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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 h p = 1. In the case of orthogonal polarizations, no energy couples to the antenna and hp = 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, hp = 1⁄2 .

The quantities Pn (u, f), En (u, f), D (u, f), G (u, f), A ef (u, f ), and c(u, f) can be given for both copolar and cross-polar fields. An ideal

antenna has no cross polarization. The cross-polar field of a practical antenna depends on the angle (u, f ) 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 = R r + R l , is divided into the radiation resistance R r and the loss resistance R l . 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, crosspolarization 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.

9.2 Calculation of Radiation from Antennas

The fields radiated by an antenna can be calculated using auxiliary quantities called the magnetic vector potential A and the electric vector potential F.

An antenna can be considered to be a sinusoidal current density distribution J in a volume V. At a point of space, the magnetic vector potential is

where R is the distance from a volume element dV to the point of observation and k = 2p/l . If the currents flow on a surface S and the surface current density is Js , (9.6) can be written as

In principle, the radiated fields can be calculated for all antennas using these equations. They are well suited for wire antennas that have a known current distribution.

An aperture antenna, such as a horn antenna, has an aperture or a surface from which the radiation seems to emanate. It may be difficult to find out the current distribution. Then it may be easier to calculate the radiated fields from the aperture fields Ea and Ha . The aperture fields are replaced with surface currents that would produce the aperture fields. The magnetic field is replaced with a surface current having a density of

where n is a unit vector normal to the surface of the aperture. The vector potential A corresponding to Js is then calculated. The radiated field components are obtained from (9.8) and (9.9). The electric field of the aperture is replaced with a magnetic surface current having a density of

214 Radio Engineering for Wireless Communication and Sensor Applications

The total radiated field is obtained by summing up the field components due to Js and Ms .

9.3 Radiating Current Element

Figure 9.5 shows a short current element at the origin. The element of a length dz along the z -axis carries an alternating sinusoidal current I 0 , which is constant along the element. This kind of current element is also called

Figure 9.5 Radiating current element.

the Hertz dipole. In his experiments, Heinrich Hertz used end-loaded dipoles. Because of the capacitive loadings, currents could flow even at the ends of the dipole making a nearly constant current distribution possible. As explained in Chapter 2, a fluctuating current produces electromagnetic waves: The current produces a changing magnetic field, the changing magnetic field produces a changing electric field, the changing electric field produces a changing magnetic field, and so on.

The volume integral of the current density is eV J dV = I0 dz uz in the case of a current element. Therefore, the vector potential at a point P (r , u, f ) is

The components of the vector potential in the spherical coordinate system are A r = A z cos u, Au = −A z sin u, and Af = 0. Equations (9.8) and (9.9) give the components of the fields:

where h is the wave impedance in free space.

Those components of the field having a 1/r 2 or 1/r 3 dependence dominate at small distances but become negligible at larger distances. Far away from the element, the fields are

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Other components are negligible. The electric and magnetic fields are in phase and perpendicular to each other, just like in the case of a plane wave.

The power radiated by the current element is calculated by integrating the Poynting vector S = (1/2) Re (E × H * ) over a sphere surrounding the element. If the radius of the sphere, r, is much larger than dz , S is perpendicular to the surface of the sphere and the outflowing power per unit area is S = | S | = (1/2) | Eu Hf | . The surface element for integration is selected as shown in Figure 9.6. The power radiated is

The radiation resistance of the current element is obtained by equating the radiated power with P = (1/2) R r I02 :

Figure 9.6 Integration of power density over a sphere.