The far end that is a derivative

The near end that stops growing put a coupled pair into the time domain, solved exactly mode by mode, and found the near end a plateau: a quarter of the sum of the two couplings on the driven line’s voltage, held for the round trip, reached once the run is longer than half the distance the edge travels while it rises. Past that knee a longer run adds width and not height, and nothing about the edge’s speed matters.

The far end is the opposite in every one of those respects, and the reason was already in the frequency domain. The far end that cancels found the far end proportional to the difference of the two couplings, so it vanishes in a homogeneous dielectric where they are equal; how wide a null is found it returning as $|1 - r|/(1 + r)$ of the near end once the couplings part. In frequency, the far end grew a decade per decade — the signature of a derivative. In time, a derivative of an edge is a pulse, and its height is the edge’s slope.

This essay measures that pulse on a pair whose inductive coupling exceeds its capacitive coupling, as on a microstrip with part of its field in air, and finds three things the near end does not have: a sign, a dependence on the edge’s rise time, and a length at which the law ends for a reason that has nothing to do with the edge.

The same pair, made inhomogeneous #

The apparatus is the one the near-end measurement used: two 50-ohm tracks in a dielectric of relative permittivity four, all four ends terminated in 50 ohms, one track driven from a 50-ohm source, and the pair solved exactly as two independent modes — even and odd — each a uniform lossless line, returned to time from the exact harmonics of a trapezoidal drive. Its agreement with a 48-section coupled ladder network below 30 megahertz was measured there at 2.3 parts in 100,000.

The one change is the ratio of the couplings. The capacitive coupling stays at five per cent and the inductive is raised to eight — a ratio of 1.6. Physically that is what happens on a microstrip: the mutual inductance is set by geometry alone, but the mutual capacitance is reduced because part of the field between the tracks runs through air rather than through the board, so the capacitive coupling falls below the inductive one. In the modal picture it means the even and odd modes no longer travel at the same speed, and that, rather than the couplings themselves, is what the far end turns out to measure.

A pulse as wide as the edge #

At the quiet track’s far end, the capacitive and inductive contributions travel with the edge rather than against it, so every millimetre’s contribution arrives at the far end at the same instant: the moment the edge does. The far end collects nothing for most of the time and then everything at once, a pulse whose shape is the shape of the edge’s time derivative. For a linear ramp that is a rectangle as wide as the rise time.

Its height is the coupling difference, the time over which it was collected — the line’s delay — and the rate at which the driven line’s voltage changes:

$$V_{\text{far}} = K_f \, T_d \, \frac{dV_{\text{line}}}{dt}, \qquad K_f = \frac12\left(\frac{C_m}{C} - \frac{L_m}{L}\right).$$

For this pair $K_f$ is −0.015, the delay is 0.676 nanoseconds, and the driven line’s voltage rises by 0.5 volt per volt of drive in 200 picoseconds. The prediction is −24.96 millivolts per volt; the modal solution gives −25.16. The sign is negative because the inductive term wins: on a microstrip a rising edge on one track produces a falling glitch at the neighbour’s far end, the opposite polarity to the near end.

Proportional to length, with no knee #

The near end stopped growing once its round trip exceeded the edge. The far end has no equivalent, because its contributions do not spread out in time; they pile up.

The measured peaks follow the derivative law to within eight per cent at every length and every edge drawn, growing in proportion to the run. At 80 millimetres the 100-picosecond edge produces 40.28 millivolts per volt — two and a half times the saturated near end of the same pair — and the near end of that run has long since stopped. So for a fast edge on a long microstrip run the far end is the larger of the two, and it is the one that keeps getting worse with every extra millimetre of parallel routing.

This is why microstrip and stripline acquire different design rules. On a stripline the far end cancels and crosstalk is a near-end problem, bounded by the plateau whatever the length. On a microstrip the far end is a derivative, unbounded in length, and long parallel runs must be limited by length rather than only by spacing.

One over the rise time #

The derivative law has the edge’s slope in it, so the far end should scale as one over the rise time. The near end, once saturated, should not care.

