The permittivity a loss forbids

The delay that is not one number found a line’s velocity moving with frequency and put the cause in the conductor: below the frequency where the series reactance overtakes the series resistance, a track is a diffusion rather than a wave, and its velocity goes as the square root of frequency. Above that crossing it stops, and the velocity becomes the constant $c/\sqrt{\varepsilon_r}$ that the rest of this field uses.

That last sentence is the one this essay is about. It assumes the permittivity is a number, which is how every datasheet presents it: a relative permittivity, and beside it a loss tangent, as though they were two independent properties a material happens to have.

They are not independent, and the dependence runs the wrong way for the assumption. A material that absorbs energy from a field does so because its polarisation lags, and a lag in the time domain is a fixed relationship between the real and imaginary parts of the response in the frequency domain. A loss tangent that is flat across a band — which is what every laminate datasheet claims — forces the real part of the permittivity to fall across that band, logarithmically, at a rate the loss tangent itself fixes. There is no material with a constant permittivity and a loss, and the model that assumes one has an edge arriving before it was sent.

Where the slope comes from, and why it is not fitted #

A dielectric’s response to a field is a convolution: the polarisation at this instant depends on the field now and on the field in the past, and never on the field in the future. That is the whole of causality as a statement about a material, and its consequence in the frequency domain is that the real and imaginary parts of the permittivity cannot be chosen separately.

The arrangement that produces a flat loss tangent is a continuum of relaxations spread uniformly in the logarithm of frequency, which is a reasonable description of a resin filled with glass and several other things. Its permittivity is $\varepsilon(\omega) = \varepsilon_\infty + K\ln((\omega_2 + j\omega)/(\omega_1 + j\omega))$ between the two frequencies $\omega_1$ and $\omega_2$ that bound the continuum, and well inside that band this is $\varepsilon_\infty + K\ln(\omega_2/\omega) - jK\pi/2$. The imaginary part is a constant, which is the flat loss. The real part falls as the logarithm of frequency, which is the price of it. Matching the pair to a quoted permittivity ε′ and loss tangent at one frequency fixes K, and the slope follows: the permittivity falls by (2/π)·ln 10 times ε′ times the loss tangent for every decade of frequency, with no free constant anywhere in it. For FR-4 at 4.4 and 0.02 that is 0.129 of permittivity a decade, and the model returns it rather than being given it — the figure above measures the slope off the computed curve and is required to match the expression to two parts in a thousand.

The proportionality is worth drawing on its own, because it is what makes dispersion something a designer already knows without being told. A material’s dispersion is not a second specification to go looking for. It is the loss tangent, read on a different axis. A laminate with a loss tangent of 0.001 loses 0.6 per cent of its permittivity over four decades; one at 0.004 loses 2.9 per cent; FR-4 at 0.02 loses 11.7 per cent. Low-loss materials are low-dispersion materials for the same reason, and not as a separate virtue.

Three laminates, and the reason the good ones are good twice #

Drawn side by side the three materials a board is likely to be made of separate by more than their loss.

PTFE-glass at ε′ = 2.2 and tanδ = 0.001 moves from 2.210 to 2.194 across five decades, seven parts in a thousand. A low-loss laminate at 3.5 and 0.004 moves from 3.562 to 3.459, about three per cent. FR-4 at 4.4 and 0.02 moves from 4.787 to 4.142, fourteen and a half.

The velocity goes as one over the square root of the permittivity, so those become 0.4, 1.5 and 7.5 per cent of velocity across the same range. A designer choosing a laminate for its attenuation is buying a second thing at the same time, and on a board where the timing budget is tighter than the loss budget the second may be the reason.

It also explains a persistent confusion about what a permittivity “is”. Two datasheets for the same laminate quoting 4.2 and 4.6 may both be right and may be measuring at different frequencies, and the difference between them is about three decades of the curve above. Neither number is a property of the material on its own; the pair (ε′, f) is, and the loss tangent is the third thing that ties the whole curve together.

What it costs a board, in picoseconds #

The delay of a length of line is its phase constant times its length over the angular frequency, and with the permittivity moving the delay moves with it.

Three hundred millimetres of FR-4 track — a long bus on a large board — has a delay of 2131.0 picoseconds at a hundred megahertz and 2068.1 at ten gigahertz. The difference is 62.9 picoseconds, just under three per cent, and it is 63 picoseconds that a timing budget written against one number does not contain.

