Lines, where a wire has a length

The via two lines can weigh

The area under a reflection is a property of the discontinuity rather than of the edge, and for a via it is one number made of two: −Z₀C/2 from its pads and +L/2Z₀ from its barrel, with opposite signs. A via of half a picofarad and a nanohenry, seen from fifty ohms, leaves 2.5 picoseconds of dip — which a reading that assumes a capacitor calls 0.100 pF, a fifth of what is there. From fifty and seventy-five ohms together the two areas give 0.5000 pF and 1.0000 nH back, and fifty femtoseconds of error on each costs 0.8 per cent of the capacitance and 1.5 of the inductance. A second line at fifty-five ohms costs five times as much.

16 min read 6 figures Two routes to a number The parasitic is the component Measured, not tabulated

Assumes: The staircase in time · Kirchhoff's own frequency

The dip whose area is fixed found the number a reflectometer can trust. A picofarad across a fifty-ohm line makes a dip whose depth depends on the edge that meets it — 0.833 volts for a six-picosecond edge, 0.0196 for a nanosecond one — and whose area does not: 25 picoseconds from both, which is $Z_0C/2$ and contains nothing about the edge. The area survives an instrument too slow to resolve the dip, two discontinuities too close to separate, and a record that nobody would call a clean exponential.

It also leaves a question the essay named and could not answer from one trace. A real via is not a capacitor. It has a pad at each end, which is capacitance to the planes it passes, and a barrel between them, which is inductance. Both reflect, both put an area under the trace, and a reflectometer sees the two areas added. One area is one equation, and a via has two unknowns.

That essay wrote the sum as $Z_0C/2 + L/2Z_0$ , as if the two added. They do not. A shunt capacitance reflects a dip and a series inductance a bump, and their areas have opposite signs, so the via’s area is a difference. That changes what a single measurement can say, and it is the reason a second measurement is worth making.

Two elements, one area

1 pF across a 50 Ω line: a dip of 0.320 V and an area of 25 ps. computed by solving, not by drawing as a cascade of two-ports, with a raised-cosine edge of 59 ps sent into it. The incident edge is the faint curve; what comes back is the shaded dip and what goes on is the third. The dip reaches -0.3202 V and its area is 25 ps, which is Z₀C/2 to a part in ten thousand. Driven by an edge fifty times faster the same cascade returns the single exponential the closed form gives, to 9.3e-6 of a volt. The transmitted edge leaves at 81.1 ps, against 59 ps arriving. — Fig. 1 One picofarad across a 50 Ω line, met by an edge of 59 ps: a dip of 0.320 V whose area is 25 ps, which is Z₀C/2 to a part in ten thousand.

The capacitor alone is the case the earlier measurement drew. Its reflection is a dip, the area under the dip is $Z_0C/2$ , and at fifty ohms a picofarad is 25 picoseconds of it. The sign is negative: a shunt capacitance momentarily looks like a lower impedance, and a lower impedance reflects a negative wave.

1 nH in series with a 50 Ω line: a bump of 0.150 V. computed by solving, not by drawing as a cascade of two-ports, with a raised-cosine edge of 59 ps sent into it. The incident edge is the faint curve; what comes back is the shaded bump and what goes on is the third. The bump reaches 0.1500 V and its area is 10 ps, which is L/2Z₀ to a part in ten thousand. Driven by an edge fifty times faster the same cascade returns the single exponential the closed form gives, to 9.3e-6 of a volt. The transmitted edge leaves at 63 ps, against 59 ps arriving. — Fig. 2 One nanohenry in series with a 50 Ω line, met by the same edge: a bump of 0.150 V whose area is 10 ps, which is L/2Z₀ to a part in ten thousand.

A series inductance is its dual. It momentarily looks like a higher impedance, it reflects a bump, and the area under the bump is $L/2Z_0$ — 10 picoseconds for a nanohenry at fifty ohms, with a positive sign.

Put them together as a via is built — a quarter of a picofarad, a nanohenry, a quarter of a picofarad — and the two areas add with their signs:

A(Z_0) = -\frac{Z_0 C}{2} + \frac{L}{2Z_0}.

From fifty ohms that is $-12.5 + 10 = -2.5$ picoseconds. A reflectometer shows a small dip. An engineer who reads it through $Z_0C/2$ , the way a lone capacitance is read, reports 0.100 picofarads: a fifth of the capacitance the pads actually have, with the barrel’s inductance hidden inside the deficit. A different via with twice the pad capacitance and half the barrel — a picofarad and half a nanohenry — reads as 0.800 picofarads from the same line. Neither reading says what the via is.

