The parasitic that moves the start

The start a step takes from infinity read the initial-value theorem as a statement about the whole start of a step: a network whose denominator is $r$ degrees above its numerator steps as $(t/\tau)^r$ at first, with the power set by the relative degree and the coefficient by the network’s gain at infinite frequency. It added a zero to a two-section cascade and watched the start turn linear, and the linear start last until about twice the zero’s time constant.

That essay ended with the uncomfortable consequence. A real layout adds parasitics, and a single pole far above a network’s bandwidth — a few picofarads against a driver’s output resistance — raises its relative degree by one, from $r$ to $r + 1$. A parasitic zero, from capacitive feed-through round a stage, lowers it by one. By the theorem a real step starts at a different power of time from its schematic’s, and which way depends on which parasitic the layout adds more of. The threshold a delay is quoted at then found that the start is what a low threshold measures. So the question is how much of a step a parasitic’s change of power actually governs, and what it does to the delays a receiver quotes.

The answer has two parts with different sizes, and the second is the one that matters.

A power of time with an expiry #

The network is the one the earlier essay used: two buffered RC sections of time constants 1 ms and 0.5 ms, a relative degree of two, a step that starts as $(t/\tau)^2$. The parasitic is either one more buffered section of time constant $\tau_p$, or a zero at $1/\tau_z$ made by adding the output’s derivative to itself, and in every figure it is a hundred to a thousand times faster than the network.

The theorem is right. With a 10 µs pole the step starts as the cube of time, a line of slope three on these axes; with a 10 µs zero it starts as the first power, slope one. Each is exactly what the relative degree says, and the coefficients are exact too: well inside the parasitic’s time constant the pole multiplies the bare step by $t/((r + 1)\tau_p)$ and the zero by $r\,\tau_z/t$, both confirmed on the expansion about infinity to two per cent at a hundredth of the time constant.

The expiry is in the same two expressions. The pole’s factor $t/((r + 1)\tau_p)$ is small only while $t$ is small against $(r + 1)\tau_p$; the zero’s $r\,\tau_z/t$ is large only while $t$ is small against $r\,\tau_z$. On the logarithmic axes the new asymptotes cross the bare one at $(r + 1)\tau_p$, 30 µs, and at $r\,\tau_z$, 20 µs, and past those points each curve bends back onto the bare step’s slope. The slider on the figure at the head of the page steps the parasitic from a microsecond to a hundred, and the crossings move with it in proportion. A parasitic governs the start’s power of time for a couple of its own time constants and for no longer — which, since it is a parasitic, is a very small fraction of the step.

Three, then two #

The same thing can be read off without asymptotes, as the local power of time the step is rising with at each instant.

Bare, the exponent is two at the start and falls as the step approaches its final value, as every step’s must. With the pole it is 2.99 a thirtieth of the pole’s time constant in, and with the zero 1.02. Each holds its value for less than a decade of time and hands back over about a decade. By thirty of the parasitic’s time constants all three curves agree to 0.067, and after that they are indistinguishable on this axis.

So the theorem’s reading of the start is exact and local. The relative degree governs the step for as long as the element that set it takes to act, and a parasitic element acts quickly. The practical form of the statement is that the power of time a real step starts with is a property of its fastest element, and the power it rises with over any interval a receiver would care about is a property of its slowest.

What the parasitic does after its time is up #

Once its time constant has passed, a parasitic no longer changes the shape of the step. It changes its timing, and by a fixed amount.

For a pole the rule is simple and the figure confirms it to six per cent at every threshold the bare step reaches later than ten of the pole’s time constants: the pole adds its own time constant to the delay. That is the convolution of the step with a short exponential, which delays anything slow by the exponential’s mean, and the mean of an exponential is its time constant. It is the same statement as the sum of time constants in a sum that is exact, where the first two coefficients of the denominator give the step’s centre of area and each pole adds its time constant to it; here it holds not only for the centre but for every threshold crossed long after the parasitic has acted.

