( z) ( ) ( )( ) ω ω. Wave equation. Transmission line formulas. = v. Helmholtz equation. Exponential Equation. Trig Formulas = Γ. cos sin 1 1+Γ = VSWR

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1 Wave equation 1 u tu v u(, t f ( vt + g( + vt Helmholt equation U + ku jk U Ae + Be + jk Eponential Equation γ e + e + γ + γ Trig Formulas sin( + y sin cos y+ sin y cos cos( + y cos cos y sin sin y + cos sin 1 sin sin cos cos cos 1 1 sin Approimations for 1 3 sin 6 for 1 cos 1 for 1 3 tan + 3 for 1 Transmission line formulas + j t { } { } (, t Re ep( + j t Re e ( ( j + R I I j C + G [ γ ] [ γ ] ep + ep I ep ep + v / β p j + R j C + G [ γ ] [ γ ] γ j + R jc + G α + jβ Γ + Γ SWR + 1+Γ ma 1 min Γ [ γ ] + ep[ + γ ] [ γ ] ep[ + γ ] ep + + ep + 1 * α ave Re cosψ P I e ossless Transmission line formulas R G, α, ψ in ( + j + j tan β tan β + 1+Γ + Γ cos( β+ φ C εµ

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7 Faraday s aw E t B t ( µ H EMF Ed Bnˆ ds t C S S Time arying Ampere s law H td+ J t( ε E + J MMF H d I + Dnˆ ds through S t S C S inside C S I J nˆ ds through S inside C S S S The curve C, the boundary of S, and the normal to the surface S are related by the right-hand rule Time Harmonic Mawell s Equations E jµ H H + jεe + J ρ ( εe ( µ H Eyt (,,, Re E ( y,, e Source free region has J, ρ j t { } ossless Plane Wave Equations Propagating along ais, inearly polaried along ais E jµ H H jε E y y ossless Plane Wave Solution jk E( y,, Ee ˆ E ˆ η jk H ( y,, e y µ k εµ, η ε Wave Speed in ossless Medium v p 1 k εµ Average Power Flow in ossless Medium 1 E S Re{ E H} ˆ η ossy Medium Plane Wave α jβ E( y,, Ee ˆ e E jφη α jβ H ( y,, e e e y η β ε µ U α ε µ µ µ 1 η ε j σ ε σ ε U ( / 1 j ( / µ 1 η ε U j 1/ 1 ( σ / ε / 1 ( σ / ε 1 + e 1/ 1/ ˆ jφη

8 Wave Speed in ossy Medium v p 1 β εµ U A lossy medium is dispersive Average Power Flow in ossy Medium 1 S Re E H E α e cos η Skin Depth skin 1 α { } ( φη Sum of Two Plane Waves ˆ Propagating along the + direction jk jk E(,, ˆ ˆ y Ee + Eye y E E ˆ ˆ η η jk y jk H ( y,, e y e Polariation Determine the motion in time of the electric field vector in a reference plane: possibilities are linear, circular, elliptical Constants: ε 1 36π µ π c εµ µ η 1π ε Plane Wave normally ident on a planar interface EE rrrrrr Γ EE iiiiii Γ η η 1 η + η 1 EE tttttttttt T EE iiiiii T 1 + Γ T η η + η 1 Average Power Transmitted into second region PP tttttttttt 1 TTTT 1 η (1 Γ EE (1 Γ PP iiiiii Standing Wave Ratio η 1 Assume interface is at and ident wave propagates along the + direction jk Total, I jk1 + 1 E ( y,, Ee 1 e 1 E (, y, E 1+Γ + Γ cos( k+ φ 1+Γ SWR 1 Γ SWR 1 Γ SWR + 1 +Γ Reflection from a PEC (Standing Wave Γ 1 pec Γ + 1 pmc Total, I Total, I E (, y, j E sin k ( φ E (, y,, t E sin k sin t + Γ

9 ossy Media ε ε j ε µ µ j µ Good Dielectric Approimations: β ε µ α 1 µ σ ε η µ ε Good Conductor Approimations: β α µ σ η µ e σ jπ /4

10 Two Dimensional Plane Wave propagating in -plane ˆ k DOP... k cosθˆ + sinθˆ k k k + k εµ k kcosθ k ksinθ η µ ε Perpendicular Polariation: jk ( + k E ( y,, Ee yˆ E jk ( + k H ( y,, e (sinθ ˆ cos θ ˆ η Parallel Polariation E jk ( + k H ( y,, e yˆ η E ( y,, Ee (cosθ ˆ sin θ ˆ jk ( + k Time-Averaged Power Flow 1 E S Re( E H kˆ η Obliquely Incident Plane Wave Scattering from a Planar Interface aw of reflection θ refl aw of refraction (Snell's aw n sinθ n sinθ θ Fresnel Coefficients η cosθ η cosθ Γ η cosθ + η cosθ T η cosθ η cosθ + η cosθ η cosθ η cosθ Γ η cosθ + η cosθ η cosθ T η cosθ + η cosθ Critical Angle ( n > n for Total Internal Reflection θ crit Brewster s Angles θ θ Brewster Brewster 1 n sin n ( µ ε 1 µε 1 ( ε ε 1 / 1 / 1 ( µε 1 µ ε1 ( µ µ 1 / 1 / 1 Parallel Plate Waveguide Quasi-TEM Solution (h is distance between plates; w is their width. All edge effects ignored. (, y, he e jk ( he w I ( y,, e h η jk Inde of refraction n εµ εµ k C εµ h η w C

11 Infinitesimal Dipole Antenna (oriented current moment I ˆ jkr e 1 j ER ( R, θφ, kηi cosθ 4π R kr ( kr jkr e j 1 Eθ ( R, θφ, jkηi 1 sinθ 4π R kr ( kr jkr e j Hφ ( R, θφ, jk I 1 sinθ 4π R kr Far Field Radiated Power Density far field 1 Re far field far field S E H η I 8 λ 1 sin θ Rˆ R Total Real Power Radiated P S R ds far field ˆ π I rad η 3 λ arge Sphere Radiation Resistance Prad π Rrad η 8 π 1 I 3 λ λ Infinitesimal Dipole Power Pattern sin θ Infinitesimal Dipole Directivity 1.5 sin θ Dipole Antenna of ength along -ais Current: I I sin k for for > k k jkr cos cosθ cos far field e Eθ jη I 4πR sinθ Circuit Impedance of an antenna R + jx ant rad ant half wavelength dipole 73 + j4.5 Ω Directivity (Power in certain direction relative to isotropic radiator far field ( ˆ ( rad 4 π Directivity S R P R Gain Gain efficiency directivity Gains of same antenna in mission and in reception G R are equal: GR GT Power received at an antenna with effective area A: received far field ( ˆ P S R A G T Relationship between Gain and Effective Area: λ A G 4π Friis Transmission aw λ P G G P 4π R rec R T input Radar Range Equation λ σ target λ P G G P 4πR λ 4π 4πR1 rec R T input