How long a track may be

The boundary that is a rise time fixed a conductor and swept the edge: ten centimetres of 50 Ω track, driven from 10 Ω into 200 Ω, leaves one per cent of overshoot on a 40.0 ns edge and more on anything faster. The load that rings with the line put a capacitance on the same track’s end and found the boundary moving with it. Both are answers to a question a layout never asks. A layout knows its edges — the driver’s data sheet states them — and asks how long a track may be.

That is the same curve read the other way, and reading it that way turns up something the rise-time reading hid. The boundary has two ingredients: the ratio of the track’s round trip to the edge, and the mismatch at its two ends. A rule written as a length carries the first and almost never the second, and the second decides not only how long the track may be but whether there is a limit at all.

The curve read the other way #

The track is the earlier essays’: 50 Ω on FR-4, where a wave travels 14.3 cm in a nanosecond. The far end is marched by the line’s characteristics as the staircase of reflections, driven by a ramp, and its overshoot read off. The difference now is that the edge is fixed and the length swept.

For a 1 ns edge the overshoot is 0.79 per cent on 2 mm of track, 1.98 on 5 mm, 3.98 on 10 mm, 8.07 on 20 mm, 22.9 on 50 mm, and 40.0 per cent from 100 mm on — the staircase’s first step, which a long enough track shows in full whatever the edge. Bisected, the longest track that stays inside one per cent is 2.51 mm. Its round trip is 35 ps, and the edge is 28.5 times that.

The check that this is the same boundary as the earlier essay’s is to feed the length back: the one-per-cent rise time of 2.51 mm of this track, bisected on the same march, is 0.994 ns. The two readings are one curve.

Two and a half millimetres is shorter than most engineers would guess for a nanosecond edge — shorter than many component pads are apart — and the reason is that the rule most of them carry is about a different overshoot, which the last section prices.

Millimetres per nanosecond #

Because the overshoot at a slow edge depends only on the ratio of round trip to edge, the longest track should be proportional to the edge.

It is. The one-per-cent length is 0.25 mm for a 0.1 ns edge, 0.75 for 0.3 ns, 2.51 for 1 ns, 7.56 for 3 ns, 25.1 for 10 ns and 75.7 for 30 ns: 2.516 mm per nanosecond of edge, constant to within half a per cent across three and a half decades. The ten-per-cent length is 23.6 mm per nanosecond, a little under ten times as much — the overshoot at a slow edge falls in proportion to the edge, so a tenfold looser tolerance buys nearly tenfold more track. Nearly and not exactly: ten times 2.52 would be 25.2 mm, and the ten-per-cent length is 23.6, because at ten per cent the track is a quarter of the way to the hundred millimetres at which the first wave alone overshoots by its full forty per cent, and the slow-edge proportionality has begun to bend towards that ceiling. The one-per-cent rule is the cleaner one precisely because it is deep inside the proportional regime; a designer extrapolating it to a generous tolerance should expect the allowance to grow a little less than in proportion.

That constant is a layout rule with every term stated: for this driver and this load, a track may be 2.5 mm long per nanosecond of the fastest edge it carries, for one per cent. It is also a statement about velocity. On a slower medium — a track under a solder mask, a denser dielectric — the millimetres shrink in proportion to the velocity, while the round trips per edge stay fixed at about twenty-nine.

The mismatch decides whether there is a limit #

The constant was measured for one driver, and the driver’s resistance turns out to be the most important term in it.

From a nearly ideal 2 Ω driver the longest track is 2.0 mm; from 10 Ω, 2.5; from 20 Ω, 3.5; from 30 Ω, 5.5; from 40 Ω, 11.5; from 45 Ω, 23.5. And from 50 Ω there is no limit at all. With the source matched to the line, the first wave reaching the far end already carries the settled voltage: it is launched at half the source, arrives, and is multiplied by $1 + \rho_L = 1.6$, which gives 0.8 of the source — exactly the divider of 50 Ω and 200 Ω. The reflection goes back to a source that absorbs it, and nothing ever returns. Above 50 Ω both ends reflect with the same sign, the far end climbs its staircase from below, and again nothing overshoots.

