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

Recap

From guide to cavity. End walls at $z = 0, d$ turn a guide mode into a standing wave with $τ = p π / d$ , $d = p λ_{g} /2$ , Eq. (7.8.6). Transverse $E$ and normal $H$ go as $sin (p π z / d)$ , the components linked to them by $\partial / \partial z$ as $cos (p π z / d)$ .
Rectangular cavity. $f_{mn p} = \frac{v}{2} (m / a)^{2} + (n / b)^{2} + (p / d)^{2}$ , Eq. (7.8.8). TM: $m, n \geq 1$ , $p \geq 0$ . TE: $p \geq 1$ , $m + n \geq 1$ . TE $_{mn p}$ and TM $_{mn p}$ are degenerate for $m, n, p \geq 1$ . The smallest side never enters the dominant resonance (TE $_{101}$ for $b < a, d$ ).
TE $_{101}$ . $E_{y} = E_{0} sin \frac{π x}{a} sin \frac{π z}{d}$ , $H$ in quadrature, $(λ /2 a)^{2} + (λ /2 d)^{2} = 1$ , $W = ε ∣ E_{0} ∣^{2} ab d /8$ with $W_{e} = W_{m}$ , and $Q_{101}$ from Eq. (7.8.24), $Q = a /3 δ$ for a cube.
Circular cavity. $f = \frac{v}{2 π} (x / a)^{2} + (p π / d)^{2}$ , Eq. (7.8.29). TM $_{010}$ : $f = 2.405 v /2 π a$ , $Q = \frac{η}{R _{s}} \frac{2.405}{2 ( 1 + a / d )}$ , the lowest mode for $d < 2.03 a$ . TE $_{011}$ : azimuthal wall currents only, very high $Q$ , used in wavemeters.
$Q$ . $Q = ω_{0} W / P_{L}$ , $Δ f = f_{0} / Q$ , $W \propto e^{- ω_{0} t / Q}$ , $1/ Q_{L} = \sum 1/ Q_{i}$ , $Q_{d} = 1/ tan δ$ , $β = Q_{0} / Q_{ext}$ , and for a symmetric two-port $S_{21} (f_{0}) = 2 β / (1 + 2 β) = 1 - Q_{L} / Q_{0}$ . Order of magnitude: $Q \approx$ size $/ δ$ times a geometry factor near $\frac{1}{3}$ , about $1 0^{4}$ for copper at $10 GHz$ .
S-parameters. $b_{i} = \sum_{j} S_{ij} a_{j}$ , $S_{ij}$ measured with matched ports. Reciprocal $S_{ij} = S_{j i}$ , lossless $[S]^{H} [S] = [I]$ , matched $S_{nn} = 0$ . $R L$ , $I L$ , VSWR from Eq. (7.8.40).
Directional coupler. Coupling $C$ , directivity $D$ , isolation $I = C + D$ . The two-hole coupler adds the forward samples and cancels the backward ones with a $λ_{g} /4$ spacing, $∣ b_{4} / b_{3} ∣ = ∣ cos τ ℓ ∣$ .

7.8.17 Recap