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161_1_Class3

# 161_1_Class3 - EE161 Electromagnetic Waves Spring 2010 Prof...

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Unformatted text preview: EE161 Electromagnetic Waves Spring, 2010 Prof. Y. Ethan Wang Electrical Engineering Dept. UCLA Lesson 3 • General Plane Wave Solutions • Field Direction and Wave Propagation Direction • Polarization of the Wave General Plane Wave Solutions θ φ θ φ θ cos sin sin cos sin ' z y x z + + = ' ) ' ( ~ jkz Ae z E − = As aforementioned, an uniform plane wave traveling along z’ axis, has an expression: x y z Z’ φ θ One can rotate the Cartesian coordinates arbitrarily, with certain span angles θ and φ : The coordinate transform defines: z jk y jk x jk jkz jky jkx z y x jk jkz z y x Ae Ae Ae Ae E − − − − − − + + − − = = = = θ φ θ φ θ θ φ θ φ θ cos sin sin cos sin ) cos sin sin cos sin ( ' ~ Therefore, a plane wave that is traveling toward an arbitrary direction is given by: (Linear phase variation in regard to x, y, z axes) Plane Wave Solutions in Vector Form ) ( ) , , ( ~ z k y k x k j x z y x Ae z y x E + + − = General solution of E field, (x- component) n k z k y k x k z y x ˆ ˆ ˆ ˆ = + + = k Wave number vector: Position vector: z z y y x x ˆ ˆ ˆ + + = r r k ⋅ Propagation direction 2 2 2 2 z y x k k k k + + = Wave number ⎪ ⎩ ⎪ ⎨ ⎧ = = = θ φ θ φ θ cos sin sin cos sin k k k k k k z y x ( θ, φ are the propagation angle) x y z k k x k y k z φ θ Observation direction Plane Wave Solutions in Vector Form r k r k ⋅ − j e Phase variation of the wave follows the factor of ) ( z k y k x k j z y x e + + − e.g....
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161_1_Class3 - EE161 Electromagnetic Waves Spring 2010 Prof...

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