Exercise 7.8.6: The cavity as a parallel RLC circuit

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 $f_{mnp}^2=f_{co,mn}^2+(pv/2d{)}^2$, $Q_{101}$ of Eq. (7.8.24) with $R_s=\sqrt{\pi f{\mu }_0/{\sigma }_c}=1/({\sigma }_c\delta )$ ($Q=a/3\delta$ for a cube), $Q_{010}$ of Eq. (7.8.33), $1/Q_L=1/Q_c+1/Q_d+1/Q_{\mathrm{ext}}$, $Q_d=1/\tan \delta$, $\beta =Q_0/Q_{\mathrm{ext}}$, the response $H(f)=1/(1+\mathrm{j}2Q{\delta }_f)$ with ${\delta }_f=(f-f_0)/f_0$, and for S-parameters $b_i={\sum }_j{S}_{ij}{a}_j$, reciprocity $S_{ij}=S_{ji}$, losslessness $[S{]}^{\mathrm{H}}[S]=[I]$, $RL=-20{\log }_{10}|S_{11}|$ and $\mathrm{VSWR}=(1+|\Gamma |)/(1-|\Gamma |)$.