The same geometry with the permittivity held constant and a conductance put beside it gives 2100.4 and 2099.1: a spread of 1.3 picoseconds, which is the conductor’s own resistive region tailing off and has nothing to do with the dielectric. Above the conductor’s crossing that model’s delay does not depend on frequency at all, to a part in ten thousand — a loss with no dispersion beside it, which is exactly the combination causality forbids.

Both curves cross the lossless delay $\ell\sqrt{LC}$ at the quoted permittivity, 2099.1 picoseconds, and that crossing is the useful way to read the figure. The number every schematic uses is not wrong; it is correct at one frequency, and that frequency is wherever the permittivity was quoted.

A femtosecond and a bit, per millimetre #

The spread is proportional to length, which makes it a per-millimetre number and therefore a design rule rather than an anecdote.

Between a hundred megahertz and ten gigahertz, FR-4 loses 209.6 femtoseconds of delay per millimetre of track. A straight line through the origin to a part in a million, because the permittivity’s variation is a property of the material and the delay is that property times the length. A hundred millimetres loses 21.0 picoseconds; four hundred loses 83.9; 1.6 metres of a long serpentine loses 335.4. For scale, a lumped discontinuity of a picofarad — the object the dip whose area is fixed measures — slows an edge by about twenty picoseconds, so a metre of ordinary track moves an arrival by as much as several vias do.

Two consequences follow immediately and neither is small.

Length matching between two nets is not affected. Two tracks of the same length on the same laminate lose the same picoseconds, so a matched pair stays matched. That is why this effect survives so long unnoticed on differential pairs, where matching is what is measured, and why the millimetre that becomes common mode can treat the two halves as sharing one velocity.

Length matching between two nets of different length is affected, and by more than it looks. A clock on a hundred-millimetre track and a datum on a four-hundred-millimetre one differ by 63 picoseconds of dispersion on top of the delay difference that was designed in, and the 63 picoseconds depend on the spectrum the signal actually has. A budget computed at one frequency and a signal with content across three decades are not the same calculation.

The shape of the edge, which is where the model gives itself away #

The delay figures are a difference between two models and a reader is entitled to ask which one is right. The answer is visible in the arriving waveform, and it does not require believing anything about materials.

A thirty-picosecond edge sent down three hundred millimetres of the dispersing line arrives taking 74.5 picoseconds to go from a tenth of its height to a half, and 221 more to reach nine tenths. A sharp toe and a long tail, which is what a causal filter does to an edge: energy that leaves at one time can arrive later and cannot arrive earlier, so a smeared edge is smeared forwards.

The same edge down the constant-permittivity line arrives taking 180 picoseconds to reach half height and 133 more to reach nine tenths — the other way round. Its lower half is slower than its upper half, which means the waveform started moving earlier than any causal filter could have made it move. Nobody put a precursor into the model; it is what a constant permittivity with a loss beside it is.

The practical consequence is not the philosophy. It is that the wrong model gives a wrong answer with the wrong sign: it reports 313 picoseconds of rise time where the causal line gives 295, so a simulation built on it says the board is worse than it is and says so in a way that cannot be calibrated out, because the error depends on the length.

The asymmetry, measured across five lengths #

The ratio of the time above half height to the time below it turns the previous figure into one number, and across five lengths it separates the two models cleanly.

The constant-permittivity line gives 0.86, 0.79, 0.70, 0.62 and 0.54 as the track grows from a hundred millimetres to 1.6 metres — always below one, and further below the longer the line. The dispersing line gives 3.06, 2.47 and finally 0.98 over the same range: above one where the dielectric dominates, and falling towards one at the longest lengths.

That last number is not a failure of the causal model, and it is worth being precise about what it is. The conductor’s loss in both lines is a resistance rising as the square root of frequency with no internal inductance beside it, which is the same non-causal shape one level down: a resistance that rises with frequency also has a reactance that rises with it, and this model has only the first half. Where the conductor’s loss dominates — which is what a 1.6-metre track at these frequencies is — the residual acausality of the conductor model pulls the ratio back towards one. Repairing the dielectric and leaving the conductor is a partial job, and the figure says so rather than hiding it.

The resistance that grows with frequency computes the resistance half of that pair. The reactance that goes with it is the same Hilbert relation applied to a conductor instead of to a dielectric, and it is the obvious next piece of this argument rather than a separate one.

What this changes, and what it leaves standing #

Three things elsewhere in this field are affected, and the first two are smaller than the argument suggests.

