325_Sp2011_14_Plane_waves_updated

325_Sp2011_14_Plane_waves_updated - © Copyright Dean P....

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Unformatted text preview: © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 1 14. Uniform plane waves EE325 Mikhail Belkin © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 2 Summary of electromagnetics: Maxwell’s equations • summarizing everything we have so far, valid even if things are changing in time • plus material properties B ∇ = r r g { r o B H μ μ μ = r r v D ρ ∇ = r r g { r o D E ε ε ε = r r J E σ = r r D H J t ∂ ∇× = + ∂ r r r B E t ∂ ∇× = - ∂ r r Faraday’s law Ampere’s law Gauss’s law © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 3 Wave equations • Let’s derive equations that will determine how E and H fields should change in space and time in case of time-varying fields • Remember that D= ε E, B= μ H, and J= σ E; assume there is no charge H ∇ = r r g E ∇ = r r g E H E t σ ε ∂ ∇× = + ∂ r r r H E t μ ∂ ∇× = - ∂ r r • Take curl of Eq. (1) to get an equation for E-field and curl of Eq. (2) to get an equation for H-field: (1) (2) (3) (4) 2 2 H E E E H t t t t μ μ μσ με ∂ ∂ ∂ ∂ ∇ ×∇ × = -∇ × = - ∇ × = -- ∂ ∂ ∂ ∂ r r r r r 2 2 E H H H E E E t t t t σ ε σ ε μσ εμ ∂ ∂ ∂ ∂ ∇×∇× = ∇× + ∇× = ∇× + ∇× = -- ∂ ∂ ∂ ∂ r r r r r r r © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 4 Wave equations (2) • We know the expression for ∇ × ∇ × F: where • We also know that and 2 2 H E E E H t t t t μ μ μσ με ∂ ∂ ∂ ∂ ∇×∇× = -∇× = - ∇× = -- ∂ ∂ ∂ ∂ r r r r r 2 2 E H H H E E E t t t t σ ε σ ε μσ εμ ∂ ∂ ∂ ∂ ∇ ×∇ × = ∇ × + ∇ × = ∇ × + ∇ × = -- ∂ ∂ ∂ ∂ r r r r r r r ( 29 2 F F F ∇×∇× = ∇ ∇ •- ∇ r r r 2 2 2 2 x y z F x A y A z A ∇ ≡ ∇ + ∇ + ∇ r ˆ ˆ ˆ H ∇ = r r g E ∇ = r r g 2 2 2 E E E t t μσ με ∂ ∂ ∇ = + ∂ ∂ r r r 2 2 2 H H H t t μσ εμ ∂ ∂ ∇ = + ∂ ∂ r r r and When σ =0 (dielectrics): 2 2 2 E E t με ∂ ∇- = ∂ r r and 2 2 2 H H t εμ ∂ ∇- = ∂ r r Wave equations: © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 5 Solutions of wave equations: use phasor form • E and H are functions of both space and time: E (t,x,y,z), H (t,x,y,z) • Assume harmonic functions to separate space and time • We can always use Fourier series to satisfy any other time form of the initial conditions • Any Fourier component of E and H field that oscillates at frequency ϖ must satisfy these equations! ( 29 ( 29 , , , , , j t s E x y z t E x y z e ϖ = r r ( 29 ( 29 , , , , , j t s H x y z t H x y z e ϖ = r r 2 2 s s s E j E E ϖμσ ϖ με ∇ =- r r r 2 2 s s s H j H H ϖμσ ϖ εμ ∇ =- r r Our wave equations become: and © Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 6 Quick review: phasor notation and complex...
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This note was uploaded on 04/03/2012 for the course EE 325 taught by Professor Brown during the Fall '08 term at University of Texas.

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325_Sp2011_14_Plane_waves_updated - © Copyright Dean P....

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