Formula
The model is the IPC-2141 form of the Hammerstad microstrip equations. u is the width divided by the height. Effective permittivity is (εr + 1) / 2 plus (εr − 1) / 2 times 1 / √(1 + 12/u). When u is under 1, that second term also adds 0.04 (1 − u)². Impedance uses the narrow-strip logarithm when u is 1 or less, and the wide-strip expression when u is greater than 1. Copper thickness is zero. There is no mask and no frequency dispersion.
εeff = (εr+1)/2 + (εr−1)/2 × (1/√(1+12/u) + narrow term)
u ≤ 1: Z0 = (60 / √εeff) ln(8/u + u/4)
u ≥ 1: Z0 = 120π / (√εeff (u + 1.393 + 0.667 ln(u+1.444)))
The width solver bisects that impedance. A published check of this model, FR-4 at εr 4.4, height 1.6 mm, width 3.0 mm, is εeff 3.325 and Z0 50.8 Ω. This page returns 3.325 and 50.82 Ω, within 0.1 Ω. The 50 Ω width is 3.083 mm, within 1% of a published 3.059 mm for the same stackup. The 1980 Hammerstad–Jensen correction is a later paper and is not what is coded here. A fabricator’s stackup replaces both. Lumped reactance is the impedance calculator. Cable after the connector is the coax loss calculator.
Worked examples
εr 4.4, h 1.6 mm, W 3.0 mm. εeff 3.325. Z0 50.82 Ω. Within 0.1 Ω of the published 50.8 Ω check.
Target 50 Ω on that board. Width 3.083 mm. The impedance at that width is 50 Ω by construction of the solver.
Reference
FR-4 at 4.4 is a planning average. Boards are quoted across roughly 4.2 to 4.8, and the value moves with resin content and frequency. Height is the dielectric, not the finished board with copper.
Questions
What impedance is a 3 mm trace on 1.6 mm FR-4?
This model gives 50.82 Ω and an effective permittivity of 3.325, for εr 4.4 and zero copper thickness. A published check of the same equations is 50.8 Ω and 3.325. The difference is under 0.1 Ω, about 0.04%.
How wide is 50 Ω on that board?
3.083 mm in this solver. A published width for the same stackup is about 3.06 mm. This result is within 1% of that figure. The bisection itself hits 50 Ω.
What did the model leave out?
Copper thickness, solder mask, dispersion, and the spread of real FR-4, which is often quoted from about 4.2 to 4.8 rather than a single 4.4. The fabricator’s stackup replaces this estimate.
Sources
- IPC-2141 microstrip equations, the Hammerstad wide and narrow pair: εeff and the two Z0 expressions above. Zero thickness, quasi-static, no solder mask.
- Check point used by published calculators of this model: εr 4.4, h 1.6 mm, W 3.0 mm, εeff 3.325, Z0 50.8 Ω. This implementation is within 0.1 Ω, and the 50 Ω width is within 1% of about 3.06 mm.