Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Recap

Lossy medium: replace $ε$ by $ε_{c} = ε - j σ / ω$ . The plane wave is $E = a_{x} E_{0} e^{- α z} e^{- j β z}$ with $γ = α + j β = j ω μ (σ + j ω ε) = j k$ , Eq. (6.2.6), $8.686 α dB$ per metre. Exact $α$ , $β$ in Eq. (6.2.11).
Impedance and power: $η_{c} = j ω μ / (σ + j ω ε) = ∣ η_{c} ∣ e^{j θ_{η}}$ , $0 \leq θ_{η} \leq 4 5^{\circ}$ , $H$ lags $E$ by $θ_{η}$ , $S_{a v} = \frac{E _{0}^{2}}{2∣ η _{c} ∣} e^{- 2 α z} cos θ_{η}$ , and $- d S_{a v} / d z = \frac{1}{2} σ ∣ E ∣^{2}$ . Useful: $γ η_{c} = j ω μ$ , $γ / η_{c} = σ + j ω ε$ .
Loss tangent $tan δ_{c} = σ / ω ε$ : low-loss below $0.01$ with $α \approx \frac{σ}{2} μ / ε = \frac{β}{2} tan δ_{c}$ , $β \approx ω μ ε$ , Eq. (6.2.22), good conductor above $100$ with $α = β = 1/ δ = π f μ σ$ , $η_{c} = (1 + j) / σ δ$ , Eq. (6.2.25). In between use the exact formulas.
Skin effect: $δ = 1/ π f μ σ$ ( $66.1 μ m / f [MHz]$ for copper), $R_{s} = 1/ σ δ$ , $R_{a c} / R_{d c} \approx a /2 δ$ , $8.686 dB$ per skin depth.
Plasma: $ε_{p} = ε_{0} (1 - f_{p}^{2} / f^{2})$ , $f_{p} \approx 9 N Hz$ . Below $f_{p}$ the wave is evanescent and totally reflected, above it propagates with $v_{p} = c / 1 - f_{p}^{2} / f^{2}$ , $v_{g} = c 1 - f_{p}^{2} / f^{2}$ , $v_{p} v_{g} = c^{2}$ . Secant law $f_{MUF} = f_{o} F_{2} sec θ_{i}$ .
Group velocity $v_{g} = d ω / d β$ , the speed of the envelope, of energy and of information. Normal dispersion $v_{g} < v_{p}$ (plasma), anomalous $v_{g} > v_{p}$ (good conductor, $v_{g} = 2 v_{p}$ ).

6.2.8 Recap