The via that is a piece of line

The via two lines can weigh found that the area under a via’s reflection is a difference, $-Z_0C/2 + L/2Z_0$, and that the difference changes sign at one impedance. Below $\sqrt{L/C}$ the barrel’s inductance wins and the via reflects a bump; above it the pads win and it reflects a dip. At $\sqrt{L/C}$ exactly the two cancel, and the area is zero from any edge.

That essay treated the zero as a warning: a reading that trusts the area concludes that a balanced via is not there. This one asks what a balanced via actually is, since it is plainly still there — half a picofarad of copper and a nanohenry of barrel have not gone anywhere. The answer turns out to be the most useful fact in the whole description of a discontinuity. A balanced via is a short piece of transmission line, and what it reflects is not a measure of how big it is but of how long.

An area of nothing #

The via here is the one the separation measured: a quarter of a picofarad at each end and a nanohenry between them. Its area as a function of the reference impedance is a curve through zero at 44.72 ohms, and the via was solved as a cascade from thirteen lines to draw it.

Put the via on a 44.72-ohm line and send edges at it. The figures below do that for edges from six picoseconds to three hundred, and at every one the area under the reflection is zero to a part in a thousand of what the pads alone would leave. A reflectometer integrating that trace reads nothing.

What it reflects is a doublet: a short dip followed immediately by a bump of the same area, or the other way round, so that the two cancel in the integral and not in the trace. For a 59-picosecond edge the larger of the two excursions is 24.4 millivolts. The via’s half picofarad of pad on its own, on the same line, would dip by 166 millivolts. So the balance has taken away six sevenths of the reflection’s height, and all of its area.

A reflection that falls as the square of the edge #

The cancellation is not a matter of degree. The doublet’s height depends on the edge in a different way from a lone element’s reflection, and the dependence is the evidence for what kind of thing the doublet is.

A lone capacitance’s reflection, once the edge is slow against its time constant, is the edge’s derivative scaled by that time constant: its height falls as the first power of the rise time, and the figure fits −0.97 between a 59-picosecond edge and a 295-picosecond one. The balanced via’s falls as −1.99 over the same range. Five times slower an edge makes the pads’ dip five times smaller and the via’s doublet twenty-five times smaller: 35 millivolts against 994 microvolts, a ratio of thirty-five where at 59 picoseconds it was seven.

The square is the signature of a second derivative. The via’s reflection coefficient, expanded in frequency, has its first-order term $j\omega(-Z_0C/2 + L/2Z_0)$ — which is exactly the area’s term, and exactly zero at the balance — and what remains starts at the next power of frequency. A term in $\omega^2$ applied to an edge is the edge’s second derivative, a shape with a positive and a negative lobe of equal area, and its height scales as the reciprocal square of the rise time. The doublet is that term made visible, and the fitted −1.99 is the measurement that says the first-order term is gone rather than merely small.

For a fast edge the advantage mostly goes. At 11.8 picoseconds the doublet reaches 273 millivolts against the pads’ 522, about half. The expansion that separates the orders assumes the edge is slow against the via, and an 11.8-picosecond edge is shorter than the via’s own delay: the edge resolves the pad at one end, the barrel, and the pad at the other end as three events. The balance is a statement about slow edges, and the rate at which it stops helping is set by the via’s delay.

The delay the via adds #

A shunt capacitance slows the edge that goes past it; the dip whose area is fixed measured how much, and found the rise times adding as a root sum of squares. What that essay did not draw is that the capacitance also delays the edge, by the same $Z_0C/2$ that is its area. For a lone element the delay and the area are one number with two signs, and nobody separates them. For a balanced via they come apart completely: the area is zero and the delay is not.

The size of what is left is worth having in plain numbers before the measurement. The via’s half picofarad of pad, alone on a 44.72-ohm line, would delay an edge by 11.2 picoseconds and put 11.2 picoseconds of area under a dip. The nanohenry of barrel, alone on the same line, would delay it by another 11.2 and put 11.2 picoseconds of area under a bump. Assembled into the via, the two areas cancel and the two delays add, to 22.4 — which is also $\sqrt{LC}$, because at the balance the two halves are equal and their sum is twice either.

Measured at the halfway point of a 295-picosecond edge, the edge leaving the via is 22.4 picoseconds behind the edge arriving, and $\sqrt{LC}$ for half a picofarad and a nanohenry is 22.4 picoseconds. The figure checks the two agree to two per cent. That is the delay of a transmission line of impedance $\sqrt{L/C}$ whose total capacitance and inductance are the via’s, and it is what a section of the line itself would have added if the via had been replaced by 22.4 picoseconds of 44.72-ohm track.

