The instrument's own rise time

The instruments field is built on one sentence: an instrument is not an observer, it is an element, it goes in the netlist, and every reading is a reading of the circuit that includes it. The probe is part of the circuit measured what a probe’s capacitance does to the circuit it is placed on. This essay measures something the probe does to the signal rather than to the circuit, and it is the error that is most often estimated with a rule and most rarely checked.

The rule is that rise times add in quadrature:

$$t_\text{measured} = \sqrt{t_\text{signal}^2 + t_\text{instrument}^2}$$

Accept ten per cent of inflation and the arithmetic gives an instrument t/√(1.1² − 1) = t/2.18. So: buy an instrument 2.18 times faster than the fastest edge of interest, and the measurement will be within ten per cent.

That 2.18 contains no instrument. There is no bandwidth in it, no pole count, no topology, nothing about what is on the bench — only the inflation accepted. A rule whose answer is independent of the thing being specified is worth examining.

Why this is the third rung and not the first #

The instruments field has had a rung about the probe since the foundation phase, and this essay sits two above it. The ordering is deliberate and says what the ladder is for.

The first rung establishes that the instrument is in the netlist at all — that a reading is a reading of a circuit that includes the instrument, which is the field’s whole premise and is not obvious to anybody who has not had it pointed out. The second rung is compensation: the instrument is not only present, it has its own internal trade to get right, and a divider with capacitance in it has two ratios that agree at one adjustment.

This rung is the one that needs both. It asks what the instrument does to a fast signal, which requires having accepted that the instrument is part of the circuit and that its own internals have a frequency response. Placed first it would read as a specification note; placed here it reads as the third thing that goes wrong, and the closing section assembles all three.

Where the rule comes from and where it stops #

Quadrature addition is not a fudge; it is exact, for Gaussian responses.

The convolution of two Gaussians is a Gaussian whose width is the quadrature sum of the two widths. If a signal’s edge were a Gaussian’s integral and an instrument’s response were a Gaussian, the measured edge would be a Gaussian’s integral of the quadrature width, and the rule would be a theorem rather than a rule.

An oscilloscope’s front end is not a Gaussian. It is a chain of poles — an attenuator, an amplifier, an interconnect, a sampler, each with corners of its own — and a chain of poles is not a Gaussian until there are a great many of them and even then only approximately. So the interesting question is not what the rule says but how wrong it is, and that is measurable in the way everything else on this site is measurable: build the two responses as networks, drive them with an edge, and read the 10–90% times off the solved output.

What the measurement says #

Everything is solved. The signal is a real edge from a real one-pole network, the instrument is a chain of identical poles, and the “measured” edge is what comes out of the second when driven by the first. The rise times are 10–90% times located on the solved waveform, not inferred from a bandwidth.

For ten per cent of inflation:

The rule is optimistic at every pole count, and by more the longer the chain. That is the wrong direction. A rule of thumb that overestimates the instrument required is a nuisance; one that underestimates it says the instrument on the bench is fast enough when it is not, and the resulting error is a systematic overstatement of every rise time measured — which then propagates into whatever was being characterised.

The size of the error is the part worth carrying away. For a four-pole front end the rule asks for 2.18× and the truth is 5.62×, a factor of two and a half in bandwidth. Bandwidth is what an oscilloscope is priced on.

The other direction, and the figure that found a dead slider #

There is a second way to ask the same question, and it is the one probe-rise-time draws: fix the speed ratio and read the inflation off it.

That figure exists for a reason worth recording, because it is a fault this collection has now met twice. It declared a slider on the source resistance, and its x-axis was the ratio of two rise times with the probe capacitance computed from that same resistance — so the resistance cancelled exactly and all five frames were the same picture. It had been that way since the figure was written, and it was invisible because both essays placing it passed nodrag. It was found only when the scale phase added a page per generator that renders every figure with its slider.

The repair gave it a slider over something the answer depends on: the number of poles in the instrument. And that is where the numbers in this essay came from — the repaired figure was the first thing in the tree to know that the pole count mattered, and this essay is the argument it was repaired for.

What “inflation” is, and why it is the right quantity #

The slider is labelled inflation and it is worth being precise, because rise-time error is quoted in at least three incompatible ways.

Inflation is what this essay uses: the measured rise time divided by the true one, minus one. Ten per cent of inflation means a 1 ns edge is reported as 1.1 ns. It is the quantity an engineer actually cares about, because it is the error in the answer.