What differs between thresholds is how much that time is worth. The bare step reaches 1 per cent at 105 µs and 50 per cent at 1.23 ms, so a 10 µs parasitic pole is 9.10 per cent of the 1% delay and 0.817 per cent of the 50% delay — a ratio of 11.1, which is the ratio of the two delays, as it has to be when the added time is the same. A 1% receiver sees a parasitic eleven times as large as a 50% receiver does, on the same signal, not because the parasitic acts differently at low thresholds but because a low threshold’s delay is short.

Only at thresholds so low that the step reaches them within a few of the pole’s time constants does the rule break, and there the pole has changed the power of time the step is rising with. A 0.1 ms pole is only ten times faster than this network, and for it the rule holds only for thresholds the step reaches after about a millisecond — above forty per cent or so; a microsecond pole satisfies it at every threshold above a hundredth of a per cent, which is every threshold a receiver uses.

The zero, which moves the other way #

A parasitic zero is the mirror image. A zero at $1/\tau_z$ adds $\tau_z$ times the step’s derivative to the step, and for anything slow that is the step shifted earlier by $\tau_z$.

The numbers are the pole’s with the sign reversed: every threshold reached well after the zero’s time constant arrives that time constant earlier, to six per cent, and the fraction it represents is again 9.1 per cent of the 1% delay and 0.82 per cent of the 50% delay for a 10 µs zero. At very low thresholds the zero’s advance departs from its time constant, where it has lowered the power of time the step starts with and the step reaches a tiny threshold far sooner than a shift alone would make it.

This is the answer to the earlier essay’s question of which a layout adds more of. It does not matter for the start’s power, which a parasitic governs only briefly. It matters for the delay, and there the two kinds of parasitic enter as time constants with opposite signs.

A pole and a zero together #

A real layout adds both: a node’s stray capacitance against its driver’s resistance is a pole, and the same capacitance coupling an input across a stage is a zero. The last figure puts one of each on the network together.

With both present the relative degree is unchanged, so the step starts at the schematic’s own power of time — but its coefficient is multiplied by $\tau_z/\tau_p$, exactly, since the pair’s gain at infinite frequency is the ratio of its two time constants. A zero slower than the pole makes the start faster than the schematic’s at the same power; a pole slower than the zero makes it slower. Past both time constants the step is shifted by $\tau_p - \tau_z$: later when the pole is the slower, earlier when the zero is, to six per cent at 50%. When the two are equal they are a pole-zero doublet and cancel exactly, at every threshold and in the coefficient — zero shift at both thresholds to four decimals.

That is the benign end of what the cancellation that leaves a tail measured at the other. A doublet that cancels to within a per cent changes a step’s delay by a per cent of its time constant, which is nothing; it changes its settling to a part in a thousand by a factor of thirty-six, because the residue of an imperfect doublet is a slow exponential and settling is a question about the slowest thing present. A parasitic pair is fast, so its imperfect cancellation leaves a fast tail, and the delay budget sees only the difference of the two time constants.

The instrument is a parasitic too #

Every measurement of a step puts at least one more pole in the path: the probe’s capacitance against the node’s source resistance, which the probe is part of the circuit measured, and the oscilloscope’s own bandwidth after it. Both are poles far above the signal’s bandwidth when the instrument is well chosen, and both obey everything above. They change the start’s power of time for a few of their own time constants, which no one reads, and then they delay every threshold by their time constants, which everyone reads.

The contrast with rise time is instructive. The instrument’s own rise time found that rise times add in quadrature, roughly, so an instrument five times faster than an edge inflates its 10-to-90 time by only two per cent. The same instrument delays the edge by its full time constant, linearly, at every threshold. A one-pole front end’s time constant is its rise time over 2.2, so an instrument with a fifth of the edge’s rise time delays the edge by about a tenth of the edge’s own 10-to-90 time — five times the inflation of the rise, and in a quantity the quadrature does nothing to reduce. A ten-to-one probe, which the probe that takes a tenth found trading bandwidth for signal, trades delay the same way: its smaller capacitance against the node is a faster pole and a smaller shift at every threshold. The shape is protected by the quadrature and the timing is not protected at all. That is why a timing measurement between two channels needs their probes and inputs matched or deskewed, and a rise-time measurement usually does not — and why a threshold low on the edge, where the delay is short, is the one an unmatched instrument misreports most as a fraction.

