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Unformatted text preview: 18 Reflecting plates, monopole antennas, corner reflectors TE reflection: k i k r k t θ 2 θ 1 x z H i k 1 cos θ 1 k 1 sin θ 1 k 2 sin θ 2 Medium 1 Medium 2 θ 1 Γ ⊥ ≡ E y r E y i = η 2 cos θ 1 η 1 cos θ 2 η 2 cos θ 1 + η 1 cos θ 2 τ ⊥ ≡ E yt E y i = 2 η 2 cos θ 1 η 2 cos θ 1 + η 1 cos θ 2 , TM reflection: k i k r k t θ 2 θ 1 x z E i k 1 cos θ 1 k 1 sin θ 1 k 2 sin θ 2 Medium 1 Medium 2 θ 1 Γ ≡  E r E i = η 2 cos θ 2 η 1 cos θ 1 η 2 cos θ 2 + η 1 cos θ 1 τ ≡ E t E i = 2 η 2 cos θ 1 η 2 cos θ 2 + η 1 cos θ 1 . • In deriving the transmission and reflection rules for TE and TM modes summarized above we assumed lossless propagation media during the last two lectures. • The equations can be easily modified as described next if either medium 1 or medium 2 or both have nonzero conductivities σ 1 and/or σ 2 . 1 In general, in the case of a noninsulating medium with a finite con ductivity σ , we expect a conduction current ˜ J = σ ˜ E , in which case the planewave form of Ampere’s law can be cast as j k × ˜ H = σ ˜ E + jω ˜ E , = jω ( + σ jω ) ˜ E . Since this equation differs from the nonconducting case only by having + σ jω in place of , propagation parameters k = ω √ μ and η = μ of nonconducting media are modified as k = ω μ ( + σ jω ) and η = μ + σ jω , respectively, in homogeneous conducting media. In other wors a con ducting medium is treated as a dielectric with a permittivity + σ jω . • Consider the wavenumber k = ω μ ( + σ jω ) in a medium with σ/ω . In that case — poor conductor approxi 2 mation — we can approximate k as k = ω μ ( j σ ω ) = ω μ (1 j σ ω ) ≈ ω √ μ (1 j σ 2 ω ) = ω √ μ  j 1 2 μ σ ≡ k j k , with k ≡ Re { k } ≈ ω √ μ Propagation constant and k ≡  Im { k } ≈ 1 2 μ σ Attenuation constant . These terms are applicable since e j k · r = e j k s = e j ( k k ) s = e k s e j k s clearly signify an attenuating planewave field with distance s measured in the direction of a unit vector ˆ k such that k introduced above relates to k = k j k as in k = ˆ k ( k j k ) ....
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 Fall '08
 Staff
 Electromagnet, Trigraph, Boundary value problem, propagation constant, Whip antenna, monopole antenna

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