Lines, where a wire has a length

The conductor made causal too

A dielectric that loses energy must disperse, and the essay before this one made FR-4's permittivity do so: the track's delay then falls 209.6 fs/mm from 100 MHz to 10 GHz. The copper has the same obligation. A conductor whose resistance rises as √f has an internal impedance (1 + j) times that rise, and putting the reactance beside the resistance more than doubles the fall, to 464.4 fs/mm — the conductor's share largest at low frequency, 418 of 584 fs/mm between 10 and 100 MHz, and the dielectric's constant at about 100 a decade. On a 30 ps edge after 300 mm the toe sharpens from 74.5 to 55.6 ps and the tail stretches from 221 to 398: the rise time is 453 ps against 295, and the tail-over-toe ratio a scope can read is 8.0 at 400 mm against 2.6.

14 min read 6 figures Every model has an edge Two routes to a number The parasitic is the component

Assumes: The delay that is not one number

The delay that is not one number found a lossy track’s delay depending on frequency, and the permittivity a loss forbids found that the dielectric cannot be modelled with a constant permittivity and a loss tangent beside it: a material that dissipates has a permittivity that falls logarithmically with frequency at a rate its own loss fixes, 0.129 of permittivity a decade for FR-4. Put into a track, that dispersion made the delay fall by 209.6 femtoseconds a millimetre between a hundred megahertz and ten gigahertz, and made an edge arriving after a long track lopsided in the causal direction — a sharp toe, a long tail — where the constant-permittivity model had it lopsided the other way, arriving before it could have.

That essay was explicit about what it had not done. The line’s other loss, the copper’s, was still a resistance rising as the square root of frequency with nothing beside it, and a resistance that rises with frequency is no more causal than a loss tangent with a constant permittivity. For a good conductor in the skin-effect regime the internal impedance is $(1 + j)$ times the same quantity: the reactance equals the resistance. This essay puts the reactance in and measures what changes.

Both halves causal

The track is the earlier one: fifty ohms, FR-4 with a permittivity of 4.4 and a loss tangent of 0.02 at a gigahertz, two ohms a metre of copper at direct current rising as $\sqrt f$ above a megahertz. Three models of it are compared: nothing dispersing, the dielectric causal, and the dielectric and the conductor causal together.

The delay of 50 Ω FR-4 track against frequency, with nothing dispersing, the dielectric causal, and the conductor causal as well. computed by solving, not by drawing at 108 frequencies on one track: 4.4 and a loss tangent of 0.02 at a gigahertz, 2 Ω/m of copper rising as √f above 1 MHz. Held constant (faint), the delay is flat above the resistance's corner. With the permittivity allowed to disperse (dashed), it falls 209.6 fs/mm from 100 MHz to 10 GHz. With the conductor's internal reactance put beside its resistance (solid), it falls 464.4 fs/mm over the same two decades: 7.389 ps/mm at 100 MHz, 7.094 at 1 GHz, 6.925 at 10 GHz. — Fig. 1 The delay of 50 Ω FR-4 track from 1 MHz to 30 GHz. Held constant (faint), flat above the resistance’s corner. Dielectric causal (dashed): it falls 209.6 fs/mm from 100 MHz to 10 GHz. Both causal (solid): it falls 464.4 fs/mm over the same two decades — 7.389 ps/mm at 100 MHz, 7.094 at 1 GHz, 6.925 at 10 GHz.

With only the dielectric causal the delay falls 209.6 femtoseconds a millimetre from a hundred megahertz to ten gigahertz, the earlier essay’s number exactly. With the conductor’s internal reactance put beside its resistance it falls 464.4 femtoseconds a millimetre over the same two decades: 7.389 picoseconds a millimetre at a hundred megahertz, 7.094 at a gigahertz, 6.925 at ten. More than half of the dispersion between those frequencies belongs to the copper.

The model with neither causal is flat above its resistance’s corner and drawn faint for that reason: it is the model in which an edge can arrive early. Below a few megahertz all three rise steeply, because the copper’s resistance is then comparable with the line’s reactance and the line is no longer a line in any useful sense.

The internal reactance, beside the external

The conductor’s reactance is not large. Its effect on the delay comes from how it varies, and the series impedance per metre makes that plain.

Fig. 2 The track’s series impedance per metre: the copper’s resistance rising as

\sqrt f

above 1 MHz (dashed), its internal reactance equal to the rising part (solid), and the external reactance ωL (faint). At 100 MHz the internal reactance is 18.3 Ω/m against 224 Ω/m external, 8.2%; at 10 GHz, 0.88%.