So on the line it balances, the via is indistinguishable at first order from a short length of that line. It has the line’s impedance, which is why nothing reflects at first order, and it has a length, which is why the edge arrives late. The pad-barrel-pad arrangement is the three-element LC ladder that a ladder is not a line builds forty of to imitate a metre of cable, reduced to one section; one section is a very good line for edges slow against it and a poor one for edges that are not, which is exactly what the doublet’s growth at 11.8 picoseconds shows.

The comparison with the track the via connects is the one a timing budget needs. A signal that changes layers crosses two vias, one on the way down and one on the way back, and on this via’s numbers that is 44.8 picoseconds of delay that the routing tool’s length-matching, which counts track, never sees. Two signals of a differential pair that change layers together pay it equally and stay matched. A single-ended bus whose members change layers different numbers of times does not.

A via on a board of 1.6 millimetres in ordinary laminate has a delay of this order — 22.4 picoseconds is about three and a half millimetres of track — so the delay is not a correction that a timing budget can ignore on a fast interface, and it is the part of the via that no adjustment of its pads can remove. Pads and clearances move its impedance. The board’s thickness sets its length.

The line section a balanced via imitates is the same object the staircase in time launched a step into: an impedance and a delay, reflecting nothing where the impedances match and everything else at its ends. A via is simply a very short one, short enough that its two ends’ reflections overlap into a doublet rather than arriving as two steps. The receiver that is a branch measured the opposite case, a stub whose length is the problem; and Kirchhoff’s own frequency is the boundary between the two descriptions, measured for a node instead of a via. The sections a wavelength needs takes the one-section LC ladder a via is and asks how many such sections make a line.

The floor under the first order #

On a real board the via does not sit on a line of 44.72 ohms. It sits on fifty, or eighty-five differential, or whatever the stack-up delivered, and the question worth asking is how close to balance a via has to be before its first-order reflection stops being the larger part.

The first-order prediction is a V: the magnitude of the area, which goes to zero at the balance, times the height a unit of area produces for this edge. The solved excursion follows it closely away from the balance — 207 millivolts from a 25-ohm line, 168 from an 80-ohm one, and the figure checks the agreement to a third wherever the reference is more than fifteen ohms from the balance — and then refuses to follow it down. It bottoms at the doublet’s 24.4 millivolts and is within twice that from 43 ohms to 52.

That band is the practical tolerance on a via’s impedance for this edge. Inside it, the via’s reflection is set by its length and no further balancing of pads against barrel helps; outside it, the reflection is set by the imbalance and the balancing is worth doing. A fifty-ohm line is inside the band for this via and this edge, so a designer who adjusted the pads to move the balance from 44.7 to fifty ohms would buy almost nothing that a reflectometer with a 59-picosecond edge could see.

With a slower edge the band narrows. At 295 picoseconds the first-order reflection is 43.1 millivolts from 25 ohms and the doublet at the balance is under a millivolt, and none of the other impedances drawn comes within twice it. The first-order term falls as the first power of the edge and the doublet as the second, so the imbalance at which the two are equal shrinks in proportion to the edge. Slow edges see a via’s impedance clearly and its length hardly at all; fast edges see its length and forgive its impedance. Taken far enough the forgiveness is complete. At an edge of about 30 picoseconds, not much longer than the via’s own 22.4, the least excursion is no longer at the balance at all: it is 98 millivolts from a 49-ohm line against 138 from 44.7, because a via that long against the edge is a line with two ends of its own rather than a doublet, and the balance was a property of the doublet. A via specification that names one tolerance on impedance without the edge it applies to has named neither.

The sum and the difference #

The sweep at 295 picoseconds carries a second measurement, and it is the one that ties the whole description of a via together. At every reference impedance the figure also measures how late the edge leaving the via is, and checks it against an expression that looks almost the same as the area’s:

$$\text{area} = -\frac{Z_0C}{2} + \frac{L}{2Z_0}, \qquad \text{delay} = \frac{Z_0C}{2} + \frac{L}{2Z_0}.$$

The reflection’s area is the difference of the pad’s term and the barrel’s, and the delay the via adds is their sum. From 25 ohms the delay is 26.2 picoseconds against a sum of 26.3; at the balance, 22.4 against 22.4; from 80 ohms, 26.2 again. Both follow from the same first-order expansion. A shunt capacitance and a series inductance each slow the phase of what passes them, in the same direction, and each reflect, in opposite directions — so a via’s two elements always cooperate in delay and always compete in reflection.

