Exercise 7.8.10: Hiding a bad antenna behind a pad or an isolator

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 7.8.10

Hiding a bad antenna behind a pad or an isolator

A source with a matched output ( $1 W$ available) feeds an antenna with VSWR $2.5$ . (a) Show that a two-port terminated in $Γ_{L}$ has $Γ_{in} = S_{11} + \frac{S _{12} S _{21} Γ _{L}}{1 - S _{22} Γ _{L}}$ and that the load receives $\frac{∣ S _{21} ∣ ^{2} ( 1 - ∣ Γ _{L} ∣ ^{2} )}{∣1 - S _{22} Γ _{L} ∣ ^{2}}$ watts. (b) A matched $6 dB$ attenuator is inserted. Find $∣ Γ_{in} ∣$ , the VSWR and the power radiated. (c) Instead, an isolator with $0.3 dB$ insertion loss, $20 dB$ isolation and $∣ S_{11} ∣ = ∣ S_{22} ∣ = 0.05$ is used. Find the range of $∣ Γ_{in} ∣$ and of the radiated power. (d) Compare with no insertion at all.

Try it yourself first.

cf. Home exercise 7.8.4