The angle two weaves allow

The skew in the glass found that a length-matched differential pair can carry tens of picoseconds of skew because a circuit board is woven glass in resin and its two conductors need not sit over the same material: over a weave of half a millimetre swinging the permittivity between 4.2 and 4.6, a quarter-millimetre pair with one conductor over a bundle and the other over a gap builds 0.318 ps of skew every millimetre. Its cure was geometric and cheap. Route the pair at an angle to the weave and each conductor crosses bundle and gap in turn, the difference oscillates, and its integral is bounded whatever the route’s length — 0.146 ps at ten degrees.

That model had one weave. A woven cloth has two: the warp, running along the panel, and the weft, across it, each a set of glass bundles with its own pitch and its own swing in permittivity. The earlier essay’s first open question was what the second does to the cure, and whether forty-five degrees — the usual advice — is the best angle in general. Turning off the warp is turning onto the weft, so the cure has a cost that grows as it is applied, and there must be an angle where the two balance.

Two weaves, and the pair across each #

The cloth is modelled as two cosines: the warp varies the permittivity across the panel with a pitch of 0.5 mm, the weft varies it along the panel with its own pitch, each by ±0.2 about 4.4. A pair routed at an angle θ to the warp has its two conductors a quarter of a millimetre apart perpendicular to the route, which puts them $d\cos\theta$ apart across the warp and $d\sin\theta$ apart across the weft. Each weave contributes a differential between the two conductors that oscillates as the route crosses its yarns: the warp’s once per $p_w/\sin\theta$ of route, the weft’s once per $p_f/\cos\theta$.

To first order in the swing the two differentials add, and each integrates to a bounded oscillation. Doing that integral over every registration of the pair to the cloth — every lateral position at which the artwork might happen to land — gives a bound with two terms,

$$B(\theta) = \frac{2}{2\pi c}\left[\frac{A_w\,p_w}{\sin\theta} + \frac{A_f\,p_f}{\cos\theta}\right],$$

with each amplitude set by the pair’s separation across that weave: $A_w \propto |\sin(\pi d\cos\theta/p_w)|$ and $A_f \propto |\sin(\pi d\sin\theta/p_f)|$. The first term falls as the route turns off the warp; the second rises as it turns onto the weft, both because the pair’s projection across the weft grows and because its yarns are crossed less often.

For two identical weaves the sum is least at 45.0°, 128.3 femtoseconds. Integrations along 80 mm of route over an eight-by-eight grid of registrations reach 587.6 fs at 5°, 215.1 at 15°, 139.3 at 30°, 126.3 at 40°, 139.2 at 60° and 304.9 at 80°, each within two per cent of the summed bound and each reaching most of it. At 10° the warp alone allows 291.4 fs and the weft adds 13.8; at 80° the roles are reversed. Turning further than forty-five degrees off the warp makes the skew worse, not better, because the route is now nearly along the weft and the weft does to it what the warp did to a route along the warp.

A factor of two, found by counting registrations #

One number above disagrees with the earlier essay. Its bound at 10° was 0.146 ps, half the warp term here, and the difference is not the weft. The earlier figure integrated a pair that starts with one conductor over a bundle and the other over a gap — the placement that is worst for a route along the weave, where the differential never changes. At an angle the differential oscillates, and its running integral swings between two extremes whose midpoint depends on where the oscillation started.

A pair that starts over bundle and gap begins at the differential’s peak, and its integral swings from zero to $A/k$ and back. A pair that starts where both conductors see the same glass begins at a zero of the differential with the differential about to grow, and its integral swings through the full peak-to-peak, $2A/k$. Since nobody chooses where the artwork lands on the cloth, the honest bound is the second: twice the earlier figure’s numbers, 2.9 ps at 1° and 0.29 at 10°. The earlier essay now says so. Its conclusion — that a few degrees turns a growing skew into a bounded one — stands; its bound was the bound for a lucky board.

The cloth decides the angle #

Forty-five degrees is the answer only for a cloth whose two weaves are the same, and few are. Glass styles commonly have different counts of yarns in the two directions, and the weft’s bundles are often flatter than the warp’s, which shows up as a smaller swing in permittivity.

