formula sheet final

# formula sheet final - Formula Sheet ECSE 352...

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1 ECSE 352 EM Waves: Formula sheet TIME-VARYING FIELDS AND MAXWELL’S EQUATIONS Constants: ε 0 =8.854 x 10 -12 F/m µ 0 =4π x 10 -7 H/m η 0 =377 c =3 x 10 8 m/s Integral Form Differential Form Constitutive Relations and Boundary Conditions 0 = = + = = S S S l S l d Q d d dt d I d d dt d d S S S l S l B D D H B E 0 = = + = × = × B D D J H B - E ρ t t D = ε E H = B a n 2 ( D 1 2 ) = ρ s 2 × ( H 1 2 ) = J E t 1 = 2 B 1 = 2 Source free wave equations: 0 1 2 2 2 2 = t u E E 0 1 2 2 2 2 = t u H H Phasors: ( ) ( ) [ ] t z y x e z y x t z y x t j ω cos ) , , ( , , , , , E E = = E Curl equations in phasor form: H E ωµ j = × E J H ωε j + = × Homogenous Vector Helmholz: 0 2 2 = + E E k 0 2 2 = + H H k Wavenumber: Phase velocity: 1/ / p rr uc µε µ ε = = Wavelength: λ = 2 π k Group velocity: u g = 1 d β / Complex permittivity: σ ε j j c = " ' Loss tangent: " tan ' εσ θ = TRANSMISSION LINES Lossless line: R = G = 0 α = 0 = LC = p = 1 Z 0 = V 0 I 0 = L C General transmission line equations: 2 Vz ( ) dz 2 = γ 2 ( ) 2 Iz ( ) 2 = 2 ( ) ( )( ) C j G L j R j α + + = + = ( ) ( ) ( ) ( ) ( ) 22 00 , cos 2 2 cos , cos zz ZZ Z VV zt e t z z e αα βθ −−  = −+ =  PP p k u = Formula Sheet, ECSE 352: Electromagnetic Waves

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2 Parameter Parallel Plate Two-wire line Coaxial line R w R s 2 R s π a 2 π 1 + 1 b L w d µ π cosh 1 D 2 2 π ln G d w σ π cosh 1 /2 ( ) 2 π / ( ) C d w ε π cosh 1 ( ) 2 π / ( ) = π f c / Wave characteristics of a finite transmission line: Reflection, transmission coefficients: 0 0 L L ZZ Γ= + 0 2 L L Z τ = + Voltage on line ( z =0 is at load): ( ) 0 2 sin jz sT VzVe j z β τβ  =  Input impedance of lossless line: Z i = 0 L + jR 0 tan l 0 + jZ tan λ /4 line: = 0 2 λ /2: = Lossy line: 2 0 2 1 1 l
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## This note was uploaded on 02/14/2011 for the course ECSE 352 taught by Professor Mi during the Fall '10 term at McGill.

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formula sheet final - Formula Sheet ECSE 352...

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