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 2 1 x z H i k 1 cos 1 k 1 sin 1 k 2 sin 2 Medium 1 Medium 2 1 E y r E y i = 2 cos 1- 1 cos 2 2 cos 1 + 1 cos 2 E yt E y i = 2 2 cos 1 2 cos 1 + 1 cos 2 , TM reflection: k i k r k t 2 1 x z E i k 1 cos 1 k 1 sin 1 k 2 sin 2 Medium 1 Medium 2 1 - E r E i = 2 cos 2- 1 cos 1 2 cos 2 + 1 cos 1 E t E i = 2 2 cos 1 2 cos 2 + 1 cos 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 1 and/or 2 . 1 In general, in the case of a non-insulating medium with a finite con- ductivity , we expect a conduction current J = E , in which case the plane-wave form of Amperes law can be cast as- j k H = E + j E , = j ( + j ) E . Since this equation differs from the non-conducting case only by having + j in place of , propagation parameters k = and = of non-conducting media are modified as k = ( + j ) and = + j , respectively, in homogeneous conducting media. In other wors a con- ducting medium is treated as a dielectric with a permittivity + j . Consider the wavenumber k = ( + j ) in a medium with / . In that case poor conductor approxi- 2 mation we can approximate k as k = (- j ) = (1- j ) (1- j 2 ) = - j 1 2 k- j k , with k Re { k } Propagation constant and k - Im { k } 1 2 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 k such that k introduced above relates to k = k- j k as in k = k ( k- j k ) ....
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350lect18 - 18 Reflecting plates, monopole antennas, corner...

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