#### Radar Technology - Primary RADAR - Height, "the RADAR equation" abnd Reflections

RADAR is can be used to track and aircrafts hieght above the gorund while in flight. One the flight decks secondary instruments if often a RAD-Alt, or RADAR Altimeter.

The height of a target over the earth's surface is called height or altitude. This is denominated by the letter H (like: Height). The altitude can be calculated with the values of distance R and elevation angle ε.

The altitude cannot be so simply calculated on a flying airplane, while refraction is caused when electromagnetic waves cross airlayers at different density and the earth's surface has a bend. The calculation of the targets altitude is not only a trigonometrically calculation. The current location grounding bend must also be taken into consideration.

In addtion the propagation of electromagnetic waves is also subject to a refraction and are also dependent on:

- the transmitted wavelength,
- the barometric pressure,
- the air temperature and
- the atmospheric humidity.

These variables are one of many reasons why modern aircraft use static pressure and Air Data Computers, rather than RADAR, to calulate their altitude.

The RADAR equation

The radar equation represents the physical dependences of the transmit power, that is the wave propagation up to the receiving of the echo-signals. Furthermore one can assess the performance of the radar with the radar equation.

Nondirectional power density diminishes as geometric spreading of the beam.

*Argumentation/Derivation*

First we assume, that electromagnetic waves propagate under ideal conditions, i.e. without dispersion.

Nondirectional power density diminishes as geometric spreading of the beam. In the above image Nondirectional power density diminishes as geometric spreading of the beam.

If high-frequency energy is emitted by an isotropic radiator, than the energy propagate uniformly in all directions. Areas with the same power density therefore form spheres (A= 4 π R²) around the radiator. The same amount of energy spreads out on an incremented spherical surface at an incremented spherical radius. That means: the power density on the surface of a sphere is inversely proportional to the to the square of the radius of the sphere.

Since a spherical segment emits equal radiation in all direction (at constant transmit power), if the power radiated is redistributed to provide more radiation in one direction, then this results an increase of the power density in direction of the radiation. This effect is called antenna gain. This gain is obtained by directional radiation of the power.

Of course in reality radar antennas aren't “partially radiating” isotropic radiators. Radar antennas must have a small beam width and an antenna gain up to 30 or 40 dB. (e.g. parabolic dish antenna or phased array antenna).

The target detection isn't only dependent on the power density at the target position, but also on how much power is reflected in the direction of the radar. In order to determine the useful reflected power, it is necessary to know the radar cross section σ. This quantity depends on several factors. But it is true to say that a bigger area reflects more power than a smaller area. That means:

A Jumbo jet offers more radar cross section than a sporting aircraft at same flight situation. Beyond this the reflecting area depends on design, surface composition and materials used. With this in mind we can say: The reflected power Pr at the radar depends on the power density, the antenna gain.

Radar Reflections from Flat Ground

The trigonometric representation shows the influence of the Earth's surface. The Earth plane surrounding a radar antenna has a significant impact on the vertical polar diagram.

The combination of the direct and re-reflected ground echo changes the transmitting and receiving patterns of the antenna. This is substantial in the VHF range and decreases with increasing frequency. For the detection of targets at low heights, a reflection at the Earth's surface is necessary. This is possible only if the ripples of the area within the first Fresnel zone do not exceed the value 0.001 R (i.e.: Within a radius of 1000 m no obstacle may be larger than 1 m!).

Specialized Radars at lower (VHF) frequency band make use of the reflections at the Earth's surface and lobing to maximize cover at low levels. At higher frequencies these reflections are more disturbing. The following picture shows the lobe structure caused by ground reflections. Normally this is highly undesirable as it introduces intermittent cover as aircraft fly through the lobes. The technique has been used in ATC ground mounted radars to extend the range but is only successful at low frequencies where the broad lobe structure permits adequate cover at higher elevations.

Raising the height of the antenna has the effect of making the lobbing pattern finer. A fine grained lobing structure is often filled in by irregularities in the ground plane. Specifically, if the ground plane deviates from a flat surface then the reinforcement and destruction pattern resulting from the ground reflections breaks down. Avoidance of lobe effects is one of the prime considerations when selecting a radar location and the height of the antenna.

Source "http://www.radartutorial.eu/"

**Copyright (C) 2009 R Colman. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License.**