350lect11 - 11 Beam pattern wave interference In this...

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Unformatted text preview: 11 Beam pattern, wave interference In this lecture we will see how antenna beams can be “patterned” by us- ing interference effects of fields radiated by multiple dipoles or dipole-like elements. • Let’s recall that the antenna beam is the shape of the antenna gain function G ( θ, φ ) that can be depicted as a surface plot in 3D. Also D = G ( θ, φ ) max = 4 π Ω o and Ω o = d Ω G ( θ, φ ) G ( θ, φ ) max = d Ω | E × H | | E × H | max as well as | E × H | = | ˜ E θ | 2 2 η o with ˜ E θ = jη o I o k sin θ e- jkr 4 πr for ˆ z-polarized antennas and elements. – With = L/ 2 the above equations would represent a short dipole. – An antenna system constructed by an array of such dipoles would also be represented by the same equations, but with a different = ( θ, φ ) (to be determined). 1 • The design and analysis of multi-element or multi-dipole arrays are facilitated by the linearity of wave solutions of Maxwell’s equations: If ˜ J 1 → ME → ˜ E 1 and ˜ J 2 → ME → ˜ E 2 then α ˜ J 1 + β ˜ J 2 → ME → α ˜ E 1 + β ˜ E 2 – If radiators ˜ J 1 and ˜ J 2 produce radiated wave solutions ˜ E 1 and ˜ E 2 , respectively, then a radiator α ˜ J 1 + β ˜ J 2 would produce a wave solution α ˜ E 1 + β ˜ E 2 with arbitrary (complex) weights α and β . – By induction, the above principle of superposition can be extended to n elements. Note that this superposition principle applies at the level of fields rather than power. This is similar to superposition principle applying at the level of voltage and currents in circuit analysis. Superposition of wave fields can produce resultant wave fields with enhanced or reduced wave amplitudes as a consequence of interference effects. – A constructive interference occurs at locations where the waves being superposed are “in phase”, meaning that the phasors repre- senting the wave fields are complex numbers having the same angle — i.e., ∠ ˜ E 2 = ∠ ˜ E 1 . – A destructive interference occurs where the waves being super- posed are “out of phase”, meaning that the phasors representing the wave fields are complex numbers having an angle difference of ± 180 ◦ — i.e., ∠ ˜ E 2 = ∠ ˜ E 1 ± 180 ◦ . Constructive interference E 1 + E 2 E 1 E 2 Destructive interference E 1 + E 2 E 1 E 2 2 With a judicious choice of the locations and relative amplitudes of the radiators ˜ J 1 and ˜ J 2 , it is possible to arrange for α ˜ E 1 + β ˜ E 2 to ex- hibit constructive interference in desired beam directions — that is the essence of antenna beam design and designing high directivity antenna systems....
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350lect11 - 11 Beam pattern wave interference In this...

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