The same via from every line

The expression has the reference impedance in it twice, in opposite places, and that is the whole of the opportunity. The capacitance’s term grows with $Z_0$ and the inductance’s shrinks with it, so the same via seen from a different line is weighted differently.

Fig. 3 A via of 0.25 pF, 1 nH and 0.25 pF, solved as a cascade and met by a 59 ps edge from reference lines of 20 to 150 Ω. The area under its reflection follows −Z₀C/2 + L/2Z₀ at every impedance and changes sign at

\sqrt{L/C}

= 44.7 Ω. From 50 and 75 Ω together the two areas give 0.5000 pF and 1.0000 nH; with 50 fs of error on each area, the worst is 0.8% on the capacitance and 1.5% on the inductance. The slider moves the second line.

Solved as a cascade from thirteen reference lines between twenty ohms and a hundred and fifty, the area follows the expression at every one, to two parts in ten thousand of the capacitance’s term. Below 44.7 ohms it is positive — the inductance’s term is the larger and the via reflects a bump — and above it negative. At exactly $\sqrt{L/C}$ it is zero, from any edge.

That last fact deserves its own measurement and gets one in the via that is a piece of line. Here it is a warning about the single trace. A via whose barrel and pads happen to balance on the line it sits on puts no area under its reflection at all, and a reading that trusts the area concludes that nothing is there.

Two lines, two equations

With two reference impedances the areas give two linear equations in the two unknowns:

A_1 = -\frac{Z_1C}{2} + \frac{L}{2Z_1}, \qquad A_2 = -\frac{Z_2C}{2} + \frac{L}{2Z_2}.

Solved, they return the capacitance and the inductance directly. From fifty ohms the via’s area is −2.5 picoseconds; from seventy-five it is −12.08. The pair gives 0.5000 picofarads and 1.0000 nanohenries back, and the figure checks both to a part in a thousand.

Physically the second line is not exotic. A via measured on a test coupon is reached through a track, and a narrower track on the same coupon is a higher-impedance line into the same via. The measurement needs the second track to deliver its edge to the via without a discontinuity of its own at the junction, which a coupon designed for the purpose can do and a production board usually cannot. What it buys is a question nobody can answer from one trace: whether a via’s pads or its barrel dominate, and therefore whether the fix is a smaller pad or a shorter barrel.

The separation is exact only while each area is exact, and a real area is not. A reflectometer’s record has noise on it, a baseline that is not quite flat, and an edge whose reflection from the line itself is not quite zero. Each of those puts an error on the area of some tens of femtoseconds. What decides whether the separation survives them is how different the two lines are.

Where fifty femtoseconds comes from

An error on an area is easier to budget once it is turned into the things that produce it. The area is a voltage integrated over time, so an error of fifty femtoseconds on a one-volt step is, for example, a baseline one millivolt high held over a fifty-picosecond integration window, or a baseline a tenth of a millivolt high over half a nanosecond. Neither is a bad instrument. A sampling reflectometer’s vertical offset drifts by fractions of a millivolt, and the window has to be long enough to contain the whole of the via’s reflection and short enough to exclude the next discontinuity along the line — which on a real coupon is the launch connector or the end of the track, a few hundred picoseconds away.

So the window, not the noise, usually sets the error, and the window is set by the coupon’s geometry. A longer run of track between the launch and the via lets the window grow without catching anything else, which makes a baseline error cost more femtoseconds rather than fewer. The practical budget runs the other way from intuition: the cleanest separation comes from a coupon short enough that the window can be narrow, with a baseline measured immediately before the via’s reflection arrives, and from two tracks whose impedances differ by a factor of one and a half or more.

That is also the reason the second line should be the higher impedance of the two when the via is being measured on a board built for fifty ohms. Below fifty ohms the tracks are wider, and a wide track arriving at a small pad is a discontinuity of its own right at the junction, reflecting into the same window. A narrower track into the same pad adds less of its own.

The two numbers a via is

The pair of areas returns a capacitance and an inductance, and there is a second way to read the same pair that says more about what a via does on a board.

Any two such numbers define a short section of transmission line: an impedance $\sqrt{L/C}$ and a delay $\sqrt{LC}$ . For the via measured here they are 44.7 ohms and 22.4 picoseconds. That is not a coincidence of notation. The dip whose area is fixed found that a short section of line of impedance $Z$ in a line of $Z_0$ reads, through its area, as an excess capacitance scaled by $(Z_0 + Z)/Z_0$ ; a via is the same object assembled from pads and a barrel instead of from a uniform track, and while it is short against the edge the two are indistinguishable from the outside.

