The edge the loss delivers

The via that is a piece of line found that a via balanced on its $\sqrt{L/C}$ line reflects only through its length: a doublet with no area, whose largest excursion falls as the square of the edge’s rise time where a lone capacitance’s falls as the first power. That makes rise time the balanced via’s best friend. Double the rise time and its reflection falls fourfold; a slow enough edge makes it vanish.

Real edges are made slow by the line they travel on. The delay that is not one number measured what a lossy trace does to an edge — its series resistance rising as the square root of frequency, its dielectric absorbing a fixed fraction of each cycle — and found the rise time growing much faster than the delay. So a balanced via sitting at the end of a length of lossy line should be hidden by the loss in front of it, and by more than a capacitive via is, since its reflection falls faster with the rise time.

The earlier essay asked whether that is so. It is, in the exponent. It is not, by a constant that the rise time does not carry.

The stakes are practical. Every via on a real board sits behind some length of trace, and the edges that reach it have come through that trace’s loss. A designer who balanced the via, computed its doublet for the rise time a simulation reports at the via, and found it small enough has made one assumption without stating it: that the arriving edge is shaped like the edges the via’s behaviour was characterised on. This page tests that assumption on one line and two vias, and finds it wrong by a factor that matters for exactly the vias that were designed most carefully.

The via, and its square law #

The clean edges in that essay, and on this page, are raised cosines: a half period of a cosine running from zero to one, a shape that is smooth, symmetric about its midpoint, and completely described by one number, its duration. Its 10–90 rise time is 0.59 of that duration. Double the duration and every derivative of the edge falls by the corresponding power of two, which is why the doublet — a reflection that reads the edge’s second derivative, its curvature — falls as the square.

That argument needs one thing to be true: that making the edge slower only stretches it. A raised cosine stretched is a raised cosine. An edge made slower by a lossy line is not.

What a lossy line delivers #

The line here is 50 Ω-class trace on a glass-epoxy laminate: a direct-current resistance of 2 Ω a metre rising as the square root of frequency above a megahertz, and a loss tangent of 0.02. A 20 ps raised-cosine edge — 11.8 ps from 10 to 90 per cent — is sent into it, and the edge that reaches the far end is computed exactly, by multiplying its spectrum by the line’s propagation factor at every frequency.

After 20 cm the edge’s 10–90 rise time is 194 ps. Drawn beside a raised cosine with exactly that rise time, the two are not the same shape at all. The lossy edge starts rising early and slowly, from a long foot; crosses its middle 3.07 times more steeply than the raised cosine; and then creeps up towards its final value along a long tail. Its 10–90 time is long because of the foot and the tail. Its middle is fast.

That shape is what loss does. Skin-effect loss attenuates high frequencies as the square root of frequency and dielectric loss as the first power, so the line removes the edge’s highest frequencies gradually rather than cutting them off. What arrives keeps a steep centre made of the frequencies that survived, and acquires slow ends from the ones that were attenuated but not removed. The 10–90 rise time measures the ends. A reflection that reads the edge’s slope or curvature sees the centre.

The two losses shape the edge differently, and both push in the same direction. The conductor’s loss, rising as the square root of frequency, makes the line behave at low frequency like a diffusion — the delay that is not one number found it doing exactly that below a megahertz — and a diffusion’s response to a step creeps towards its final value as one over the square root of time: the long tail. The dielectric’s loss, rising in proportion to frequency, attenuates each frequency by the same number of decibels a decade and so rounds the edge more evenly. Neither removes the frequencies that make the middle of the edge steep; both spread energy into the foot and the tail. The longer the line, the more of the edge’s 10–90 time is foot and tail.

The reflection that comes back #

The via’s reflection, seen from the line’s input after the return journey through the same loss, has a largest excursion of 9.1 mV. The raised cosine with the same 194 ps rise time draws 2.29 mV from the same via on a lossless line. By the measure a signal-integrity budget uses — the rise time at the discontinuity — the two edges are identical, and the lossy one excites the via 3.97 times as much.

