The delay that is not one number

Every line in this field so far has had a delay. A length, a velocity, and a number: 7 nanoseconds per metre on FR-4, the same for every frequency, and the edge that comes out is the edge that went in with everything shifted by that number. Nine essays are written on top of that assumption and it is correct for the lines they are about.

It is a property of a lossless line, and it is the first thing to go.

A real conductor has a series resistance. Below the frequency at which the series reactance overtakes it, the line is not a wave at all: it is a diffusion, of the kind heat is, and a diffusion has no velocity in the usual sense — its phase velocity is proportional to the square root of frequency, so the faster part of a signal arrives first and the slow part arrives whenever.

Two losses, entering in different places #

The propagation constant of any uniform line is

$$\gamma(\omega) = \sqrt{(R + j\omega L)(G + j\omega C)}$$

and the whole of this essay is what happens when $R$ and $G$ are not zero and are not constants either.

The series resistance rises as $\sqrt f$. Above a frequency set by the conductor’s own dimensions, the current stops filling it and flows in a layer whose depth falls as $1/\sqrt f$. This collection already measures that, in the magnetics field, where a winding’s resistance grows with frequency for the same reason and by the same expression — it is the same physics arriving in a different component.

The shunt conductance rises as $f$. The board material dissipates in proportion to the electric field’s rate of change, and the coefficient is the loss tangent, so $G = \omega C\tan\delta$ by definition rather than by model.

Two mechanisms with two different frequency dependences, and the rest of the essay is about telling them apart from the outside.

Below the crossing, nothing in this field applies #

At low frequency the series term is nearly real — $R \gg \omega L$ — and $\gamma$ becomes $\sqrt{j\omega RC}$. Both its real and imaginary parts go as $\sqrt\omega$, so:

• the phase velocity is $\omega/\beta \propto \sqrt\omega$, measured at the 0.50 power exactly;
• the characteristic impedance $\sqrt{R/j\omega C}$ rises without bound as the frequency falls — 1,508 Ω at a kilohertz on this trace against 50 Ω at ten gigahertz;
• and there is no delay, because there is no velocity.

The crossing is at $R/2\pi L$, which for two ohms per metre and 350 nH/m is 910 kHz — and it does not depend on the length at all, because both quantities are per metre.

That number is the boundary this whole field has been standing above without saying so. Every essay about a staircase in time, a termination, a stub or a quarter-wave section is written at frequencies far above it. The one thing in the field that lives below it is the return path essay’s low-frequency regime, where a current comes back the way of least resistance rather than least inductance — and that crossing is the same crossing, in the same quantity, arrived at from the other side.

What the crossing means for a board, and what it does not #

It is worth being careful about what 910 kilohertz is a boundary of, because the number is easy to misread in either direction.

It is not a frequency below which signals do not travel. Direct current travels down a trace perfectly well; what it does not do is travel as a wave with a characteristic impedance and a delay. Below the crossing the trace is a resistance in series with a capacitance to the plane, which is a low-pass filter, and it behaves as one — which is exactly how anybody would model a trace at those frequencies without thinking about it.

It is not a boundary that moves with length, either. Both $R$ and $L$ are per metre, so their ratio is a property of the cross-section — the copper’s width and thickness against its height above the plane — and a trace ten times longer has the same crossing frequency. What length does change is how much attenuation there is at any given frequency, which is a different quantity.

And it is not one boundary but two, because there is a second frequency in the same expression. The skin effect has its own corner, where the current stops filling the conductor, and above it the resistance is rising while the reactance is rising faster. On the trace here the two are a decade apart; on a thick conductor or a thin one they can be much closer, and then there is no clean diffusive regime at all — the line goes from resistive straight into skin-effect-limited without ever being the textbook RC line.

The edge, built by multiplying a spectrum #

To see what a line does to an edge there is no filter to march. The propagation factor is $e^{-\gamma(\omega)\ell}$, which is a different complex number at every frequency and has no rational form, so there is no difference equation that is it and no ladder of a finite number of sections that is it either.

So the edge is built in the frequency domain: transform it, multiply by the propagation factor at each bin, transform back.

