## Fundamental Concepts of Antennas

Antennas are reciprocal devices. That means that the properties of an antenna are similar both in the transmitting mode and in the receiving mode. For example, if a transmitting antenna radiates to certain directions, it can also receive from those directions—the same radiation pattern applies for both cases. The reciprocity does not apply if nonreciprocal components such as ferrite devices or amplifiers are integrated into the antenna. Also, a link between two antennas is reciprocal: The total loss is the same in both directions. However, magnetized plasma, such as in the ionosphere, between the antennas may cause Faraday rotation, making the link nonreciprocal.

The space surrounding an antenna can be divided into three regions according to the properties of the radiated field. Because the field changes smoothly, the boundaries between the regions are more or less arbitrary. The reactive near-field region is closest to the antenna. In this region, the reactive field component is larger than the radiating one. For a short current element, the reactive and radiating components are equal at a distance of

A/(2w) from the element. For other current distributions this distance is

shorter. As the distance increases, the reactive field decreases as1/r or1/r and becomes negligible compared to the radiating field. In the radiating near-field region or Fresnel region, the shape of the normalized radiation pattern depends on the distance. As the distance of the observation point changes, the difference in distances to different parts of the antenna changes essentially compared to the wavelength. In the far-field region or Fraunhofer region, the normalized radiation pattern is practically independent of the distance and the field decreases as 1/r. The boundary between the near-field and far-field regions is usually chosen to be at the distance of

2D 2

where D is the largest dimension of the antenna perpendicular to the direction of observation. At the boundary, the edges of a planar antenna are A/16 farther away from the observation point P than the center of the antenna, as illustrated in Figure 9.1. This difference in distance corresponds to a phase h

A/16

Figure 9.1 At the boundary of near-field and far-field regions.

difference of 22.5°. Because antennas are usually operated at large distances, the far-field pattern is of interest. It should be noted that at lower frequencies in case of small antennas, the outer limit of the reactive near-field region, A/(2w), is larger than the distance obtained from (9.1).

The coordinate system used for antenna analysis or measurements should be defined clearly. In analysis, the complexity of equations depends on the system. Figure 9.2 shows the spherical coordinate system that is often used. The elevation angle d increases along a great circle from 0° to 180°. The azimuth angle (f is obtained from the projection of the directional vector in the xy-plane, and it is between 0° and 360°.

An antenna can be described by several properties, which are related to the field radiated by the antenna, for example, directional pattern, gain, and polarization. Due to the reciprocity, these properties also describe the ability of the antenna to receive waves coming from different directions and having different polarizations. The importance of different radiation properties depends on the application. As a circuit element an antenna also has an impedance, efficiency, and bandwidth. Often mechanical properties such as the size, weight, and wind load are also very important.

Figure 9.2 Spherical coordinate system used for antenna analysis and measurements.

Figure 9.2 Spherical coordinate system used for antenna analysis and measurements.

An isotropic antenna that radiates at an equal strength to all directions is a good reference antenna but is not realizable in practice. A real antenna has a certain radiation pattern, which describes the field distribution as the antenna radiates. Often the radiation pattern means the same as the directional pattern. The directional pattern describes the power density P(0, <) in watts per square meter or the electric field intensity E(0, <) in volts per meter as a function of direction. Usually, the directional pattern is normalized so that the maximum value of the power density or electric field is 1 (or 0 dB). The normalized field En (0, <) is equal to the square root of the normalized power density Pn (0, <).

Often the antenna radiates mainly to one direction only. Then one main beam or the main lobe and possibly a number of lower maxima, sidelobes, can be distinguished, as in Figure 9.3(a). The directions of the lobes and nulls, the width of the main lobe, the levels of the sidelobes, and

Main lobe Sidelobes

Main lobe Sidelobes

Figure 9.3 Different representations of the directional pattern: (a) rectangular; (b) polar; (c) three-dimensional; and (d) constant-value contours.

Figure 9.3 Different representations of the directional pattern: (a) rectangular; (b) polar; (c) three-dimensional; and (d) constant-value contours.

the depths of nulls can be obtained from the directional pattern. The halfpower beamwidth, d^dB, or (f 3dB, is often used as the measure of the main lobe width.

Figure 9.3 shows different representations of directional patterns. The rectangular representation is suitable for directive antennas having a narrow main beam. The polar representation is natural for an antenna radiating over a wide range of angles. Both rectangular and polar plots are two-dimensional cuts of the three-dimensional pattern. The directional patterns are often 0-cuts or (f-cuts. For a 0-cut, for example, the angle 0 is constant and the angle (f is variable. The most important cuts are the cuts in the principal planes. The principal planes are orthogonal planes that intersect at the maximum of the main lobe, that is, at the boresight. For example, for a linearly polarized antenna, the principal planes are the E-plane and ^-plane, which are the planes parallel to the electric field vector and magnetic field vector, respectively. The whole pattern can be represented as a three-dimensional or contour plot. The scale of different plots may be a linear power, a linear field, or a logarithmic (decibel) scale.

The number of different shapes of directional patterns is countless. A pencil beam antenna has a narrow and symmetrical main lobe. Such highly directional antennas are used, for example, in point-to-point radio links, satellite communication, and radio astronomy. The directional pattern of a terrestrial broadcasting antenna should be constant in the azimuth plane and shaped in the vertical plane to give a field strength that is constant over the service area. The directional pattern of an antenna in a satellite should follow the shape of the geographic service area.

