Maxwell's Equations and Electromagnetics

• In high frequency regimes, voltages and currents take the form of waves, causing interference.
• Electromagnetics equations underlie circuit analysis in these regimes.
• The propagation of electromagnetic waves through different media is examined using these equations.

Maxwell's Equations in Free Space

• Maxwell unified electricity and magnetism theories in 1864, adding the displacement current term to Ampère's Law.
• Oliver Heaviside reformulated Maxwell's twenty equations into four, using vector calculus.
• Under the assumptions that fields exist in a simple, linear, homogeneous, and isotropic medium the flux densities and conduction current are defined as D = ∈E, B = μH, and J = σΕ.
  • Here ∈ is the permittivity, μ is the permeability, and σ is the medium's conductivity.
• The impressed charge ρv,s and impressed current density Js are source terms that generate the fields.
• Maxwell's Equations:
  • Faraday's Law: ∇ × E = -∂B/∂t
  • Ampère's Law: ∇ × H = Js + ∂D/∂t
  • Gauss's Law for Electric Fields: ∇ ⋅ E = ρv/ε
  • Gauss's Law for Magnetic Fields: ∇ ⋅ H = 0
• Divergence represents the outward flow of a field, while curl represents the rotation of a field.
• Maxwell's equations relate field quantities locally throughout space in their differential form.

Plane Wave Equations

• In a source-free region with an electric field oriented purely in the x direction and a magnetic field in the y direction, the fields can only vary in the z direction.
• The equations can be written as:
  • ∂Ex/∂z = -μ ∂Hy/∂t
  • ∂Hy/∂z = -σEx - ε ∂Ex/∂t
• These equations are similar in form to the Telegrapher's Equations.
• The FDTD transmission line simulator can simulate 1D electromagnetic wave propagation by renaming variables.
  • V ↔ Ex
  • I ↔ Hy
  • L ↔ μ
  • C ↔ ε
  • G ↔ σ
  • R ↔ σm (magnetic conductivity)
• The phase velocity and characteristic impedance for an electromagnetic wave propagating in a vacuum can be determined using the analysis from Module 3:
  • vp = 1/√(με) = 3.0x10⁸ m/s
  • Z₀ = √(μ/ε) = 377Ω

Time Harmonic Form

• In time harmonic form using phasors the Maxwell equations become:
  • ∇ × E = -jωμH
  • ∇ × H = J + jωεE
  • ∇ ⋅ E = 0
  • ∇ ⋅ H = 0
• Material losses can be represented through the negative imaginary components of ε and μ.
• Conduction losses from non-zero conductivity σ can be completely absorbed into a complex ε.
  • ∇ × H = (σ + jωε)E = jω(ε - j(σ/ω))E
  • The complex (lossy) permittivity is defined as ε = ε' - jε''.
  • In MatLab:
    • eps = ε'
    • sig = ωε"
    • mu = μ'
    • sig_m = ωμ"
• Material losses can be accounted for by simply replacing ε and μ in lossless equations with complex numbers.
  • The imaginary part of ε represents conduction and polarization losses.
  • The imaginary part of μ represents magnetization losses.

Deriving Wave Equations

• Solutions to Maxwell's equations can be derived similarly to transmission-line wave solutions, but using vector calculus due to 3D vector fields.
• The wave equation can be derived by taking the curl of Faraday's Law:
  • ∇ × (∇ × E) = -jωμ(∇ × H)
  • Using the vector identity ∇ × (∇ × E) = ∇(∇ ⋅ E) - ∇²E:
    • ∇(∇ ⋅ E) - ∇²E = -jωμ(jωεE)
    • Since ∇ ⋅ E = 0:
      • ∇²E + ω²μεE = 0 (Helmholtz equation)
      • In Cartesian coordinates: ∇²E = x̂∇²Ex + ŷ∇²Ey + ẑ∇²Ez
• Solutions can take many forms, from plane waves (complex exponentials) to spherical waves (Bessel functions).
• Only plane waves will be studied. They have the simplest solutions and Fourier analysis can be used.
• The plane wave solution to the wave equation is represented by a complex exponential with a vector coefficient and a scalar phase term.
  • E = E₀e⁻ʲk⋅r
    • E₀ indicates the electric field's direction.
    • k indicates the phase variation of E.
• In the time domain, the solution is:
  • E(r, t) = Re{E₀e⁻ʲk⋅r eʲωt}
  • E(r, t) = |E₀|cos(ωt - k⋅r + θ₀)

Understanding Wave Vectors

• The wave vector k indicates the direction of maximum phase change and attenuation if complex.
  • k = kxx̂ + kyŷ + kzẑ
• k is in direction of max. phase change (propagation).
• Let us verify that this solution satisfies Maxwell's equations by solving for the wave number k.
  • ∇²E₀e⁻ʲk⋅r = E₀∇²e⁻ʲk⋅r, using a vector identity
  • ∇²E₀e⁻ʲk⋅r = E₀(∂²/∂x² + ∂²/∂y² + ∂²/∂z²)e⁻ʲ(kxx + kyy + kzz)
  • ∇²E₀e⁻ʲk⋅r = E₀(-kx² - ky² - kz²)e⁻ʲk⋅r = -k²E
  • -k²E + ω²μεE = 0
  • Valid if k² = ω²με
• When k is real:
  • k = ω/vp = 2π/λ = Ω spatial frequency
• From Gauss' Law, E is perpendicular to k.
  • ∇ ⋅ E = 0
  • ∇ ⋅ E₀e⁻ʲk⋅r = E₀ ⋅ ∇e⁻ʲk⋅r using a vector identity
  • ∇ ⋅ E₀e⁻ʲk⋅r = E₀ ⋅ (-jk)e⁻ʲk⋅r = -jk ⋅ E₀e⁻ʲk⋅r
  • Therefore, k ⋅ E₀ = 0

Faraday's Law

• Using Faraday's Law:
  • The corresponding magnetic field H can be determined.
  • The electric field phasor can be related to the magnetic field phasor through the intrinsic impedance η.
    • η = √(μ/ε)
  • Similarly, H ⋅ k = 0.
• ∇ × E = -jωμH
  • ∇ × E₀e⁻ʲk⋅r = -jωμH
  • ∇ × E₀e⁻ʲk⋅r = ∇e⁻ʲk⋅r × E₀ using vector identity
  • ∇ × E₀e⁻ʲk⋅r = -jk × E₀e⁻ʲk⋅r
  • Therefore, -jk × E₀ = -jωμH
• H = (k/ωμ) × E
  • Note: k̂ = k/k and η = ωμ/k = ωμ/ω√(με) = √(μ/ε)