Consider a radar antenna located at the origin of a spherical coordinate system as shown in Fig. 2. The observation point is on a sphere having a very large radius R, which is located in the far field of the antenna. Assuming that the antenna is reciprocal, so that the transmit and receive patterns are identical. When the antenna is transmitting, its radiation efficiency is defined as

where P0 is the total power consumed by the antenna, and Pr is the power radiated by the antenna. The later can be expressed by the radiation intensity Ф(в, ф)

in which ф is the azimuth angle and в is the elevation angle, as shown in Fig. 2. From Eq. (2), the average radiation intensity is easily obtained

(3)

From Eqs. (1)-(3), we can define some basic parameters of an antenna. Directivity is a measure of the ability of the antenna to concentrate the radiated power in a particular direction. Thus it can be defined as the

Fig. 2. Spherical coordinate system. The azimuth angle ф and elevation angle в are adopted by most antennas. |

ratio of the achieved radiation intensity in the direction to the average:

Ф{&, ф)

(4i |

0(в, ф) =

m

In practice, one is usually interested in the maximum directivity of the main lobe.

Gain is another important parameter of an antenna. It represents the ability to concentrate the power accepted by the antenna in a particular direction:

Thus antenna gain is always less than directivity except for a lossless and reflectionless antenna (n = 1). Again, the peak value of the gain, G0, is of more interest in practice.

The aperture of an antenna is its physical area projected on a plane perpendicular to the desired direction. If the antenna is lossless and the aperture of area A is uniformly illuminated with equiphase, the directivity is given by

which is the maximum available gain from an aperture A. Practical antennas are not uniformly illuminated, but they have a maximum in the center of the aperture and are less tapered toward the edges in order to reduce the sidelobes of the pattern. In this case, an effective aperture is defined from the directivity

Г7) |

Ae(9, ф) = -^—D(9, ф) 4 л-

The concept of effective aperture is very useful when considering the antenna in its receiving mode, which measures the effective absorption area of the antenna to an incident plane wave. Let Ae be the peak value of

Ae(e, ф). Clearly, the effective aperture Ae is always less than the physical aperture A by a factor na

which is usually called the aperture efficiency. From the definitions in Eqs. (6)-(8), we can see that the aperture efficiency measures only how effectively a given aperture is used, but does not involve the EM energy loss. Therefore, the efficiency of the antenna should be the multiplication of na and antenna losses nL.

When the power radiation intensity Ф(в, ф), the directivity D(e, ф), the gain G(e, ф), and the effective

aperture Ae(e, ф) are normalized to their peak values, they will be identical and are called the antenna radiation pattern, which in fact represents the EM-energy distribution in three-dimensional (3-D) angular space. There are several ways to plot the radiation pattern: rectangular or polar, voltage intensity or power density, absolute value or dB value, or power per unit solid angle. However, the 3-D plots of radiation pattern require extensive data. Thus two-dimensional (2-D) plots are usually adopted as a result of the ease in measurement and plotting. Generally, the two cuts of 3-D plots in the principal elevation and azimuth planes are sufficient to describe the pattern performance of an antenna, which is much less costly.

Instead of azimuth and elevation, E — and H-plane patterns are usually used in actual radar antennas because the terms azimuth and elevation imply the earth-based reference coordinates, which are not applicable to some space-based systems like airborne and satellite.

The main lobe of the radiation pattern is in the direction of maximum gain; all other lobes are called sidelobes. In addition to the peak gain of the main lobe, two other important features of an antenna pattern are the beamwidth of the main lobe, which is usually specified at the half-power level (3 dB), and the maximum sidelobe level. The half-power beamwidth (HPBW) is usually a measure of the resolution of an antenna. Therefore, if two identical targets at the same range can be separated by the HPBW, they are said to be resolved in angle.

The beamwidth of an antenna is determined by the size of the antenna aperture as well as the amplitude and phase distributions across the aperture. For a given distribution, the half-power beamwidth in a particular plane is inversely proportional to the size of the aperture in that plane

(9) |

hpbw = k-;

JL_i>

where L is the aperture dimension and K is a constant for the given distribution, which is known as the beamwidth factor. Accurate estimate of the beamwidth factor must take into account the aperture illumination function. However, K = 70° can give a rough estimate in most cases. For reflector-type antennas, a good estimation to K is given by (6)

‘ 101 |

К = 1.05238/+ 55.9486 (deg)

in which I is the absolute value of edge illumination in decibels.

From the half-power beamwidths in two orthogonal principle planes, one can obtain a practical formula for predicting the gain of a relatively lossless antenna

where C is a unitless constant and в1 and в2 are HPBWs in degrees. The accurate value of C depends on the antenna efficiency, but the rough estimate can be taken from 26,000 to 35,000 (1,7).

Fig. 3. Common reflector antenna types. (a) Paraboloid; (b) parabolic cylinder; (c) shaped reflector; (d) multiple beam; (e) monopulse; (f) cassegrain. |

(d) (e) (f) |

Ideally, an antenna radiation pattern would consist of a single main lobe and no sidelobes. However, for all practical radar antennas, it contains numerous sidelobes. These sidelobes can be a source of problems for the radar system. For example, a radar for detecting low-flying aircraft targets can receive strong ground or ocean echoes, which is called the clutter, through the sidelobes. This signal will interfere with the desired echoes coming from the low radar cross-section targets through the main lobe. Another fatal problem of the sidelobes is coming from jamming, which threatens most military radars. Therefore, it is often (but not always) desirable to design radar antennas with sidelobes as low as possible to minimize such problems. However, low sidelobes and high gain are competing requirements. Thus, the trade-off between sidelobes level, gain, and beamwidth is an important consideration for designing radar antennas.