Read this way, a single reflection from a fifty-ohm line gives only the sign of the difference between the via’s impedance and the line’s. A small dip says the via is a little below fifty ohms, a small bump that it is a little above, and the size of the area mixes how far below with how long the via is. The via here, at 44.7 ohms and 22.4 picoseconds, and a via at 49 ohms and a much longer barrel could leave the same 2.5 picoseconds of dip. Two lines separate the impedance from the length.

That reading also says which number a designer is trying to change. A via’s delay is fixed by the board’s thickness and the dielectric; its impedance is what the size of its pads and the clearance around its barrel adjust. The design goal is not a small capacitance or a small inductance but an impedance close to the line’s, and the measurement that confirms it is the one that finds $\sqrt{L/C}$ — which is where the via’s area vanishes, and which two lines locate without either of them having to be at it.

How different the two lines must be

With 50 fs of error on each area, a second line at 55 Ω recovers the inductance to 5.5%; at 75 Ω, to 1.5%. computed by solving, not by drawing. The worst error in the capacitance and inductance of a 0.5 pF, 1 nH via recovered from two reflection areas, one from 50 Ω and one from the second impedance drawn, when each area carries an error of 50 fs in the direction that does most damage. With 55 Ω: C to 4%, L to 5.5%. With 75 Ω: C to 0.8%, L to 1.5%. With 110 Ω: C to 0.33%, L to 0.92%. With 200 Ω: C to 0.13%, L to 0.67%. The error falls as the two impedances part, roughly as 1/|Z₂/Z₁ − Z₁/Z₂|, and the inductance — the smaller term in the area near fifty ohms — is always the worse determined. — Fig. 4 The worst error in the recovered capacitance and inductance, with 50 fs of error on each area in the direction that does most damage, against the second line’s impedance, the first at 50 Ω. With 55 Ω: 4% and 5.5%. With 75 Ω: 0.8% and 1.5%. With 110 Ω: 0.33% and 0.92%. With 200 Ω: 0.13% and 0.67%.

Fifty femtoseconds of error on each area, applied in the four combinations of sign and the worst one kept, is a modest error: a fiftieth of the via’s whole area from fifty ohms. With the second line at fifty-five ohms it costs four per cent of the capacitance and five and a half of the inductance. At seventy-five ohms, 0.8 and 1.5 per cent. At a hundred and ten, 0.33 and 0.92. At two hundred, 0.13 and 0.67.

The reason is visible in the two equations. When $Z_2$ is close to $Z_1$ they are nearly the same equation, and the difference between them — which is what carries the information about how the area divides — is small beside the error on either. The conditioning of the pair goes as the reciprocal of $|Z_2/Z_1 - Z_1/Z_2|$ , and the figure checks that the capacitance’s error follows that factor to within the spread a four-sign worst case leaves.

A via's area changes sign at 44.7 Ω, and two impedances give its 0.5 pF and 1 nH back. computed by solving, not by drawing, as a cascade of two-ports: a via of 0.25 pF, 1 nH and 0.25 pF, met by an edge of 59 ps from reference lines of 20 to 150 Ω. The area under the reflection is −Z₀C/2 + L/2Z₀ at every impedance: a bump below 44.7 Ω, where the inductance's term is the larger, nothing at it, and a dip above. From 50 Ω the area is 2.5 ps of dip, which a single-capacitance reading calls 0.100 pF. From 50 and 55 Ω together the two areas give 0.5000 pF and 1.0000 nH. An error of 50 fs on each area moves them by up to 4% and 5.5%. — Fig. 5 The same via from 50 and 55 Ω. The two areas still give 0.5000 pF and 1.0000 nH exactly, and 50 fs of error on each moves them by up to 4% and 5.5%.

The inductance is always the worse determined, and not by a small margin at the wide end: at two hundred ohms the capacitance is at 0.13 per cent and the inductance at 0.67. Near fifty ohms the inductance’s term is the smaller of the two in the area, $L/2Z_0$ against $Z_0C/2$ , and it shrinks further as the second line’s impedance rises — so moving the second line up improves the conditioning of the pair and simultaneously weakens the inductance’s signal in the second area. Past a hundred ohms or so the first effect has mostly been spent and the second has not, and the inductance’s error falls only slowly.

The error scales directly with the error on each area. At ten femtoseconds a second line at seventy-five ohms recovers the inductance to 0.3 per cent and at two hundred to 0.13; at two hundred femtoseconds the same lines give 6 and 2.7 per cent, and a line at fifty-five ohms is out by 22. So a laboratory that trusts its areas to ten femtoseconds can separate a nanohenry from half a picofarad with lines ten per cent apart; one that trusts them to a fifth of a picosecond needs a factor of two between its lines and still gets the inductance to a few per cent at best.