The two waveforms look different as well as being different in size. The clean edge’s doublet is broad, spread across the edge’s whole duration, with its sharpest features at the edge’s two ends, where a raised cosine’s curvature jumps from zero. The lossy edge’s doublet is narrow and tall and sits at the edge’s centre, where its curvature is concentrated. The via is reading the same thing in both — the edge’s curvature — and the lossy edge has more of it, in less time.

The capacitive via, which reads the edge’s slope rather than its curvature, is affected too but less: 68 mV behind the loss against 53.1 mV from the clean edge, a factor of 1.28. The steeper middle raises the slope by about the factor the edge figure measured, but the capacitive via’s reflection integrates the slope over its own time constant and so feels the average more than the peak.

Across lengths of line #

A single length could be an accident of this line’s loss at this frequency. So the length is swept, and each via’s reflection is drawn against the rise time of the edge that reaches it — the axis on which the square law was stated.

Behind the loss, from 10 to 50 cm, the balanced via’s reflection falls as the −1.97 power of the arriving rise time: the square law survives. Loss does hide a balanced via faster than it hides a capacitive one, whose reflection falls only as the −0.82 power. What loss does not do is hide it as fast as the rise time says. The lossy curve runs parallel to the clean one and three to four times above it, the gap staying almost constant from 78 to 735 ps of arriving rise time.

The exception is the shortest line. After 5 cm the edge has slowed from 12 to 35 ps but has not yet taken on the lossy shape, and the balanced via’s reflection is what a clean edge of that rise time would give, within three per cent. The factor of three appears between five and ten centimetres, which is where the line’s loss stops merely stretching the edge and starts reshaping it, and then it stays.

At five centimetres, in other words, the edge has been slowed by a factor of three and has kept its shape: its rise time has grown mostly because its highest frequencies have been attenuated evenly, and an evenly attenuated raised cosine is still close to a raised cosine. The long foot and tail need more line to grow. By fifty centimetres they dominate the edge.

After 50 cm the lossy edge’s rise time is 735 ps and its steepest slope is 4.00 times a raised cosine’s of the same rise time — steeper, relative to its own rise time, than after 20 cm. The foot and the tail grow faster with length than the middle does, so the 10–90 time increasingly measures the ends and increasingly misrepresents the centre. The ratio of reflections at 50 cm is 3.04, a little lower than the 3.97 at 20 cm, though the ratio of slopes has grown: the via’s reflection follows the edge’s curvature at its centre, and the slope ratio and the curvature ratio of two different shapes need not move together. The measurement says the factor sits between three and four across the whole range; it does not say it is any one simple property of the edge.

The capacitive via, in the same terms #

The capacitive via’s exponent is worth a sentence of its own. On clean edges its reflection falls as the −0.97 power of the rise time, very nearly the first power a lone capacitance’s area-driven reflection should have. Behind the loss it falls as the −0.82 power. A capacitive via’s reflection is set by the edge’s slope smoothed over the via’s own time constant, and the lossy edge’s slope is concentrated in its middle, which shrinks more slowly than its 10–90 time grows. So the capacitive via, too, is hidden less by the loss than the rise time says, and by a margin that grows with length — 1.07 at 10 cm, 1.54 at 50.

None of this reverses the ordering. Behind 20 cm the balanced via reflects 9.1 mV and the capacitive one 68 mV, still more than seven times apart. Balancing a via is worth doing behind a lossy line as much as anywhere; what changes is how small its remaining reflection is, and whether a budget written in rise times has got that number right.

What a budget built on rise time gets wrong #

Signal-integrity budgets are often written in rise times: the rise time at a discontinuity, compared with the discontinuity’s own delay, decides whether it is “electrically small” and how large its reflection will be. For a discontinuity that reads the edge’s slope, like a capacitive via, that is nearly right — a factor of 1.1 to 1.5 on this line. For one that reads the edge’s curvature, like a balanced via or any structure designed to cancel its first-order reflection, it is wrong by a factor of three to four, in the unsafe direction. The better the discontinuity has been balanced, the more of its remaining reflection depends on the edge’s shape rather than its rise time, and the more a lossy line’s edge surprises it.

