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

Recap

Not TEM: in two media a TEM wave would need $τ = k_{0}$ and $τ = k_{0} ε_{r}$ at once. For $h ≪ λ$ the wave is quasi-TEM: static transverse field, uniform effective medium.
Effective permittivity: $ε_{eff} = C^{'} / C_{0}^{'}$ , $L^{'} = 1/ (c^{2} C_{0}^{'})$ , $v = c / ε_{eff}$ , $Z_{0} = Z_{0, air} / ε_{eff}$ , Eq. (7.7.3). Hammerstad's fit, Eq. (7.7.4), between $(ε_{r} + 1) /2$ and $ε_{r}$ , a function of $w / h$ only.
Impedance: $Z_{0} = (60/ ε_{eff}) ln (8 h / w + w /4 h)$ for $w / h \leq 1$ , $120 π / (ε_{eff} F)$ for $w / h \geq 1$ , Eq. (7.7.5). Synthesis by Wheeler's formulas, Eq. (7.7.7). $50 Ω$ : $w \approx 2 h$ on FR-4, $w \approx h$ on alumina.
Line constants: $β = k_{0} ε_{eff}$ , $λ_{g} = λ_{0} / ε_{eff}$ , $C^{'} = ε_{eff} / (c Z_{0})$ , $L^{'} = Z_{0} ε_{eff} / c$ .
Losses: $α_{c} \approx R_{s} / (Z_{0} w) \propto f$ , $α_{d} = q k_{0} ε_{eff} tan δ /2 \propto f$ with $q = (1 - 1/ ε_{eff}) / (1 - 1/ ε_{r})$ . FR-4 at $2.4 GHz$ : $7.9 dB/m$ , mostly dielectric.
Limits: Getsinger's dispersion, Eq. (7.7.12), the TE $_{1}$ surface wave at $c / (4 h ε_{r} - 1)$ and the transverse resonance at $c / [ε_{r} (2 w + 0.8 h)]$ . Keep $h ≪ λ_{g}$ .

7.7.9 Recap