At a hundred megahertz the internal reactance is 18.3 ohms a metre against 224 of external reactance, 8.2 per cent of it; at ten gigahertz it is 0.88 per cent. The external reactance is $\omega L$ and contributes a fixed inductance; the internal reactance grows as $\sqrt f$ and so contributes an inductance that falls as $1/\sqrt f$ . A small share of the line’s inductance that varies with frequency is enough, because the delay is the square root of the inductance times the capacitance, and a share that changes by ten per cent over two decades changes the delay by five.

The physics is the skin effect seen from the other side. The resistance that grows with frequency measured the current crowding to a conductor’s surface and its resistance rising as the conducting layer thins. The same crowding removes magnetic field from the conductor’s interior, and the field that was there is inductance that the line loses as frequency rises. The resistance’s rise and the inductance’s fall are one phenomenon, which is why they come as a pair with equal magnitudes: $(1 + j)$ .

The edge, lopsided further

The place the earlier essay found its model giving itself away was the shape of an edge arriving after a long track. The same measurement, with the conductor made causal too:

A 30 ps edge after 300 mm of FR-4 track, with nothing dispersing, the dielectric causal, and both halves causal. computed by solving, not by drawing: each line's transfer applied to the edge's spectrum and inverted. Held constant (faint): 180 ps from a tenth to a half and 133 ps from a half to nine tenths — the wrong way round. Dielectric causal (dashed): 74.5 ps and 221 ps, rise time 295 ps. Both causal (solid): 55.6 ps and 398 ps, rise time 453 ps. The conductor's internal inductance, largest at the low frequencies an edge's tail is made of, delays the tail most: the toe sharpens and the tail stretches. — Fig. 3 A 30 ps edge after 300 mm of FR-4 track. Held constant (faint): 180 ps from a tenth to a half and 133 ps from a half to nine tenths, the wrong way round. Dielectric causal (dashed): 74.5 and 221 ps, rise time 295 ps. Both causal (solid): 55.6 and 398 ps, rise time 453 ps.

After three hundred millimetres, with only the dielectric causal, the edge takes 74.5 picoseconds from a tenth to a half and 221 from a half to nine tenths: the causal shape, toe sharp and tail long, rise time 295. With the conductor causal as well it takes 55.6 and 398: the toe sharper still, the tail nearly twice as long, and the rise time 453 picoseconds. The model with nothing dispersing has it the wrong way round, 180 against 133, and the earlier essay explained why that is impossible for a physical line.

The reason the tail stretches is where the conductor’s dispersion lives. An edge’s toe is made of its highest frequencies and its tail of its lowest, and the internal inductance is largest at low frequency, so the low frequencies travel slowest and arrive last. The copper’s causality is therefore a statement mostly about the tail of an edge — which is where a receiver’s threshold sits when a signal is attenuated, and where inter-symbol interference comes from in a long serial link.

The edge the loss delivers found that a lossy line makes an arriving edge steep in the middle and slow at both ends, three times steeper at its centre than a raised cosine of the same rise time, and that a discontinuity downstream answers the steep middle rather than the rise time. Making both halves causal keeps the steep middle and moves the slowness from the toe to the tail: an edge whose first half arrives in 55.6 picoseconds is an edge that a via or a connector sees as fast, whatever its ten-to-ninety figure says. A reflection budget written against the rise time of 453 picoseconds would be written against the wrong number by a factor near eight.

The tail in a serial link

Put the tail beside a unit interval. At five gigabits a second a bit lasts 200 picoseconds; after 300 millimetres the dielectric-causal model has an edge finish its last forty per cent in 221 picoseconds, a little over one bit, and the model with both halves causal takes 398, very nearly two. The difference is a whole bit of residue: the energy of one transition still arriving while the receiver decides the bit after next. That is what an equaliser’s post-cursor taps exist to subtract, and their number and weights are set from a model of the channel. A channel model with a non-causal conductor tells the designer the tail is half as long as it is, and an equaliser trained on that model is trained on a different line.

At ten gigabits a second the unit interval is 100 picoseconds, and the comparison changes character. The tail is then four bits long in either causal model, and a measured channel is what sets the taps; but the toe, 74.5 against 55.6 picoseconds, is now most of a bit, and it decides how much of each transition lands inside its own interval. Over the decade where that link’s energy lives, the conductor’s share of the delay’s fall is 189 of 295 femtoseconds a millimetre, so it is not the small correction to a laminate’s dispersion it is usually treated as.