The instrument’s own rise time as a fraction of the signal’s is what a specification usually states, and it is the reciprocal of the speed ratio in the table above. It is a statement about the equipment rather than about the error, and converting between the two is exactly the step this essay says the standard rule gets wrong.

Bandwidth ratio is what a purchase order contains, and it is related to rise time by another rule of thumb — tᵣ ≈ 0.35/BW — that is itself exact only for a single pole. For a two-pole front end the constant is nearer 0.4 and for four poles nearer 0.45, so a second approximation is stacked on the first in the same direction. Both err optimistic, and the errors multiply.

Which is the practical reason the numbers in this essay are larger than most people expect. The chain from a 1 ns edge, to be measured to one per cent to a scope of so many gigahertz passes through two rules of thumb, both exact only for a single pole, both wrong the same way, and the product of the two errors is what shows up on the invoice.

Why more poles need more speed #

The direction of the error has an explanation that does not require any arithmetic.

A single pole’s step response starts at its steepest and decays. Two poles in series start at zero slope, build, and then decay — the response has an inflection, and the 10–90% transit takes longer relative to the response’s own time constant. Each additional pole adds more of that initial flatness. So for a given −3 dB bandwidth, a longer chain has a slower 10–90% edge, and a specification written in bandwidth understates the chain’s sluggishness by more the longer it is.

The quadrature rule is blind to this because it works in rise times and assumes the combination law that only Gaussians obey. What it misses is not the individual rise times — those are measured correctly — but how they combine. A chain of poles combines its rise times more than quadratically, and the excess grows with the count.

This is the same observation the transients field made about settling: the fastest-settling second-order response is at a damping ratio near 0.95, not at 0.707 and not at critical damping. Both are cases where a quantity everybody quotes for one order is silently extrapolated to another, and both were found by solving rather than by consulting.

The measurement is a measurement, not an inversion #

One detail of how the numbers were obtained is worth stating, because it is the difference between a figure and a calculation.

The requirement — “how fast must the instrument be for ten per cent” — is found by solving the forward problem repeatedly and searching, not by inverting an expression. The generator drives the combination with an edge, reads the 10–90% time off the solved output, compares it with the signal’s own, and bisects on the speed ratio until the inflation is exactly what was asked for.

That matters because the forward problem is the one with no approximations in it. There is no expression for the 10–90% time of a chain of poles driven by another chain of poles; the times are read off a waveform, in the way an oscilloscope reads them off a screen, and the search is around that. So what the figure reports is not a closed form that happens to disagree with the quadrature rule but a measurement that does.

The check that the search converged is the one the site uses everywhere: the ratios come out monotonic in pole count, the rule’s own answer is reproduced independently from its closed form to six digits, and the rule returns the same number at every pole count — which the figure asserts explicitly, because “the rule contains no instrument” is the essay’s central complaint and it should be a check rather than a remark.

What this does not include, and what that costs #

Three things are outside the model here, and one of them cuts the other way.

The instrument is a chain of identical poles. Real front ends are not; they are a mixture of corners, some of them deliberately placed to shape the response. A scope specified as “Gaussian response” is one whose designers have spent effort making the rule closer to true, and for such an instrument the numbers here are pessimistic. A scope specified as “flat response” — maximally flat magnitude, which is a Butterworth and rings — is a different chain again, with overshoot the 10–90% time does not capture at all.

The signal is a single-pole edge. A real edge from a real driver has its own shape, and if it too has multiple poles the combination is worse still.

There is no interconnect. The probe, the cable and the input have been treated as a response rather than as a transmission line, which the lines field has spent three essays establishing is only valid below a frequency set by the length. At a 100 ps edge, a metre of cable is emphatically not a lumped element, and the whole framework of this essay stops applying before its numbers do.

That last one is the boundary this site’s rule asks for, and it is worth stating in the units a reader has: the lumped model gives out at about a tenth of a wavelength, and Kirchhoff’s own frequency measured the conventional criterion to be already 30% wrong where it is usually drawn.

Three errors, one measurement #

This field now has three separate ways an instrument is wrong about the thing it is measuring, and it is worth assembling them because they compose rather than compete.

Loading. The probe’s capacitance is across the circuit, so the circuit is slower with the probe on it than without. The signal being measured is genuinely different because the measurement is happening. This is the rung below, and it is a change in the circuit.

