The skew in the glass

The millimetre that becomes common mode measured what a length mismatch between a pair’s halves costs: a millimetre and a half of FR-4 is ten and a half picoseconds, and a tenth of the signal is common mode by 3 GHz. The common mode that has three slopes added the driver’s two other imbalances and found the skew leading from a few gigahertz up, and it ended on a skew that no length-matching rule can see: two conductors of exactly equal length arriving apart because they ran over different material.

A printed circuit board’s dielectric is not uniform. It is glass cloth — bundles of fibres woven in a grid, a few tenths of a millimetre apart — set in resin, and glass has a higher permittivity than resin. A conductor running over the centre of a bundle sees more glass beneath it than one running over the gap between bundles, and so a higher effective permittivity and a slower signal. For a single track this is a small change in delay. For a pair it is a skew, and it has the property that makes skew expensive: it accumulates with length for as long as the two conductors stay in their different places.

A permittivity with a pitch #

The model is as simple as the effect allows. The effective permittivity under a conductor varies across the weave as a cosine, from 4.2 over the middle of a gap to 4.6 over the middle of a bundle, with the weave’s pitch of 0.5 mm; along the weave it does not vary. The pair’s two conductors are 0.25 mm apart — half the weave’s pitch, the arrangement a quarter-millimetre spacing on a common glass style produces. Each conductor’s delay is the integral of $\sqrt{\varepsilon_r}/c$ along its path, and the skew is the difference of the two integrals.

Routed exactly along the weave, each conductor stays at its lateral position for its whole length, and the skew is the length times the difference in $\sqrt{\varepsilon_r}$ under the two, over $c$. With one conductor over a bundle’s centre and the other over a gap’s, that difference is its largest: 0.318 ps a millimetre, which the integration reproduces to six figures. In 20 cm the pair has 63.6 ps of skew, and in 30 cm 95.4 ps — the equivalent of 13.6 mm of routed length mismatch, on a pair whose two lengths are identical to the last micrometre.

That is several times the whole skew budget of a fast serial link, and nothing in the layout database knows about it. A design rule that checks the difference between the two conductors’ lengths is checking the wrong difference. What converts the signal is the difference between their delays.

A lottery on position #

The worst case assumed the pair lands with one conductor over glass and the other over resin. Where a pair actually lands across the weave is set by nothing a designer controls.

Over 100 mm along the weave, the skew is ±31.8 ps at the worst positions and zero where the two conductors sit symmetrically about a bundle’s centre or a gap’s, and see the same glass. Between, it follows the difference of the cosine at two points a quarter of a millimetre apart. The board’s fabrication places the pair wherever the artwork’s registration to the cloth happens to put it, so a design with a hundred pairs has a hundred draws from this curve, and its worst pair is near the extreme.

This is what makes the effect notorious in practice: two boards from the same files measure differently, and two pairs on one board that should be identical are not. The skew is not a property of the design but of the design’s accidental registration to a pattern, and a measurement on one board says little about the next.

Rotating the route #

The skew accumulates because each conductor stays over the same material. Route the pair at an angle to the weave and each conductor crosses bundle and gap in turn.

At an angle $\theta$ each conductor’s lateral position moves across the weave as it goes, so the difference in $\sqrt{\varepsilon_r}$ between the two oscillates, once per weave pitch of lateral travel — every $p/\sin\theta$ of route. The skew is the integral of an oscillating difference, and the integral of an oscillation is bounded: its largest value is the difference’s amplitude times the oscillation’s length over $2\pi$,

$$\Delta\tau_{max} = \frac{p\,\Delta(\sqrt{\varepsilon_r})}{2\pi c\,\sin\theta},$$

whatever the route’s length. The integration over 300 mm finds it to two per cent at every angle measured: 1.45 ps at 1°, 0.725 at 2°, 0.290 at 5°, 0.146 at 10° and 0.036 at 45°, against 95.4 ps for the same pair along the weave. A tenth of a picosecond needs 14.7°. Those figures are for a route that begins with the pair over bundle and gap, the placement that is worst along the weave. Rotated, the worst start is a different one: a route that begins where both conductors see the same glass, and diverges from there, swings through the integral’s full peak-to-peak, twice the bound, so for an arbitrary registration of the artwork to the cloth the numbers above should be doubled — 2.9 ps at 1°, 0.29 at 10°.