On a metre of cable the same physics acts on a slower scale. The staircase in time followed a source into a metre of cable and back, its steps set by the impedance and the 4.83-nanosecond delay; the corners of those steps are rounded by exactly this dispersion, and their tails are the copper’s, because a cable’s dielectric is a low-loss plastic and nearly all of its loss is in its conductors.

A shape a scope can read

The ratio of the tail to the toe is a measurement with no free parameters: it needs only the arriving edge and two crossings either side of half height. The earlier essay drew it against length for its two models; here is the third.

How lopsided the arriving edge is, against the length of track, for the three models. computed by solving, not by drawing for a 30 ps edge at five lengths. A symmetric edge reads one. Held constant (faint) the ratio is 0.98, 0.92, 0.85, 0.79, 0.73. Dielectric causal (dashed): 1.34, 2.07, 3.04, 3.48, 2.59. Both causal (solid): 1.63, 2.97, 5.09, 6.86, 8.03. The measurement a scope can make on a long track — whether the tail is longer than the toe, and by how much — separates the three models at every length. — Fig. 4 The arriving edge’s tail over toe, $(t_{90} - t_{50})/(t_{50} - t_{10})$ , for a 30 ps edge at 25 to 400 mm. Held constant (faint): 0.98 to 0.73, below one. Dielectric causal (dashed): 1.34 to 2.59. Both causal (solid): 1.63, 2.97, 5.09, 6.86 and 8.03.

A symmetric edge reads one. The constant model reads 0.98 at 25 millimetres and 0.73 at 400 — below one, the backward smear. The dielectric-causal model reads 1.34 at 25 millimetres and peaks near 3.5 at 200. The model with both halves causal reads 1.63, 2.97, 5.09, 6.86 and 8.03 from 25 to 400 millimetres, rising at every length. At every length the three are distinct by a margin a scope with a twenty-gigahertz bandwidth would resolve, so a single measurement of an edge through a long test coupon — a toe and a tail, read off the screen — says which model a laminate and a copper surface are better described by.

That is the measurement the earlier essay asked for, and it has become sharper rather than blurrier with the conductor added. The dielectric model alone would predict a tail-over-toe of about 2.6 at 400 millimetres; a measured eight would say that the copper’s dispersion dominates the tail on that board, which a model fitted to insertion loss alone cannot say, since both mechanisms lose energy in proportion to frequency over most of the band.

The impedance has a slope

The characteristic impedance moves too, because the internal inductance is part of the series impedance it is a ratio of.

The magnitude of the track's characteristic impedance against frequency, dielectric causal alone and both halves causal. computed by solving, not by drawing. With the dielectric alone causal (dashed) the impedance falls from 61.2 Ω at 1 MHz to 50.01 Ω at 1 GHz; with the conductor's reactance in as well (solid), from 65.8 Ω to 50.70 Ω and 50.98 Ω at 10 GHz. A termination chosen at one frequency is half an ohm out at another, which is a reflection of half a per cent nobody would design. — Fig. 5 The magnitude of the track’s characteristic impedance from 1 MHz to 30 GHz, dielectric causal alone (dashed) and both causal (solid). The first falls from 61.2 Ω at 1 MHz to 50.01 Ω at 1 GHz; the second from 65.8 Ω to 50.70 Ω, and 50.98 Ω at 10 GHz.

With the dielectric alone causal the impedance falls from 61.2 ohms at a megahertz to 50.01 at a gigahertz; with the conductor’s reactance in as well, from 65.8 to 50.70, and it is 50.98 at ten gigahertz. A termination chosen to match the line at one frequency is about half an ohm out at another, a reflection of half a per cent that nobody would design and that a time-domain reflectometer reads as a slow drift of the impedance along the trace. The mismatch that the cable hides found loss hiding a reflection from an instrument; here the loss’s own partner, the internal inductance, makes a small one.

Whose dispersion, decade by decade

The two causal halves disperse by different laws, and the split between them changes across the band.

Fig. 6 The fall in the track’s delay over each decade, split between the conductor’s dispersion (darker) and the dielectric’s (lighter). 10 to 100 MHz: 583.9 fs/mm, the conductor 418.2. 100 MHz to 1 GHz: 295.3, the conductor 188.9. 1 to 10 GHz: 169.1, the conductor 65.9 and the dielectric 103.3.

From ten to a hundred megahertz the delay falls 583.9 femtoseconds a millimetre, 418 of them the conductor’s; from a hundred megahertz to a gigahertz, 295.3, of which 189 the conductor’s; from one to ten gigahertz, 169.1, of which only 66 are the conductor’s and 103 the dielectric’s. The dielectric’s share is nearly the same in every decade, because a flat loss tangent makes the permittivity fall logarithmically, by the same amount a decade; the conductor’s falls as $1/\sqrt f$ , so it dominates below a gigahertz and fades above it.

