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

Three short calculations tied to the applications, with resonant frequencies, $Q$ , bandwidth, stored energy and wall loss. They use only the results of this section. A good way to use them: read the given quantities, produce the formula from the corresponding application yourself, and only then put the numbers in. Constants: $c = 3 \times 1 0^{8} m/s$ , $ε_{0} = 1 0^{- 9} /36 π F/m$ , $η_{0} = 120 π Ω$ , copper $σ_{c} = 5.8 \times 1 0^{7} S/m$ , $c /2 = 15 GHz cm$ , $c /2 π = 4.775 GHz cm$ .

Worked example 7.8.1

The modes of a microwave oven

cf. Application 7.8.1

Given. An oven cavity $a = 32 cm$ wide ( $x$ ), $b = 20 cm$ high ( $y$ ), $d = 30 cm$ deep ( $z$ ), fed at $2.45 GHz$ . With food inside the loaded $Q$ is about $100$ .

Find. (a) $λ$ and the lowest resonance of the box, (b) the resonances between $2.425$ and $2.475 GHz$ , compared with the estimate $Δ N \approx \frac{8 π V f ^{2}}{c ^{3}} Δ f$ , (c) the hot spots of TE $_{304}$ on the floor, (d) how many modes the magnetron drives at once.

Try it yourself first.

Strategy

$f_{mn p} = 15 (m /32)^{2} + (n /20)^{2} + (p /30)^{2} GHz$ , Eq. (7.8.28). A large box has a dense spectrum: the estimate $Δ N$ counts the modes per band (both polarisations), and the loaded bandwidth $f_{0} / Q_{L}$ tells how many of them are excited together.

$λ = c / f = 12.24 cm$ . $b$ is the smallest side, so the lowest mode is TE $_{101}$ : $f_{101} = 15 3 2^{- 2} + 3 0^{- 2} = 0.685 GHz$ . At $2.45 GHz$ the box measures $2.6 λ \times 1.6 λ \times 2.5 λ$ .

A search over $m, n, p$ gives, in GHz,

TM_{230} TE_{304} TE_{421}, TM_{421} TM_{510} TE_{032} : 2.4375, : 2.4449, : 2.4527, : 2.4608, : 2.4622,

six modes in $50 MHz$ . With $V = 0.32 \times 0.20 \times 0.30 = 0.0192 m^{3}$ ,

Δ N = \frac{8 π \times 0.0192 \times ( 2.45 \times 1 0 ^{9} ) ^{2}}{2.7 \times 1 0 ^{25}} \times 5 \times 1 0^{7} = 5.4.

TE $_{304}$ has $E_{y} \propto sin \frac{3 π x}{a} sin \frac{4 π z}{d}$ ( $n = 0$ : uniform in height). Its maxima lie $a /3 = 10.7 cm$ apart across and $d /4 = 7.5 cm$ apart in depth: $3 \times 4 = 12$ hot spots with cold lines between them (Figure 7.8.10b).

$Δ f = 2450/100 = 24.5 MHz$ , that is $\pm 12.25 MHz$ around $2.45 GHz$ . TE $_{304}$ , TE $_{421}$ , TM $_{421}$ and TM $_{510}$ lie inside, TM $_{230}$ ( $12.5 MHz$ below) and TE $_{032}$ ( $12.2 MHz$ above) at the edges: four to six modes at once (Figure 7.8.27).

Answer

λ = 12.24 cm, f_{101} = 0.685 GHz 6 modes in 50 MHz (Δ N = 5.4) TE_{304} hot spots 10.7 cm \times 7.5 cm apart 4 to 6 modes at once

Worked example 7.8.1. The resonances of the 32\times20\times30\,\mathrm{cm} oven between 2.40 and 2.50\,\mathrm{GHz} (bars, height proportional to the number of degenerate modes, labelled above) and the response 1/[1+(2Q_L\delta_f)^2] of the loaded cavity (Q_L=100) to the magnetron at 2.45\,\mathrm{GHz}. The shaded band is f_0\pm f_0/2Q_L. — **Figure 7.8.27.** Worked example 7.8.1. The resonances of the $32 \times 20 \times 30 cm$ oven between $2.40$ and $2.50 GHz$ (bars, height proportional to the number of degenerate modes, labelled above) and the response $1/ [1 + (2 Q_{L} δ_{f})^{2}]$ of the loaded cavity ( $Q_{L} = 100$ ) to the magnetron at $2.45 GHz$ . The shaded band is $f_{0} \pm f_{0} /2 Q_{L}$ .

Check

Over $2.40$ to $2.50 GHz$ the estimate gives $10.7$ , and the search finds $11$ (counting TE $_{024}$ , which falls exactly on $2.500 GHz$ ), with the mean spacing $1/ (Δ N /Δ f) = 9.3 MHz$ .

Remark

The food sees a sum of several standing waves, each with its own hot spots. Their spacing ( $10.7$ and $7.5 cm$ here) is never smaller than $λ /2 = 6.1 cm$ , because each of the three wave-number components is at most $k$ . The turntable moves the food through the pattern, and the mode stirrer shifts the resonances so that the pattern itself moves.

Worked example 7.8.1: The modes of a microwave oven