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The values of gain range between zero and in nity, whereas for directivity the values range between unity and in nity. However, though directivity can be found either theoretically or experimentally, the gain of an antenna is almost always determined by a direct comparison of measurement against a reference, usually the standard gain antenna. Gain is expressed also in decibels, that is G 10 log10 g dB: 2:17
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When we use the isotropic antenna as a reference then we use the dBi notation, which means decibels over isotropic. 2.5. POLARIZATION 2.5.1. Wave and Antenna Polarization Polarization refers to the direction of the electric eld component of an electromagnetic wave. The wave is called linearly or plane polarized, that is the locus of oscillation of the electric eld vector within a plane perpendicular to the direction of propagation forms a straight line. On the contrary, when the locus of the tip of an electric eld vector forms an ellipse or a circle, the wave is called an elliptically or circularly polarized wave, respectively. There is a tendency to refer to antennas as vertically or horizontally polarized, though it is only their radiations that are polarized. Next, we discuss the mechanics of various polarizations that we encounter in antenna communication systems. 2.5.2. Linear, Circular, and Elliptical Polarization Consider a plane wave traveling in the positive z direction, with the electric eld component along the x direction as shown in Figure 2.7(a). This wave is linearly polarized, in the x direction, and its electric eld can be expressed as: Ex Ex0 sin ot bz :
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FIGURE 2.7. Polarization of a wave: (a) linear, (b) circular, and (c) elliptical.
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FIGURE 2.8. Polarization ellipse at z 0 of an elliptically polarized electromagnetic wave.
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In Figure 2.7(b) the wave has both a x and a y electric eld component. If the two components Ex and Ey have the same magnitude, then the total (vector) electric- eld rotates as a function of time with the tip of the vector forming a circular trace, and the wave is thus called circular polarized. Generally, the wave consists of two electric eld components, Ex and Ey , of different amplitude ratios and relative phases that can yield an elliptically polarized wave, as shown in Figure 2.7(c). The polarization ellipse may have any orientation, which is determined by its tilt angle, as depicted in Figure 2.8. The ratio of the major to minor axes of the polarization ellipse is called the axial ratio (AR). For any wave traveling in the positive z direction, the electric eld components in the x and y directions can be written as: Ex Ex0 sin ot bz 2:19 2:20
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where Ex0 and Ey0 are the amplitudes in x and y direction, respectively, and d is the time-phase angle between them. By manipulating these two components we can show that [1,11]
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2 2 Ex 2Ex Ey cos d Ey 2 sin2 d: 2 E 1 E2 E1 E2
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Depending on the values of Ex , Ey , and d, this equation can be expressed as the equation of an ellipse or of a circle. The sense of rotation of a circularly or elliptically polarized wave plays an important role in a communication link. It is de ned by the direction of rotation of the wave as it propagates towards or away from an observer along the direction of propagation. If, for example, a wave is moving away and its rotation is clockwise
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then we say that the wave has a clockwise sense of rotation. The most common notation used today is that of the IEEE by which the sense of rotation is always determined observing the eld rotation as the wave travels away from the observer. If the rotation is clockwise, the wave is right-handed or clockwise circularly polarized (RH or CW). If the rotation is counterclockwise, the wave is left-handed or counterclockwise circularly polarized (LH or CCW). The same applies to elliptically polarized waves. The polarization state of an antenna is de ned as the polarization state of the wave transmitted by the antenna. It is characterized by the sense of rotation and the spatial orientation of the ellipse, if it is elliptically polarized. If the receiving antenna has a polarization that is different from that of the incident wave, a polarization mismatch will occur. A polarization mismatch causes the receiving antenna to extract less power from the incident wave. Polarization loss factor (PLF) is used as a gure of merit to measure the degree of polarization mismatch. It is de ned as the square power of the cosine angle between the polarization states of the antenna in its transmitting mode and the incoming wave (see Fig. 2.9). PLF jcos gj2 Generally, an antenna is designed for a desired polarization. The component of the electric eld in the direction of the desired polarization is called the co-polar component, whereas the undesired polarization, usually taken in orthogonal direction to the desired one, is known as cross-polar component. The latter can be due to a change of polarization characteristics during the propagation of or scattering of waves that is known as polarization rotation. An actual antenna does not completely discriminate against a cross-polarized wave due to structural abnormalities of the antenna. The directivity pattern obtained over the entire direction on a representative plane for cross-polarization with respect to the maximum directivity for the desired (co-polar) polarization is called antenna cross-polarization discrimination and plays an important factor in determining the antenna performance.