Electromagnetics

Syllabus

Maxwell's equations (differential and integral forms, boundary conditions, wave equation, Poynting vector); plane waves (reflection and refraction, polarization, phase and group velocity, propagation through media, skin depth); transmission lines (equations, characteristic impedance, impedance matching and transformation, S-parameters, Smith chart); rectangular and circular waveguides; light propagation in optical fibres; dipole and monopole antennas; linear antenna arrays.

Key concepts and formulas

Maxwell's equations and boundary conditions. ∇·D = ρ, ∇·B = 0, ∇ × E = −∂B/∂t, ∇ × H = J + ∂D/∂t. At an interface: tangential E is continuous; normal D jumps by the surface charge; tangential H jumps by the surface current; normal B is continuous.

Plane waves.

• Phase velocity v = 1/√(με) = c/√(μ_rε_r); intrinsic impedance η = √(μ/ε), η₀ ≈ 377 Ω (120π) in free space.
• Poynting vector S = E × H; average power density |E|²/(2η) for a peak field |E|.
• In a good conductor, the skin depth δ = √(2/(ωμσ)) = 1/√(πfμσ); fields fall by e⁻¹ over δ.
• Normal incidence on a boundary: reflection coefficient Γ = (η₂ − η₁)/(η₂ + η₁), transmission τ = 2η₂/(η₂ + η₁) = 1 + Γ.
• Polarization: linear, circular (equal amplitudes, 90° phase difference), elliptical. Brewster angle tan θ_B = √(ε₂/ε₁) (no reflection of parallel polarization).

Transmission lines.

• Characteristic impedance Z₀ = √((R + jωL)/(G + jωC)) ≈ √(L/C) for a lossless line.
• Reflection coefficient at the load Γ_L = (Z_L − Z₀)/(Z_L + Z₀); VSWR = (1 + |Γ|)/(1 − |Γ|).
• Input impedance of a lossless line of length l: Z_in = Z₀(Z_L + jZ₀ tan βl)/(Z₀ + jZ_L tan βl).
• A quarter-wave transformer gives Z_in = Z₀²/Z_L, so a section with Z₀ = √(Z_inZ_L) matches two resistances. A half-wave line repeats the load impedance. A short-circuited stub of length λ/4 looks like an open circuit.
• Return loss = −20 log₁₀|Γ| dB.

Waveguides. Rectangular guide (a > b): cut-off frequency f_c,mn = (c/2)√((m/a)² + (n/b)²); the dominant mode is TE₁₀ with f_c = c/(2a). Guide wavelength λ_g = λ/√(1 − (f_c/f)²). Below cut-off, the mode does not propagate.

Optical fibre. Numerical aperture NA = √(n₁² − n₂²) = sin of the acceptance angle (in air). Number of modes V = (2πa/λ)NA; single mode if V < 2.405.

Antennas. Half-wave dipole: radiation resistance about 73 Ω, directivity 1.64 (2.15 dBi). Quarter-wave monopole over a ground plane: about 36.5 Ω. Hertzian dipole: R_rad = 80π²(dl/λ)². Friis equation: P_r = P_tG_tG_r(λ/(4πd))². A uniform linear array of N elements has its main beam where the progressive phase cancels the path difference.

Worked example

A 50 Ω line feeds a 100 Ω load: find Γ, the VSWR and the quarter-wave transformer impedance for matching. Find the skin depth of copper (σ = 5.8 × 10⁷ S/m) at 1 MHz, the TE₁₀ cut-off of a guide with a = 2.286 cm, and the numerical aperture of a fibre with n₁ = 1.48 and n₂ = 1.46.

import math

Z0, ZL = 50, 100
G = (ZL - Z0) / (ZL + Z0)
print(f"Gamma = {G:.4f}, VSWR = {(1 + abs(G)) / (1 - abs(G)):.2f}, quarter-wave Z = {math.sqrt(Z0 * ZL):.2f} ohm")
mu0, sigma, f = 4 * math.pi * 1e-7, 5.8e7, 1e6
print(f"skin depth of copper at 1 MHz = {1 / math.sqrt(math.pi * f * mu0 * sigma) * 1e6:.2f} um")
c, a = 3e8, 2.286e-2
print(f"TE10 cut-off = {c / (2 * a) / 1e9:.3f} GHz")
print(f"NA = {math.sqrt(1.48 ** 2 - 1.46 ** 2):.4f}")

Gamma = 0.3333, VSWR = 2.00, quarter-wave Z = 70.71 ohm
skin depth of copper at 1 MHz = 66.09 um
TE10 cut-off = 6.562 GHz
NA = 0.2425

Practice questions