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

Recap

No TEM wave in a hollow guide, only TM $_{mn}$ ( $H_{z} = 0$ ) and TE $_{mn}$ ( $E_{z} = 0$ ) modes. $E_{z}$ or $H_{z}$ obeys $\nabla_{T}^{2} ψ + h^{2} ψ = 0$ , and the transverse fields follow by differentiation, Eqs. (7.3.8) and (7.3.21).
Separation of variables $ψ = X (x) Y (y) e^{- j τ z}$ gives harmonic functions with $h^{2} = k_{x}^{2} + k_{y}^{2}$ . TM: $E_{z} = 0$ on the walls, $E_{z} = C sin \frac{mπ x}{a} sin \frac{nπ y}{b}$ , $m, n \geq 1$ . TE: $\partial H_{z} / \partial n = 0$ , $H_{z} = C cos \frac{mπ x}{a} cos \frac{nπ y}{b}$ , $m, n \geq 0$ not both zero.
Cutoff, Eq. (7.3.28): $f_{co, mn} = \frac{v}{2} (m / a)^{2} + (n / b)^{2}$ , the same for TE $_{mn}$ and TM $_{mn}$ (degenerate modes). The dominant mode is TE $_{10}$ , $f_{co, 10} = v /2 a$ . Single-mode band $v /2 a < f < v / a$ when $b \leq a /2$ . A mode chart shows all cutoffs at a glance.
TE $_{10}$ , Eq. (7.3.29): $E_{y} = E_{0} sin \frac{π x}{a}$ , $H_{x} = - E_{y} / Z_{T E}$ , $H_{z} = j \frac{π E _{0}}{ω μ a} cos \frac{π x}{a}$ . Two plane waves bouncing between the narrow walls at $sin θ = f_{c} / f$ , hence $λ_{g} = λ / cos θ$ , $v_{p} = v / cos θ$ , $v_{g} = v cos θ$ .
Walls: $J_{s} = a_{n} \times H$ , longitudinal on the centre line of the broad wall (slotted line), crosswise near the edges and vertical on the narrow walls (slot antennas). $ρ_{s} = ε a_{n} \cdot E$ .
Power, Eq. (7.3.37): $P = ∣ E_{0} ∣^{2} ab /4 Z_{T E}$ , limited by breakdown ( $3 MV/m$ in dry air). Loss, Eq. (7.3.42): $α_{c} = \frac{R _{s}}{η b s} [1 + \frac{2 b}{a} (f_{c} / f)^{2}]$ with $s = 1 - (f_{c} / f)^{2}$ , large near cutoff and $\propto 1/ b$ .
Standard guides: WR- $N$ has $a = N /100 in$ , $b \approx a /2$ , band $\approx 1.25$ to $1.9 f_{co, 10}$ .

7.3.10 Recap