University study material for ELECTROMAGNETICS AND SIGNAL PROCESSING FOR SPACEBORNE APPLICATIONS in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Politecnico di Milano — Electromagnetics Formula Sheet Electromagnetics - Aayush Golwala 0 ▶ Contents 1 Phasor Representation 2 1.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Phasor Domain . . . . . . . . . . . .
University study material for ELECTROMAGNETICS AND SIGNAL PROCESSING FOR SPACEBORNE APPLICATIONS in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Politecnico di Milano — Electromagnetics Formula Sheet Electromagnetics - Aayush Golwala 0 ▶ Contents 1 Phasor Representation 2 1.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Phasor Domain . . . . . . . . . . . .
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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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