The time constant one trimmer cannot reach

A divider with two ratios put capacitance into a resistive divider and found it dividing by its resistors at direct current and by its capacitors at high frequency, two different numbers unless one equation holds: $R_1C_1 = R_2C_2$. The trimmer on every oscilloscope probe exists for that equation, and the square wave on the instrument’s front panel displays which of the two ratios is winning — an overshoot for too much capacitance on top, a slow climb for too little. Nine megohms over one, with 115 picofarads of instrument and cable beneath, the equation asks for 12.78 picofarads, and the window inside which the edge lands within two per cent was a few per cent wide.

That essay’s lower arm was one resistor and one capacitor. It had one time constant, and the trimmer had one time constant to copy. Real lower arms are not like that. A probe cable has its capacitance distributed along a length of resistive wire, an instrument’s input attenuator has capacitances that charge through resistances of their own, a dielectric absorbs charge late — and each of those is a part of the lower arm’s capacitance that is not reached at once. The question this essay asks is what happens to a divider compensated by its one trimmer when the arm beneath it has two time constants.

The lower arm here is a lumped stand-in for any of those: 100 picofarads the divider reaches directly, and 15 more reached through a kilohm. The total is the 115 of the earlier essay, so at the calibrator’s time scale nothing has changed. What changes is everything faster.

Flat on the calibrator, and not on a fast edge #

Set the trimmer where a kilohertz square wave reads flat. The upper arm then matches the lower arm’s total capacitance, 12.78 picofarads exactly as before, because on a time scale of half a millisecond the late fifteen picofarads are charged and indistinguishable from the rest. Now give the same divider an edge that rises in a nanosecond.

The output leaps to 1.128 times its settled value and falls back over fifteen nanoseconds, the time constant of the kilohm and the late fifteen picofarads. It is within one per cent of settled only after 35.5 nanoseconds. The reason is visible in the ratios: for the first few nanoseconds the late capacitance has not charged and is not part of the divider, so the lower arm is a hundred picofarads, and the upper arm’s 12.78 is too much for it — the divider is over-compensated, briefly, by exactly the fraction of the lower arm that arrives late. Then the late branch charges, the lower arm becomes 115, and the divider is the balanced one the calibrator found.

On a time axis that runs in decades the response is a hook: a step to the wrong value, a relaxation to the right one, and nothing after that at all. The slider moves the resistance in front of the late branch: 300 ohms gives a 4.5-nanosecond hook, three kilohms a 45-nanosecond one. The height of the hook barely moves with the resistance — 11.8, 12.8 and 13.1 per cent — and approaches 13.3 per cent as the branch becomes slow against the edge: the ratio of the trimmer to the hundred picofarads present at once, against the ratio to all hundred and fifteen. What the resistance sets is the hook’s duration.

How slow an edge hides it #

Most of the edges a probe is shown are not a nanosecond. A slow edge gives the late branch time to charge while the edge is still rising, and then the hook is buried in the edge itself.

Measured as the largest departure from the settled value anywhere after the edge has finished, the hook falls below one per cent for edges slower than 63.1 nanoseconds behind 300 ohms, 200 behind a kilohm and 631 behind three kilohms — about thirteen times the late branch’s own time constant in each case. The response to a linear edge is the step response averaged over the edge’s duration, and averaging a hook of height 0.128 and time constant τ over an edge of length $t_r$ leaves roughly $0.128\,\tau/t_r$; one per cent of it needs $t_r$ near thirteen τ.

That is the reason the calibrator can be trusted for what it does. Its square wave has edges of a few hundred nanoseconds at most and a period of a millisecond, so it sees the divider’s slow time constant clearly and its fast one not at all; the trimmer it adjusts is the right trimmer for every signal slower than the hook. The divider is wrong only for edges a decade or so faster than the late branch — and for a general-purpose probe that is precisely the regime its bandwidth specification advertises. The instrument’s own rise time measured how much a probe’s bandwidth inflates an edge; this hook is a separate error that adds a shape to the edge rather than a delay.

The best one knob can do #

If the fast edge matters, the obvious response is to adjust the trimmer while watching it. Judged on a nanosecond edge, by its worst departure anywhere from the end of the edge to a millisecond, every trimmer setting can be scored.

At the calibrator’s setting the score is 12.8 per cent, the hook’s height. Smaller trimmer values lower the hook and introduce an error of the other kind: with less capacitance on top than the full lower arm needs, the divider is under-compensated once the late branch has charged, and its output then creeps up over the slow time constant. The best score is 6.1 per cent, at 11.9 picofarads, where the fast overshoot and the slow undershoot are equal. That is the floor for one knob. No setting of a single trimmer is flat on this divider, because there is no single number of picofarads that is both a ninth of a hundred and a ninth of a hundred and fifteen.

