Effective Dielectric Constant Calculator (Microstrip)

A microstrip is a mixed-dielectric structure. The trace sits on the laminate with a reference plane below it, but above the trace is air. The field lines therefore split between two media of different permittivity, and the wave propagates as if it were travelling in a single uniform medium of some intermediate value $\varepsilon_{eff}$, strictly bounded by $1 < \varepsilon_{eff} < \varepsilon_r$. Every number that depends on velocity — propagation delay, guided wavelength, stub length, electrical length of a matching section — depends on $\varepsilon_{eff}$ and not on the laminate $\varepsilon_r$ printed in the datasheet.

Hammerstad and Jensen fitted $\varepsilon_{eff}$ to the exact quasi-static field solution as a function of a single shape parameter $u = w/h$, with an $\varepsilon_r$-dependent exponent. Their closed form holds to roughly 1 % over $0.05 \le u \le 20$ and $\varepsilon_r \le 12$, which covers essentially all printed-circuit work. Copper thickness enters first as an effective width increase $\Delta w = (t/\pi)\ln(1 + 2h/t)$, because a conductor with real thickness presents extra sidewall to the field; ignoring it overstates $Z_0$ noticeably on heavy copper.

Two limits confirm the model is doing the right physics. As $w/h \to 0$ the trace barely disturbs the air-dielectric interface and $\varepsilon_{eff} \to (\varepsilon_r + 1)/2$, the arithmetic mean of the two half-spaces. As $w/h \to \infty$ the field is squeezed entirely into the substrate and $\varepsilon_{eff} \to \varepsilon_r$ — the parallel-plate limit, where a microstrip behaves like a stripline. Between them the filling factor $q = (\varepsilon_{eff} - 1)/(\varepsilon_r - 1)$ states directly what fraction of the field energy sits inside the laminate.

The quasi-static result is frequency-independent, which is not quite true. As frequency rises the field concentrates further into the higher-permittivity substrate and $\varepsilon_{eff}$ climbs monotonically toward $\varepsilon_r$. Getsinger's model captures this as $\varepsilon_{eff}(f) = \varepsilon_r - (\varepsilon_r - \varepsilon_{eff})/(1 + G(f/f_p)^2)$ with $f_p = Z_0/(2\mu_0 h)$ and $G = 0.6 + 0.009 Z_0$. The correction scales as $(f h)^2$, so it is under 1 % on thin laminate below about 10 GHz and becomes several percent on thick substrate or above roughly 15 GHz.