Circuit Response
every model has an edge, and the figure carries it
Concept

Skew — where it appears

The difference in arrival time between two signals that were launched together, such as the two halves of a differential pair. Skew turns a clean difference into a common-mode pulse at each edge, whose size is the edge's slope times the skew.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

What 1.5 mm of length mismatch does to a differential pair. computed by solving, not by drawing. Two lines of the same impedance and different lengths, driven differentially. The solid rising curve is what arrives as common mode; the dashed one beside it is sin(ωΔτ/2), which is what a lossless pair gives and is the same curve until the null. The flat curve at the top is the differential signal, and it is the point: at 3.04 GHz a tenth of the launched amplitude is common mode and the differential has lost 5011 parts per million of itself. The conversion is first order in the skew and the loss is second order, so the error is not missing from the signal — which is why a pair can pass its own eye and fail an emissions test. At 95.3 GHz the closed form has a null and the real pair does not: the longer conductor is also the lossier one, and an amplitude imbalance has no null in it.

The millimetre that becomes common mode

A pair carries two modes rather than two signals, and a length mismatch between its halves converts one into the other. A millimetre and a half of skew is ten picoseconds, a tenth of the signal is common mode by three gigahertz, and the differential signal has lost five thousand parts per million of itself getting there — so the error is not missing from the signal, which is why a pair can pass its own eye and fail an emissions test. The product in the answer is ωΔτ, which is what the instruments field's rejection corner is one over.

lines · Differential line Three imbalances, three slopes: a 2% amplitude imbalance is 1% common mode at every frequency, 1 mm of skew overtakes it above 455 MHz, and a 5 ps rise-time mismatch climbs at forty decibels a decade. computed by solving, not by drawing, from the transforms of the two conductors' edges and checked against the sampled edges transformed by FFT to two per cent: the common mode at the far end of a pair, as a fraction of the differential, for three separate imbalances between its halves. An amplitude imbalance of 2% gives 1.00% at every frequency. A skew of 1 mm on FR-4, 7.00 ps, gives sin(ωΔτ/2), rising at twenty decibels a decade: 2.20% at 1 GHz. Edges of 50 and 55 ps, perfectly timed and equal in size, give a common mode rising at forty decibels a decade: 0.043% at 1 GHz and 5.38% at 10 GHz. The skew passes the amplitude imbalance near 455 MHz, which is a/(2πΔτ).

The common mode that has three slopes

A differential pair turns into common mode wherever its two halves differ, and they can differ in three ways — in size, in timing and in edge rate — each with its own slope. A 2 per cent amplitude imbalance is 1 per cent common mode at every frequency. A millimetre of skew rises at twenty decibels a decade and overtakes it above 455 MHz, a/(2πΔτ). Edges of 50 and 55 picoseconds, perfectly timed, rise at forty: 0.043 per cent at a gigahertz and 5.4 at ten. In time they are a step that stays, a pulse with an area, and a doublet with none — each slope is a moment that vanishes. And an amplitude imbalance and an edge-rate mismatch of the right signs cancel at one frequency, down 74 dB, which half a picosecond of skew fills back to −25.

lines · Differential line Routed along the glass weave, a length-matched pair with one conductor over a bundle and one over a gap builds 63.6 ps of skew in 20 cm — the skew nobody routed. computed by solving, not by drawing by integrating each conductor's delay along its route: a length-matched pair of 0.25 mm pitch over a glass weave of 0.5 mm pitch whose effective permittivity swings between 4.2 and 4.6 across it, positioned so that one conductor runs over a bundle and the other over the gap. Routed along the weave (dashed) the skew grows in proportion to the length, 0.318 ps a millimetre, to 63.62 ps in 200 mm. The slider rotates the route.

The skew in the glass

Match a differential pair's two lengths exactly and it can still have tens of picoseconds of skew, because a circuit board is woven glass in resin and the two conductors need not sit over the same one. Over a weave of half a millimetre whose effective permittivity swings from 4.2 to 4.6, a pair of quarter-millimetre pitch with one conductor over a bundle and the other over a gap builds 0.318 ps of skew every millimetre — 63.6 ps in 20 cm, the equivalent of 13.6 mm of routed mismatch by 30 cm, and 68 per cent common mode at 2.5 GHz. Where the pair falls across the weave is a lottery from nothing to that. Rotate the route by ten degrees and the skew never exceeds 0.146 ps, however long the pair: each conductor crosses bundle and gap in turn, and the difference averages out.

lines · Differential line With the weft counted as well as the warp, the skew a pair can reach is least at 45.0° to the warp — 128 fs — and turning further off the warp makes it worse again. computed by solving, not by drawing, to first order in the permittivity's swing, the summed bound held against an integration over 64 registrations of the pair to the cloth at six angles. A 0.25 mm pair over a cloth whose warp has a 0.5 mm pitch and whose weft has a 0.5 mm pitch, each swinging the permittivity by ±0.2 about 4.4. The warp's contribution falls as the route turns off it, as the earlier figures found; the weft's rises, because the pair's projection across the weft grows as d·sin θ while its yarns are crossed less often. The worst skew is least at 45.00°, 128.3 fs. At 10°, where the warp alone allows 291.4 fs over every registration — twice the earlier figure's 146 fs, which was for a route starting over bundle and gap — the weft adds 13.8 fs. Integrated: 5° 587.6 fs, 15° 215.1 fs, 30° 139.3 fs, 40° 126.3 fs, 60° 139.2 fs, 80° 304.9 fs. The slider is the weft's pitch.