The edge it is judged on matters. On a 0.35-nanosecond edge the best is worse; on a ten-nanosecond edge it is better, and the best setting moves back towards the calibrator’s as the edge slows and the hook is averaged away. A single-trimmer divider therefore has a best setting that depends on the edge it is judged on, which means it has no best setting at all — only a choice of which signals to be wrong about.

Trimming the wrong time constant #

The worst outcome is an adjustment made carefully on the wrong evidence. A technician who sets the trimmer so that a fast edge lands flat — so the first few nanoseconds after it sit exactly on the settled level — has matched the upper arm to the part of the lower arm that is present at once.

The trimmer that makes the fast edge land flat is 11.4 picofarads — a ninth of a hundred and three, the capacitance present in the first three nanoseconds. Shown the calibrator, the same divider starts 9.9 per cent low and creeps to its settled value over the slow time constant, 114 microseconds: a square wave with a sloping top, the signature of under-compensation that the earlier essay identified. Fixing the fast edge with the only knob has broken the slow one, by about the share of the capacitance that arrives late.

This is the mistake the earlier essay’s figure would have predicted had its lower arm been honest. It is also a familiar one on the bench, where a probe adjusted against a fast pulse generator and then used on a slow signal gives amplitude errors of several per cent at frequencies of tens of kilohertz — far below any frequency at which a probe is supposed to have errors.

What a hook does to a reading #

A twelve-per-cent hook lasting thirty nanoseconds is easy to dismiss as an aberration at the edge of a trace, and for some measurements it is. For others it is the whole of the error, and which is which depends on what is being read off the screen.

A pulse amplitude. A twenty-nanosecond pulse shown through this divider never reaches the part of the response where the late branch has charged. Its top sits on the hook, and the amplitude read off it is several per cent high for its whole duration — not as an overshoot that settles but as a level that is simply wrong, on a pulse whose width is comparable with the hook. Narrow pulses are exactly the signals for which a reading is taken at the top rather than after settling, and they are read high.

A rise time. The 10-to-90-per-cent rise of a nanosecond edge through the hook is barely changed — the hook lifts the top of the edge rather than slowing it — but the trace now overshoots and settles, and an overshoot on a measured edge is normally read as ringing in the circuit under test. The arrow that goes past where it settles is a reminder that a resonant circuit’s real overshoot has the same appearance; the two are told apart only by changing the probe.

A settling time. A converter driver or an amplifier specified to settle to a tenth of a per cent in a hundred nanoseconds is measured through a probe whose own hook takes 35 nanoseconds to come within one per cent and longer to come within a tenth. The probe’s settling is added to the circuit’s, and below a few per cent the reading is a measurement of the probe.

A low-frequency amplitude after a bad adjustment. The mis-trimmed divider of the previous section is nearly ten per cent low on its square wave’s first edge and creeps up over a hundred microseconds; shown a ten-kilohertz sine it reads about nine and a half per cent low — above about 1.4 kHz the slow time constant has given up and the divider is its capacitive ratio, a ninth of 103 against a ninth of 115 — at a frequency where nobody expects a probe to be anything but exact. The probe is part of the circuit found a probe one per cent wrong at 6.8 kHz through its loading; a mis-trimmed one is wrong at lower frequencies still, through its compensation.

In each case the hook’s height decides how wrong the reading is and its duration decides which readings it reaches. That is the reason a probe’s specification includes an aberration figure — a percentage of overshoot within a stated time after a fast edge — beside its bandwidth, and the reason the aberration figure is quoted only after the probe has been adjusted on a fast edge as well as on the calibrator. The corner that says nothing about an edge made the general point about an AC-coupled input: a corner frequency describes sinusoids, and an edge needs its own number.

A copy, not a match #

The repair follows from asking what “compensated” has to mean when the arm beneath has more than one time constant. A divider’s ratio is constant at every frequency only if its two arms’ admittances are in a fixed ratio at every frequency: the upper arm must be a scaled copy of the lower one, element for element. For one resistor and one capacitor that is the earlier equation. For this lower arm it is three equations — the resistors in the ratio nine to one, the directly reached capacitors in the ratio one to nine, and a second branch in the upper arm whose resistance is nine times the late branch’s and whose capacitance is a ninth of it.

