The threshold a delay is quoted at

The start a step takes from infinity applied the initial-value theorem again and again and found that a step through a network whose denominator is $r$ degrees above its numerator starts as $(t/\tau)^r$: the first $r - 1$ derivatives are zero, and the $r$-th is fixed by the network’s behaviour at infinite frequency. It measured the consequence on a cascade of buffered sections. The time to reach one per cent moves later much faster than the time to reach half as sections are added, from 1.4 per cent of the 50% time with one section to a third of it with eight.

That essay ended by naming where the result matters. A digital receiver defines propagation delay at a threshold — usually half the swing, sometimes a tenth — and the power law says a delay quoted at 10% should move with the number of poles in the path far more than one quoted at 50%. A real interconnect is not a cascade of buffered sections; it is a line, and a line modelled as a ladder network has as many poles as the model has sections. The question is how much of a quoted delay is a property of the threshold and how much of the path, and whether the answer survives making the model finer. It depends on which kind of line it is, completely.

A lossless line, cut into sections #

The first line is a lossless one: a microhenry and a hundred picofarads in total, which is a 100 Ω line with a time of flight of ten nanoseconds, driven through 100 Ω and terminated in 100 Ω so that an ideal line would deliver a clean half-volt step at exactly one time of flight and nothing before. Cut into $N$ equal sections of inductance and capacitance, it becomes a ladder network with $2N$ poles and no zeros, and each ladder network is marched in time.

With eight sections the ladder network passes 10% of its final value at 0.863 of the time of flight and 50% at 1.043. So a delay quoted at 10% is 18 per cent of a flight time shorter than one quoted at 50%, for the same path. The step starts as the sixteenth power of time, as the theorem requires of sixteen poles, which is why it is so flat before its edge; and it rings after the edge, which is a separate failing of lumped models that the comparison of a ladder network with a line measured at length. The slider on the figure at the head of the page steps the section count from one to thirty-two.

Both delays close on the time of flight #

As the line is cut finer, the power law’s exponent grows without limit and the step’s start becomes flatter than any power. The question is what the two thresholds do.

They converge on each other, and on the time of flight. The 10% delay rises through 0.357, 0.610, 0.770, 0.863, 0.917 and 0.949 of a flight time as the section count doubles from one to thirty-two; the 50% delay comes down from its early overshoot to 1.019. The spread between them falls from 0.656 of a flight time to 0.070, and it falls as a power of the section count: $-0.681$ fitted over the four finest ladder networks.

That exponent has a reason. A ladder network of many sections carries a step as a front whose width grows with the distance travelled, and near the front the waveform is an Airy function of the distance behind it scaled by the cube root of the section count, which puts the rise time of an $N$-section ladder network in proportion to $N^{1/3}$ sections’ worth of delay, and the whole delay in proportion to $N$. Their ratio is $N^{-2/3}$. The measured $-0.68$ is that $-2/3$, and it says the lumped model’s threshold dependence is an artefact of cutting the line up. A continuous lossless line has a front: at its time of flight the output jumps, and every threshold between zero and one is crossed at the same instant. A finer model finds the front and so finds that the threshold does not matter.

Why one delay converges from below and the other from above #

The two thresholds approach the time of flight from opposite sides, and each side has its own cause. The 10% delay is early because of the precursor: a lumped ladder network starts its output at time zero, however slowly, and with few sections that slow start has reached a tenth of the swing well before the flight time. Every doubling of the section count adds that many more powers of time to the start and pushes the precursor back towards the flight time, which is why the 10% delay rises monotonically.

The 50% delay is late for a different reason, and it is a statement about the ladder network’s phase rather than its start. The sections a wavelength needs found that a ladder network’s phase delay is too long by $\arcsin(x)/x - 1$ at a frequency where it has $\pi/x$ sections per wavelength: every component of the step travels slower through the ladder network than through the line it imitates, and the higher the frequency the slower. The middle of the edge is built from those components, so it arrives late — 6.6 per cent late with two sections, 1.9 per cent with thirty-two — and comes in as the model resolves more of the edge’s spectrum.