So the question “how long may this track be?” has three kinds of answer depending on the driver. A strong driver, much below the line’s impedance, gives a short fixed limit set by the edge. A driver near the line’s impedance gives a long limit that grows without bound as the match improves. A matched driver gives none. The resistor at the wrong end and terminated at both ends measured the terminations themselves; this is what they are for, stated as a length.

The rule in closed form #

The bisected lengths have a formula behind them, and it is short enough to be the rule itself. Past the end of a slow ramp the far end’s excess over its settled value is the staircase’s area beyond its first interval, divided by the edge. Each step of the staircase differs from the settled value by the settled value times a power of $q = \rho_S\rho_L$, the product of the two ends’ reflection coefficients, and each lasts a round trip, so the area is a geometric series and the overshoot at a slow edge is

$$\text{overshoot} \approx \frac{2T}{t_r}\cdot\frac{-q}{1-q}.$$

For a tolerance $\varepsilon$ the longest track is therefore

$$\ell_{max} = \varepsilon\,\frac{t_r v}{2}\cdot\frac{1-q}{-q},$$

with $v$ the velocity on the board. From 10 Ω into 200 Ω, $q$ is −0.667 × 0.6 = −0.4, the factor $(1-q)/(-q)$ is 3.5, and a 1 ns edge at one per cent allows 71.5 mm × 0.01 × 3.5 = 2.50 mm — the bisected 2.51. The same expression gives 23.3 mm from 45 Ω against the bisected 23.5, 12.5 mm into 60 Ω against 12.6, and 1.79 mm into an open end against 1.80; every finite length in the figures matches it to within two per cent, and the figures are required to agree with it.

The formula also says when there is no limit. If $q$ is positive — both ends reflecting with the same sign, a weak driver into a high impedance or a strong one into a low — the staircase climbs from one side and never overshoots at a slow edge. And if even the first wave, which is all an infinitely long track shows, overshoots by less than the tolerance, no length can exceed it: from a 10 Ω driver that never happens, but from 49 Ω into 200 Ω the first wave overshoots by only 0.6 per cent, and above 48.4 Ω the limit is gone well before the source is matched.

Written this way the rule has three factors a designer controls separately. The edge and the velocity set a length scale, $t_r v/2$ — the track whose round trip is the edge, which is exactly where the rule of thumb stops. The tolerance scales it down, one per cent to a hundredth of it. And the two ends’ mismatch scale it up, by $(1-q)/(-q)$: 3.5 for this driver and load, 2.5 into an open end, 32 from a 45 Ω driver. The rule of thumb is the first factor alone.

A board, worked #

The formula makes a worked case quick. A logic family with 25 Ω outputs and 2 ns edges, driving CMOS inputs that are nearly open: $\rho_S = -1/3$, $\rho_L = 1$, $q = -1/3$ and $(1-q)/(-q) = 4$. The length scale is 2 ns × 143 mm/ns ÷ 2 = 143 mm. At one per cent that allows 5.7 mm of track, and at ten per cent 57 mm. So on such a board every net longer than about six millimetres rings by more than one per cent, and every net longer than about six centimetres by more than ten — which is why a board of that family laid out by the round-trip rule, 143 mm, rings visibly on any net that approaches it.

The same board with a 22 Ω series resistor at each output brings the source to 47 Ω, $\rho_S$ to −0.03 and $(1-q)/(-q)$ to 33: a one-per-cent length of 48 mm, 143 mm at three per cent, and at four per cent no limit at all, since the first wave from 47 Ω into an open end overshoots by only 3.1 per cent. The resistor at the wrong end is the measurement of where such a resistor belongs; this is the length it buys.

Where the formula stops #

The closed form is the slow-edge limit, and it has a ceiling it does not know about. However long the track, the far end cannot overshoot by more than its first wave does — the staircase’s first step, arriving before the source’s reflection has had time to come back — and that is $(1+\rho_L)Z_0/(R_s + Z_0)$ over the settled value, less one: 40 per cent from 10 Ω into 200 Ω, which is where the hero figure’s curve flattens from 100 mm on. The formula keeps rising past it. So it is exact where it is needed, at lengths short against the edge, and it overstates the overshoot of a long track, which no one designs to anyway.

It also assumes the edge is a ramp. A real edge is rounded at both ends, and a rounded edge hides a staircase better than a ramp of the same 10–90 per cent time, so the formula errs on the safe side there too. The staircase in time drew the staircase against a step, where every step is visible; the formula is that staircase’s area, which is what a ramp of any shape integrates it into once the ramp is slow.

