# 350lect18 - 18 Reflecting plates monopole antennas corner...

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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 non-zero conductivities σ 1 and/or σ 2 . 1 In general, in the case of a non-insulating medium with a finite con- ductivity σ , we expect a conduction current ˜ J = σ ˜ E , in which case the plane-wave 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 non-conducting case only by having + σ jω in place of , propagation parameters k = ω √ μ and η = μ of non-conducting 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 plane-wave 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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350lect18 - 18 Reflecting plates monopole antennas corner...

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