ece107_set8_nbg

ece107_set8_nbg - ECE 107 Electromagnetism Set 8 Plane...

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1 ECE 107: Electromagnetism Set 8: Plane waves Instructor: Prof. Vitaliy Lomakin Department of Electrical and Computer Engineering University of California, San Diego, CA 92093
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2 Wave equation Source-free lossless Maxwell’s equations Apply curl Helmholtz equation (HE) In Cartesian coordinates 0 0 j j ωμ ε ωε μ ×= = = EH E HE H ±± ± ± N N 2 0 22 () 0 j j ω με ∇∇ ⋅ −∇ ∇×∇× =− ∇× ⇒∇ + = E EE E E ± ± ± ²³´ ³µ 0 0 k k += HH wavenumber k ωμε = ,, 0 xyz kE ±
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3 Plane waves in free space (1) Radiation from a uniform surface current Maxwell’s eqs. reduce to TL equations! Helmholtz equations reduce to Solutions 00 ˆ | s zs J = = Jx ˆˆ 0, x y xy E H ⇒∂ ∂ =∂ ∂ = ⇒ = = Ex H y ±± ± ± x y y x dE j H dz dH j E dz ωμ ωε −= ± ± ± ± 2 2 22 0 y x dH dE kE kH dz dz += + = ± ± z x y s J 0 0 , jkz jkz x xx y E EE e H e η + +− == ± ± ,2 xs EJ ημ ε
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4 Plane waves in free space (2) Plane waves solutions are similar to TL solutions! Frequency domain expressions Time domain expressions Parameters ± Phase velocity ± Wavelength ± Characteristic (intrinsic) impedance ( ) 00 ˆ () ˆ jkz jkz xx jkz jkz zE e E e EE zee ηη +− =+ ⎛⎞ =− ⎜⎟ ⎝⎠ Ex Hy ±± ± ± 0 (= 1 2 0 ) ημ ε η μ π = ( ) ˆ ( , ) | | cos( ) | | cos( ) || ˆ ( , ) cos( ) cos( ) t z Et k z k z tz t k z z ωφ ++ + + + + + + + 8 1( 1 2 1 0 ) p vk με ω μ ε == = × 2 p kv f λ = =
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5 Plane waves in free space (3) Equivalence between plane waves and TL waves Replace The resulting waves will have the same behavior x E ± ˆ z 1 yx H E η = ±± V ± 0 1 I V Z = 0 ,, VE IH Z →→
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6 Plane waves in free space (4) Relation between and propagating direction Let be the propagation direction of a plane wave with being angles General solutions Field-direction relations Plane waves are TEM waves form a right-handed triple Plane waves satisfy Maxwell’s and Helmholtz equations 00 1 ˆ ˆ η =− × Hk E Ek H ±± & EH ˆ ˆˆ ˆ sin cos sin sin cos θ ϕθ ϕ =+ + kx y z ˆ ,, EHk ⊥⊥ ±±± ± , jk jk ee ⋅− == kr EE HH ± ± 1 ˆ ˆ = × = −× E H or &
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7 Plane waves in free space (5) Examples Any combination of the two above Parallel polarization (TM field) Perpendicular polarization (TE field) ˆ ˆ , jkz xx yx EE e H E η +− == = kz ±± ± ± ˆ ˆ , jkz yy x y e H E = ± ± (s i n c o s) i n c o ˆ ˆˆ ˆ sin cos , (c o s s i n) jk x z y jk x z y He θ θθ ηθ + + =+ = ⇒= kx z Hy Ex z ˆ ˆ = ˆ x E + = ± ˆ ˆ y H + = ˆ z ˆ y E + = Ey ± ˆ x H + = Hx ± i n c o i n c o ˆ ˆ sin cos , o s s i n ) jk x z y y jk x z Ee E e + + −+ = + z z ± ± ˆ k ˆ y E + = ± H ± ˆ x ˆ z ˆ k E ± ˆ y H + = ± ˆ x
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8 Plane waves in free space (6) Plane waves in lossy media wave equation Consider Skin depth Dielectrics vs. conductors ± ± c jj σ ε εε ω ′′ →= = + 22 0 γ −= EE ±± ˆ ˆˆ , x E == kz Ex 1 good dielectric (large ) s εδ ² 1 good conductor (small ) s ³
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9 Plane waves in free space (7) Power flow (1)
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ece107_set8_nbg - ECE 107 Electromagnetism Set 8 Plane...

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