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Figure 6.6 (a) A radiation pattern. (b) The beamwidth in the H-plane. (c) The beamwidth in the E-plane.
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Because the radiation pattern is defined in terms of radiated power, the normalized electric field strength pattern will be given by 2g( , ). 6.8 Beam Solid Angle and Directivity Plane angles are measured in radians, and by definition, an arc of length R equal to the radius subtends an angle of one radian at the center of a circle. An angle of radians defines an arc length of R on the circle.
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Surface area R2 R R R Unit steradian R
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Steradians
(b) Figure 6.7
(a) Defining the radian. (b) Defining the steradian.
This is illustrated in Fig. 6.7a. The circumference of a circle is given by 2 R, and hence the total angle subtended at the center of a circle is 2 rad. All this should be familiar to the student. What may not be so familiar is the concept of solid angle. A surface area of R2 on the surface of a sphere of radius R subtends unit solid angle at the center of the sphere. This is shown in Fig. 6.7b. The unit for the solid angle is the steradian. A solid angle of steradians defines a surface area on the sphere (a spherical cap) of R2 . Looking at this another way, a surface area A subtends a solid angle A/R2 at the center of the sphere. Since the total surface area of a sphere of radius R is 4 R2, the total solid angle subtended at the center of the sphere is 4 sr. The radiation intensity is the power radiated per unit solid angle. For a power Prad radiated, the average radiation intensity (which is also the isotropic value) taken over a sphere is Ui Prad 4 W/sr (6.7)
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From the definition of directivity d, the maximum radiation intensity is Umax
(6.8)
The beam solid angle, A, for an actual antenna is defined as the solid angle through which all the power would flow to produce a constant radiation intensity equal to the maximum value. Thus Umax Prad
(6.9)
Combining Eqs. (6.7), (6.8), and (6.9) yields the important result
(6.10)
This is important because for narrow-beam antennas such as used in many satellite communications systems, a good approximation to the solid angle is
> HPBWE
HPBWH
(6.11)
where HPBWE is the half-power beamwidth in the E plane and HPBWH is the half-power beamwidth in the H plane, as shown in Fig. 6.6. This equation requires the half-power beamwidths to be expressed in radians, and the resulting solid angle is in steradians. The usefulness of this relationship is that the half-power beamwidths can be measured, and hence the directivity can be found. When the half-power beamwidths are expressed in degrees, the equation for the directivity becomes
6.9 Effective Aperture
41253 HPBW8E 3 HPBW8H
(6.12)
So far, the properties of antennas have been described in terms of their radiation characteristics. A receiving antenna has directional properties also described by the radiation pattern, but in this case it refers to the ratio of received power normalized to the maximum value. An important concept used to describe the reception properties of an antenna is that of effective aperture. Consider a TEM wave of a given power density at the receiving antenna. Let the load at the antenna terminals be a complex conjugate match so that maximum power transfer occurs and power Prec is delivered to the load. Note that the power delivered to the actual receiver may be less than this as a result of
Antennas
feeder losses. With the receiving antenna aligned for maximum reception (including polarization alignment, which is described in detail later), the received power will be proportional to the power density of the incoming wave. The constant of proportionality is the effective aperture Aeff which is defined by the equation Prec Aeff (6.13)
For antennas which have easily identified physical apertures, such as horns and parabolic reflector types, the effective aperture is related in a direct way to the physical aperture. If the wave could uniformly illuminate the physical aperture, then this would be equal to the effective aperture. However, the presence of the antenna in the field of the incoming wave alters the field distribution, thereby preventing uniform illumination. The effective aperture is smaller than the physical aperture by a factor known as the illumination efficiency. Denoting the illumination efficiency by I gives Aeff
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