Where Your Differential Pair Actually Sits

Three places a differential pair can sit, and only one has a textbook

Edge-coupled differential pairs get drawn one way in every reference: two traces, side by side, exactly halfway between two ground planes. That geometry has an exact solution, which is presumably why it is the one everybody draws.

Your pair is probably not there.

It is on an inner layer with a thin prepreg to one plane and a thicker core to the other. Or it is on the surface, under soldermask. Those are different problems with different answers, and the difference is larger than the tolerance your fabricator works to.

The centred case really is exact

This one deserves its reputation. Two zero-thickness coplanar strips centred between two ground planes can be mapped conformally onto a parallel-plate capacitor, and the even- and odd-mode capacitances fall out as complete elliptic integrals:

k_e = tanh(pi*W/2b) tanh(pi(W+S)/2b) k_o = tanh(pi*W/2b) / tanh(pi*(W+S)/2b)

with C/(eps0*epsr) = 4*K(k)/K(k') per line. Cohn published it in 1955 and nothing has improved on it, because there is nothing to improve — it is not a fit.

That matters more than it sounds. Almost every impedance formula you use daily is a curve fit to solver data with an error band. This one has no error band. We checked it against a 2-D method-of-moments solver anyway and got agreement to 0.008%, which is the solver's own discretisation error rather than the formula's.

Take a 0.15 mm pair on 0.2 mm spacing, centred in a 0.6 mm plane gap, 1 oz copper, FR4 at 4.2:

• odd mode 49.18 ohm, even mode 62.31 ohm
• Z_diff = 98.36 ohm, Z_common = 31.15 ohm
• the same trace with its partner deleted: 56.07 ohm

That last number is the one to keep. Twice it is 112.15 ohm — what Z_diff would be if the two traces ignored each other completely. The real answer is 13.8 ohm below that, and the gap is the coupling. Anyone who tells you differential impedance is "just twice the single-ended value" is quoting the uncoupled limit.

The coupling coefficient here is 0.118. It is the honest measure of how much the two traces are actually talking, and it is worth watching when you compare stack-up candidates: a weakly coupled pair is more tolerant of etch variation, because less of its impedance depends on the spacing your fabricator actually achieves.

Off-centre is the normal case

Core and prepreg come in the thicknesses your vendor stocks. Unless the layer arrangement divides evenly, the pair lands nearer one plane.

Same idea as a single offset stripline, with the same consequence: closer to a plane means more capacitance means lower impedance. A 0.13 mm pair on 0.2 mm spacing sitting 0.2 mm above one plane and 0.5 mm below the other gives Z_diff = 103.02 ohm against a single-trace reference of 59.44 ohm. The offset ratio is 0.43 — well off centre.

Move the same pair to the middle and the impedance rises. Ten to twenty percent is typical for a pair at a quarter of the plane spacing, which is several times what a controlled-impedance shop will hold. Treating an offset pair as centred is a routine way for a board to come back out of specification.

There is no exact solution here, so the model is built as the exact centred result multiplied by a correction fitted to the solver. The useful property is that the correction is constructed to vanish identically when the two gaps are equal — set them equal and the offset calculator returns the centred calculator's answer to the last digit, not merely close to it. A model that misses its own degenerate case is wrong rather than inaccurate.

The surface case has a problem the others do not

Put the pair on an outer layer under soldermask and something appears that no stripline has: the two modes travel at different speeds.

A stripline sits in one dielectric. Both modes see the same permittivity, both propagate at the same velocity, and a pair that starts differential stays differential. A surface pair has air above it, and the odd mode drives its field through the gap between the traces while the even mode spreads sideways. They sample different amounts of air, so they see different effective permittivities.

A 0.2 mm pair on 0.2 mm spacing over 0.2 mm of FR4, under 25 um of soldermask:

• eps_eff,odd = 3.122, eps_eff,even = 3.472
• delays 5.894 and 6.215 ps/mm — 0.321 ps/mm apart

Over a 100 mm route that is 32 ps of differential-to-common-mode conversion, accumulated purely by geometry. Nothing is wrong with the board.

The mask is not the villain, incidentally. Strip it off and the skew gets worse — 0.458 ps/mm — because the mask fills the gap the odd mode works hardest in and pulls its permittivity up toward the even mode's. Soldermask is doing you a small favour here, which is not the intuition most people carry.

It does cost impedance: Z_diff drops from 113.38 ohm bare to 107.52 ohm under 25 um of mask. Six ohms, from something thinner than a human hair, because it sits exactly where the field is.

What these models claim

Worth stating plainly, because a number with no error bar invites a decision it cannot support.

Outside those ranges the calculators still return a number, but they flag it as extrapolated. A fit outside the geometry it was fitted over has no stated accuracy, and pretending otherwise is how a tolerance becomes decoration.

Practical order of operations

Set trace width for the single-ended impedance you want. Then open the spacing until Z_diff reaches target. The two interact, but that order converges in a couple of iterations where the reverse does not.

Watch the coupling coefficient while you do it. Once it is small, further spacing buys almost nothing — you are past the useful range of the knob and should be adjusting width instead.

And if the pair can live on an inner layer, put it there. Not for the impedance, which you can hit either way, but for the skew you then do not have to length-match away.