The skew that moves with frequency

The skew in the glass put the two halves of a differential pair over different parts of a board’s weave — one over a glass bundle, one over the resin between bundles — and found 0.318 picoseconds of skew building every millimetre, 63.6 in twenty centimetres, with the lengths matched exactly. It took each region’s permittivity as a constant: 4.6 over the glass and 4.2 over the resin. That is the model the permittivity a loss forbids showed cannot be right for a material that dissipates, and the conductor made causal too has since made both halves of a single track’s model causal.

This essay puts the two together. Each half of the pair becomes its own causal line, its permittivity dispersing at the rate its own loss fixes and its copper carrying its internal reactance, and the skew is the difference of the two delays. The two regions do not lose equally — resin is the lossier of the two, a loss tangent of 0.025 against glass’s 0.015 — and that one difference is the whole of what this essay measures.

A skew with a shape #

With both permittivities held at their gigahertz values the skew is 0.318 picoseconds a millimetre at every frequency, the earlier essay’s number. With each half causal it is 0.272 at a hundred megahertz, 0.319 at a gigahertz and 0.366 at ten — the same at the frequency where the two models were made to agree, and fifteen per cent either side of it two decades apart. The skew is not one number, in exactly the way the delay that is not one number found a single line’s delay is not.

The reason is a rate. A dielectric with a flat loss tangent has a permittivity that falls by a fixed amount every decade of frequency, and the amount is $\varepsilon \tan\delta$ times $2 \ln 10 / \pi$, about 1.47 times their product: for the FR-4 of the earlier essay, 4.4 and 0.02, that is the 0.129 a decade it measured. For the glass-rich region it is about 0.10 a decade and for the resin-rich about 0.15. The two permittivities are 0.4 apart at a gigahertz by construction; below it the lossier resin climbs faster and the gap narrows, above it the resin falls faster and the gap widens. A narrower gap is less skew and a wider one more.

That makes the direction of the bend a consequence of which region is lossier, not of anything about the pair. If the glass were the lossier material the skew would fall with frequency instead. For ordinary laminates the resin carries most of the loss, so the skew of a pair caught across the weave grows toward the frequencies where a fast link’s energy is.

What a deskew corrects #

A pair with a known skew can be deskewed, by adding length to the faster half — usually as a serpentine near the receiver — until the two arrive together. The faster half here is the one over resin, with its lower permittivity. A deskew is a length, and a length is a delay that is the same at every frequency, so it can cancel the skew at one frequency only.

On fifty millimetres of pair the skew is 13.6 picoseconds at a hundred megahertz, 16.0 at a gigahertz and 18.3 at ten. A deskew set at a gigahertz leaves 2.35 picoseconds uncorrected at ten gigahertz and the same amount, of the opposite sign, at a hundred megahertz. On two hundred millimetres those residues are 9.4 picoseconds. For a twenty-five-gigabit link, whose unit interval is forty picoseconds, a residue of nine is not small, and it is a residue that no amount of length matching removes, because it is a property of the materials rather than of the route.

A deskew also assumes that the weave is known, which in production it is not: the angle two weaves allow found that where a pair falls across the cloth is a matter of how the artwork registers to it, and a deskew that corrects one board makes the next one worse. That essay’s remedy, routing at an angle to the weave, is the one this essay returns to at the end.

Two ways to convert #

The millimetre that becomes common mode found that a skew turns differential signal into common mode, in proportion to the product of frequency and skew. The common mode that has three slopes separated three ways the halves can differ — in amplitude, in timing and in edge rate — each with its own slope against frequency. The pair over glass and resin differs in timing and in amplitude at once, because the region that is faster is also the region that loses more.

Along fifty millimetres the conversion is 2.1 millivolts per volt at a hundred megahertz, 24 at a gigahertz and 198 at ten. From a hundred megahertz to a gigahertz it grows by 11.4 times, more than the tenfold a fixed skew would give, because the skew itself has grown by seventeen per cent over the decade. From one gigahertz to ten it grows by 8.3, less than tenfold, because by then the two halves are far enough out of phase that their difference is no longer proportional to it, and both have lost amplitude on the way.