The fitted exponent is −1.000 across a thirty-two-fold range of rise time, and the near end does not move until the edge becomes longer than the round trip. So a faster driver on the same board is a larger far end and the same near end, exactly. At 50 picoseconds the far end of a 100-millimetre microstrip run is 100.5 millivolts per volt, six times the near end; at 1.6 nanoseconds it is 3.14, a fifth of it.

That makes the far end the crosstalk that ages badly. A design whose crosstalk was budgeted with one logic family’s edges and then rebuilt with faster parts — the same board, the same routing — keeps its near end and multiplies its far end by the ratio of the edge rates. The parts are faster for good reasons, but the board did not change, and nothing about its layout says that its far-end crosstalk just went up threefold.

The sign belongs to the dielectric #

The far end’s sign and size are set by the coupling difference, which on a real board is set by how much of the field is in air.

In time as in frequency the far end is zero in a homogeneous dielectric — the time-domain pulse is absent to the last digit — and negative and growing on the microstrip side. The figure is the time-domain restatement of how wide a null is, and adds only the one thing the frequency-domain ratio did not show: on a microstrip the far-end pulse is inverted relative to the edge that caused it. A rising aggressor makes the victim’s far end dip, which at a receiver looking for a high level is the dangerous direction, since it eats margin from a signal that should be high.

A buried stripline sits at the left of this axis and a surface microstrip to the right; an embedded microstrip under a solder mask or a thin layer of resin sits between them, which is why covering a microstrip lowers its far-end crosstalk by pulling the effective permittivities of the two modes closer together.

The ratio of the two ends, read in time #

The frequency-domain essays gave the far end as a fraction of the near end, $|1 - r|/(1 + r)$ with $r$ the ratio of the couplings, and that fraction was independent of frequency because both ends grew a decade per decade. In time the two ends have different shapes — a plateau and a pulse — so the useful comparison is of their heights, and it is not independent of anything.

The saturated near end is $K_b V_{\text{line}}$ and the far end is $|K_f|\,T_d\,V_{\text{line}}/t_r$, so their ratio is

$$\frac{V_{\text{far}}}{V_{\text{near}}} = \frac{|K_f|}{K_b}\cdot\frac{T_d}{t_r} = \frac{2|1 - r|}{1 + r}\cdot\frac{T_d}{t_r}.$$

The first factor is the frequency-domain ratio doubled, a property of the stack-up; the second is the run’s delay counted in rise times, a property of the route and the driver together. For this pair the first factor is 0.46, so the far end overtakes the near end once the run’s delay is about two rise times. At 100 millimetres with a 200-picosecond edge the delay is 3.4 rise times and the far end is 1.55 times the near end, which is what the opening figure shows.

That splits the responsibility cleanly. The stack-up decides the first factor and cannot be changed after the board is made; the routing and the logic family decide the second and change every time a faster part is used. A board that was safe because its far end was small on its original parts can become unsafe by the second factor alone, and the near-end measurement, which was the one usually made, will not show it.

The same factor of the delay over the rise time also says when the lumped picture of crosstalk — a mutual capacitance and a mutual inductance between two nodes — is enough. While the run’s delay is a small fraction of the rise time, the coupled section is electrically short, both ends see the same small coupling and the far-end pulse is a fraction of a slow edge. That is the regime the comparison of a ladder network with a line locates for a single track; once the delay is comparable to the rise time the coupled pair has to be treated as a line, and the far end becomes the derivative measured here.

Mode skew connects the law to its own end. The far-end pulse is the skew between the even and odd modes, read through the edge’s slope: a skew $\Delta\tau$ that is short against the edge gives a far end of the line voltage times $\Delta\tau/2t_r$, and that skew grows in proportion to the run at a rate set by how unequal the modes’ permittivities are — the same quantity the skew in the glass measures inside a single differential pair, where it converts a differential signal into a common-mode one. Once the skew is as long as the edge, the expression has nothing left to multiply.

The law ends when the modes part #

A derivative law that grows without limit in length must fail somewhere, and the near-end measurement named where. The far-end pulse exists because the even and odd modes travel at different speeds, so their edges arrive at slightly different times; for short runs that difference is a small fraction of the edge, and the far end is the edge’s slope times that difference. When the difference exceeds the edge, the two modes arrive as separate steps.