With the swing equal and the weft’s pitch varied, the best angle is 68.7° for a 0.3 mm weft, 52.1° at 0.4, 45.0° at 0.5, 42.1° at 0.6 and 40.7° at 0.7. A finer weft is crossed more often at any angle, so its term is smaller and the route can turn further onto it to escape the warp. With the weft’s swing halved the best angle at a 0.5 mm pitch is 57.5°; quartered, 67.0°. A flatter weft contributes less skew wherever the route goes, so the route should go further towards it.

That makes the best angle a property of the glass style and not a constant of layout, and it is not small: a quarter-swing weft wants the route twenty-two degrees further round than the rule of thumb says. For a 0.4 mm weft the best angle gives 119 fs against 128 for the symmetric cloth at 45°, and a route at 45° on that cloth would give more than either.

Traced along 60 mm on that cloth, the worst-registered pair at 10° swings within 305 fs, the warp’s slow oscillation — one cycle every 2.9 mm of route — carrying the weft’s faster ripple. At the best angle, 52.1°, it swings within 118 fs: both weaves are crossed every millimetre or so, neither term dominates, and their sum is as small as the cloth allows. Neither trace grows with length, which is the earlier essay’s point; the best angle makes the bound about two and a half times tighter than ten degrees.

Why a finer weft lets the route turn further #

The direction in which the best angle moves follows from the two terms’ shapes, and it is worth having as an intuition rather than a search. The weft’s term at an angle θ is its amplitude, which grows with the pair’s projection across the weft, times its period along the route, $p_f/\cos\theta$. A finer weft shortens that period in proportion, so at every angle its term is smaller and the sum’s balance point moves to where the weft’s term is larger — towards the weft. A flatter weft lowers the amplitude instead, with the same effect. The warp’s term, meanwhile, is fixed, and falls as $1/\sin\theta$ all the way to ninety degrees. So the best angle is always pulled towards whichever weave is finer or flatter, by an amount that the two terms’ ratio decides.

That gives a quick check on a glass style’s data sheet. If its two yarn counts differ — more ends per inch one way than picks per inch the other — the finer direction is the one to route nearer to; if its two directions are listed with different thicknesses or resin contents, the flatter one is. The best angle can then be read off the figure above or computed from the two terms, and it will rarely be exactly forty-five degrees.

A link, worked #

Put the numbers against a link. A unit interval at 25 gigabits a second is 40 picoseconds, and a skew budget of a few per cent of it is a picosecond or two. On the square cloth a 300 mm pair routed along the warp, with its conductors over bundle and gap, accumulates 95 ps — more than two unit intervals. Rotated ten degrees it is bounded at 0.305 ps whatever its length; at the best angle, 0.128 ps. Both are comfortably inside the budget, and the angle within the rotation is worth little to this link.

The skew is not only a timing error. The millimetre that becomes common mode found that a millimetre and a half of skew, ten picoseconds, puts a tenth of the signal into common mode by three gigahertz, and the conversion rises with frequency as $\omega\Delta\tau$. The 95 ps the unrotated pair builds is nearly ten times that, so along the warp the board is an emitter before it is a timing problem; at a bounded 0.3 ps the conversion falls by more than two orders of magnitude with it.

At ten times the rate, with a unit interval of 4 ps, the budget is a tenth of a picosecond, and now the angle matters. Ten degrees allows three times the budget; the best angle a little more than the budget; and a pair routed on the exact diagonal, growing at 2.2 fs a millimetre, has spent the whole budget in 45 mm. The faster the link, the more of the cure is in the choice of angle rather than in the decision to rotate.

The pair’s own pitch #

The amplitudes in the bound depend on how far apart the two conductors are across each weave, so the pair’s own pitch enters as much as the cloth’s.

At the best angle the worst skew is 61.5 fs for a 0.1 mm pair, 111.1 for 0.2, 128.3 for 0.25, 139.1 for 0.3, 138.8 for 0.4 — where the best angle has moved to 30° — and 4.2 fs for a 0.5 mm pair, whose conductors are a whole weave pitch apart across the warp when routed along it. At 10° the same pairs give 174.7, 286.6, 305.3, 296.2, 201.9 and 40.6 fs. The best angle beats ten degrees by a factor of between two and three across every ordinary pitch.

The 0.5 mm case is the earlier essay’s “pitch matching” seen in two dimensions. A pair whose pitch equals the warp’s sees the same warp under both conductors when routed along it, and the weft is then the only contribution, small because the pair’s projection across the weft is small at a small angle. It is a cure that works only when the pair’s pitch and the cloth’s are both known and fixed, which in production they rarely are.

