How the tables were checked
Each check below covers a stated set of cases. Differences are published, including the ones that don't flatter the tables.
Built by AI agents; confirm with your fab. None of this was measured on a physical board, and no fab's tool or data was used.
What produced each number
- All table rows (324 cross-sections, 4869 rows): 2D field solver. Our own finite-volume Laplace solver on a graded grid. It solves the cross-section twice, with and without dielectrics, to get C and Cair, then Z = 1 / (c0 √(C·Cair)), with Richardson extrapolation across grid sizes. Zdiff = 2·Zodd. Every printed row was solved again at its printed, rounded geometry, and dropped if it missed its target by more than 0.5%.
- Beside each row: a closed form. Formula only, for comparison.
- 1 solved 3D (openEMS) row (section below). Every other row has no 3D result, and its page says “not run”.
The solver against published closed forms
These compare the solver with published formulas that don't depend on it. They cover the bare line only: no published reference here checks the solder mask model, and none covers two different Dk values. The formulas were coded from the literature by the agents that built this site and weren't re-checked line by line against the original papers, so a formula slip would show up as a difference here.
| Reference | Inputs, mm | Reference, Ω | Solver, Ω | Solver vs reference | Kind |
|---|---|---|---|---|---|
| Cohn 1954, centred stripline, t = 0 | w 0.25, b 1.0, Dk 4 | 70.01 | 69.96 | -0.1% | exact |
| Cohn 1954, centred stripline, t = 0 | w 0.5, b 1.0, Dk 4 | 50.25 | 50.18 | -0.1% | exact |
| Cohn 1954, centred stripline, t = 0 | w 1.0, b 1.0, Dk 4 | 32.70 | 32.66 | -0.1% | exact |
| Cohn 1955, coupled stripline, t = 0, Zdiff | w 0.3, gap 0.2, b 1.0, Dk 4 | 98.80 | 98.53 | -0.3% | exact |
| Cohn 1955, coupled stripline, t = 0, Zdiff | w 0.15, gap 0.15, b 1.0, Dk 4 | 116.91 | 116.38 | -0.5% | exact |
| Wheeler 1978, centred stripline with thickness | w 0.3, b 1.0, t 0.035, Dk 4 | 58.83 | 59.29 | +0.8% | about 0.5% |
| Wheeler 1978, centred stripline with thickness | w 0.5, b 1.0, t 0.07, Dk 4 | 43.59 | 43.94 | +0.8% | about 0.5% |
| Hammerstad and Jensen 1980, microstrip, t = 0, no mask | w 3.06, h 1.6, Dk 4.4 | 50.06 | 50.09 | +0.1% | fit |
| Hammerstad and Jensen 1980, microstrip, t = 0, no mask | w 0.35, h 0.2, Dk 4.0 | 55.05 | 55.01 | -0.1% | fit |
| Hammerstad and Jensen 1980, microstrip, t = 0, no mask | w 1.0, h 1.0, Dk 2.2 | 95.03 | 95.07 | +0.0% | fit |
| Hammerstad and Jensen 1980, microstrip with thickness, no mask | w 0.35, h 0.2, t 0.035, Dk 4.0 | 53.52 | 52.60 | -1.7% | fit with a thickness correction |
| Hammerstad and Jensen 1980, microstrip with thickness, no mask | w 3.06, h 1.6, t 0.035, Dk 4.4 | 49.83 | 49.76 | -0.1% | fit with a thickness correction |
| Garg and Bahl 1979, coupled microstrip, t = 0, no mask, Zdiff | w 0.2, gap 0.2, h 0.2, Dk 4.0 | 124.33 | 125.97 | +1.3% | about 3% |
| Garg and Bahl 1979, coupled microstrip, t = 0, no mask, Zdiff | w 0.4, gap 0.3, h 0.4, Dk 4.0 | 118.48 | 120.07 | +1.3% | about 3% |
Against a 3D solver (openEMS)
Each row below was solved at 1 GHz with openEMS, an independent time-domain 3D method, on a straight line at least 16 mm long. Ports read three voltage probes and two current loops around the whole trace. A differential row uses the odd mode with a conducting wall on the plane of symmetry. Solder mask on bare laminate is rounded up to copper height to keep the time step usable, so the 3D model has a little more mask than the 2D one. The two columns show the same row on two meshes, which tells you roughly how far the mesh is from converged: treat a difference of a few percent as the limit of this check, not as a correction to the tables.
| Cross-section | Target | W / gap, mm | 2D solver, Ω | Formula, Ω | FDTD, normal mesh, Ω | FDTD, finer mesh, Ω | FDTD vs 2D |
|---|---|---|---|---|---|---|---|
| microstrip, 0.20 mm dielectric, Dk 4.0, 35 µm copper | 50 Ω single-ended | 0.362 | 49.98 | 48.8 (-2.4%) | 51.7 | 50.6 | +1.3% |
The same 3D setup on a reference line
The 1.6 mm FR-4 line from the published table above, with 35 µm copper and no mask.
2D solver 49.76 Ω. Hammerstad and Jensen with thickness 49.83 Ω.
| Line length, mm | Mesh multiplier | FDTD at 1 GHz, Ω | FDTD at 2 GHz, Ω | εeff at 1 GHz | FDTD vs 2D |
|---|---|---|---|---|---|
| 16 | 1 | 53.72 | 54.65 | 3.79 | +7.9% |
| 64 | 1 | 50.65 | 50.82 | 3.37 | +1.8% |
| 64 | 1.5 | 50.45 | 50.65 | 3.37 | +1.4% |
The 16 mm run came first and read high with an εeff well above the 2D value. On a line this tall the probes sit too close to the feed, so the setup now uses a line 40 times the plane height where that exceeds 16 mm. The grid rows above all use the setup as it stands now, chosen on this reference line and not adjusted to match them.
Separately, a different openEMS setup on the same line, with a 100 mm line and a thin strip, read 50.36 Ω against 50.06 Ω from Hammerstad and Jensen. That is the only other 3D reference case, not a general validation.
Automated tests
The build's tests repeat the comparisons above with fixed limits, and add hand-worked values for the current and via formulas and agreement between the browser solver and the one that made the tables. A build also fails if a published row's impedance is more than 0.5% from its target.
References
- S. B. Cohn, “Characteristic impedance of the shielded-strip transmission line”, IRE Trans. MTT, 1954.
- S. B. Cohn, “Shielded coupled-strip transmission line”, IRE Trans. MTT, 1955.
- H. A. Wheeler, “Transmission-line properties of a strip line between parallel planes”, IEEE Trans. MTT, 1978.
- E. Hammerstad and O. Jensen, “Accurate models for microstrip computer-aided design”, IEEE MTT-S Digest, 1980.
- R. Garg and I. J. Bahl, “Characteristics of coupled microstriplines”, IEEE Trans. MTT, 1979.
- IPC-2141A, Design Guide for High-Speed Controlled Impedance Circuit Boards, 2004.
- D. Brooks and J. Adam, “Trace Currents and Temperatures Revisited”, UltraCAD/ADAM Research, 2015.
- H. Johnson and M. Graham, High-Speed Digital Design, Prentice Hall, 1993.
- openEMS, an open-source FDTD solver.