Transmitting and Receiving Antennas
740 16. Transmitting and Receiving Antennas The total radiated power is obtained by integrating Eq. (16.1.4) over all solid angles 16 dΩ = sin θ dθdφ, that is, over 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π : π 2π Prad = U(θ, φ) dΩ (total radiated power) (16.1.5) Transmitting and Receiving Antennas 0 0 A useful concept is that of an isotropic radiator—a radiator whose intensity is the same in all directions. In this case, the total radiated power Prad will be equally dis- tributed over all solid angles, that is, over the total solid angle of a sphere Ωsphere = 4π steradians, and therefore, the isotropic radiation intensity will be: π 2π dP Prad Prad 1 UI = = = = U(θ, φ) dΩ (16.1.6) dΩ I Ωsphere 4π 4π 0 0 Thus, UI is the average of the radiation intensity over all solid angles. The corre- 16.1 Energy Flux and Radiation Intensity sponding power density of such an isotropic radiator will be: dP UI Prad The flux of electromagnetic energy radiated from a current source at far distances is = = (isotropic power density) (16.1.7) 2 2 given by the time-averaged Poynting vector, calculated in terms of the radiation fields dS I r 4πr (15.10.4): 16.2 Directivity, Gain, and Beamwidth 1 1 e−jkr ejkr PP= × ∗ = − ˆ + ˆ × ˆ ∗ − ˆ ∗ Re(E H ) jkη jk Re (θθFθ φφFφ) (φφFθ θθFφ) 2 2 4πr 4πr The directive gain of an antenna system towards a given direction (θ, φ) is the radiation intensity normalized by the corresponding isotropic intensity, that is, Noting that θˆ × φˆ = ˆr, we have: ˆ + ˆ × ˆ ∗ − ˆ ∗ = | |2 +| |2 = 2 U(θ, φ) U(θ, φ) 4π dP (θθFθ φφFφ) (φφFθ θθFφ) ˆr Fθ Fφ ˆr F⊥(θ, φ) D(θ, φ)= = = (directive gain) (16.2.1) UI Prad/4π Prad dΩ Therefore, the energy flux vector will be: It measures the ability of the antenna to direct its power towards a given direction.
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