7.7.8 Dispersion and higher-order modes: the frequency limit

The quasi-TEM picture holds while $h ≪ λ$ . As the frequency rises the field concentrates in the substrate, under the strip, and $ε_{eff}$ rises from its static value towards $ε_{r}$ : the line becomes dispersive. Getsinger's model describes this with two parameters:

ε_{eff} (f) f_{p} G = ε_{r} - \frac{ε _{r} - ε _{eff} ( 0 )}{1 + G ( f / f _{p} ) ^{2}}, = \frac{Z _{0}}{2 μ _{0} h}, = 0.6 + 0.009 Z_{0} (Z_{0} in Ω) . (7.7.12)

At $f = f_{p}$ the correction has done roughly half of its work. For the $50 Ω$ FR-4 line $f_{p} = 12.5 GHz$ (Figure 7.7.7).

Upper limits

Two mechanisms end the useful range of a line, both set by the cross-section dimensions compared with the wavelength in the substrate.

The TE $_{1}$ surface wave of the substrate. The grounded substrate is the coated conductor of Section 7.6.9, and by Eq. (7.6.32) in Section 7.6 with $t = h$ and $n_{2} = 1$ its TE $_{1}$ mode starts at

$f_{T E_{1}} = \frac{c}{4 h ε _{r} - 1} . (7.7.13)$

Above it the line couples strongly to a wave that spreads over the whole board.
The transverse resonance of the strip: a field that is half a wavelength across the strip and its fringe, at

$f_{tr} \approx \frac{c}{ε _{r} ( 2 w + 0.8 h )} . (7.7.14)$

The TM $_{0}$ surface wave has no cutoff at all (Section 7.6.9), and the coupling to it grows with $h / λ$ . The design rule is to keep $h ≪ λ_{g}$ : thick FR-4 below a few gigahertz, $0.1 mm$ GaAs at $60 GHz$ .

$Getsinger's dispersion, Eq. (7.7.12), of the 50\,\Omega FR-4 line, f_p=12.5\,\mathrm{GHz}. Dotted: \varepsilon_r=4.4 and the static value 3.33. Dashed: the transverse resonance at 19.3\,\mathrm{GHz} and the TE_1 surface wave at 25.4\,\mathrm{GHz}, which close the useful range (shaded).$

Figure 7.7.7. Getsinger's dispersion, Eq. (7.7.12), of the

50 Ω

FR-4 line,

f_{p} = 12.5 GHz

. Dotted:

ε_{r} = 4.4

and the static value

3.33

. Dashed: the transverse resonance at

19.3 GHz

and the TE

_{1}

surface wave at

25.4 GHz

, which close the useful range (shaded).

Application 7.7.1 · Microstrip on RF circuit boards

Controlled impedance. Every trace that carries a radio signal, or a digital edge faster than its travel time along the board, is a transmission line. The board house is asked for " $50 Ω \pm 10 %$ ": on the usual $1.6 mm$ FR-4 that means $w = 3.06 mm$ , on a $0.2 mm$ inner prepreg $0.38 mm$ . The tolerance comes from $ε_{r}$ (the glass weave), from $h$ and from the etched width (Exercise 7.7.2).

\frac{w}{h} \approx 1.91 for 50 Ω on FR-4, λ_{g} = \frac{λ _{0}}{1.825} . (7.7.15)

Patch-antenna feeds. A patch antenna is a wide microstrip section about $λ_{g} /2$ long that radiates from its two open edges. Its edge resistance is a few hundred ohms, and a quarter-wave transformer $Z_{T} = 50 R_{edge}$ (or an inset feed) matches it to the $50 Ω$ line (Figure 7.7.8 and Worked example 7.7.1). GPS, Wi-Fi and phased-array radar tiles use it.

Application 7.7.1. A 2.45\,\mathrm{GHz} patch antenna on FR-4, drawn to scale: a 50\,\Omega line (3.06\,\mathrm{mm} wide) from the edge connector, a 100\,\Omega quarter-wave transformer (0.71\,\mathrm{mm} wide, 17.6\,\mathrm{mm} long) and the patch, 28.8\times37.3\,\mathrm{mm}, about half a guide wavelength long. — **Figure 7.7.8.** Application 7.7.1. A $2.45 GHz$ patch antenna on FR-4, drawn to scale: a $50 Ω$ line ( $3.06 mm$ wide) from the edge connector, a $100 Ω$ quarter-wave transformer ( $0.71 mm$ wide, $17.6 mm$ long) and the patch, $28.8 \times 37.3 mm$ , about half a guide wavelength long.

When FR-4 is not good enough. $tan δ = 0.02$ costs $7.2 dB/m$ at $2.4 GHz$ and $30 dB/m$ at $10 GHz$ . Radio front ends and radar boards therefore use PTFE or hydrocarbon laminates ( $tan δ \leq 0.003$ ), bonded in the same multilayer board to cheap FR-4 layers for the digital part. Above $30 GHz$ thin laminates, alumina and GaAs chips take over.