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

Recap

Uniform guide: $E = E_{⊥} (x, y) e^{- j τ z}$ , so $\partial / \partial z \to - j τ$ . $τ$ is the phase constant along the guide ( $β$ ), complex below cutoff or with losses, $γ = j τ$ .
Transverse fields, Eq. (7.1.11): all four follow from $E_{z}$ and $H_{z}$ , with $h^{2} = k^{2} - τ^{2}$ . The generating fields obey $\nabla_{T}^{2} E_{z} + h^{2} E_{z} = 0$ , $\nabla_{T}^{2} H_{z} + h^{2} H_{z} = 0$ , Eq. (7.1.12).
Families: TM ( $H_{z} = 0$ ), TE ( $E_{z} = 0$ ), TEM ( $E_{z} = H_{z} = 0$ , $h = 0$ , $τ = k$ ). TEM needs two conductors: a hollow pipe has none.
Cutoff, Eq. (7.1.14): $f_{c} = h / (2 π μ ε)$ . Above it $τ = k 1 - (f_{c} / f)^{2}$ , below it $e^{- α z}$ with $α = h 1 - (f / f_{c})^{2}$ (evanescent, no power).
Guide wavelength and velocities: $λ_{g} = λ / 1 - (f_{c} / f)^{2}$ , $v_{p} = v / 1 - (f_{c} / f)^{2}$ , $v_{g} = v 1 - (f_{c} / f)^{2}$ , $v_{p} v_{g} = v^{2}$ . Dispersion relation $ω^{2} = ω_{c}^{2} + v^{2} τ^{2}$ : hyperbolas above the light line.
Wave impedances, Eq. (7.1.22): $Z_{T M} = η 1 - (f_{c} / f)^{2}$ , $Z_{T E} = η / 1 - (f_{c} / f)^{2}$ , $Z_{T E M} = η$ , $H_{T} = a_{z} \times E_{T} / Z$ . Reactive below cutoff.
Dispersion: the group delay $t_{g} = L / v_{g}$ depends on frequency, a pulse of bandwidth $B$ spreads by $∣ d t_{g} / df ∣ B$ , Eq. (7.1.24).

7.1.9 Recap