The source the hook hides behind

The time constant one trimmer cannot reach found a ten-to-one divider that reads flat on a 1 kHz calibrator with its trimmer at 12.8 pF, and overshoots 12.8 per cent on a 1 ns edge, because 15 of the 115 pF in its lower arm are reached through a kilohm. A second branch in the upper arm, a scaled copy of the late branch, makes the whole thing flat to rounding. That essay drove the divider from an ideal source throughout and said so, and it asked what would happen otherwise: a source with resistance sees the divider’s input capacitance as a load, the late branch then charges through the source as well as through its own resistance, and 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.

The source’s resistance is not a small correction. It is the variable that the real use of a probe has and the calibration of one does not. A probe is trimmed on a calibrator or on a generator, both of them stiff sources, and it is then put on a node whose whole interest may be in its being a kilohm or more. The first thing a reader expects is that the fast hook gets rounded away, and that does happen. What the rounding does to the reading, and to the correction that was built to remove the hook, is less obvious.

What the source has to charge #

The netlist is the earlier essay’s with one resistor added in front of the tip: a step, a source resistance $R_s$, then the divider. The input capacitance the source sees is the trimmer in series with the whole lower arm, 11.5 pF, and that gives a lag of $R_sC_{in}$ whatever else the divider does. Without a late branch, a compensated divider’s ratio is the same at every frequency, so its output would be exactly the tip’s voltage divided by ten, and the tip’s voltage would be a plain first-order rise. That is the expected result, and it is the one a probe’s data sheet offers: a capacitance at the tip, forming a low-pass with the source, nothing more. The probe is part of the circuit measured it on the one-to-one probe.

A late branch changes it because it gives the divider two ratios on a fast time scale. For the first few nanoseconds the lower arm is 100 pF, and the upper arm’s 12.78 pF is a little too much for that, so the output is 13.2 per cent above where it will settle. Over the next tens of nanoseconds, the late 15 pF charges through its kilohm and the output comes down. Those nanoseconds were a property of the divider alone. A source with resistance now puts a lag of its own on the same time axis, and the two processes are no longer separable on the screen.

The overall time constants come out of the same netlist, by residues. It has only three of them worth talking about: the source against the input capacitance, the late branch’s own, and a very long one from the resistors, at about a hundred microseconds, which is the one a calibrator’s flat top reads.

How much hook gets through #

The direct question has a direct answer.

The overshoot is 12.5 per cent from 10 Ω, and 10.5 from a fifty-ohm generator, which is why the earlier essay’s numbers were representative of a generator bench. It falls to half at about 200 Ω, and it is gone by a kilohm. The late branch’s own time constant is 13 ns; the source’s lag equals that at about 1.1 kΩ. So the hook disappears from the display almost exactly where a simple comparison of two time constants would put it.

This has an uncomfortable reading. A kilohm is an ordinary node. A probe with a hook in it, on an ordinary node, shows no hook at all, and it would pass any inspection by eye of the edge. The trimmer is not compensated at the fast end, the second adjustment inside the probe body may be well out, and the display is clean. Only by moving the probe onto something stiffer does the defect appear. So a probe whose compensation has been checked on the circuit it is measuring has not had its compensation checked. The check needs a source much stiffer than the node.

The edge that is too fast #

The natural conclusion is that nothing is wrong at a kilohm: the hook has gone. Looking at the rise rather than the peak shows that it has not gone anywhere, because a hook that has merged into a lag has changed the rise.

A first-order rise from a source through 11.5 pF is $\ln 9\,R_sC_{in}$. From a kilohm that is 25.3 ns, and it is the fastest edge this probe could honestly show on that node, since the probe’s own capacitance is slowing it by that much. The one-trimmer divider displays 21.4 ns. So the screen is showing an edge faster than the node can have with this probe on it, by 15 per cent, with no overshoot to warn of it. The excess of the fast ratio is being spent on steepening the lag, which is what a lead network does, and in effect the mis-trimmed upper arm is a lead.