The directivity D of an antenna is obtained by integrating the normalized power pattern Pn (0, (f) over the whole solid angle

where dft is an element of the solid angle. Because Pn (0, 4) = P(0, 4)/ Pmax, the directivity is the maximum power density divided by the average power density.

Example 9.1

The beam of an antenna is rotationally symmetric. Within the 1°-wide beam, the pattern level is Pn = 1, and outside the beam the pattern level is

Pn = 0. (In practice, this kind of a beam is not realizable.) What is the directivity of this antenna?

### Solution

Because the beamwidth is small, the section of the sphere corresponding to the beam can be approximated with a circular, planar surface. The beamwidth is 1° = t/180 = 0.01745 radians. The solid angle of the beam is ft a = //4^Pn (0, 4) dft = t X 0.017452/4 = 2.392 X 10"4 steradians (square radians). The directivity is D = 4t/&a = 52,500, which in decibels is 10 log (52,500) = 47.2 dB.

The gain G of an antenna is the ratio of the maximum radiation intensity produced by the antenna to the radiation intensity that would be obtained if the power accepted by the antenna were radiated equally in all directions. For an antenna having no loss, the gain is equal to the directivity. In practical antennas there are some conductor and dielectric losses. All the power coupled to the antenna is not radiated and the gain is smaller than the directivity:

where rjr is the radiation efficiency. If the power coupled to the antenna is P, the power radiated is 7]rP, and the power lost in the antenna is (1 — rjr) P. Losses due to impedance and polarization mismatches are not taken into account in the definition of gain. The directivity and gain can also be given as functions of direction: D (0, 4) = Pn (0, 4) • D, G(0, 4) = Pn (0, 4) • G.

The effective area Af is a useful quantity for a receiving antenna. An ideal antenna with an area of A ef receives from a plane wave, having a power density of S, the same power, AfS, as the real antenna. As shown in Section 9.6, the effective area is directly related to the gain as a2

Thus, the effective area of an isotropic antenna is A /(4t). For an antenna having a radiating aperture, the aperture efficiency is defined as

where Apy is the physical area of the aperture.

The phase pattern ip(0, <) is the phase difference of the constant phase front radiated by the antenna and the spherical phase front of an ideal antenna. The position of the reference point where the spherical wave is assumed to emanate must be given. The phase center of an antenna is the reference point that minimizes the phase difference over the main beam. For example, the phase center of the feed antenna and the focal point of the reflector that is illuminated by the feed should coincide.

The polarization of an antenna describes how the orientation of the electric field radiated by the antenna behaves as a function of time. We can imagine that the tip of the electric field vector makes an ellipse during one cycle on a plane that is perpendicular to the direction of propagation (Figure 9.4). The polarization ellipse is defined by its axial ratio Emax /Emin, its tilt angle t, and its sense of rotation. The special cases of the elliptical polarization are the linear polarization and the circular polarization. The polarization of an antenna is also a function of angle (d, <).

The field radiated by the antenna can be divided into two orthogonal components: the copolar and cross-polar field. Often the copolar component is used for the intended operation and the cross-polar component represents an unwanted radiation or an interference. Linear polarizations that are perpendicular to each other, as for example the vertical and horizontal polarizations, are orthogonal. The right-handed and left-handed circular polarization are orthogonal to each other as well.

Generally, the polarization of an incoming wave and the polarization of the antenna are different, which causes a polarization mismatch. If the polarizations are the same, there will be no mismatch and the polarization efficiency is T]p = 1. In the case of orthogonal polarizations, no energy couples to the antenna and T]p = 0. If the wave is circularly polarized and the antenna is linearly polarized, one-half of the power incident on the effective area couples to the antenna, that is, T]p = V2.

The quantities Pn (6, 0), En (6, 0), D (6, 0), G(6, 0), Af (6, 0), and ip(6, 0 can be given for both copolar and cross-polar fields. An ideal antenna has no cross polarization. The cross-polar field ofa practical antenna depends on the angle (6, 0 and is often at minimum in the direction of the main beam. An antenna should have a low level of cross-polarization, for example in such applications where two channels are transmitted at the same frequency using two orthogonal polarizations.

All the power couples from the transmission line to the antenna and vice versa, if the impedance Z of the antenna is equal to the characteristic impedance of the transmission line (note that the characteristic impedance of a transmission line is real). A part of the power reflects back from an impedance mismatch. The impedance Z = R + jX has a resistive part and a reactive part. The resistive part, R = Rr + R[, is divided into the radiation resistance Rr and the loss resistance R[. The power ''absorbed'' in the radiation resistance is radiated and the power absorbed in the loss resistance is transformed into heat in the antenna. The impedance of an antenna depends on its surroundings. The reflections coming from nearby objects, such as the head of a mobile phone user, alter the impedance. Due to the mutual couplings of elements in an antenna array, the impedance of an element embedded in the array differs from that of the element alone in free space.

The bandwidth of an antenna can be defined to be the frequency band in which the impedance match, gain, beamwidth, sidelobe level, cross-polarization level, or some other quantity is within the accepted limits.

The parameters of an antenna may also be adjustable. In case of an adaptive antenna, its impedance, radiation pattern, or some other characteristic can adapt according to the electromagnetic environment.

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