The loss difference’s own contribution is nine to ten per cent of the delay difference’s above a gigahertz. It is not the larger mechanism, but it is not one a design can neglect either, and it is invisible to a model in which the two regions share a loss tangent — which most field solvers’ default material libraries assume, since they give a laminate one loss tangent and model the weave, if at all, as a permittivity variation alone.

Two halves, two losses #

The amplitude half of the difference is easiest to see in the loss each half takes over a long route.

Along two hundred millimetres the half over glass loses 0.40 decibels at a hundred megahertz and the half over resin 0.44; at a gigahertz, 1.71 and 2.07; at ten, 9.6 and 13.3 — a gap of 3.7 decibels. At low frequency the copper’s loss dominates and is the same for both halves, so the gap is small; the dielectric’s loss is proportional to frequency and to the loss tangent, and it takes over above a gigahertz.

A 3.7-decibel imbalance is an amplitude ratio of about 0.65. The common mode that has three slopes found an amplitude imbalance making common mode of half its size at every frequency, which would be seventeen per cent here — except that this imbalance is not at every frequency. It is close to zero at a hundred megahertz and large at ten gigahertz, so it has a slope of its own against frequency, and in a signal’s spectrum it acts on the fast edges rather than on the slow ones. An equaliser at the receiver sees the two halves as one differential channel and corrects their average loss; the difference between them it cannot reach, because it has already become common mode before it arrives.

Conversion against length #

A fixed skew converts in proportion to length. The dispersing pair does too, at low frequencies, and stops at high ones.

At a hundred megahertz the conversion grows from 0.43 to 4.19 millivolts per volt from ten millimetres to a hundred — proportional, as a skew is. At a gigahertz it is very nearly proportional, 4.97 to 45. At ten gigahertz it grows from 54 to 242, only 4.5 times over a tenfold length, and between eighty millimetres and a hundred it barely moves. By then the skew is past thirty picoseconds and the phase difference between the halves at ten gigahertz is past two radians: further skew turns the difference round rather than enlarging it. And both halves are losing amplitude together, so the difference between them is a difference of smaller signals.

That flattening is not a reprieve. It says that on a long route the common mode at ten gigahertz is already as large as it will be, about a quarter of the differential swing, and that the effort to reduce it has to go into the cause rather than into shortening the route by a few millimetres.

The edges, side by side #

In time the two differences — in arrival and in loss — show up as two edges that differ in both where they are and how steep they are.

After two hundred millimetres the half over resin reaches half height 63.7 picoseconds before the half over glass — the earlier essay’s flat estimate, 63.6, to a tenth of a picosecond, because the half-height crossing is set by the frequencies near a gigahertz where the two models were made to agree. What the flat model could not show is the shape: the resin half rises in 257 picoseconds and the glass half in 202. The faster half arrives with the slower edge.

That inversion matters downstream of the pair as well as within it. The edge the loss delivers found that a discontinuity answers the steep middle of an arriving edge rather than its ten-to-ninety time, and that a lossy line makes the middle steep while slowing both ends. Here the two halves reach a via or a connector with middles of different steepness, so a discontinuity that is balanced for the differential mode reflects them unequally, and the reflection carries a common-mode part that neither the skew nor the pair’s own conversion accounts for. The two causal halves therefore matter at every point along the route where the pair meets something, not only at its end, and a channel model that treats the pair as one line with one loss will place all of that conversion at the receiver, where it is easiest to see and too late to remove.

The two halves therefore differ in timing and in edge rate together, the second and third of the three slopes, and the common mode they leave is a pulse peaking at 151 millivolts for each volt of swing. The shape of the pulse is not the symmetric bump a pure delay difference makes: it is lopsided toward the end of the transition, where the resin’s slower tail is still rising after the glass half has finished. A common-mode choke at the receiver, the inductor one mode cannot see, would be asked to absorb a pulse whose energy sits at higher frequencies than the skew alone predicts — which is where that essay found the choke turning into a capacitor and stopping working.

The angle that removes both #

The remedy the earlier essays found for weave skew was to route the pair at an angle, so that each half crosses glass and resin alike and the difference averages away. With each half causal, it is worth asking whether the averaging removes the dispersion too, or only the skew at the frequency where the two models agree.