One detail decided the answer and it is worth stating. The input is a pulse, not a step. Multiplying a spectrum and transforming back is a circular convolution, so a record whose two ends do not match wraps its own discontinuity into the answer — and a step does not match its own ends. The first version of this used a step and what came back was the wrapped falling edge sitting on top of the rising one, at a level near one everywhere and a delay of 539 nanoseconds on a 42-nanosecond record.

The two quantities, and only one of them is a delay #

Down a metre of this trace, a 60 picosecond edge:

• arrives at 6.989 nanoseconds against the lossless line’s 6.997 — the same number to three figures;
• has become 1,927 picoseconds wide, thirty-two times the edge that was sent;
• and reaches 96 per cent of the level it started at rather than 100.

The first of those is the finding. However badly the loss smears the edge, the fifty per cent point arrives when $\ell\sqrt{LC}$ says it will. The dispersion is not moving the signal; it is spreading it, symmetrically enough about the half point that the half point does not move.

So “the delay of this line” is still a quantity, and it is still the one every other essay in this field uses. What is not a quantity is the idea that the edge arrives — because there is a second number, and it does not scale the way the first one does.

The left-hand column doubles when the length doubles. The right-hand column more than triples.

Two exponents, and the diagnosis they are #

The rise time’s growth with length is not one law, and separating it is the sharpest result here.

Send the same edge down a line whose only loss is the conductor, and down one whose only loss is the laminate, and fit the exponent over two decades of length:

A loss that rises as the square root of frequency smears an edge as the square of the length. A loss that rises in proportion to frequency smears it linearly. Those are different laws rather than different constants, and the reason is one line of arithmetic: the frequency at which the line has attenuated by a fixed amount is where $\alpha(f)\ell$ is constant, so for $\alpha\propto\sqrt f$ the usable bandwidth falls as $1/\ell^2$ and for $\alpha\propto f$ it falls as $1/\ell$.

That makes the exponent a diagnosis. Double the length of a trace and measure four times the rise time and the copper is the problem — a wider trace or a thicker foil helps. Measure twice and the laminate is the problem, and no amount of copper will do anything: the material has to change.

The combined exponent of 1.624 is where an ordinary board sits, and it moves with the length: at short lengths the laminate’s linear term is the larger of the two and at long ones the copper’s quadratic term overtakes it. Which regime a given design is in is a question about its own dimensions rather than about the materials, which is why the number is measured here rather than quoted.

Where the design rule sits #

The practical form of all of this is a length, and it comes out of the two exponents rather than out of either.

A designer has an edge — say a hundred picoseconds, which is an ordinary logic output — and a tolerance for how much of it may be lost, say twenty per cent. On this trace the added rise time reaches twenty per cent of the edge somewhere around a decimetre, and that is the length beyond which the receiver is no longer seeing the transmitter’s edge.

That number is not the same as the length at which the trace has to be treated as a transmission line, which this collection measured two fields ago and put at a fraction of the rise time times the velocity — a few centimetres for the same edge. The two are close enough to be confused and they are about different failures: the first is a reflection, which a termination fixes, and the second is a smear, which nothing at the ends can fix at all.

That is the sharpest way to put what this rung adds. Everything else in this field is a boundary a terminator can move. This one is not: the loss is distributed along the trace, it acts on the signal continuously, and there is no component that can be added at either end to undo it — only an equaliser, which is a filter shaped like the inverse of the line and is a different subject.

What this does to everything else in the field #

Three of this field’s results have a boundary here that they do not carry, and it is worth being explicit about which.

A staircase in time is drawn on a lossless line, so each step is the previous one multiplied by a reflection coefficient and nothing else. On a real trace each round trip also smears, so a staircase of many steps has steps that are not steps by the end of it — and the number of visible steps is set by the loss rather than by the reflection coefficients.

A termination’s plateau assumes the far end sees the same waveform the near end sent. Over a metre of this trace with a hundred-picosecond edge, it does not: it sees a two-nanosecond one, and a termination mismatch measured on that edge is being measured on a different signal.