The attenuation is barely touched, so everything the cable that hides two things measures about loss stands as it is. At three hundred millimetres the two models give 2.794 decibels at a gigahertz — the same to four figures — and 16.69 against 16.67 at ten. Every loss budget in this field stands. It is the phase the two disagree about, so anything measured in decibels survives and anything measured in picoseconds does not, which is a clean division and not a coincidence: the loss tangent is what both models were given.

The crossing between diffusion and wave is unaffected, because it is set by the conductor’s resistance against the series inductance and the dielectric is not in it. The delay that is not one number puts that at 910 kilohertz for ordinary copper, and it stays there. What changes is what happens above it: that essay says the velocity becomes constant, and the honest statement is that it stops going as the square root of frequency and starts falling slowly and for a different reason.

And the characteristic impedance is a function of frequency too, by the same arithmetic and by about half as much — it goes as one over the square root of the permittivity, so it rises where the permittivity falls — which is the quantity the mismatch that the cable hides takes as a constant. The line designed at fifty ohms at a gigahertz is 49.38 ohms at a hundred megahertz and 50.75 at ten gigahertz, a spread of 2.8 per cent across two decades. That sits inside most impedance tolerances, which is why nothing has ever complained about it, and it means a reflectometer measuring a board with one rise time and a network analyser measuring it at one frequency are not measuring the same number. The constant-permittivity model gets this backwards as well: its impedance falls with frequency, from 51.10 ohms at ten megahertz to 50.00 at ten gigahertz, because with the permittivity pinned the only thing left moving is the conductor’s resistance.

That last comparison is the neatest summary of the whole difference. Both models are given the same geometry and the same loss tangent, and everything either of them says about magnitude agrees to four figures. Everything either of them says about time — the delay, the impedance’s trend, which half of an edge is the slow one — differs, and one of the two answers is the one causality permits.

Where the quoted number actually belongs #

One practical question falls out of all of this and is worth answering plainly, because it is the thing a designer has to do on Monday.

If a laminate is quoted at 4.4 and a board’s signals have content from ten megahertz to five gigahertz, what number goes into the field solver? The honest answer is that no single number works for both the impedance and the delay, because the two are sensitive to different parts of the band. A line’s characteristic impedance is set by the geometry and the permittivity at the frequencies where the reflection matters, which is near the edge’s own knee; a net’s delay is set by the permittivity across whatever the signal’s spectrum is, weighted by the energy in it.

For a single-frequency design the two coincide and the quoted permittivity at that frequency is exactly right. For an edge, they do not, and the practical rule that follows is to take the permittivity at the knee frequency — roughly 0.35 divided by the rise time — for the impedance, and to accept that the delay will be a few per cent longer than that value predicts because the lower half of the spectrum sees a larger permittivity. On FR-4 with a 30-picosecond edge that is a knee near twelve gigahertz, a permittivity near 4.13 for the impedance, and a delay computed at 4.4 that is about three per cent short.

None of that is a substitute for solving with the frequency-dependent permittivity, and the point of writing the rule down is to say how large the error it replaces is. Three per cent of a two-nanosecond bus is sixty picoseconds, which on a fast interface is most of a timing budget and on a slow one is nothing at all. The rule is worth what the budget is.

Still open: the conductor’s other half, a measurement, and the pair #

The conductor’s reactance. The half of this argument that is not done here is the internal inductance that must accompany a resistance rising as the square root of frequency. For a good conductor in the skin-effect regime the internal impedance is (1 + j) times the same quantity, so the reactance equals the resistance — and putting that into the series term would make the conductor’s contribution causal too. The measurement it would make is the one the asymmetry figure could not: whether a long line’s arriving edge is asymmetric in the causal direction once both losses are causal, and by how much the rise time moves.

The comparison a measurement could actually settle. Everything here compares two models. What would settle it is a measured phase across a decade on one board — the delay falling by 209.6 femtoseconds a millimetre between a hundred megahertz and ten gigahertz is a prediction a network analyser can check, and the prediction has no free parameters in it once the loss tangent is known. The same measurement would say whether the uniform continuum is the right description of a laminate or merely the simplest one with a flat loss tangent; a material with structure in its loss has structure in its permittivity, in exactly the places the loss has it.

The pair, where the two halves see different glass. A differential pair on woven laminate has one track over glass and one over resin, which is the local-permittivity problem the millimetre that becomes common mode treats as a skew. If the two regions have different loss tangents as well as different permittivities — and glass and resin do — then the skew between the halves is itself a function of frequency, and the mode conversion it produces has a shape rather than a size. That is a distinct measurement from either essay and it needs both.