Materiale universitario di ELECTROMAGNETICS AND SIGNAL PROCESSING FOR SPACEBORNE APPLICATIONS per il corso di Aerospace Engineering presso Politecnico di Milano. Il documento tratta: Politecnico di Milano — Electromagnetics Formula Sheet Electromagnetics - Aayush Golwala 0 ▶ Contents 1 Phasor Representation 2 1.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Phasor Domain . . . . . . . . . . . .…
Materiale universitario di ELECTROMAGNETICS AND SIGNAL PROCESSING FOR SPACEBORNE APPLICATIONS per il corso di Aerospace Engineering presso Politecnico di Milano. Il documento tratta: Politecnico di Milano — Electromagnetics Formula Sheet Electromagnetics - Aayush Golwala 0 ▶ Contents 1 Phasor Representation 2 1.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Phasor Domain . . . . . . . . . . . .…
Qualità dell’importazione: il testo è stato estratto direttamente dal documento originale.
Passaggi rappresentativi riconosciuti nelle diverse parti del materiale. Il testo completo resta presente nella pagina per la ricerca, mentre l’anteprima compatta rende più semplice la lettura.
Politecnico di Milano — Electromagnetics Formula Sheet Electromagnetics - Aayush Golwala 0 ▶ Contents 1 Phasor Representation 2 1.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Phasor Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Material Parameters 2 2.1 Conductivity σ [S/m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Permittivity ε [F/m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Permeability µ [A/m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Uniform Plane Wave 3 3.1 Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Key Wave Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.3 Approximation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 Power Flow — Poynting Vector 4 5 Polarisation 4 5.1 Linear Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5.2 Circular Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.3 Elliptical Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 Reflection and Transmission at Interfaces 5 6.1 Normal Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6.2 Oblique Incidence — Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.3 TE and TM Polarisations (Lossless Media) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.4 Brewster Angle…
Politecnico di Milano — Electromagnetics Formula Sheet Classification by εr Value of εr Material Description εr = 1 Free space Vacuum εr > 1 (real) Perfect dielectric Lossless εr = ε′r − jε′′r Real dielectric Lossy (complex ε) 2.3 Permeability µ [A/m] µ = µ0 µr , µ0 = 4π × 10−7 A/m (5) Classification by µr Value of µr Material Description µr = 1 Free space Non-magnetic µr ≈ 1 (real) Perfect non-magnet e.g. most dielectrics µr ≫ 1 (real) Ideal ferromagnet High permeability µr = µ′r − jµ′′r Real ferromagnet Lossy magnetic 3 ▶ Uniform Plane Wave 3.1 Wave Equations A uniform plane wave propagating in the +z direction with x̂-polarised electric field: Time-Domain and Phasor Forms ⃗ t) = E0 e−αz cos(ωt − βz) ûx E(z, (time domain) (6) Ẽ(z) = E0 e−αz e−jβz ûx (phasor domain) (7) n o ⃗ t) = Re Ẽ(z) ejωt ûx E(z, (recovery) (8) Right-Hand Rule for Wave Direction ⃗ points in x̂ and H If E ⃗ points in ŷ, the wave propagates in ẑ: Point fingers along x̂, curl towards ŷ, thumb points in ẑ. 3.2 Key Wave Parameters Wave Parameters — Definitions Wavelength & Phase Velocity 2π λ= [m] (9) β ω vph = = fλ [m/s] (10) β 3