Inflation. Whatever edge arrives at the instrument’s input is slowed further by the instrument’s own response before it is displayed. This essay, and it is a change in the reading.

Compensation. A divider with capacitance in it has two ratios — a resistive one at direct current and a capacitive one at high frequency — and they agree only at one setting of the trimmer. Off that setting the reading has an overshoot or a droop that is neither of the above.

The three are independent and they multiply. A 10× probe on a 2 kΩ source, into a four-pole front end of nominally adequate bandwidth, mis-trimmed by a few per cent, is wrong about a fast edge by an amount none of the three explains on its own. That is the honest version of the field’s opening sentence: the instrument is an element, and it is several elements, and each of them has its own way of being in the answer.

What makes this tractable rather than hopeless is that all three are computable and none of them is random. Each is a systematic error with a sign, and a measurement whose three systematic errors are known is a measurement that can be corrected. The reason to draw them is that a reader who has seen the numbers will not confuse an instrument’s error with a circuit’s behaviour, which is the failure mode this field exists to prevent.

What a specification would have to say #

If the quadrature rule is a lower bound, the natural question is what a datasheet would have to publish for the true requirement to be computable, and the answer is short: the number of poles, or equivalently the shape of the step response.

Instruments do publish something adjacent. A scope specified as having a “Gaussian” or “maximally-flat” response has told the buyer which family it belongs to, and from the family the combination law follows. What is rarely published is the order, and the order is what the table in this essay varies.

There is a second quantity that would settle it and is published even less often: the overshoot. A single pole has none. A maximally-flat chain of two or more has a few per cent and rises with order. A Gaussian-approximating chain has essentially none by design. So overshoot is a direct observable that distinguishes the families, and an instrument that quotes it has said more about how its rise times combine than one that quotes only bandwidth.

The practical advice that falls out is unglamorous and is what experienced people already do: do not specify from the rule, measure the instrument. Feed it an edge much faster than its own — a fast pulse generator, or a reflection off a shorted line — and read its own 10–90% time off the screen. That number, combined with the signal’s, gives the inflation directly and needs no combination law at all, because the two responses have been put in series physically rather than arithmetically.

Which is, in the end, the same move this whole collection makes. The rule is a substitute for a measurement, it is exact for a circuit that is not on the bench, and the measurement is cheap.

The rule this model stops being true under #

2.79× for one pole and 5.62× for four, against a rule that says 2.18× for everything.

The rule is exact for a response nothing on a bench has, it errs optimistic, and the error grows with the length of the chain — so the instrument that is comfortably adequate by the rule is inadequate by a factor that depends on a specification most instruments do not publish. The honest summary is that the quadrature rule is a lower bound on what is needed, and this essay’s contribution is to measure how much of a lower bound.

The three other ways an instrument is part of its own measurement #

A rise-time inflation is one of four in this field, and the four together are its argument.

The probe is part of the circuit is the same failure in impedance: a hundred and fifteen picofarads across a two-kilohm source is one per cent wrong at 6.8 kHz, with nothing inaccurate anywhere — the reading is correct about a circuit that exists only while the probe is attached.

The corner that says nothing about an edge is the same failure at the other end of the band: a corner frequency is a statement about steady sinusoids, and what an instrument is usually shown is a pulse, for which the number is a sag — 9.5 per cent on a hundred-millisecond pulse through a 0.159 hertz corner, with a one per cent flat top needing a pulse rate 625 times the corner.

And the corner the instrument has no part in is the one where the instrument is blameless: a 95 dB instrumentation amplifier delivers 84 dB at a kilohertz because its source has a kilohm of imbalance, and the corner at 290 hertz belongs to the connection rather than to the part.

Four specifications — a bandwidth, an input impedance, a low-frequency corner, a rejection — each true about the instrument alone and none of them a statement about a measurement. Which is this field’s whole content, and the reason its numbers are computed on two solves rather than quoted from one.

The other three errors an instrument makes #

A rise time is one of four things an oscilloscope changes about what it is shown. The probe is part of the circuit is the loading, which is a change to the circuit rather than to the reading. A divider with two ratios is the compensation, which is exact at one trimmer setting and wrong in two directions either side of it. The current the instrument draws and The current that does not reach the input are the two direct-current errors this page’s rise time says nothing about. All four rest on One step, computed twice, and The step that is too big is the boundary outside which none of the four applies, because the instrument has stopped being linear.