It removes both, in proportion. With neither half crossing any of the other region the skew is 0.271 picoseconds a millimetre at a hundred megahertz and 0.366 at ten gigahertz; at twenty per cent mixture, 0.163 and 0.219; at forty, 0.054 and 0.073; at an equal mixture, zero at both. The ratio of the two, 1.35, holds all the way down. A route that averages the regions averages their loss tangents as well as their permittivities, so the two halves’ rates of dispersion converge at the same time as their delays do, and the frequency dependence of the skew is never left behind as a residue when the skew itself is removed.

That makes the angle the remedy and the deskew the stopgap. A deskew cancels the skew at one frequency and leaves the dispersion intact; the angle removes the cause, and the dispersion goes with it. The angle essay’s residual skew, a worst case of 128 femtoseconds set by the cloth’s two weaves and no longer growing with length, carries its own dispersion in the same ratio: about 110 femtoseconds at a hundred megahertz and 147 at ten gigahertz, still far smaller than the skew a pair caught along one bundle builds in a single millimetre.

A twenty-five-gigabit route, worked #

Put the numbers on a route a designer would meet. A twenty-five-gigabit lane has a unit interval of forty picoseconds and most of its energy below about twelve gigahertz. Route it 150 millimetres with one half along a glass bundle and the other along a resin gap, the worst registration the cloth allows.

At a gigahertz the skew is 0.319 picoseconds a millimetre, 47.9 picoseconds in the route — more than a whole unit interval, which no receiver tolerates. A deskew set at a gigahertz removes that and leaves 0.047 picoseconds a millimetre at ten gigahertz, 7.1 picoseconds in the route: eighteen per cent of a bit, in the band where the lane’s edges live, and a common-mode residue there that the deskew was supposed to have removed. Set at ten gigahertz instead, the deskew leaves the same 7.1 picoseconds at a gigahertz, where the lane’s pattern-dependent low-frequency content sits.

The skew that is left is a delay that varies with frequency between the halves, and a delay that varies with frequency is what an all-pass network corrects: flat delay bought with more delay measured an all-pass section changing a filter’s delay without touching its magnitude. A frequency-dependent deskew on one half of a pair is that idea applied to a line, and it is not something a serpentine can be. The angle needs no such network, because it never lets the difference form.

How the pair was solved #

Each half is the single-track model of the essay before this one: fifty ohms, its own permittivity and loss tangent at a gigahertz, the permittivity dispersing by the wide-band relaxation form, the copper’s resistance rising as $\sqrt f$ with its internal reactance beside it. The two halves are treated as uncoupled lines, which is the approximation the earlier pair essays made too: a tightly coupled pair has odd and even modes that see a mixture of both regions, and the skew between the halves is then smaller than the uncoupled estimate by a factor set by the coupling. Conversion is half the difference of the two transfers, divided once more by two to put it per volt of differential drive, and its delay and loss parts are the same difference with one or the other of the two halves’ differences removed. Edges are the transfers applied to a 30-picosecond edge and inverted. The angled route is modelled as each half seeing a weighted average of the two regions’ permittivity and loss.

Still open: coupling, a measured coupon, and the weave’s own frequency #

Coupling. The halves here are uncoupled. A pair close enough to share its field sees the two regions through its odd and even modes, each of which averages them by a different weight. Solving the coupled pair with each region causal would say how much the coupling reduces the skew and whether it reduces the skew’s frequency dependence by the same factor or by a different one.

A coupon. The prediction that a deskew set at a gigahertz leaves ±2.35 picoseconds in fifty millimetres at a hundred megahertz and ten gigahertz is a measurement waiting for a board: two lines laid along a bundle and a gap, measured with a vector network analyser, would give the skew as a curve directly. Its slope would give the difference of the two regions’ loss tangents, which is the number nobody quotes.

The weave’s own period. At frequencies where the weave’s pitch is a sizeable part of a wavelength, half a millimetre against about seven millimetres at twenty gigahertz in FR-4, a conductor crossing the weave no longer sees an average: it sees a periodic structure, and periodic structures have bands where they reflect. That is a different model from the one this essay used, and it sets the upper frequency at which any of these averages, angled or not, can be trusted.