So the 10%-to-50% spread shrinks from both ends. Neither end reaches the flight time at any finite count, and the ringing after the edge that the comparison of a ladder network with a line measured does not shrink the same way, because it lives at the ladder network’s cut-off rather than in its passband. A delay converges long before a waveform does.

A resistive line never finds a front #

The second line is resistive: a kilohm and a nanofarad in total, driven with no source resistance and left open at the far end, the model of an on-chip wire or a long high-resistance trace. Cut into $N$ sections it becomes an RC ladder with $N$ poles, and the power law says its start flattens as $N$ grows exactly as the LC ladder’s does.

The three delays converge again. One section is a single pole — 0.105, 0.693 and 2.303 of the time constant, which are $\ln(10/9)$, $\ln 2$ and $\ln 10$ — and thirty-two sections give 0.134, 0.391 and 1.064 of R·C, changing by a few thousandths per doubling. But they converge to three different numbers. The 10% delay stays at about a third of the 50% delay, 0.343 of it, however finely the line is cut.

The difference from the lossless line is that an RC line does not propagate; it diffuses. Its continuous limit is the heat equation, whose step response at the far end has no front: every point of the line starts to rise at once, slowly, and the output crosses each threshold at its own time because it is climbing a smooth curve rather than jumping. A finer model resolves that curve better and does not change its shape. The power law at the very start still holds for any finite ladder network and still gets steeper with $N$, but it governs an ever smaller part of the waveform and has no influence on where the thresholds fall.

What the diffusion’s numbers are #

The resistive line’s limits are the continuous RC line’s, and they are worth knowing as numbers. At thirty-two sections the 50% delay is 0.391 of the line’s R·C and still falling by about a hundredth per doubling, towards a continuous-line limit a little below 0.39; the 10% delay is 0.134, the 90% delay 1.064. None of them is the line’s total R·C, and none is the half of it that the sum of the time constants gives. The single number usually carried about a resistive wire, R·C/2, overstates its 50% delay by nearly a third and its 10% delay nearly fourfold.

The reason a diffusion gives such different delays at different thresholds is that its output rises in a shape with no characteristic edge. A lossless line has one time and every threshold is crossed at it. A diffusing line’s output is a smooth sigmoid that spends a third of its 50% time getting to 10% and more than its whole 50% time again getting from 50% to 90%. Where Kirchhoff’s own frequency sets the boundary at which a lumped model of a propagating conductor fails, a diffusing one has no such boundary: the lumped RC ladder is a good model of the distributed line at a modest section count, and what it models accurately is a delay that depends on the threshold.

How much of the delay is the threshold #

The two lines’ behaviour is clearest set side by side, with the spread expressed as a fraction of the delay a receiver would quote.

With one section the two are the same, since one section of either is a single pole, and the 10% delay is a third of the 50% one: 64.7 per cent of the quoted delay is the threshold. With thirty-two sections the lossless line is down to 6.9 per cent and still falling; the resistive line has settled at 65.7 per cent. On a line that propagates, a delay is a property of the path and becomes one more so the better the model is. On a line that diffuses, two-thirds of a 50% delay is the choice of threshold, and a receiver switching at 10% sees the signal arrive three times sooner than one switching at 50%, on the same wire.

That also says what the 50% convention is worth on each. For a propagating line it is harmless: any threshold gives nearly the same number. For a diffusing line it is a definition rather than a measurement, and a timing budget that mixes gates with different switching thresholds on RC wires is mixing different delays under one name. The sum of the time constants, which is what a sum that is exact found the first two coefficients of a denominator deliver, is R·C/2 for the continuous RC line — larger than both of its measured delays, and closest to neither.

What arrives before the edge #

The lossless ladder network’s convergence has one more reading, and it is the one the power law speaks to most directly. An ideal lossless line delivers nothing before its time of flight. A lumped ladder network always delivers something, because a finite network’s step starts at zero time.

