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FIGURE 2.5. De nition of a solid angle and a steradian.
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steradian, which is de ned as the solid angle with its vertex at the center of a sphere of radius r, subtended by a spherical surface area equivalent to that of a square of size r2 (see Fig. 2.5). But the area of a sphere of radius r is given by A 4pr 2, so in a closed sphere there are 4pr2 =r 2 4p steradian (sr). For a sphere of radius r, an in nitesimal surface area dA can be expressed as: dA r2 sin y dy df m2 and hence the element of solid angle dO of a sphere is given by: dO dA sin y dy df sr : r2 2:6 2:5
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The total power that can be radiated is given by:
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2p p  I y; f dO I y; f sin y dy df: O 0 0
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Let us consider an isotropic radiator as an example. An isotropic antenna refers to a hypothetical antenna radiating equally in all directions and its power pattern is uniformly distributed in all directions. That means the radiation intensity of an isotropic antenna is independent of the angles y and f and the total radiated power will be:
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2p p  Ii dO Ii sin y dy df Ii  dO 4pIi O 0 0 O
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or Ii Prad, which is the radiation intensity of an isotropic antenna. 4p
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Dividing I y; f by its maximum value Imax, leads to the normalized antenna power pattern, that is, In y; f I y; f Imax y; f dimensionless : 2:9
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2.4. DIRECTIVITY AND GAIN An important parameter that indicates how well radiated power is concentrated into a limited solid angle is directivity D. The directivity of an antenna is de ned as the ratio of the maximum radiation intensity to the radiation intensity averaged over all directions (i.e., with reference to the isotropic radiator). Thus, the average radiation intensity is found by dividing the total antenna radiated power r by 4p sr, or D Imax y; f Imax y; f Imax y; f 4pImax y; f Iav Ii Prad =4p Prad dimensionless : 2:10 The narrower the main lobe of the antenna radiation pattern, the larger the directivity of the antenna. Obviously, the directivity of an isotropic antenna is unity. Any other antenna will have a directivity larger than unity (i.e., larger than the isotropic), as shown in Figure 2.6. Let us consider the directivity of a very short dipole, as an example. The average pointing vector for the dipole is given by [11]   Z I0 Lb 2 2 sin y Sav 2 4pr
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FIGURE 2.6. Directive pattern versus an isotropic one.
where L is the length of the short dipole, I0 is the current owing through the dipole, Z is the wave impedance in free space, and b 2p=l. Using Equation (2.3) we can solve for the radiation intensity, and then we can use Equation (2.10) to obtain a directivity of 1.5. This occurs at the y 90 direction (see Fig. 2.2). Thus, in this direction, a very short dipole can radiate 1.5 times more power than the isotropic radiator. This is often expressed in decibels such that D 10 log10 d dB 10 log10 1:5 1:76 dB: 2:12
The gain of an antenna is closely associated with directivity, and it is de ned as the ratio of the maximum radiation intensity in a given direction to the maximum radiation intensity produced in the same direction from a reference antenna with the same power input. Any convenient type of antenna can be taken as a reference antenna. Usually, the type of reference antenna is determined by the speci c application, but the most commonly used one is the isotropic radiator, and thus we can write: G Imax y; f Imax y; f Ii Pin =4p dimensionless 2:13
where the radiation intensity of the isotropic radiator is equal to the input, Pin, of the antenna divided by 4p. As the gain of an antenna depends on how ef cient it is in converting input power into radiated elds, we need to take into consideration its ef ciency before we determine the actual gain. In general, antenna ef ciency (e) is de ned as the ratio of the power radiated by the antenna to the input power at its terminals: e Prad Rr Pin Rr Rloss dimensionless 2:14
where Rr is the radiation resistance of the antenna; Rr is an equivalent resistance in which the same current owing at the antenna terminals will produce power equal to that radiated by the antenna. Rloss is the loss resistance due to any conductive or dielectric losses of the materials used to construct the antenna. So, if we include these losses, a real antenna will have radiation intensity I y; f eI0 y; f 2:15
where I0 y; f is the radiation intensity of the same antenna with no losses. Using Equation (2.15) into (2.13) yields the de nition of gain in terms of the antenna directivity: G Imax y; f eImax0 y; f eD: Ii Ii 2:16