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

7.3.11 Problems and solutions

Throughout, the guide has the cross-section $0 < x < a$ , $0 < y < b$ ( $a \geq b$ ), perfectly conducting walls (unless losses are asked for) and a lossless homogeneous filling, and the fields vary as $e^{j ω t} e^{- j τ z}$ . The problems use

f_{co, mn} = \frac{v}{2} (\frac{m}{a})^{2} + (\frac{n}{b})^{2}, s = 1 - (f_{c} / f)^{2}, τ = k s, λ_{g} = \frac{λ}{s}, v_{p} = \frac{v}{s}, v_{g} = v s, Z_{T M} = η s, Z_{T E} = \frac{η}{s}, E_{y} = E_{0} sin \frac{π x}{a} e^{- j τ z}, H_{x} = - \frac{E _{y}}{Z _{T E}}, H_{z} = j \frac{π E _{0}}{ω μ a} cos \frac{π x}{a} e^{- j τ z}, P = \frac{∣ E _{0} ∣ ^{2} ab}{4 Z _{T E}}, α_{c} = \frac{R _{s}}{η b s} [1 + \frac{2 b}{a} (\frac{f _{c}}{f})^{2}], R_{s} = π f μ_{0} / σ_{c}, P_{T M} = \frac{ab}{8} \frac{ω ε τ}{h ^{2}} ∣ C_{mn} ∣^{2},

TM $_{mn}$ with $m, n \geq 1$ , TE $_{mn}$ with $m, n \geq 0$ not both zero, and below cutoff $α = h 1 - (f / f_{c})^{2}$ . The constants are those of this course, $ε_{0} = 1 0^{- 9} / (36 π) F/m$ , $μ_{0} = 4 π \times 1 0^{- 7} H/m$ , $c = 3 \times 1 0^{8} m/s$ , $η_{0} = 120 π = 377 Ω$ , with copper $σ_{c} = 5.8 \times 1 0^{7} S/m$ , the breakdown field of dry air $3 MV/m$ and $1 Np = 8.686 dB$ .

7.3.11 Problems and solutions

In this part

Problems · 5

Home exercises · 4

Further exercises · 10

Worked examples · 3