The diagonal of a square weave #

The first-order model says forty-five degrees on a square weave is the best route. The full model says it is the one route to avoid.

The permittivity’s square root, expanded beyond first order, has a term proportional to the product of the two weaves. That product oscillates at the sum and the difference of the rates at which the route crosses the warp and the weft. On a square weave at exactly 45° those two rates are equal, the difference is zero, and part of the product does not oscillate at all: it is a steady differential, small — second order in the swing — but steady, and a steady differential integrates to a skew that grows with length, exactly as the warp alone did along the weave.

With the full square root, the worst-registered pair at exactly 45° accumulates skew at 2.2 fs a millimetre, reaching 631 fs in 200 mm — five times the first-order bound and still growing. At 44° and 46° it stays within 128 fs over the whole route, the first-order bound, because a degree’s difference in the two crossing rates is enough to make the product oscillate again.

So the rule of thumb’s angle is, on a square weave, the single angle at which the cure fails. The failure is slow — 2.2 fs a millimetre against the 318 the same pair builds along the weave, under one per cent of it — but it does not stop, and a long route on a square weave’s diagonal returns to the problem the rotation was meant to remove. The model’s product term is a property of the square root of a sum of cosines, and a real cloth’s permittivity pattern, with bundles crossing at knuckles, has products of its own; the conclusion that the exact diagonal is special is the robust part, because any interaction between two weaves crossed at the same rate fails to average there. The cure is the same as before: a degree or two off.

What a layout rule should say #

The figures reduce to a rule with three clauses. Route differential pairs at an angle to the glass, not along it, because along it the skew grows with length. Choose the angle from the cloth: 45° for a square weave, further towards the finer or flatter yarns for a cloth whose two weaves differ, and the bound above says how much further. And never exactly on a square weave’s diagonal, because there the two weaves’ interaction does not average.

The size of what is being bought is worth keeping in view. At the best angle the worst skew is about 120 to 130 femtoseconds for an ordinary pair, against tens of picoseconds along the weave: the rotation is worth two to three orders of magnitude, the choice of angle within it a further factor of two or three. For a serial link whose skew budget is a few picoseconds the rotation is essential and the choice of angle a refinement; for one whose budget is a few hundred femtoseconds, the refinement is the margin.

How the numbers were obtained #

The cloth’s permittivity is the sum of two cosines about 4.4, one across the panel with the warp’s pitch and one along it with the weft’s, each of the stated swing. Each conductor’s delay per metre is the square root of the permittivity under it over the speed of light — to first order for the bounds and the hero’s integrations, in full for the diagonal figure — and the two conductors are placed exactly, a quarter of a millimetre apart perpendicular to the route. The skew is the running difference of the two delays, integrated along the route from each of a grid of starting positions, and the worst over the route and the grid is compared with the bound. The best angle is searched on the bound in steps of a twentieth of a degree.

What it leaves out #

The cloth’s real structure. Bundles are not cosines: they are flattened ellipses with resin between, crossing over and under each other at knuckles, and a pair can sit over a knuckle for its whole length on some angles. The two-cosine model captures the two periodicities and their interaction to second order; a real cloth has higher harmonics, each with its own resonant angle like the diagonal.

Layers. A pair has glass above and below it, often of different styles, and the two weaves need not be aligned. Each layer contributes its own bound, and the best angle for one need not be best for the other.

Still open: a route that zig-zags, the receiver’s view, and the pair over two cloths #

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 conductors’ positions on the glass and reverses the skew’s growth. How often it must jog to hold the skew under a budget, what each jog’s own discontinuity costs, and whether a zig-zag along the warp can beat a straight route at the best angle are questions about the same integral with a different path.

What the receiver sees in time. A skew of a hundred femtoseconds against edges of tens of picoseconds is a small conversion; one of tens of picoseconds against the same edges smears the differential edge itself. How much eye opening a weave-induced skew costs, compared with a routed mismatch of the same size, is what a link budget actually needs. The common mode that has three slopes separated a pair’s imbalance into size, timing and edge rate; a weave contributes to the first and third as well as the second, because the permittivity under each conductor also sets its impedance and its loss, and the three terms from one cloth have signs fixed by the same registration.

Glass above and below. With two cloths of different styles around the pair, the bound has four terms and the best angle is a compromise between two compromises. Whether there is a single angle that serves both, or whether the stack-up should pair styles so that one angle does, is the question a board designer choosing the glass could answer before routing anything.