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The spherical coordinate system.
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which gives the distance between point P and the antenna; is the angle measured from the x axis to the projection of r in the xy plane; and is the angle measured from the z axis to r. It is important to note that the x, y, and z axes form a right-hand set. What this means is that when one looks along the positive z direction, a clockwise rotation is required to move from the positive x axis to the positive y axis. (This is the same as the right-hand set introduced in Sec. 5.1) The right-hand set rotation becomes particularly significant when the polarization of the radio waves associated with antennas is described. 6.4 The Radiated Fields There are three main components to the radiated electromagnetic fields surrounding an antenna: two near-field regions and a far-field region. The field strengths of the near-field components decrease rapidly with increasing distance from the antenna, one component being inversely related to distance squared, and the other to the distance cubed. At comparatively short distances these components are negligible compared with the radiated component used for radio communications, the field strength of which decreases in proportion to distance. Estimates for the distances at which the fields are significant are shown in Fig. 6.4a.
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(a) The electromagnetic-field regions surrounding an antenna. (b) Vector diagrams in the far-field region.
Here, D is the largest dimension of the antenna (e.g., the diameter of a parabolic dish reflector), and l is the wavelength. Only the far-field region is of interest here, which applies for distances greater than about 2D2/l. In the far-field region, the radiated fields form a transverse electromagnetic (TEM) wave in which the electric field is at right angles to the magnetic field, and both are at right angles (transverse) to the direction
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of propagation. The vector relationship is shown in Fig. 6.4b, where E represents the electric field, H the magnetic field, and k the direction of propagation. These vectors form a right-hand set in the sense that when one looks along the direction of propagation, a clockwise rotation is required to go from E to H. An important practical point is that the wavefront can be assumed to be plane; that is, E and H lie in a plane to which k is a normal. In the far field, the electric field vector can be resolved into two components, which are shown in relation to the coordinate system in Fig. 6.5a. The component labeled E is tangent at point P to the circular arc of radius r. The component labeled E is tangent at point P to the circle of radius r sin centered on the z axis (this is similar to a circle of latitude on the earth s surface). Both these components are functions of and and in functional notation would be written as E ( , ) and E ( , ). The resultant magnitude of the electric field is given by E 2E
(6.1)
If E and E are peak values, E will be the peak value of the resultant, and if they are rms values, E will be the rms value of the resultant. The vector E0 shown at the origin of the coordinate system represents the principal electric vector of the antenna itself. For example, for a horn antenna, this would be the electric field vector across the aperture as shown in Fig. 6.5b. For definiteness, the E0 vector is shown aligned with the y axis, since this allows two important planes to be defined: The H plane is the xz plane, for which The E plane is the yz plane, for which 0 90
Magnetic field vectors are associated with these electric field components. Thus, following the right-hand rule, the magnetic vector associated with the E component will lie parallel with E and is normally denoted by H , while that associated with E will lie parallel (but pointing in the opposite direction) to E and is denoted by H . For clarity, the H fields are not shown in Fig. 6.5, but the magnitudes of the fields are related through the wave impedance ZW. For radio waves in free space, the value of the wave impedance is (in terms of field magnitudes) ZW E H E H 120 (6.2)
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