The dip whose area is fixed

The staircase in time sends a step down a metre of cable and finds that what the source drives into is decided by the line’s characteristic impedance and nothing else, until the far end answers. Everything in that essay happens at the ends: a source at one, a load at the other, and a uniform line between them.

A real line has things on it. A via, a connector pin, a test point, a package lead, a short piece of track that is wider than the rest — each of them is a small lump of capacitance or inductance sitting in the middle of an otherwise uniform line, and each of them reflects. What they reflect is not a staircase. It is a single pulse, and the pulse has two numbers in it that behave completely differently.

The depth of the pulse is a property of the edge that produced it. Its area is a property of the discontinuity and nothing else. That separation is why a reflectometer can measure a picofarad it cannot resolve, and it is why the depth — which is what anybody looks at — is the number least worth quoting.

One pole, and the two things a pole has #

A shunt capacitance between two lines of the same impedance is the simplest discontinuity there is, and it is worth doing exactly before anything is measured.

A wave arriving at the capacitance sees the onward line in parallel with the capacitance itself, because a matched line of impedance $Z_0$ looks, to a wave travelling towards it, exactly like a resistor of $Z_0$. The two in parallel give $Z_0/(1 + j\omega C Z_0)$, and the reflection coefficient that follows is $-j\omega\tau/(1 + j\omega\tau)$ with $\tau = Z_0 C/2$. The transmission is 1/(1 + jωτ). One pole, one time constant, and the factor of two in it is the parallel combination rather than anything about the capacitor.

Driven by an ideal step of one volt, that reflection coefficient gives exactly −exp(−t/τ). It starts at the full height of the incident wave — a capacitor is a short circuit at the first instant, so the reflection is momentarily that of a short — and decays with τ. Its area is therefore exactly −τ, which is $-Z_0 C/2$, and there is nothing in that expression about how the step was made.

Nothing in the solve is written in those terms. The chain is built as a cascade of two-port matrices, a raised-cosine edge is transformed, multiplied by the reflection the cascade gives at each frequency and transformed back; the closed form is then available as an independent second route. Driven fifty times faster than its own time constant, the cascade returns the exponential to 1.4 × 10⁻⁴ of a volt, which is the agreement two routes that share only the value of the capacitor can reach.

Forty-two times the depth, and the same area #

Sweeping the edge is what turns the algebra into a working rule.

A picofarad has a time constant of 25 picoseconds. Met by an edge whose 10–90 rise time is 5.9 picoseconds the reflection reaches 0.833 volts; met by one of 1.18 nanoseconds it reaches 0.0196 volts. That is a factor of forty-two in the thing a reader of an oscilloscope trace looks at, produced by changing nothing about the board.

The capacitance read back from the area, across the whole of that range, is 0.99994 picofarads. It moves by six parts in a hundred thousand, and what moves it is the trapezoidal integration of a sampled record rather than the physics.

So the working rule has a shape worth stating plainly: a discontinuity that looks negligible on a slow instrument is exactly as large as it ever was. The depth is the product of the discontinuity and the instrument, and a measurement that reports only the depth has not separated them. The same board measured on a faster reflectometer has not got worse.

That is the reflectometer’s version of a complaint made about instruments elsewhere. The instrument’s own rise time measures a scope reporting its own front end rather than the signal, and the probe is part of the circuit measures the same confusion in the other direction. Here the instrument does not corrupt the measurement at all — it changes what the measurement is a measurement of, and the area is the part that is about the board.

The same statement with the sign reversed #

A series inductance is the dual and it is worth drawing, because the duality is exact rather than approximate.

A wave arriving at a series inductance sees the onward line in series with it, which gives a reflection coefficient of $+j\omega\tau/(1 + j\omega\tau)$ with $\tau = L/2Z_0$. Same pole, opposite sign, and the same consequence: the reflected pulse is a bump rather than a dip, its area is $L/2Z_0$ whatever the edge, and its height is whatever the edge could reach. Five nanohenries on a fifty-ohm line has a time constant of fifty picoseconds and a bump of 0.4933 volts to the same 59-picosecond edge, and the inductance read back from the area is 5.0000 nanohenries.

A real via is both, and that is the point at which the two numbers stop being separable from one reflection. The barrel is an inductance and the pads are a capacitance, and what the reflection’s area gives is one combination of them — the signed area $L/2Z_0 - Z_0 C/2$, a bump’s area less a dip’s — with no way to split it. A single reflectometer trace of a single via measures one number, not two, and calling that number “the via’s capacitance” is a choice about which of the two is assumed to be zero.