A via that is mostly pad

The via so far has its balance just below fifty ohms, so the two terms in its area are nearly the same size and the separation is a matter of weighing two comparable quantities. A via that is mostly pad is a harder case for one of them.

A via's area changes sign at 22.4 Ω, and two impedances give its 1 pF and 0.5 nH back. computed by solving, not by drawing, as a cascade of two-ports: a via of 0.5 pF, 0.5 nH and 0.5 pF, met by an edge of 59 ps from reference lines of 20 to 150 Ω. The area under the reflection is −Z₀C/2 + L/2Z₀ at every impedance: a bump below 22.4 Ω, where the inductance's term is the larger, nothing at it, and a dip above. From 50 Ω the area is 20 ps of dip, which a single-capacitance reading calls 0.800 pF. From 50 and 75 Ω together the two areas give 1.0000 pF and 0.5000 nH. An error of 50 fs on each area moves them by up to 0.4% and 3%. — Fig. 6 A via of 0.5 pF, 0.5 nH and 0.5 pF from the same reference lines. Its area changes sign at $\sqrt{L/C}$ = 22.4 Ω. From 50 Ω it leaves 20 ps of dip, which a single-capacitance reading calls 0.800 pF. From 50 and 75 Ω the two areas give 1.0000 pF and 0.5000 nH; 50 fs of error on each moves them by up to 0.4% and 3%.

With a picofarad of pad and half a nanohenry of barrel the balance is at 22.4 ohms, far below the line, and from fifty ohms the via leaves twenty picoseconds of dip — read as a capacitor, 0.800 picofarads, which is at least the right order. Separated from fifty and seventy-five ohms, the pair returns 1.0000 picofarads and 0.5000 nanohenries exactly, and with fifty femtoseconds on each area the capacitance is good to 0.4 per cent and the inductance only to 3.

The capacitance has improved and the inductance has got worse, and both for one reason. The inductance’s term in the area is now a fifth of the capacitance’s from fifty ohms and smaller still from seventy-five, so the equations are separating a small number from a large one, and a fixed error on the area is a larger fraction of the small one. The rule that falls out is unsurprising once stated and easy to forget when planning a measurement: the separation recovers well whichever term dominates the area and recovers poorly the term that does not, and the worse-determined quantity is always the one that matters less to the reflection on the line being measured. A capacitive via’s barrel inductance is the number a coupon measures least well and a designer of that via needs least.

What the single trace can and cannot be asked

The measurement sorts the uses of one reflection into two.

The area is still the right number for what the reflection does. A receiver further down the line sees the via’s reflection, and the area is what sets how much of the edge it robs, whatever the via is made of. The receiver that is a branch is about that use, and it does not need the separation.

The area is the wrong number for what the via is. It cannot say whether a small dip is a small capacitance or a large capacitance mostly cancelled, and a via optimised by shrinking its dip on one line can be optimised into balancing its pads against its barrel — a via that is invisible on that line and reflects from every other. The separation needs two lines, and the two lines need to differ by more than the measurement’s own error can blur.

The model is two elements and a delay-free barrel. A via is a short piece of vertical transmission line with a discontinuity at each end, and the three-element model holds only while its round trip is short against the edge — the same fifth-of-the-edge boundary the dip whose area is fixed found for a section of wide track. Past it the area is still additive but the via is no longer two numbers, and two reference lines cannot recover more than two.

The staircase in time described what a line does with its ends; this and the essay before it describe what a line does with the things in its middle. And Kirchhoff’s own frequency is the reminder that every lumped element in this essay is lumped only because the edge is slow against its size.

Still open: the via on the line it balances, the loss in front of it, and the stub below it

The via whose area is zero. At $\sqrt{L/C}$ the area vanishes for every edge, and the first-order reflection with it. The via is still there: it has delay, and a reflection of the next order whose depth should fall faster with the edge than a lone element’s. The via that is a piece of line measures how much faster, and how much delay a balanced via adds.

The loss in front of the via. The areas here are measured on a lossless line. A real edge arrives at a via already slowed by the loss that the delay that is not one number measures, and comes back through the same loss. The area should survive a loss that is flat at low frequency, since the reflection coefficient is unchanged and the energy only rearranged in time, but the separation divides one small difference by another, and whether a loss that is not flat biases the two areas differently is unmeasured.

The stub below the via. A via that passes through a board and is used only to a middle layer leaves a length of barrel hanging below the connection, which is an open stub rather than an inductance. It reflects as a capacitance at low frequency and resonates at a quarter wave, and whether two reference lines still separate a via with a stub — or measure a stub’s capacitance as part of the pad’s — is the case that production vias actually present.