For this pair the modes drift apart by 200.6 femtoseconds a millimetre, so a 50-picosecond edge is fully separated after about 249 millimetres. Below that the far end follows the derivative law. Above it, the far end is the difference of two complete steps, each a quarter of a volt per volt of drive, and the quiet line sees half the driven line’s voltage for the duration of the skew — 250 millivolts per volt, where the derivative law would have predicted more than three volts at 3.2 metres.

That limit is a ceiling rather than a relief. Half the driven line’s voltage is an enormous crosstalk; a run that reaches it has turned the pair into two lines carrying the same signal at two different speeds, which is the mode-conversion problem the millimetre that becomes common mode measures in a differential pair, here appearing on two single-ended tracks. The derivative law is not a small- signal approximation that improves at long length; it is the first part of a curve that ends at a disaster.

What the far end costs a design #

The derivative law gives a design rule in one line: the far end, as a fraction of the line voltage, is the coupling difference times the delay over the rise time, $|K_f|\,T_d/t_r$. To keep it under five per cent with this pair’s $|K_f|$ of 0.015, the delay may be at most $0.05/0.015 = 3.3$ rise times — 333 picoseconds, or 50 millimetres of microstrip, for a 100-picosecond edge. The measured curve agrees: at 80 millimetres the same edge gives 8.1 per cent of the line voltage. Halve the rise time and the allowed run halves with it, which is why fast microstrip buses are routed with their parallel sections short, or moved to stripline where the limit vanishes.

The rule also tells a designer what not to bother with. Increasing the spacing between tracks reduces both couplings and so both ends. Covering the microstrip with a dielectric reduces only the far end, by bringing the ratio toward one, and costs nothing at the near end. Slowing the driver reduces the far end in proportion and the near end not at all, past its knee. And termination at the victim’s far end matters: the pulse is measured into 50 ohms, and a high-impedance receiver at the far end sees it doubled, with a reflection then travelling back to the near end — the arrangement terminated at both ends is about for a single line.

What the model leaves out #

The lines are lossless. Loss attenuates the high-frequency content of the edge as it travels, so the edge reaching the far end of a long run is slower than the edge that left, and the far-end pulse — which is the edge’s derivative — is lower and wider than the lossless law says. Over long runs this softens the derivative law well before mode skew ends it, and the delay that is not one number measures the dispersion that does it.

The two couplings are constants along the run and frequency-independent. On a real board the effective permittivity varies with frequency, so the mode skew is itself frequency-dependent, and the skew-limited regime is reached at different lengths for the different harmonics of an edge.

And only two conductors are present. With several aggressors the far-end pulses from each add at the victim’s far end, and since they all arrive with their own edges, aggressors switching together produce a far-end glitch several times the single-aggressor height — the worst case on a parallel bus.

The number to keep #

Where the couplings are unequal the far end is the edge’s derivative: a pulse as wide as the rise time, arriving after one delay, of height $K_f T_d\, dV/dt$ — −25.16 millivolts per volt for 100 millimetres of microstrip-like coupling and a 200-picosecond edge, against −24.96 predicted. It grows in proportion to length with no knee, and as one over the rise time with a fitted slope of −1.000 while the saturated near end sits still; on a microstrip it is inverted relative to the edge. The law ends when the modes separate by a whole edge, 249 millimetres for a 50-picosecond edge on this pair, and the far end then stops at half the driven line’s voltage.

Still open: the lossy pair, several aggressors, and the receiver’s view #

A lossy coupled pair. With conductor and dielectric loss the edge slows along the run, and the far end, as its derivative, should fall below the lossless law at a length set by the loss rather than the skew. Which of the two limits arrives first, for a given board and edge, is a measurement on a lossy modal solution.

Several aggressors. A victim between two aggressors switching together receives both far-end pulses at once; switching in opposite directions, they cancel. The worst and best patterns on a bus, and how much a stagger in their switching times buys, follow from the single-aggressor pulse and its timing.

What the receiver sees. A far-end pulse as narrow as the edge may be shorter than a receiver can respond to; a receiver with a finite bandwidth filters it, and the filtered height is what decides whether a bit is misread.