For estimating and education only. Results come from the simplified models described below, are provided as is with no warranty, and are not a substitute for engineering analysis, simulation or measurement. Check them before relying on them in a design, purchase or safety decision. See the terms of use.
Equations
u = \frac{w_\mathrm{eff}}{h}\varepsilon_\mathrm{eff} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}\left[\left(1 + \frac{12}{u}\right)^{-1/2} + 0.04\,(1 - u)^2\right]last term only for u < 1u ≤ 1
Z_0 = \frac{60}{\sqrt{\varepsilon_\mathrm{eff}}}\,\ln\!\left(\frac{8}{u} + \frac{u}{4}\right)Ωu ≥ 1
Z_0 = \frac{120\pi}{\sqrt{\varepsilon_\mathrm{eff}}\,\left[u + 1.393 + 0.667\ln(u + 1.444)\right]}Ωw_\mathrm{eff} = w + \frac{t}{\pi}\left(1 + \ln\frac{2h}{t}\right)w/h ≥ 1/2π\varepsilon_\mathrm{eff}(t) = \varepsilon_\mathrm{eff} - \frac{\varepsilon_r - 1}{4.6}\cdot\frac{t/h}{\sqrt{w/h}}\lambda_g = \frac{c}{f\sqrt{\varepsilon_\mathrm{eff}}}Variables
| Symbol | Meaning | Unit |
|---|---|---|
w | Trace width | m |
h | Substrate height | m |
t | Copper thickness (1 oz = 35 µm) | m |
\varepsilon_r | Substrate relative permittivity | — |
\varepsilon_\mathrm{eff} | Effective permittivity | — |
Z_0 | Characteristic impedance | Ω |
\lambda_g | Guided wavelength | m |
Common substrates (nominal)
| Material | εr | Note |
|---|---|---|
| FR-4 | 4.2 – 4.5 | Varies by supplier, weave and frequency |
| Rogers RO4003C | 3.38 ± 0.05 | Datasheet value at 10 GHz |
| Rogers RO4350B | 3.48 ± 0.05 | Datasheet value at 10 GHz |
| Rogers RT/duroid 5880 | 2.20 ± 0.02 | Datasheet value |
| Alumina (96 %) | 9.4 – 9.9 | Varies by grade |
Worked example
| Input | Value |
|---|---|
| Target impedance | 50 Ω |
| Substrate | FR-4, εr = 4.4, h = 1.6 mm |
| Copper | 1 oz (0.035 mm) |
| Frequency | 2.4 GHz |
| Result | Value |
|---|---|
| Required trace width | 3.021 mm |
| Characteristic impedance | 50 Ω |
| Effective permittivity | 3.321 |
| Width / height | 1.888 |
| Velocity factor | 0.5488 |
| Guided wavelength | 68.55 mm |
| Quarter-wave length | 17.14 mm |
w/h ≈ 1.9 for 50 Ω on FR-4, a common rule of thumb. εr and dielectric height both shift the result, so use your fabricator’s stack-up.
Load this example in the calculatorValidity
- Quasi-static model: accurate to about 1–2 % for 0.05 ≤ w/h ≤ 20 and 1 ≤ εr ≤ 16.
- Dispersion is ignored. Expect εeff to rise with frequency, more on thick substrates.
- No solder mask, no cover, no nearby ground pours. Use a field solver for coplanar or tight-pitch designs.
Sources
- E. Hammerstad and O. Jensen, “Accurate Models for Microstrip Computer-Aided Design,” IEEE MTT-S Int. Microwave Symp. Dig., 1980, pp. 407–409.
- D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. §3.8 (microstrip).
- B. C. Wadell, Transmission Line Design Handbook, Artech House, 1991.
- Substrate values: manufacturer datasheets (Rogers Corporation).