7.7.2 The quasi-TEM field and the effective permittivity

In the quasi-TEM approximation the line is described by its static capacitance and inductance per unit length. Let $C^{'}$ be the capacitance per unit length of the real line, with its substrate, and $C_{0}^{'}$ that of the same conductors with the substrate replaced by air. The magnetic field does not see $ε$ : all materials have $μ = μ_{0}$ , so the inductance $L^{'}$ is the same as that of the air-filled line. For any TEM line in a homogeneous medium $L^{'} C^{'} = μ ε$ (Section 7.5). For the air-filled line this gives $L^{'} C_{0}^{'} = 1/ c^{2}$ , hence

L^{'} = \frac{1}{c ^{2} C _{0}^{'}} . (7.7.2)

The real line has $L^{'}$ and $C^{'}$ , so its phase velocity and characteristic impedance are $v = 1/ L^{'} C^{'}$ and $Z_{0} = L^{'} / C^{'}$ .

Effective permittivity, phase velocity and impedance

ε_{eff} Z_{0} = \frac{C ^{'}}{C _{0}^{'}}, v = \frac{1}{L ^{'} C ^{'}} = \frac{c}{ε _{eff}}, = \frac{L ^{'}}{C ^{'}} = \frac{1}{c C ^{'} C _{0}^{'}} = \frac{Z _{0, air}}{ε _{eff}}, (7.7.3)

where $Z_{0, air} = 1/ (c C_{0}^{'})$ is the impedance of the same strip with the substrate removed.

In plain words

Fill all of space with a fictitious uniform dielectric of permittivity $ε_{eff}$ , chosen so that it gives the same capacitance as the real substrate plus air. The line then is an ordinary TEM line in that medium. $ε_{eff}$ is a weighted average: the field in the substrate counts with $ε_{r}$ , the fringing field in the air with $1$ .

Figure 7.7.2 shows the computed cross-section field of a $50 Ω$ line on FR-4. The electric field lines are drawn with the same charge per line, so their density shows where the charge sits: most of it is on the underside of the strip, and most lines run straight down to the ground. The lines refract at the surface of the substrate, exactly as the boundary conditions of Section 3.4.6 require. The magnetic field lines close around the strip. Because $L^{'}$ does not see the substrate, they are the equipotentials of the air-filled line.

The quasi-TEM field of a 50\,\Omega microstrip line on FR-4 (w/h=1.91), computed by finite differences. Orange: electric field lines, drawn with equal charge per line, so most start on the underside of the strip and end on the ground below it. They bend at the surface of the substrate. Blue, dashed: magnetic field lines, the equipotentials of the same strip without substrate, closing around the strip. The wave travels out of the page. — **Figure 7.7.2.** The quasi-TEM field of a $50 Ω$ microstrip line on FR-4 ( $w / h = 1.91$ ), computed by finite differences. Orange: electric field lines, drawn with equal charge per line, so most start on the underside of the strip and end on the ground below it. They bend at the surface of the substrate. Blue, dashed: magnetic field lines, the equipotentials of the same strip without substrate, closing around the strip. The wave travels out of the page.