Short note: Loss tangent

In a medium with conductivity σ and permittivity ε, Maxwell's equation is ∇×H = σE + jωεE = jω(ε − jσ/ω)E. So the medium has a complex permittivity ε_c = ε − jσ/ω.

Loss tangent is the ratio of the conduction current density to the displacement current density:

tan δ = |J_c| / |J_d| = σ / (ωε)

(δ is the loss angle, tan δ = σ/(ωε); it measures how far the medium departs from a perfect dielectric, for which δ = 0.)

Uses and meaning:

• It shows how lossy a medium is at a given frequency. It changes with frequency.
• Good dielectric: tan δ ≪ 1 (σ ≪ ωε); small attenuation, α ≈ (σ/2)√(μ/ε).
• Quasi-conductor: tan δ ≈ 1.
• Good conductor: tan δ ≫ 1 (σ ≫ ωε); α = β = √(πfμσ).
• The intrinsic impedance becomes complex when tan δ ≠ 0, so E and H are out of phase by an angle θ_η = δ/2 (E leads H).
• Example: dry soil (σ = 10⁻⁴ S/m, ε_r = 3) has tan δ = 1 at about 599 kHz.