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

About the further exercises of Section 7.8

Ten further exercises in the spirit of Problems 7.8.1–7.8.5 and the home exercises, graded from easy to harder. Exercises 7.8.1–7.8.6 are about cavity resonators, Exercises 7.8.7–7.8.10 about microwave devices and scattering parameters. They use only the results of this section, in particular $f_{mn p}^{2} = f_{co, mn}^{2} + (p v /2 d)^{2}$ , $Q_{101}$ of Eq. (7.8.24) with $R_{s} = π f μ_{0} / σ_{c} = 1/ (σ_{c} δ)$ ( $Q = a /3 δ$ for a cube), $Q_{010}$ of Eq. (7.8.33), $1/ Q_{L} = 1/ Q_{c} + 1/ Q_{d} + 1/ Q_{ext}$ , $Q_{d} = 1/ tan δ$ , $β = Q_{0} / Q_{ext}$ , the response $H (f) = 1/ (1 + j 2 Q δ_{f})$ with $δ_{f} = (f - f_{0}) / f_{0}$ , and for S-parameters $b_{i} = \sum_{j} S_{ij} a_{j}$ , reciprocity $S_{ij} = S_{j i}$ , losslessness $[S]^{H} [S] = [I]$ , $R L = - 20 lo g_{10} ∣ S_{11} ∣$ and $VSWR = (1 + ∣Γ∣) / (1 - ∣Γ∣)$ .

Exercise 7.8.4

Measuring a liquid with a pillbox

The copper TM $_{010}$ cavity of Problem 7.8.3 ( $a = 2.296 cm$ , $d = 2 cm$ , $f_{0} = 5 GHz$ , $Q = 11440$ when empty) is filled completely with a nonmagnetic silicone oil. Weakly coupled, the resonance moves to $3.043 GHz$ and its $Q$ becomes $900$ . (a) Find $ε_{r}^{'}$ of the oil. (b) Find $Q_{c}$ of the walls with the oil inside. (c) Find $Q_{d}$ , $tan δ$ and $ε_{r}^{''}$ . (d) How much does a $20 %$ error in $Q_{c}$ change $tan δ$ ? What would neglecting the walls give?

Try it yourself first.

cf. Problem 7.8.3

Exercise 7.8.4: Measuring a liquid with a pillbox