This is the same mechanism a compensation network uses deliberately. A divider with two ratios showed overcompensation as an overshoot on a stiff source, and on a soft source an overcompensated divider looks clean and fast. A measured rise time from such a probe would be recorded in good faith. The instrument’s own rise time corrects a reading for the instrument by quadrature, on the assumption that the instrument can only slow things. A probe can do the opposite here, so the correction has the wrong sign.

The exact copy is not faster than the lag, and that is not because it is doing nothing. Its input impedance still has two time constants, because the upper arm’s copy is nine times the lower arm’s and the sum is ten times the lower arm, so its input is a scaled version of a two-time-constant arm. So the copy’s rise from a kilohm is 25.9 ns, 3 per cent slower than the plain lag: at the input, the fast 10 pF is charged first, and the rest arrives late. What the copy guarantees is that the output is a tenth of the tip at all frequencies. It says nothing about what the tip does, because it cannot.

A trimmer set in the wrong place #

There is a further trap here that the earlier essay’s result implies, and that a source makes concrete. A user with a fast generator and a probe with one adjustment will naturally set the trimmer until the first peak disappears. On a stiff source that means moving away from the calibrator’s setting.

Nulling the peak from 50 Ω needs 11.42 pF, and that leaves the divider undercompensated: the fast part of the edge is right and it is then 9.65 per cent short, climbing to its true value on the slow 115 µs time constant. That is a big error. A user reading a pulse of a few microseconds on it would take the shortfall as real. It also makes no difference what the divider is subsequently read on, as the figure says: 9.65 per cent at 50 Ω, 9.66 at a kilohm, and 9.88 at 30 kΩ. The undercompensation is a property of the ratio at the slow time scale, where a source’s resistance against a few tens of nanoseconds of lag is irrelevant.

Set on a kilohm, on the other hand, the trimmer comes out at 12.79 pF, within a part in a thousand of the calibrator’s. So here is a surprising fact to go with the first one. The soft source gives the right trimmer setting, because it hides the hook that would have misled the eye. The stiff generator, the instrument that seems the better test, is the one that suggests the wrong adjustment. The right approach, as the earlier essay showed, is to use the slow calibrator for the trimmer and leave any fast edge to the second adjustment. This shows why, in one number.

Which divider settles first #

If the purpose of a divider is to report a settled voltage as soon as possible, then a settling time is what counts, and it reverses the expected ranking.

On a stiff source the second branch is worth what it was built to be worth: 5.98 ns against 36 from 100 Ω, a factor of six, since one trimmer has to wait out its own hook and the copy does not have one. By 500 Ω they are level, at 39.8 and 36.9. And from a kilohm the order has reversed, with the one-trimmer divider settled in 44 ns and the exact copy in 65.6. Its lag and its hook have cancelled, so that its output reaches its final value early, whereas the copy has only a lag followed by a slower tail. Beyond a few kilohms the source’s lag dominates both and they converge: 272 and 279 from 5 kΩ, each near $\ln(100)R_sC_{in}$.

There is a broad design lesson. The arrangement that is exact for a voltage source is not fastest on a source of comparable impedance, because exactness of the ratio was never the question a source of that kind is asking. A probe’s fast adjustment has a best setting for a given source impedance, and the factory has to pick one. A probe made for a 50 Ω world picks the setting that is exact there. On a kilohm node a probe with some hook in it actually settles faster than the exact probe.

The ring the ground lead adds reached a similar place from another mechanism: a probe’s fast behaviour is in part a statement about what it is connected to, and any reading of it is a reading of the pair.

Why the calibration cannot see any of this #

The calibrator’s square wave is a slow, stiff source, typically a low-impedance output at 1 kHz, on which the probe’s long time constant is all that shows. On that test both the one-trimmer divider and the two-branch divider look the same, because on a time scale of a hundred microseconds the late branch has fully charged. That is what makes it a correct calibration of the trimmer, since the slow ratio is what the trimmer controls, and that is also what makes it no test of anything on the fast end.

So the usual routine, in which a user checks a probe by looking at the calibrator, tells nothing about any of the four results above. Nor does a generator with a fast edge, for the reason just shown: it reveals a hook and invites the wrong adjustment. To check the probe’s fast end honestly, the right tool is a fast and stiff source with the trimmer locked at its slow setting, and any remaining hook read off as a property of the probe. A probe maker does this on a test jig, and a bench user almost never does. What a hook does to a reading gave the arithmetic of what it costs not to.