So the cure is geometric and cheap. A pair routed a few degrees off the weave’s axes, or with the whole artwork rotated a few degrees on the panel, turns a skew that grows with length into one that does not. The angle needed is small because the bound falls as one over the sine: a degree takes the 20 cm skew from 63.6 ps to 1.45, and ten take it to a seventh of a picosecond.

Pitch against pitch #

The worst case also assumed the pair’s pitch is half the weave’s. The skew per millimetre along the weave depends on how the two pitches compare.

The skew per millimetre is the difference in $\sqrt{\varepsilon_r}$ between two points a pair’s pitch apart on a periodic pattern. It is largest when the pitch is half the weave’s — 317 fs/mm, one conductor at a peak and the other at a trough — and it falls to almost nothing when the pitch equals the weave’s, since two points a whole period apart see the same thing. At one and a half weave pitches it is largest again.

That suggests a second cure and bounds its value. A pair whose pitch matches its board’s glass pitch is immune along the weave — but glass styles differ, the pitch of a given cloth is not held tightly, and a pair’s pitch is chosen for its impedance, not for the weave. The rotation works for any pitch; the matching works only when both pitches are known and fixed, which is why the rotation is the rule and the matching a curiosity.

What it does to the signal #

The skew is only a delay until it is converted, and the earlier essays supply the conversion.

At 2.5 GHz — the fundamental of a five-gigabit link — the conversion from a skew $\Delta\tau$ is $\sin(\pi f \Delta\tau)$. Along the weave the worst-placed pair reaches 68.1 per cent common mode by 300 mm: more than two-thirds of the signal is travelling in a mode the receiver rejects and the termination reflects, which is no longer a matter of emissions but of the link closing at all. The differential signal, at cos(πfΔτ) of its amplitude, has lost nearly half of its power. Rotated ten degrees, the same pair never converts more than 0.115 per cent.

The comparison makes the point sharper than any number. A pair of exactly equal lengths, routed along the weave, can be as badly skewed as a pair routed with 13.6 mm of mismatch; the same pair ten degrees off the weave is better than a pair length-matched to a fiftieth of a millimetre. The length-matching rule that design tools enforce is necessary and, on a woven board at gigahertz rates, very far from sufficient.

A budget in millimetres #

The earlier essay’s budget for a ten-gigabit link — each source of common mode held 30 dB below the differential at the 5 GHz fundamental — allowed 2.0 ps of skew. Along the weave at the worst position the glass supplies 0.318 ps of it for every millimetre, so the whole budget is spent in 6.3 mm of route. At a typical position, the root-mean-square of the lottery, the pair gets about 9 mm. No high-speed pair on a real board is that short, so along the weave the glass alone breaks the budget on essentially every link, and no routing tolerance recovers it.

Rotated, the budget is not spent at all. At 5° the bound is 0.29 ps whatever the length, a seventh of the allowance; at 10°, 0.146 ps, a fourteenth. The rotation does not merely reduce the skew. It changes it from a quantity that grows with the route to one that does not, which moves it from the budget’s largest item to a rounding error, and it leaves the routed length mismatch — the thing the design rule checks — as the budget’s real constraint again.

That also says when the effect can be ignored. The skew along the weave grows with the length at a rate set by the glass, and the budget shrinks with the link’s rate. A link at a hundred megabits has a skew budget of a nanosecond, three metres of worst-case weave; nobody meets the effect there. The weave becomes a design constraint in the band where a link’s skew budget in picoseconds, divided by a third of a picosecond a millimetre, falls below the length of its route — from a few gigabits a second on a board of ordinary size.

Finding it on a bench #

The skew hides from the usual measurements for the same reason it hides from the design rule. A length check on the artwork reports the two conductors equal. A time-domain reflectometer launched into each conductor separately measures each one’s delay correctly — and a technician comparing them sees two delays that differ by the skew, attributes the difference to the fabricator’s etching or to a mismatch in the fixture, and remeasures on another board, where the difference is different. The dip whose area is fixed measured what a reflectometer can resolve about a single discontinuity; a uniform difference in velocity along a whole line is the one thing it reports as a delay rather than as a feature, and it has no feature to look for.

The diagnostic that works is a mixed-mode measurement of the pair: the differential-to-common conversion against frequency. A skew from the glass has the earlier essays’ signature — rising at twenty decibels a decade, with a null where $\omega\Delta\tau$ reaches $2\pi$ — and the null’s frequency gives the skew directly. Two pairs of identical layout with nulls at different frequencies are the weave’s fingerprint: routed mismatch is the same on every copy of a design, and the glass is not.