The fix in the budget is not to use a different rise time but to use a different number for the edge. What the balanced via responds to is the edge’s peak curvature, and what the capacitive via responds to is closer to its peak slope. Both are properties of the arriving waveform that a 10–90 time does not carry; both can be read directly off the computed edge. A clean edge and a lossy edge with the same 10–90 time differ in peak slope by a factor of three here, and a factor of that order in peak curvature is what the via’s reflection shows.

The pad the stub comes out of found a via repaired by trimming the pad beside its stub, a repair that works to four per cent at every clean edge. Behind a lossy line the edge arriving at that via is steeper in the middle than its rise time says, and the four per cent is four per cent of a reflection three times larger than the rise time predicts. The permittivity a loss forbids found the dielectric’s loss inseparable from its permittivity’s fall with frequency; this page finds the loss inseparable from the edge’s shape. Loss is not one number, and the rise time is not one number either.

Every cancellation reads a higher derivative #

The balanced via is one instance of a general pattern. Any structure designed to cancel its own first-order reflection — a via balanced on its line, a connector compensated with a little extra capacitance, a bend whose corner is chamfered to match — is left with whatever reflection comes from the next order, and the next order reads a higher derivative of the edge. A lone capacitance reads the slope; a balanced via reads the curvature; a structure balanced to second order as well would read the derivative after that.

Each derivative is more sensitive to the concentrated middle of a lossy edge than the one before, because differentiating a steep centre and a long tail makes the centre dominate more at each step. So the better a discontinuity is compensated, the larger the factor by which a lossy edge excites what is left, relative to a clean edge of the same rise time. That is an argument from the shape of the edges measured here rather than a measurement of a second-order structure, and the second-order case is not solved on this page. What is measured is the first two members of the sequence — a factor of 1.1 to 1.5 for slope and 3 to 4 for curvature — and they rise in the direction the argument says.

How the numbers were obtained #

The lossy line is a transmission line with a series resistance of 2 Ω a metre at direct current rising as $\sqrt{1 + f/1\,\text{MHz}}$, a shunt conductance of ωC·0.02, and the inductance and capacitance per metre of a 44.72 Ω line in a relative permittivity of 4.4. A raised-cosine pulse with a 20 ps edge is transformed, multiplied at each frequency by the line’s propagation factor to give the arriving edge, and by the via’s reflection coefficient and the propagation factor twice to give the reflection at the line’s input; both are transformed back on one record of 32,768 samples. The via’s reflection coefficient is the same cascade of shunt capacitances and series inductance the earlier essays used, referred to the line’s nominal impedance, so that the line’s own small mismatch against that reference is not counted as the via’s. The clean route gives the raised cosine exactly the arriving edge’s measured 10–90 time and computes the via’s reflection on a lossless line. The exponents are fitted over lengths from 10 to 50 cm.

What it leaves out #

The line’s own reflection. A lossy line’s characteristic impedance is complex and varies with frequency, so a real line reflects a little at its own ends; it is left out here so that the via is the only thing being measured.

The edge’s other half. Only the rising edge is measured. A falling edge through the same line has the same shape inverted, and a data pattern is a sequence of both, whose reflections from a via overlap at the rates a real channel runs.

And the receiver. The reflection is measured at the line’s input; what matters at a receiver is what the via does to the transmitted edge, which a doublet distorts with the same dependence on the edge’s shape.

Still open: a number for the edge, the transmitted distortion, and a pair of vias behind the loss #

A rise time that means curvature. If the balanced via’s reflection is proportional to the arriving edge’s peak curvature, then a curvature-based rise time — the duration of a raised cosine with the same peak curvature — should put the lossy and clean routes on one line. Computing it from the arriving edges and testing the collapse would say whether one number can stand in for the edge’s shape for curvature-reading discontinuities.

The transmitted edge. A doublet in reflection is a small extra delay and a small overshoot in transmission. Measuring the transmitted edge’s distortion behind the loss, against the clean route, would say whether a receiver is surprised by the same factor as the reflectometer.

Two vias behind the loss. A signal that changes layers and changes back crosses two vias a short distance apart, and their doublets overlap. Behind a lossy line, whose edge is narrow in the middle, the spacing at which the two doublets reinforce should be set by the edge’s steep centre rather than by its rise time — and so should be shorter than a budget written in rise times would place it.