With one section the step has reached 17.7 per cent by half the time of flight, a precursor that no real line would show. With four sections it is 0.6 per cent; with thirty-two, $2.4 \times 10^{-14}$. Each section adds two to the power of time the step starts with, and the precursor falls faster than any power of the section count, which is the power law’s statement about the start extended to a statement about where the start ends. For the resistive line the same measurement would show nothing so dramatic, since its output at any fixed fraction of R·C converges to the diffusion’s own nonzero value.

What a designer should take #

Quote the threshold with the delay, and know which kind of line it is. On a controlled-impedance board trace, where the line propagates, the threshold matters only to the extent that the edge is slow — and a model of the trace needs enough sections that the 10%-to-50% spread is small against the budget, since the spread it shows is the model’s rather than the trace’s. That spread falls as $N^{-2/3}$, so halving it needs about 2.8 times the sections. The sections a wavelength needs gives the frequency-domain version of the same count.

On a resistive interconnect — an integrated circuit’s long wires, a polysilicon line, a high-value resistor chain with stray capacitance — the threshold is most of the answer. A 10% receiver and a 50% receiver on the same RC wire see delays in the ratio 0.34, and a model with more sections will not reconcile them. Timing on such a path has to be stated at the threshold the receiving gate actually switches at, and a change of logic family on the receiving end is a change of delay. At the extremes, a receiver switching at 10% and one switching at 90% of the swing see delays of 0.134 and 1.064 of the wire’s R·C on the same wire — eight to one — which is a larger ratio than most designers would expect between two wire lengths, let alone two gates.

And the general point is the one the initial-value theorem was pointing at: a power of time at the start is a statement about a finite network. How much of the waveform it controls depends on whether the network is approximating something with a front or something without one.

How the numbers were obtained #

Each ladder network is a netlist of $N$ equal sections — series inductance and shunt capacitance for the lossless line, series resistance and shunt capacitance for the resistive one — with the totals fixed, the lossless line driven through its characteristic impedance and terminated in it, the resistive line driven from an ideal source and left open. Each is marched by the trapezoidal rule at 8000 steps over four flight times or five R·C, and every threshold crossing is interpolated linearly between the two steps that straddle it. The power of the spread is a least-squares fit of its logarithm against the section count’s over the four finest ladder networks. The early outputs are read at the first step at or after the stated fraction of the flight time.

What it leaves out #

A lossy line. A real board trace has both inductance and resistance, and at a few centimetres and nanosecond edges it behaves as a propagating line whose front is smoothed by loss. The edge the loss delivers measured that smoothing; how much of a delay becomes the threshold’s as resistance is added to the lossless line, between the 6.9 per cent here and the RC line’s 66, is the transition this essay’s two cases bracket.

The source’s own edge. Every step here is ideal. A driver with a rise time of its own adds its own threshold dependence, which on the lossless line dominates once the line is modelled finely and on the RC line adds to a spread that was already large.

The receiver’s threshold varying with supply and temperature. The delay on an RC wire is steep in the threshold near the bottom of the swing, so a receiver whose switching point wanders by a few per cent of the swing moves its delay by a fraction that this measurement’s slopes would give.

Still open: the lossy line, the driver’s edge, and the sampled step #

The line with loss. Adding series resistance to the lossless ladder network moves it from a front to a diffusion. The threshold spread, as a fraction of the delay, should rise from the lossless line’s few per cent towards the RC line’s two-thirds as the ratio of the line’s resistance to its characteristic impedance grows; finding where it crosses a third would say which board and package interconnects need a stated threshold and which do not.

A driver with its own rise time. Convolving each line’s response with a ramp moves every threshold by different amounts. On the lossless line the ramp should dominate the spread; on the RC line it should add to it. The ramp at which the lossless line’s spread is the driver’s rather than the model’s is the practical form of the section count above.

A sampled step. A step applied by a digital-to-analogue converter holds its first value for a whole clock, and the start of its response is the network’s response to a pulse. Whether the hold changes the power of time a reconstructed step starts with, and so its low-threshold delay, is the question the earlier essay left and this one did not take up.