Worked example 5.6.2: Wavelengths in air and on a board

About the worked examples of Section 5.6

Two short calculations tied to the applications. They use only $λ = v_{p} / f$ , $k = ω μ ε_{c}$ and the heat density $p_{a v} = \frac{1}{2} σ_{eff} E_{0}^{2}$ , and no new physics. 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: $ε_{0} = 1 0^{- 9} / (36 π) F/m$ , $μ_{0} = 4 π \times 1 0^{- 7} H/m$ , $c = 3 \times 1 0^{8} m/s$ and $η_{0} = 120 π Ω$ .

Worked example 5.6.2

Wavelengths in air and on a board

cf. Application 5.6.1

Given. FM radio $100 MHz$ , GPS L1 $1575.42 MHz$ , Wi-Fi $2.45 GHz$ , car radar $77 GHz$ . A $2.45 GHz$ signal on a $10 cm$ track buried in FR-4, $ε_{r} = 4.4$ (losses neglected here, see Home exercise 5.6.2).

Find. (a) $λ_{0}$ and $λ_{0} /2$ for each service, (b) $T$ , $v_{p}$ and $λ$ on the board, (c) the phase delay along the track. Is the track electrically short?

Try it yourself first.

Strategy

$λ_{0} = c / f$ in air. In the board $v_{p} = c / ε_{r}$ and $λ = v_{p} / f$ , and a length $l$ delays the phase by $β l = 2 π l / λ$ .

	$λ_{0}$	$λ_{0} /2$
FM	3.00 m	1.50 m
GPS L1	19.04 cm	9.52 cm
Wi-Fi	12.24 cm	6.12 cm
radar	3.90 mm	1.95 mm

$T = 1/ f = 408 ps$ , $v_{p} = 3 \times 1 0^{8} / 4.4 = 1.43 \times 1 0^{8} m/s$ and $λ = v_{p} / f = 5.84 cm$ .

β l \frac{l}{λ} = \frac{2 π \times 10}{5.84} = 10.76 rad = 61 7^{\circ}, = 1.71, \frac{l}{v _{p}} = 0.70 ns = 1.71 T .

Answer

λ = 5.84 cm on the board, β l = 10.76 rad = 61 7^{\circ} l = 1.71 λ : not electrically short

Check

$λ_{0} / λ = 12.24/5.84 = 2.10 = 4.4$ , and the delay is the same whether counted in wavelengths ( $l / λ$ ) or in periods ( $l / v_{p} T$ ).

What the numbers say

The voltage at the far end of the track is not the voltage at its near end (Figure 5.6.28): the track is a transmission line, not a wire. A common rule calls a track electrically short below $λ /10$ , here $5.8 mm$ . The same factor $ε_{r} = 2.1$ shrinks antennas printed on the board.

Worked example 5.6.2. A 2.45\,\mathrm{GHz} signal along a 10\,\mathrm{cm} track: in FR-4 the track holds 1.71 wavelengths, and the phase at its far end lags that at its near end by 617^\circ. — **Figure 5.6.28.** Worked example 5.6.2. A $2.45 GHz$ signal along a $10 cm$ track: in FR-4 the track holds $1.71$ wavelengths, and the phase at its far end lags that at its near end by $61 7^{\circ}$ .