The scale of every number here is set by the network, which makes the rule portable. Divide every time by a thousand and the same figures describe a cascade with a microsecond time constant and a nanosecond parasitic: a 1% delay of 105 ns, a 50% delay of 1.23 µs, and a 10 ns stray pole costing 9.1 and 0.82 per cent of them. The ratio between the two shares is the ratio of the two delays at every scale, and the delay a parasitic adds is its time constant at every scale.

What a designer should take #

A parasitic’s effect on a step is a delay equal to its time constant — added for a pole, removed for a zero, and the difference for a pair — at every threshold the step reaches well after the parasitic has acted. Budget it as a time, not as a fraction: it is the same number at 1% and at 50%, and therefore eleven times as large a fraction of a 1% delay as of a 50% one on this network, and in general the ratio of the two delays.

Do not budget the change of power. The theorem’s statement that a parasitic pole makes a step start as $t^{r+1}$ is exact and lasts for about $(r + 1)$ of the pole’s time constants; for any parasitic small enough to be called one, that is a sliver at the very bottom of the waveform that no receiver’s threshold is set in.

And for an estimate of a whole layout’s parasitics, add the poles’ time constants and subtract the zeros’, as a sum of time constants would, and apply the total to every threshold reached after them. For a threshold reached long after the parasitics, the sum is not an estimate of the shift but the shift itself, to the six per cent the figures here hold it to.

How the numbers were obtained #

The cascade is two buffered RC sections of 1 ms and 0.5 ms; the parasitic pole is a third buffered section of time constant $\tau_p$, and the parasitic zero is an ideal differentiator of time constant $\tau_z$ summed with the output, so that the transfer gains a factor $1 + s\tau_z$. Each transfer function is recovered from its netlist. The step is evaluated from its expansion about infinity — the power series whose leading term is the theorem’s — at times below a third of the fastest pole’s time constant, and from its residues after; switching on the network’s slow time constant rather than the parasitic’s, as the first version of this did, summed thirty-four terms of a series far outside its radius and gave numbers that were not the step. Local exponents are central differences of the logarithm across two per cent of time. Every threshold is bisected in log time, a hundred iterations.

What it leaves out #

Parasitics that are not first order. A layout’s parasitics include inductance, which with the stray capacitances makes resonances rather than single poles; a parasitic resonance far above the bandwidth raises the relative degree by two and adds ringing inside its own time scale, and its effect on later thresholds is again a delay, equal to the sum of its time constants.

Loading. Each parasitic here is buffered, so adding it changes nothing else in the network. A real stray capacitance loads the node it hangs on and moves the network’s own poles a little as well as adding its own; the shift then includes that movement.

The receiver. A threshold is an idealisation of a comparator, which has its own delay, its own bandwidth and its own dependence on how fast the input crosses its threshold. A slow-rising input spends longer near the threshold, which makes a comparator’s own delay larger at the low thresholds where a parasitic’s share was already largest.

Still open: the loaded parasitic, the parasitic resonance, and a sampled step #

A parasitic that loads. An unbuffered stray capacitance at an internal node both adds a pole and moves the others. Whether the threshold shift is still the sum of the time constants of the loaded network’s poles — which by the exact sum it must be at the centre of area — and how far each threshold’s shift departs from that, would tell a designer whether the buffered picture is safe to budget with.

A parasitic resonance. Lead inductance with a stray capacitance is a pair of poles. Above the bandwidth it adds two to the relative degree and rings; how much of that ringing reaches a threshold crossing, and whether its delay contribution is still the sum of its time constants, is a measurement the same expansion can make.

The step a converter makes. A step from a digital-to-analogue converter is a held staircase, and its first sample’s hold acts like a pulse at the start. Whether the hold changes the power of time a reconstructed step starts with, or only its delay by half a clock as the staircase on the way out found for a sinusoid, is the earlier essay’s third question and still unanswered.