The same fingerprint distinguishes the weave from the driver’s imbalances. An amplitude imbalance is flat and has no null; a rise-time mismatch rises at forty decibels a decade; only a skew makes the null, and only the weave makes the null move from pair to pair.

The medium is not homogeneous, twice over #

The weave is the second way these essays have found a pair’s medium failing to be the uniform dielectric the textbook assumes. The first was vertical: a microstrip has air above and glass below, so its even and odd modes see different mixtures and travel at different speeds, which the permittivity a loss forbids and the dispersion essays traced to the same material properties. That inhomogeneity is symmetric between the two conductors and converts nothing; it separates the modes in time without mixing them.

The weave is lateral and asymmetric — the two conductors see different material — and it converts. The distinction is exactly the one the delay that is not one number drew for a single line, where the delay depended on frequency through the dielectric’s loss; here it depends on position through the dielectric’s composition, and it is the difference in position between two conductors that matters rather than either one’s absolute delay.

Where it sits among the pair’s other failures #

The weave’s skew joins the pair’s other ways of turning into common mode, and they interact in ways worth naming. It adds to the routed mismatch as a delay: the two are one skew, and a routed mismatch that happens to oppose the glass’s reduces it, which is one reason a board can measure better than its artwork. It meets the pair’s loss, which the edge the loss delivers measured slowing edges, and a slower edge makes a given skew convert less at the edge’s own time scale. And its common mode, once made, leaves the pair by the routes the far-end crosstalk of the far end that cancels describes, coupling into neighbours far more strongly than the differential signal would.

What it does not share with the others is its randomness. Every other imbalance in these essays is a property of a design — a length, a driver, a termination — and is the same on every copy. The weave’s skew is a property of the copy, and a design verified on one prototype has been verified on one draw. That is the distinction the shape a tolerance does not print draws for resistors — a specification describes a population, and what a single build does is decided by where in the population it landed — and a board’s glass is a component whose tolerance nobody prints.

How the numbers were obtained #

Each conductor’s delay is integrated along its path in 4,000 to 12,000 steps: at each point its lateral position fixes the effective permittivity through the cosine model, and the delay increment is $\sqrt{\varepsilon_r}/c$ times the step. The skew is the difference of the two running integrals. Along the weave the integral is checked against the length times the difference in $\sqrt{\varepsilon_r}$ over $c$, to six figures; at an angle, the largest excursion is checked against the closed-form bound to two per cent. The position and pitch figures evaluate the per-millimetre difference directly over a fine grid of positions. The common mode is $\sin(\pi f \Delta\tau)$, the conversion the earlier essay measured on the solved pair.

The permittivity swing — 4.2 to 4.6 — is a stated assumption, representative of a coarse glass style under a thin trace; finer or spread glass styles swing less, and a wide trace averages over more of the weave. Every skew here scales in proportion to the swing.

What it leaves out #

The weave in two directions. Glass cloth has warp and weft at right angles with slightly different pitches, and a route at an angle to one is at the complementary angle to the other. The bound applies to each, and the larger governs.

A weave that is not a cosine. Real bundles have flat tops and sharp edges, which puts harmonics into the permittivity pattern. The along-weave skew is unchanged in character; the angled route’s bound gains terms from each harmonic, each smaller and each averaging out faster.

The trace’s width. A trace wider than a bundle averages the permittivity beneath it and sees a smaller swing; the effect is strongest for the narrow traces of dense fast boards, which is where it is seen.

Still open: the two weaves, a zig-zag route, and the receiver’s view #

Warp and weft together. With two weave directions of different pitch, the angle that minimises the larger of the two bounds is not forty-five degrees in general, and a route at an angle to one weave is nearly along the other when the pitches differ enough. Which angle is best for a given cloth is a small optimisation over the two bounds.

A route that zig-zags. Instead of rotating the whole route, a pair can jog sideways by half a weave pitch every so often, which interchanges its two conductors’ positions on the glass and reverses the skew’s growth. How often it must jog to hold the skew under a budget, and what each jog costs in its own discontinuity, is the same integral with a different path.

What the receiver sees in time. A skew of tens of picoseconds against edges of tens of picoseconds is no longer a small conversion; the differential edge itself is smeared and delayed. How much eye opening a weave-induced skew costs, compared with a routed one of the same size, is a question the time-domain march of the earlier essay could answer by feeding it the integrated skew.