That pair of expressions turns every earlier statement in this essay into arithmetic. The two terms are $Z_0C/2$ and $L/2Z_0$; their product is $LC/4$, fixed by the via whatever the line; and for a fixed product the sum is least when the two are equal. So the balance is not only where the reflection’s area vanishes. It is also where the via adds the least delay it can, $\sqrt{LC}$, and every imbalance, in either direction, costs delay as well as reflection. From 25 ohms and from 80 ohms, which sit on opposite sides of the balance by roughly the same factor, the via adds the same 26.2 picoseconds, 17 per cent more than at the balance.

The practical reading runs against a common intuition about vias. Shrinking a via’s pad to reduce its capacitance is often described as making it faster. On a line below the via’s balance impedance it does make it faster, because it moves the balance towards the line. On a line above the balance it moves the balance away, and the via reflects more and delays the edge more at the same time. There is no direction in which a pad is simply smaller-is-better; there is only the balance, and the pad’s job is to put the via’s balance on the line.

At 59 picoseconds the same measurement is less clean — 25.9 picoseconds of delay from 25 ohms against a sum of 26.3, and 23.1 at the balance against 22.4 — because a half-way crossing on an edge that is not slow against the via is already shaped by the second-order terms that make the doublet. The figure makes the check only at edges slow enough for the first-order statement to be the statement being tested, and says so.

A reflectometer’s view of a well-made via #

Put the pieces together from the side of the instrument, since that is where a via is usually judged. A reflectometer with an edge of a few tens of picoseconds, looking at a via designed to balance on its fifty-ohm line, shows a doublet of a few tens of millivolts that integrates to nothing. A technician trained to integrate reports a perfect via; one trained to look at the depth reports a small discontinuity of uncertain sign. Both are describing the same thing, and neither is describing the part of the via that matters most on a fast link, which is the delay — a quantity a reflection measurement does not show at all, because the delay is in the transmitted wave.

A reflectometer with a slower edge makes the doublet vanish into the noise long before it makes a first-order dip of the same via-size vanish: five times the rise time costs a lone discontinuity a factor of five in visibility and a balanced via a factor of twenty-five. That is the reason a well-balanced via reads as clean on a slow instrument and faintly visible on a fast one, and it is a property of what the via is rather than a flaw in either instrument.

What this changes about describing a discontinuity #

The dip whose area is fixed established that the area is the number worth quoting and the depth the number least worth quoting, and for a discontinuity that is one element that remains true. For a via it needs a second clause.

The area measures the via’s imbalance against the line, and nothing else. It is zero for a via of any size that happens to balance, and it is proportional to how far the via’s own impedance is from the line’s, weighted by its length.

The doublet measures the via’s length. At the balance it is all that is left, it falls as the square of the edge, and it is set by $\sqrt{LC}$ rather than by either element — which makes it the reflection a well-designed via cannot design away.

And the delay is first order even when the reflection is not. A balanced via reflects at second order and delays at first, so on a fast interface its timing cost can exceed its signal-integrity cost by a wide margin. The staircase in time and the essays after it measured what lines do with their ends and their lumps; this is the case where a lump behaves exactly like more line.

Still open: the via with a stub, the pair of vias, and the loss that shapes the doublet #

A via with a stub. A via used to a middle layer leaves barrel hanging below the connection, an open stub that looks like extra capacitance at low frequency and resonates at its quarter wave. On the line it would otherwise balance, the stub unbalances it, and its contribution grows as the edge gets faster rather than slower. Whether a stub can be absorbed into the balance by a smaller pad, or only for edges slower than its resonance, is the question back-drilling exists to avoid asking.

Two vias close together. A signal that changes layers and changes back crosses two vias, and two balanced vias are two short pieces of line with a stretch of track between them. Their doublets add with a delay between them, and at a particular spacing the second’s dip lands on the first’s bump. The spacing at which a pair of vias is quieter than one is a measurement the cascade here can make directly.

The loss in front of the via. A doublet is a second derivative of the edge, and a lossy line in front of the via rounds the edge before it arrives. That should suppress the doublet more strongly than a first-order reflection, since it falls as the square of the rise time, so a lossy line may make a balanced via disappear where a lossless one shows it. The delay that is not one number measures the rounding; putting the two together is unmeasured.