350lect15 - 15 Plane-wave form of Maxwells equations,...

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15 Plane-wave form of Maxwell’s equations, prop- agation in arbitrary direction Having seen how EM waves are generated by radiation sources and how spher- ical TEM waves develop a “planar” character over increasingly large regions as they propagate away from their sources, it is time to shift our attention to propagation and guidance phenomena using a plane-wave formalism. x y z D x 0M HPBW = λ D x 2 D x 2 D 2 x λ Fresnel region r o x Perhaps the most “practical” rationalization of this switch from spherical to plane-wave emphasis is that waves produced by compact sources invariably “look” planar at the scales of practical receiving systems (that will study near the end of this course) situated afar. We wish to study wave solutions of Maxwell’s equations exhibiting the planar phasor form ˜ E = E o e - j k · r eE o e - j k · r and time-domain variations Re { ˜ E e jωt } = Re { E o e j ( ωt - k · r ) } e | E o | cos( - k · r + E o ) where wave vector k is to be found in compliance with ω and Maxwell’s equations according to some speciFc “dispersion relation” including the details of the propagation medium. 1
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For simplicity, the above phasor has been declared to be linearly polarized. Circular or elliptic polarized wave felds can be con- structed later on via superposition methods. k z x y z k r =( x, y, z ) k x k y k · r =const . Linearly polarized wave feld phasor above can be expanded as ˜ E = E o e - j k · r = E o e - j ( k x x + k y y + k z z ) assuming a wave vector k k x ,k y z )=ˆ xk x yk y zk z expressed in terms o± its projections ( k x y z ) along the Cartesian coordinate axes ( x, y, z ) . Aspec ia lcaseweare±am i l iarw ithis k x = k y =0 z > 0 , when k = k z ˆ z = k ˆ z and e - j k · r = e - jkz as in plane TEM waves travelling in + z direction having a wavelength λ = 2 π k and propagation speed v p = ω k .
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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350lect15 - 15 Plane-wave form of Maxwells equations,...

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