What goes past, and a rule that happens to hold #

The wave that is not reflected is transmitted, and it leaves slower than it arrived.

One pole of time constant τ has a 10–90 step-response rise time of τ·ln 9, which is 2.197τ. The rule everybody uses for putting two rise times together is the square root of the sum of their squares, and here it is right: across a hundredfold of capacitance, from a tenth of a picofarad to ten, the transmitted edge’s measured rise time is within 1.3 per cent of $\sqrt{t_\mathrm{in}^2 + (2.197\tau)^2}$.

The rule’s accuracy here is not evidence that the rule is general. It holds because both shapes in the convolution are close to single poles, so their variances add and the 10–90 time is close to proportional to the standard deviation for both. Neither condition is guaranteed: a response with overshoot has a rise time that is not a fixed multiple of its own spread, and one step, computed twice draws exactly such a response. Where that rule breaks is the question this field hands to the transients field rather than answering here.

What the transmitted edge does say about a board is the usual thing said backwards. A discontinuity small enough that its reflection is invisible can still be the largest single contributor to an edge’s rise time, because the reflection’s depth falls as the edge slows and the rise-time contribution does not.

The same pole, said as a frequency #

A designer who specifies boards rather than reading reflectometer traces meets this discontinuity as a return-loss number, and the translation is worth doing because it makes the capacitance a bandwidth.

The reflection coefficient’s magnitude is $\omega\tau/\sqrt{1+\omega^2\tau^2}$, which rises at twenty decibels a decade and flattens at one. So a capacitance has a frequency below which it meets any given return-loss specification, and the frequency is inversely proportional to it. A picofarad on a fifty-ohm line holds −20 dB of return loss to 640 megahertz and −10 dB to 2.12 gigahertz. Half a picofarad doubles both; a fifth of a picofarad reaches 3.20 gigahertz at −20 dB.

Those numbers are small beside the edge that produced the picture above. A 59-picosecond edge has significant content to something like six gigahertz, which is where the same discontinuity is reflecting more than half of what arrives. That is the frequency-domain statement of the same fact the depth sweep makes in time: the discontinuity is not a small effect that a fast edge exposes, it is a large effect whose time-domain appearance shrinks when the edge is slow because a slow edge has nothing up there to reflect.

The two descriptions carry different things well. The frequency one gives a specification that composes — several discontinuities on one line have return losses that combine, badly but predictably — and it hides the fact that the reflection is localised in time. The time one shows where on the board the reflection came from, which is the whole reason a reflectometer exists, and hides the bandwidth. Neither is the more fundamental; the pole is, and both are readings of it.

Two of them, and the distance at which they become two #

The most useful consequence of the area being an invariant is what happens when two discontinuities are too close to tell apart.

Two half-picofarad capacitances with a picosecond of line between them, met by a 59-picosecond edge, give one dip 0.3185 volts deep — almost exactly the 0.3202 a single picofarad gives. Two hundred picoseconds apart they give two dips of 0.1831 volts each, which is what a half-picofarad discontinuity gives on its own. The depth has halved, and neither reading is more correct than the other; they are two different pictures of the same board.

The area reads 1.00000 picofarads at every spacing, to four parts in a hundred thousand. Whether the instrument can resolve the two discontinuities or not, integrating the reflection gives the total capacitance on that stretch of line, and the resolution limit — which is what the depth is about — is irrelevant to it.

That is a genuinely useful property, and it has an obvious limit which is worth stating so that the result is not over-read. It is a total, over whatever length is integrated. It says nothing about how the capacitance is distributed, so a board with one bad via and a board with four mediocre ones integrate to the same answer, and a design rule written against the total is a different rule from one written against the worst single feature. The depth, which is what everyone quotes, is the number that distinguishes them — and it is the number that depends on the instrument.

A short piece of wider track is not exactly a capacitor #

Everything above treats the discontinuity as a lumped element. On a board it is usually not: it is a short length of line whose impedance is different, and the lumped element is a model of it.

A section of line of impedance Z and delay $d$, sitting in a line of $Z_0$, carries $d/Z$ of capacitance where the line it replaced carried $d/Z_0$, and $dZ$ of inductance where the line carried $dZ_0$. So it has an excess capacitance and — for $Z$ below $Z_0$ — a deficit of inductance, and both of them reflect. The reflection’s area is $(L_\mathrm{excess}/Z_0 - Z_0 C_\mathrm{excess})/2$, and both terms have the same sign when the section is the low-impedance kind.