Politecnico di Milano — Electromagnetics Formula Sheet 6.2 Oblique Incidence — Snell’s Law Snell’s Law General (lossy) media: γ1 sin θt = sin θi (23) γ2 Lossless media (no surface waves): n1 √ sin θt = sin θi , n= εr µr (refractive index) (24) n2 6.3 TE and TM Polarisations (Lossless Media) ⃗ orthogonal to the plane of incidence Transverse Electric (TE) — E Incident field: ⃗e TE −jβ1 cos θi z −jβ1 sin θi y E i (z, y) = −E0 e e ûx (25) Effective impedance & coefficients: η η2TE − η1TE η TE = , ΓTE = , T TE = 1 + ΓTE (26) cos θ η2TE + η1TE Reflected & transmitted fields: TE ⃗e E r (y, z) = −Γ TE E0 e jβ1 cos θr z e−jβ1 sin θr y ûx (27) ⃗e TE E t (y, z) = −T TE E0 e−jβ2 cos θt z e−jβ2 sin θt y ûx (28) ⃗ orthogonal to the plane of incidence Transverse Magnetic (TM) — H Incident field: ⃗e TM E i = E0 (cos θi ûy − sin θi ûz ) e−jβ1 cos θi z e−jβ1 sin θi y (29) Effective impedance & coefficients: η2TM − η1TM η TM = η cos θ, ΓTM = , T TM = 1 + ΓTM (30) η2TM + η1TM Reflected & transmitted fields: ⃗e TM E r (y, z) = ΓTM E0 (cos θr ûy + sin θr ûz ) e jβ1 cos θr z e−jβ1 sin θr y (31) ⃗e TM TM cos θi −jβ2 cos θt z −jβ2 sin θt y E t (y, z) = T E 0 cos θ û i y − û z e e (32) tan θt 6.4 Brewster Angle (TM) When θi = θB , there is total refraction for TM polarisation (ΓTM = 0, no reflected wave): Brewster Angle r ε2 θB = sin−1 (33) ε1 + ε2 6.5 Critical Angle and Total Internal Reflection Applies when ε1 > ε2 . Since sin θt cannot exceed 1, a critical angle is defined: 6
Politecnico di Milano — Electromagnetics Formula Sheet General Voltage and Current Solutions V (z) = V0+ e−γz + V0− e+γz (49) I(z) = I0+ e−γz + I0− e+γz (50) The characteristic impedance ZC relates the progressive voltage and current amplitudes: Characteristic Impedance s r V+ V− r z r + jωl l ZC = 0+ = − 0− = = ≈ [Ω] (51) I0 I0 y g + jωc c The approximation on the right holds for low-loss lines (the typical practical case). 9.3 Reflection at a Load — Reflection Coefficient Consider a transmission line of length L and characteristic impedance ZC terminated by a load ZL . Enforcing continuity of voltage and current at section z = L, and applying the definition of ZC , yields: Load Reflection Coefficient V− ZL − Z C ΓL = = − 1 ≤ ΓL ≤ +1 (52) V + z=L ZL + Z C • ZL = ZC ⇒ ΓL = 0 (matched load, no reflection) • ZL ̸= ZC ⇒ mismatch; a reflected wave is generated The reflection coefficient at an arbitrary position z (distance d = L − z from the load) evolves as: Reflection Coefficient Along the Line Γ(z) = ΓL e−2γ(L−z) = ΓL e−2γd (53) In particular, at the input section z = 0: ΓIN = Γ(z = 0) = ΓL e−2γL (54) 9.4 Impedance Transformation The equivalent impedance seen looking into the line at any position z is: Impedance at Generic Position z V (z) 1 + Γ(z) Z(z) = = ZC (55) I(z) 1 − Γ(z) At the input section (z = 0), this gives the input impedance: 1 + ΓIN ZIN = ZC (56) 1 − ΓIN Lossless Line Special Cases (α = 0) Three important simplifications arise for a lossless line: • ZL = ZC ⇒ ZIN = ZL • L = nλ/2 (n ∈ Z) ⇒ ZIN = ZL (half-wave transformer) • L = λ/4 + nλ/2 (n ∈ Z) ⇒ ZIN = ZC 2 /ZL (quarter-wave transformer) 9
Politecnico di Milano — Electromagnetics Formula Sheet 10 ▶ Quick Reference — Physical Constants Constant Symbol Value Speed of light in vacuum c 3 × 108 m/s Permittivity of free space ε0 8.854 × 10−12 F/m Permeability of free space µ0 4π × 10−7 A/m Intrinsic impedance (free space) η0 377 Ω 12
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