The angle two weaves allow

Turning a differential pair a few degrees off a board's glass weave stops its skew growing with length — but a cloth has two weaves, and turning off the warp turns onto the weft. Counting both, the worst skew over every registration of the artwork to the cloth is a sum of two bounds, one falling as the route leaves the warp and one rising as it meets the weft, and it is least at an angle the cloth decides: 45° and 128 fs for two identical half-millimetre weaves and a quarter-millimetre pair, against 305 at 10°; 52° for a finer weft; 67° for a flatter one. Doing the count properly also finds the earlier bound a factor of two low, and finds the obvious answer a trap: on a square weave's exact diagonal the two weaves' product stops averaging and the skew grows again, 2.2 fs a millimetre, while a degree either side it stays at 128.

lines · Differential line With a 300 ps skew and 5 mV of noise a core with 100 ns of headroom saturates after 502 periods on average — the 500 the skew alone gives — and one drive in a hundred by 184. Seeded: 2,000 drives of an ideal core by the event map at β = 0.5, 5 mV of threshold noise and a comparator skew of 300 ps, each run until the summed high-minus-low time leaves ±100 ns. A histogram of the counts on logarithmic bins, against the closed-form first-passage distribution, between two barriers, of a walk with drift 0.200 ns a period and spread 1.89 ns per root period. Mean 501.9 against 499.9; quantiles 1% 184, 10% 275, 50% 463; the empirical and closed-form cumulative distributions differ by at most 0.014.

The count a core survives

A relaxation oscillator with a 300 ps comparator skew and 5 mV of threshold noise saturates an ideal core with 100 ns of volt-second headroom after 502 periods on average — the 500 the skew alone gives — and one drive in a hundred by 184. The mean does not see the noise at all until the noise is large enough to saturate the core the other way first; the early tail sees nothing else. Every skew and noise level falls on one curve in one ratio, the skew's count over the count at which its ramp overtakes the walk. Read as a burst, a thousand periods with one failure in a thousand allows 150 ps of skew with no noise, 93.5 ps at a millivolt, and none at all above 2.41 mV.

applied · Hysteresis A pair with one conductor over glass-rich laminate (4.6, tan δ 0.015) and one over resin-rich (4.2, tan δ 0.025): the skew per millimetre against frequency. computed by solving, not by drawing: each conductor as its own causal line — permittivity dispersing with its own loss tangent, copper with its internal reactance — and the skew as the difference of their delays. With both permittivities held at their 1 GHz values the skew would be 0.318 ps/mm at every frequency (level line). Dispersing, it is 0.272 ps/mm at 100 MHz, 0.319 at 1 GHz and 0.366 at 10 GHz: the resin, lossier, loses permittivity faster with frequency, and the gap between the two halves widens.

The skew that moves with frequency

A pair with one half over glass-rich laminate and one over resin-rich builds 0.318 ps of skew a millimetre when each permittivity is held at its 1 GHz value. But the two regions lose differently — tan δ 0.015 against 0.025 — so their permittivities fall at different rates, and the skew is a curve: 0.272 ps/mm at 100 MHz, 0.319 at 1 GHz, 0.366 at 10 GHz. A deskew set at one frequency leaves ±2.35 ps in 50 mm at the other two. The resin half also loses 3.7 dB more by 10 GHz over 200 mm, so its edge arrives 63.7 ps early and rises in 257 ps against 202; the halves differ in timing and in edge rate at once, and leave a common-mode pulse of 151 mV for each volt.

lines · Dispersion Both ends of the quiet line, 100 mm of microstrip-like coupling, edge 200 ps. computed by solving, not by drawing. The near end (pale) is the saturated plateau, 16.28 mV per volt. The far end (dark) is a single pulse arriving with the edge after one delay, 0.676 ns, as wide as the edge and 25.16 mV per volt deep — negative, because the inductive coupling exceeds the capacitive. Its depth is the coupling difference, the delay and the edge's slope multiplied: 24.96 predicted.

The far end that is a derivative

Where the two couplings are unequal the far end does not cancel, and in time it is the derivative of the edge: a single pulse as wide as the rise time, arriving after one delay, its height the coupling difference times the delay times the edge's slope. On 100 mm of microstrip-like coupling a 200 ps edge gives −25.16 mV per volt against 24.96 predicted, and the height falls as one over the rise time with a fitted slope of −1.000 while the saturated near end sits still. It grows with length without a knee until the two modes separate by a whole edge — 249 mm for a 50 ps edge — and then stops at half the driven line's voltage.

lines · Crosstalk

Named alongside it

The objects these essays reach for when they reach for this one.

Differential lineMode conversionPermittivityHomogeneous dielectricRise timeCrosstalkEmissionsPropagation velocityCharacteristic impedanceClosed formCommon-mode rejectionDesign tradeoff

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