A second look at the node #

The most useful implication may be for the reading rather than for the probe. If a probe has a hook, a source’s lag conceals it near a kilohm and changes the displayed rise there by 15 per cent. Two probe settings then read differently on the node in ways that say something about the node. A user who has a probe with an accessible fast adjustment can move it and watch the edge. On a stiff node the peak moves. On a soft node only the rise moves. So how much a node’s displayed shape responds to a change in the probe’s fast adjustment is an estimate of the node’s source resistance, read off the probe rather than the circuit.

That is comparable to the ring-and-decay reading of the ring the ground lead adds, which found the node’s output resistance from an artefact of the probe. A source’s resistance appears in the probe’s misbehaviour in more than one way, and each is a measurement available with no additional equipment.

What a lighter probe changes #

The kilohm is not a constant of nature. It is where the late branch’s 13 ns equals the source’s lag against this probe’s 11.5 pF, and the probe’s input capacitance is on the same side of that equation as the source. The probe that takes a tenth cut the tip capacitance by ten in exchange for noise, and the same arithmetic moves the boundary. A one-to-one probe at 115 pF, if it had a late branch at all, would lose it at about 110 Ω. A probe with a tip of 1 pF keeps its hook visible up to 13 kΩ if its late branch is the same, since the source now has much less to charge.

So every improvement to a probe’s loading exposes more of its fast compensation to the reading. A very light active probe has to be exact at its fast end in a way a heavy passive one on the same node does not, because nothing on the node will hide the error. The same logic applies when moving a probe to a stiffer point in a circuit, and it is a reason a hook that was never seen may appear when a design moves to lower impedances.

What the netlist leaves out #

The late branch is one lumped kilohm and 15 pF, for continuity with the earlier essay and because everything here turns on one late time constant. A real cable has a continuum of them, and the next essay in this group takes that up. The source is a pure resistance. An actual node of a kilohm is usually a transistor’s output or a resistive divider, with its own capacitance at the node, which would add to the probe’s and would move the lag’s size but not the mechanism. There is no ground lead either, and on a 50 Ω source that would dominate everything here on a nanosecond scale. A step is the only waveform used; a slower or noisier edge shows less of all of this, and how slow an edge hides it applies to the source’s lag exactly as it did to the edge’s own rise.

The number worth carrying #

Through a source resistance, a divider’s hook is 10.5 per cent from 50 Ω, 1.81 from 500 Ω and gone at a kilohm, where the source’s lag against 11.5 pF equals the late branch’s time constant. It has not gone anywhere: the same divider then shows an edge 15 per cent faster than that source and capacitance allow. A trimmer set by nulling the peak on a fast generator is 11.42 pF against the true 12.78, and it reads 9.65 per cent short after every edge on every source, while a soft source gives the correct setting. The second branch is worth a factor of six on a stiff source and is half as slow again near a kilohm, 65.6 ns against 44. A probe’s compensation is a statement about the probe and the source together.

Still open: many late branches, a late branch that drifts, and the source’s own capacitance #

A continuum of late charge. A resistive cable’s contribution is a distributed line and not a single branch. How many branches a copy needs in order to hold an edge flat, and whether the source’s lag hides a continuum the way it hides one branch, is the measurement this essay sets up.

A late branch that drifts. The late branch’s resistance is resistance wire, and its capacitance a dielectric, and both move with temperature, the first by hundreds of parts per million per kelvin. Since the hook’s visibility depends on the ratio of its time constant to the source’s lag, the temperature at which a probe’s error is largest depends on the node it is reading.

The source’s own capacitance. A kilohm node with 10 pF of its own has a lag before the probe is attached. The probe then adds a parallel capacitance across it, and the node’s displayed rise is a two-pole affair in which the late branch is one of three time constants. Where the hook then goes, and whether the conclusion about a soft source’s correct trimmer setting survives, is a small extension of this netlist.