7.7.3 Effective permittivity: Hammerstad's formula

Accurate numerical solutions of the static problem have been condensed into closed-form curve fits. For a thin strip, Hammerstad's fit is

• Narrow strip, $w\ll h$: ${\epsilon }_{\mathrm{eff}}\to ({\epsilon }_r+1)/2$ (plus a small correction): half of the field is in the air, as for a thin wire lying on the surface.
• Wide strip, $w\gg h$: ${\epsilon }_{\mathrm{eff}}\to {\epsilon }_r$, the parallel-plate limit.
• ${\epsilon }_{\mathrm{eff}}$ depends on the ratio $w/h$ only: scale the cross-section up or down and nothing changes. A design for one substrate thickness is a design for all.

The fit is accurate to about $1\%$ (Hammerstad, 1975). The same leading terms are Sadiku's Eq. (11.70), and Orfanidis (Section 11.3) gives the refined Hammerstad–Jensen fits, which yield ${\epsilon }_{\mathrm{eff}}=3.331$ and $Z_0=50.07\Omega$ for the FR-4 line with $w/h=1.91$ (with ${\eta }_0=120\pi \Omega$), against $3.330$ and $50.2\Omega$ from Eq. (7.7.4) and the formulas of the next subsection. Figure 7.7.3 shows ${\epsilon }_{\mathrm{eff}}$ for three common substrates.