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

Hammerstad's closed forms for the characteristic impedance use $ε_{eff}$ of Eq. (7.7.4):

Characteristic impedance of a microstrip line

\frac{w}{h} \leq 1 : Z_{0} = \frac{60}{ε _{eff}} ln (\frac{8 h}{w} + \frac{w}{4 h}) [Ω], \frac{w}{h} \geq 1 : Z_{0} = \frac{120 π}{ε _{eff} F} [Ω], F = \frac{w}{h} + 1.393 + 0.667 ln (\frac{w}{h} + 1.444) . (7.7.5)

In plain words

A narrow strip is a high-impedance line and a wide strip a low-impedance line. Like $ε_{eff}$ , $Z_{0}$ depends on $w / h$ and $ε_{r}$ only, and it changes only logarithmically with the width for narrow strips.

The two branches have simple limits (Sadiku's Eq. (11.71) lists the same pair of formulas).

Narrow strip: a strip of width $w$ carries charge like a round wire of radius $w /4$ . A wire of radius $r$ at the height $h$ above a ground plane has, with its image, $Z_{0} \approx (60/ ε_{eff}) ln (2 h / r)$ , which with $r = w /4$ is the leading term $(60/ ε_{eff}) ln (8 h / w)$ .
Wide strip: the parallel-plate line of Section 7.2, $Z_{0} = η h / w$ with $η = η_{0} / ε_{eff}$ , and $F = w / h + \dots$ adds the fringing at the edges.

Characteristic impedance, Eq. (7.7.5), against w/h for the three substrates of Figure 7.7.3. Dotted: 50\,\Omega. On FR-4 the 50\,\Omega line has w/h=1.91 (dot). — **Figure 7.7.4.** Characteristic impedance, Eq. (7.7.5), against $w / h$ for the three substrates of Figure 7.7.3. Dotted: $50 Ω$ . On FR-4 the $50 Ω$ line has $w / h = 1.91$ (dot).

Common mistake · Ignoring the fringing field

Using the parallel-plate formula $Z_{0} = η_{0} h / (ε_{r} w)$ for a microstrip line. For the $50 Ω$ FR-4 line ( $w / h = 1.91$ ) it gives $377/ (2.098 \times 1.91) = 94 Ω$ , almost twice the true value: half of the capacitance sits in the fringing field at the edges and in the air. On $1.6 mm$ FR-4, $Z_{0}$ runs from $154 Ω$ at $w / h = 0.1$ to $15 Ω$ at $w / h = 10$ .

Quick check

Find $ε_{eff}$ and $Z_{0}$ of a strip with $w / h = 0.5$ on FR-4 ( $ε_{r} = 4.4$ ).

$p = 0.04 \times 0.25 = 0.01$ and $(1 + 12/0.5)^{- 1/2} = 0.2$ , so $ε_{eff} = 2.7 + 1.7 \times 0.21 = 3.057$ . Then $Z_{0} = (60/1.748) ln (16 + 0.125) = 34.32 \times 2.780 = 95.4 Ω$ : a strip half as wide as the substrate is thick is close to a $100 Ω$ line.

7.7.4 Characteristic impedance