Performing the necessary integrals yields a jkμ iπa

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Performing the necessary integrals yields A = jkμ Iπa 2 sin θ e jkr 4 πr ˆ φ The resulting magnetic field is obtained by taking the curl of the vector potential. The electric field in the far zone is normal to the magnetic field and to the direction of propagation (radially away) with an amplitude governed by the impedance of free space: H = 1 μ ∇ × A = Mk 2 sin θ e jkr 4 πr ˆ θ E = Z ˆ r × H = MZ k 2 sin θ e jkr 4 πr ˆ φ where M = Iπa 2 is the magnetic moment of the current loop. The radiation pattern for the small current loop is given by P r = sin 2 θ , just like the elemental electric dipole. The total radiated power and radiation resistance can be calculated in the same way, leading to the result R a ( a/λ ) 4 . 42
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In view of the small radius limit assumed here, the radiation resistance of the small current loop is necessarily very small, making the antenna an inefficient one. The radiation resistance of an antenna made of multiple ( n ) current loops grows as n 2 , which can improve the efficiency to some degree. In some situations, such as the short dipole AM car radio antenna discussed earlier, receiving antennas with poor efficiency represent acceptable engineering solutions, as we will see after analyzing the effects of system noise on signal detectability. Poor efficiency is seldom acceptable for transmitting antennas, however. 2.3.8 Duality 2.4 Polarization The chapter would be incomplete without a discussion of electromagnetic wave polarization. The antennas discussed above all transmit and receive waves with a specific polarization, and polarizations must be “matched” in order to optimize communication and radar links. Below, wave polarization is introduced in an elementary way. E x y kz- ω t α ω t-kz z Figure 2.10: Illustration of linear (dashed lines) and circular (solid lines) polarized waves. The direction or propagation ( ˆ z ) is toward the right. The polarization of a wave refers to variations in the direction of the wave’s electric field. We restrict the discussion to transverse electromagnetic (TEM) plane waves, in which surfaces of constant phase are planes oriented normally to the direction of propagation. Taking the direction of propagation to be ˆ z , we can express the electric field of the wave as: E = ( E 1 ˆ x + E 2 e ˆ y ) e j ( ωt kz ) where E 1 , E 2 , and ψ are taken to be real constants. The plane wave nature of the wave is contained in the trailing exponential, which is assumed if not always explicitly written. The ψ term controls the relative phase of the vector components of the transverse electric field. Depending on the ratio of E 1 to E 2 and the value of ψ , the wave can be linearly, elliptically, or circularly polarized. 2.4.1 Linear polarization If ψ =0, the two transverse components of E are in phase, and in any z = const plane, they add to give a vector with the same direction specified by α = tan 1 E y /E x = tan 1 E 2 /E 1 . Since the electric field always lies in the same plane, we refer to such waves as linear or plane polarized. Examples are vertically ( α = π/ 2
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