

Horn antenna theory: flare vs gain

The angle of the flare on the horn antenna has a marked effect on the gain and beamwidth. The actual mathematical proofs and theory for this are complicated and not applicable here.

The gain of the horn antenna will varies with frequency and also the angle of the flare of the horn itself. Without delving deep into horn antenna theory and mathematics, it can be imagined that there is an optimum flare angle.

The theory shows that there are two areas where the impedance changes abruptly: the mouth of the horn antenna, and the point where the sides begin to flare outwards. It is possible to gain an understanding of the operation of the horn by looking at the two extremes where the angle of flare is 0° and 90° and at the case between the two extremes.

  •   This form of horn might be considered a narrow horn. These antennas have small levels of gain because the antenna appears like an open ended waveguide, and there is little conditioning of the radiated beam as the horn antenna flares out.
  • Increasing angles:   As the flare angle is increased, the reflection at the mouth decreases rapidly and as a result the gain of the horn antenna increases. Theory also states that the amount of reflection at the point of the antenna where the sides start to flare drops, and this also results in an increase in the level of gain.
  • 90°   In contrast, for horn antennas with wide angles, it is found that most of the reflection occurs at the area of the horn where the antenna sides flare out, but again the horn antenna gain is low because the throat approximates to an open ended waveguide.

In view of this there is an optimum horn flare for given requirements.

Formulas & theory for horn antenna apertures

For a rectangular horn antenna the formulas are:

The formulas or equations for the optimal rectangular horn antenna: ae=sqrt(2 lambda Le) and ah=sqrt(3 lambda Lh)

Then for a conical horn antenna the formula is:

The formula or equation for the optimal conical horn antenna: diameter=sqrt(3 lambda L)

      ApertureE is the width of the aperture in the E-field direction.
      ApertureH is the width of the aperture in the H-field direction.
      LE is the slant length of the side in the E-field direction.
      LH is the slant length of the side in the H-field direction.
      diameter is the diameter of the cylindrical horn aperture.
      L is the slant length of the cone from the apex.
      λ is the wavelength of the signal.

Horn antenna gain formulas

It is easy to calculate the gain of a horn antenna with the knowledge of a few of its parameters.

Pyramidal horns are normally constructed to provide optimal gain. The gain of a pyramid horn antenna over an isotropic source, i.e. one that radiates equally in all direction can be derived from the formula:

The formula or equation for the gain of a pyramid horn antenna over an isotropic source is 4 pi A / lambda squared, all times the aperture efficiency

Then for a conical horn the gain formula can be shown to be:

The formula or equation for the gain of a conical horn antenna over an isotropic source is ( pi d / lambda) squared times the aperture efficiency

      A is the physical area of the aperture
      d is the physical diameter of a conical horn aperture
      λ is the wavelength
      eA is the aperture efficiency and is a figure between 0 and 1

By Ian Poole

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