7.7.6 Wavelength, phase velocity and electrical length

In the quasi-TEM range the microstrip is an ordinary transmission line in a medium of permittivity $ε_{eff}$ .

Microstrip as a transmission line

β v_{p} = k_{0} ε_{eff}, λ_{g} = \frac{λ _{0}}{ε _{eff}}, = \frac{c}{ε _{eff}}, (7.7.8)

and, from $Z_{0} = L^{'} / C^{'}$ and $v_{p} = 1/ L^{'} C^{'}$ ,

C^{'} = \frac{ε _{eff}}{c Z _{0}}, L^{'} = \frac{Z _{0} ε _{eff}}{c} . (7.7.9)

(The phase constant along the line is $β = τ$ in the notation of Section 7.1. For a line, the transmission-line letter $β$ is used.)

The electrical length $β ℓ$ fixes what a section does: stubs and impedance transformers are cut to $λ_{g} /4$ or $λ_{g} /2$ .
The delay per unit length, $ε_{eff} / c$ , is the "propagation delay" of digital design. Lines of different width have different $ε_{eff}$ and therefore different delays.

For the $50 Ω$ line on $1.6 mm$ FR-4, $ε_{eff} = 3.33$ and $ε_{eff} = 1.825$ : $v_{p} = 0.548 c$ and the delay is $60.8 ps/cm$ . At $2.4 GHz$ , $λ_{g} = 68.5 mm$ , $λ_{g} /4 = 17.1 mm$ and $β = 91.7 rad/m$ . The line constants are $C^{'} = 121 pF/m$ and $L^{'} = 306 nH/m$ . A stripline fully embedded in the same FR-4 has $ε_{eff} = ε_{r} = 4.4$ and is slower: $70 ps/cm$ instead of $61$ .

The guide wavelength. Along the strip (orange) the wave repeats after \lambda_g=\lambda_0/\sqrt{\varepsilon_{\rm eff}}, 1.825 times shorter than in free space (dotted) for the 50\,\Omega FR-4 line. — **Figure 7.7.5.** The guide wavelength. Along the strip (orange) the wave repeats after $λ_{g} = λ_{0} / ε_{eff}$ , $1.825$ times shorter than in free space (dotted) for the $50 Ω$ FR-4 line.

Quick check

How long is a quarter-wave section of the $50 Ω$ FR-4 line at $5.8 GHz$ ?

$λ_{0} = c / f = 51.7 mm$ , $λ_{g} = 51.7/1.825 = 28.3 mm$ and $λ_{g} /4 = 7.09 mm$ . At higher frequencies the static $ε_{eff}$ must be corrected for dispersion (Section 7.7.8).