That split decides which half of the model matters for which signal. A one-gigabit link, whose energy is mostly below a gigahertz, sees a line whose dispersion is mostly the copper’s; a twenty-five-gigabit link sees mostly the laminate’s. Board materials are chosen for their loss tangent, which is the right decision for the fast link and only half of the right decision for the slow one.

The two halves also answer to different parts of a board’s construction, which is what makes the split useful rather than merely descriptive. The dielectric’s dispersion is a property of the material and is the same for every track in it, whatever its width. The conductor’s is proportional to the resistance per metre, so at a fixed fifty ohms it falls as the track is widened — which a fifty-ohm track can be only by putting the reference plane further away, in a thicker dielectric. A stack-up with thicker cores carries wider tracks, less copper loss and less copper dispersion in proportion, and leaves the laminate’s share untouched. The decade-by-decade split is therefore a split between the choice of material and the choice of geometry.

A differential pair adds a third consideration, because its two conductors need not see the same dielectric. The skew in the glass measured a pair over glass and resin building 0.318 picoseconds of skew a millimetre with each half’s permittivity held constant. With each half dispersing at its own rate the skew becomes a function of frequency, and the millimetre that becomes common mode measured what skew turns into once it arrives. The conductors of the pair are the same copper, so the conductor’s dispersion is common to both and very nearly cancels in the skew; the dielectric’s does not, and that difference is the next measurement.

Why the conductor’s half was left out, and what that cost

A model with a $\sqrt f$ resistance and no internal reactance is common, and it is not unreasonable: the reactance is a few per cent of the line’s, and in insertion loss — the measurement most models are fitted to — it does nothing at all, because a reactance stores rather than dissipates. A model fitted to loss alone will therefore never discover it is missing. It shows up only in timing: in the delay’s dispersion, the edge’s tail and the impedance’s slope, which are exactly the quantities this essay measured, and each of them moves by a factor of one and a half to two.

The copper that makes it worse found proximity effect raising a winding’s resistance beyond what the skin effect alone gives; on a track, surface roughness does the same, raising the resistance at high frequency by a factor that saturates near two. A rough conductor’s internal reactance rises with it, by the same causality, so a roughness model added to the resistance alone repeats this essay’s omission at a larger scale.

How the lines were solved, and what the conductor model assumes

Each line is the earlier essays’ model — a series resistance and inductance, a shunt capacitance and conductance, per metre — with the propagation constant computed at each frequency; the delay is the phase constant’s length times the line over the angular frequency, and the impedance the square root of the series over the shunt. The dielectric is the wide-band Debye form, its permittivity and loss one complex quantity. The conductor’s internal reactance is taken as the resistance’s frequency-dependent part, $R(f) - R_\text{dc}$ , which is exact in the skin regime and rises linearly in frequency below it, as an internal inductance does. A full solution of the conductor’s own field — a Bessel-function solution for a round wire, a numerical one for a rectangular track — would differ from this in the transition around a megahertz and agree above it. Edges are the line’s transfer applied to the edge’s spectrum and inverted on 65,536 points.

Still open: roughness, the measurement on a coupon, and a ground plane’s own skin

Roughness with its reactance. A rough copper surface raises the resistance by a factor that grows with frequency and saturates. Put in with its causal partner, it adds dispersion in the decade where the roughness factor changes fastest, a few gigahertz for ordinary foil, and the conductor’s share of the delay’s fall in that decade would rise rather than fade. How much, for a stated surface, is a solve of this model with the roughness factor and its Hilbert partner in it.

A measured coupon. Everything here compares models. A coupon of 400 millimetres measured with a fast edge would put a number on the tail-over-toe ratio directly, and the three models predict 0.73, 2.6 and 8.0 for it. A laminate supplier’s own data would supply the loss tangent; the measurement would then say whether the conductor’s causality is visible on a real board, with no fitting at all.

The return conductor. The ground plane under a track carries the return current in its own skin and contributes its own resistance and internal reactance, usually less than the track’s because the current spreads wider. Where the current comes back found the return gathering directly beneath the track above a corner near a hundred kilohertz, so in the band this essay measures the plane’s current is concentrated, and its internal reactance is not negligible beside the track’s. Splitting the conductor’s share between the track and the plane would say how much of the dispersion a heavier plane removes, and whether the plane’s share is worth a model of its own.