With a branch of 9.0 kilohms and 1.7 picofarads beside a main trimmer of 11.1, the nanosecond edge is flat to seven parts in a hundred billion from the edge to a millisecond. The two capacitors add at low frequency, 11.1 and 1.7 making the calibrator’s 12.8, so the adjustments are not independent: the second branch is set on a fast edge first, and the main trimmer then on the calibrator, in that order. That is the arrangement real high-bandwidth probes carry — one low-frequency adjustment that every user is shown how to make, and one or more high-frequency adjustments inside the compensation box that a calibration laboratory makes.

The second knob has its own tolerance. With its resistor twenty per cent high the edge departs 0.84 per cent at worst, near 16 nanoseconds — inside the late branch’s time constant, which is the only place a second-branch error can live.

The second branch’s two parts have very different windows. For a one-per-cent flat edge the resistor may be 18 per cent low to 24 per cent high; the capacitor only six per cent either way. The capacitor sets how much of the late charge the branch copies, which is the hook’s height, while the resistor sets only how fast the copy charges, which matters less. So the resistor can be a fixed part of ordinary tolerance and the capacitor is the one to trim — the same logic that made the main trimmer a trimmer in the first place, applied to the second time constant.

Where the late branches come from #

A resistive probe cable is the commonest source and the reason the effect is famous. A probe’s cable is deliberately lossy — its centre conductor is resistance wire — so that reflections along it are damped; its capacitance is therefore reached through a resistance that grows with distance along it, and the lumped late branch here stands in for a continuum of them. The staircase in time described what an unterminated line does to an edge; the resistive cable trades those reflections for a smear of late charge, and the probe’s high-frequency adjustments shape that smear.

An instrument’s input attenuator is another. Switching an oscilloscope from one volt per division to ten inserts a compensated divider of its own, with its own trimmer and its own stray time constants, and it is why a probe compensated on one input range can show a hook on another.

A capacitor’s dielectric is a third, and the least obvious. The capacitor that remembers measured dielectric absorption: part of a capacitor’s charge goes into the dielectric slowly, through what behaves as a chain of resistances and capacitances inside it. A divider whose lower capacitor absorbs has exactly this structure, with late branches whose time constants run to seconds, and a trimmer can no more copy them than it could copy the cable. Using a low-absorption dielectric in the lower arm is the same repair as a second branch, made by removing the late charge rather than copying it.

How it was solved, and what the lumped branch leaves out #

The divider is a netlist of six elements, or eight with the second branch, and its response is taken from the poles of its transfer function and their residues rather than marched. The two time constants are four decades apart — fifteen nanoseconds and 114 microseconds — and a fixed-step march fine enough for the first would need a hundred million steps to reach the end of the second. From the residues, the step response is a sum of exponentials, and the response to an edge rising linearly over $t_r$ is its integral over the edge, which is in closed form. The settled value is held against $R_2/(R_1 + R_2)$ to a part in a million, and the exact-copy divider is held flat to a part in a million everywhere after the edge.

The late branch is one resistance and one capacitance. A real resistive cable is a continuum of them, with inductance too, and its hook has a shape — a sum of many time constants — rather than one exponential, and a real probe’s high-frequency network has several adjustments to copy several of them. The numbers above belong to the lumped branch; the conclusion that one trimmer copies one time constant does not depend on how many there are. The instrument’s input is a resistor and a capacitor with no inductance and no bandwidth limit of its own, and the source driving the probe is ideal. The probe that takes a tenth is the essay in which the source’s own resistance enters, and with it the divider’s input capacitance becomes a load on the circuit being measured.

Still open: many late branches, the source’s resistance, and a temperature #

A continuum of late charge. A resistive cable is a distributed RC line, and its contribution to the lower arm is an impedance that goes as $1/\sqrt{s}$ over a band rather than a single exponential. A finite number of copy branches can approximate it only over a band, and the number of branches needed to hold an edge within one per cent from a nanosecond to a millisecond is a question about how well a ladder network of sections fits a square root — the same question a line’s own sections answer for its delay.

A source with resistance. Everything here drives the divider from an ideal source. A source with resistance sees the divider’s input capacitance — 11.1 picofarads plus the branch — as a load, and the late branch then charges through the source as well as through its own resistance. Whether a probe compensated on a fifty-ohm generator shows a different hook on a kilohm source, and which way, is a two-source solve of this same netlist.

A late branch that drifts. The late branch’s resistance is resistance wire and its capacitance is a dielectric, and both move with temperature, the first by hundreds of parts per million per kelvin. A probe whose second adjustment was set warm will show a hook of a few tenths of a per cent cold; the window above says how much drift the resistor tolerates, and the capacitor’s six per cent says the dielectric’s drift is the one to watch.