That is what makes a reflectometer’s reading of an impedance step overstate its capacitance. Reading the area back through $Z_0 C/2$, as though the section were purely capacitive, gives the section’s true excess capacitance times $(Z_0 + Z)/Z_0$: 1.20 for a ten-ohm section, 1.50 for a twenty-five-ohm one, 1.80 for a forty-ohm one, exactly, at every impedance drawn. A ten-picosecond patch of twenty-five ohm line has 0.2 picofarads of genuinely extra capacitance and reads as 0.3.

The excess is not an error in the instrument and it is not an error in the reading either, so long as the reading is used for what it is. What the area measures is the reflection, and the reflection is what a receiver further down the line will see. The number is right for predicting that and wrong for telling a field solver how much copper there is.

Where the lumped element stops #

The lumped model has an edge and it is a ratio, in the way every edge in this collection turns out to be.

Comparing the reflection of a twenty-five-ohm section with that of a single shunt capacitance of the value its own area gives, waveform by waveform: the two agree to 7 × 10⁻⁴ of a volt when the section’s round trip is seven per cent of the arriving edge, to 1.6 × 10⁻² at thirty-four per cent, and to 0.17 at a hundred and seventy. The two-picosecond section is a capacitor. The fifty-picosecond one is two reflections with a delay between them, and drawing it as an element loses the delay entirely.

A fifth of the edge is a workable boundary — under it the disagreement is a few parts in a thousand of the incident step — and it is the same shape as Kirchhoff’s own frequency, which asks how long a piece of copper may be before its two ends stop being one node. Both are statements that a length is negligible when its round trip is small against the thing travelling over it, and both give a number rather than a slogan.

The two boundaries also stack in a way worth noticing. A discontinuity that is lumped by this rule is still a discontinuity: the section being electrically short is what makes it one element, not what makes it no element. Everything about the area survives the section being long — a longer section has more excess capacitance and reflects more area — and it is only the shape of the reflection that stops being a single exponential.

What this does to the rest of the field #

Three things elsewhere in this field are affected by the area being the measurable quantity, and one of them is a correction rather than an addition.

A receiver on a stub is a capacitance on a line, which is what the receiver that is a branch measures as a branch and what this essay measures as a lump. The two descriptions agree while the stub’s round trip is short against the edge, by exactly the boundary above, and the branch description is the one that survives past it.

A termination is not a discontinuity and a connector is. The staircase of the staircase in time has reflections that never decay away on their own, because a mismatched resistive termination reflects a constant fraction for ever. A lumped discontinuity’s reflection is a pulse that ends, so its effect on a settled level is nothing at all: it costs a board timing margin and eye height and it costs it no direct-current accuracy.

And an impedance tolerance is not the same statement as a discontinuity budget. A line held to ±10 per cent of fifty ohms reflects a tenth of the wave continuously along its length. A single via of a picofarad reflects a third of the wave for twenty-five picoseconds. Neither number bounds the other, and a board specified only on impedance tolerance has said nothing at all about the second.

Still open: a via that is both, the loss the record does not have, and the crossing #

Separating the L and the C of one feature. The area gives $L/2Z_0 - Z_0 C/2$, which is one number for two unknowns, and no single measurement on one line resolves it. Two measurements do: the same feature measured from a line of a different characteristic impedance weights the two terms differently, so a pair of solves at, say, 50 and 75 ohms would separate them — and would say whether a via’s inductance or its pads dominate, which is the design question nobody can answer from one trace.

The record has no loss in it. Everything here is a lossless line with the discontinuity between two matched sections. On a real board the edge arriving at a via has already been slowed by the loss that the delay that is not one number measures, so the depth at the via is smaller than the launched edge suggests and the area is unchanged. That last clause is a claim rather than a measurement: the area should survive a dispersive line in front of the discontinuity, because the reflection coefficient is unchanged and the incident energy is only rearranged in time, and it is worth a solve that puts the two together to find whether the record can still be integrated cleanly once the reflection comes back through the same loss.

A discontinuity in a plane rather than in a line. The features measured here sit in the signal conductor. A track crossing a gap in its return plane is a discontinuity in the return, and where the plane runs out says a cross-section cannot describe it. What it would look like from a reflectometer — whether it reads as an inductance, with the inductance’s sign, and whether the area is still an invariant when the discontinuity is in the geometry rather than in an element — is a question both fields have now reached from different directions.