The load decides it too #

The far end’s mismatch enters the same way, from the other side.

Into a load matched to the line there is again no limit: the first wave is absorbed where it arrives, whatever the source did to it. Into 60 Ω the longest track is 12.6 mm; into 75 Ω, 6.1; into 100 Ω, 3.95; into 200 Ω, 2.51; into 500 Ω, 2.03; and into an open end — a CMOS input, to a first approximation — 1.80 mm. The limit shortens as the far end’s reflection coefficient grows, steeply at first and then slowly, because past a few hundred ohms the load reflects nearly everything anyway.

The open end is the case a digital board presents, and it is the tightest. A 1 ns edge from a strong driver into a bare input may cross less than two millimetres of track before the far end overshoots by one per cent. Every longer net on such a board is, by this measure, a transmission line, and the choice is between accepting the overshoot, slowing the edge, or matching one end — the three options the load that rings with the line priced for a capacitive input.

The rule of thumb, priced #

Layout guides commonly say a track is electrically short while its round trip is shorter than the edge, and more cautious ones use a third or a tenth of that length. Each of those is a statement about the ratio of round trip to edge. None mentions a mismatch.

For a 1 ns edge the track whose round trip equals the edge is 71.5 mm long. Driven from 5 Ω into 200 Ω it overshoots by 49.1 per cent; from 10 Ω, 40.0; from 25 Ω, 20.0. A third of that length, 23.8 mm, overshoots by 12.3, 10.1 and 5.6 per cent. A tenth, 7.1 mm, still overshoots by 3.3, 2.9 and 1.7 per cent. The one-per-cent length for this driver and load, 2.51 mm, is about a twenty-eighth of the rule’s.

The rule is not wrong about its variable. The overshoot does fall in proportion as the ratio of round trip to edge falls, and the three rule lengths are three points on the proportional part of the curve. What the rule leaves out is the other factor, which sets the scale: a strong driver into a high impedance makes every ratio count for twice what it would from a driver closer to the line’s impedance. A rule that states a length and no mismatch is half a boundary, and for the case a digital board presents it is the less important half.

The edges that are lengths gathered the handful of boundaries in these essays that are sizes rather than frequencies or amplitudes. This one belongs among them, with its second variable stated.

How the numbers were obtained #

The far end is marched by the line’s characteristics — forward and backward waves each read back one transit after they are launched — with the source end solved from the source’s ramp, its resistance and the arriving wave, and the resistive load end algebraic, so that the march is the staircase of reflections exactly, at forty steps a transit. Each case runs for at least sixty transits and three edges. The longest track is bisected in the logarithm of the length between 0.1 mm and 30 m, and a case with no length reaching the target within that range is reported as having no limit. The rule-of-thumb lengths are the edge times the velocity, halved, and then divided by three and by ten. The round-trip check feeds the found length back into the rise-time bisection of the earlier essay’s boundary.

What it leaves out #

Capacitance at the load. Every load here is resistive, or open. With an input capacitance the boundary becomes a rise time and a capacitance together, as the previous essay found, and the length allowed for a given edge falls as the capacitance grows.

Loss. A lossy track damps its own staircase and so allows a longer track than this for the same overshoot — and delays the edge, which is a cost of its own.

Branches. A net with a stub or several loads along it is not one track, and each branch has its own round trip and its own reflection.

Still open: capacitive loads as a length, branches, and the stub #

A length rule for a capacitive input. The capacitive boundary of the previous essay is a rise time that grows with the load’s capacitance; read as a length it should shrink with the capacitance at a fixed edge, and at heavy loads it should be set by the ring rather than by the round trip, so that a longer track costs less than proportionally. A layout rule for a bus of inputs would come from exactly that curve.

Branches. A track that forks sends two waves on and reflects part of one back, and the longest branch a given edge tolerates depends on where the fork is. The pad the stub comes out of measured a stub’s own resonance; the same stub as a boundary on length is the missing reading.

Where the mismatch comes from. A driver’s output resistance changes with its load current and its supply, so a source that is 20 Ω at one level may be 10 at another. The limit, which depends on it steeply, then depends on the logic level being driven — a boundary that moves with the signal it bounds.