And the characteristic impedance a designer terminates into is a high-frequency limit. Fifty ohms is what this trace presents above about ten megahertz. Below that it is not fifty, and a termination is correct for the part of the spectrum that matters and wrong for the rest — which is invisible in a reflection measurement because there is nothing down there to reflect.

What a frequency-dependent delay changes #

Nine essays in this field quote a delay as a length over a velocity, and this one makes it a function of frequency. The mismatch that the cable hides is where the same loss hides a reflection from the instrument end. The staircase in time is the lossless picture the whole field is drawn on, and The resistance that grows with frequency is the mechanism behind one of this page’s two exponents.

What is checked #

The high-frequency velocity is asserted against $c/\sqrt{\varepsilon_r}$ to two per cent, so the model reduces to the field’s own lossless line where it should. The low-frequency velocity is asserted to be several times below it, and the exponent between them fitted over the points below the crossing and required to be a half — a diffusion’s law, rather than merely “slower”.

The arrival is asserted against $\ell\sqrt{LC}$ to two per cent at every length and on every one of the three loss cases, which is the claim that the loss does not move the signal. The rise time is asserted to grow, and the two exponents are asserted separately: between 1.85 and 2.1 for the conductor, between 0.9 and 1.1 for the laminate, and the combination strictly between them. A single assertion on the combination would have passed for a model with one mechanism in it.

The edge’s own rise time is removed in quadrature before the exponents are fitted, because at the shortest length the source dominates and a fit that includes it measures the source. The lossless comparison line is required to deliver the edge it was given — 0.59 of the raised cosine’s own width, which is that shape’s 10–90 — so a bug in the synthesis would fail before any dispersion was attributed to a line.

One residual is bounded rather than removed. The lossy record starts at 3.4 per cent of its plateau rather than at nothing, and that is the diffusive tail — a diffusion’s response has no end, so no finite record contains it and what does not fit wraps round. The lossless comparison line’s record starts at six parts in ten million, which is what says the pedestal belongs to the physics rather than to the transform. It is asserted to be between one and six per cent, so a version whose wrap grew would fail rather than be absorbed.

What is not modelled: the reflections, since both ends are matched to the line’s own impedance at every frequency, which is not a terminator anybody has; the surface roughness of the copper, which makes the skin-effect exponent larger than a half in real laminates; the frequency dependence of $\varepsilon_r$ itself, which is a third dispersion mechanism and is small; and any coupling to a neighbouring trace, which this field measures elsewhere on a lossless line.

What nine essays assumed, and how much it cost them #

A delay that is a length divided by a velocity is assumed by every other essay in this field, and it is worth saying which of their results survive the correction and which do not, because the answer is not uniform.

The results about arrivals survive intact. This essay’s own second finding is that the arrival is still exactly linear in the length even in the diffusive regime, so the staircase in time’s 4.83 nanoseconds for a metre, and every lattice-diagram interval built on it, are unaffected: the wave arrives when it arrives. The resistor at the wrong end’s $2(1-x)$ delays and the receiver that is a branch’s two-thirds-for-two-stub-delays are both ratios of times, so a common factor in the velocity cancels out of them entirely.

The results about edges do not survive, and they are the ones a digital designer reads the field for. A rise time growing as the square of the length is a different statement from a rise time preserved, and it means the half-swing interval that the resistor at the wrong end measures as a clean plateau is in reality a plateau with a soft edge whose softness depends on how far along the net it is being looked at. On a short board that is negligible; over a backplane it is the reason the same net is readable at one connector and not at another.

And a ladder is not a line is the essay whose convergence argument this one qualifies most sharply. Forty lumped sections still ring by five per cent and about nine hundred and sixty would be needed for one per cent — but that is a count against a lossless line. A real trace below 910 kilohertz is not the thing the chain is converging on, so the extrapolation is to a limit that ordinary copper does not reach at low frequency and does reach above it.

The one result the correction strengthens rather than qualifies is the mismatch that the cable hides, whose whole argument is about a loss that this essay’s mechanism supplies. Its hiding length falling as one over the root of frequency is the skin effect measured here, read through a different question — so the two essays are one measurement of copper, used once to explain a delay and once to explain a flattered reading.