Ten seconds, and fifteen minutes

The capacitor that remembers builds a dielectric as a distribution of relaxation times, marches the test that defines dielectric absorption, and gets the number a data sheet quotes: 0.2000 per cent for the part modelled there. On the way it says something in passing and does not measure it.

Change the short to two seconds and the answer rises, because fewer branches were emptied. Change the reading to an hour and it rises again, because more of them have returned.

Both sentences are true and neither has a number attached, which leaves the reader with the impression that the durations are a second-order correction to a figure that mostly belongs to the dielectric. They are not a correction. They are most of the figure, and the arithmetic that says so has been sitting in the machinery since the day it was written: the marched test takes the short’s duration and the reading’s duration as arguments, and every caller has left them at the ten seconds and fifteen minutes the specification names.

What the specification actually contains #

The test is a procedure before it is a measurement. Charge the part for an hour. Short the terminals for ten seconds. Open the short, wait fifteen minutes, and read what has come back as a fraction of the charging voltage.

Three durations, and they do not all do the same job. The charging hour has to be long against the slowest relaxation in the dielectric, and once it is, making it longer changes nothing — every branch is fully polarised and there is nothing left to polarise. It is a sufficiency condition and it drops out of the answer.

The other two do not drop out. The short discharges every relaxation faster than itself and leaves every relaxation slower than itself alone. The wait collects whatever has come back by the time the reading is taken. So what the terminals show is the charge held on relaxation times between the two durations, which makes the quoted figure a statement about an interval of time rather than about a quantity of dielectric.

That distinction is the whole essay, and it is worth stating before any of it is drawn, because it predicts something specific: if the answer is an interval, then two tests with the same interval should agree however far apart they sit in absolute time, and two parts with different intervals should disagree however carefully they were matched at one point.

The calibration, which comes first #

A model that is about to be asked an unfamiliar question is first asked one whose answer is already fixed. The branch fraction here is not chosen; it is bisected until the specified test returns exactly the specified absorption, so the reproduction below is not evidence about the dielectric — it is the instrument reading zero on a known zero.

The line being straight on a logarithmic time axis is the reason the durations matter as much as they do, and it is the same absence of a characteristic time that the corner where averaging stops working measures in a noise density. A quantity that grows by a fixed amount per decade has no settling time and no plateau, so there is no reading time far enough out to be safely past the effect and no short brief enough to be safely inside it. Every choice of the two durations lands somewhere on a straight line, and the specification’s choice is not distinguished except by being written down.

The departure #

Holding one duration and moving the other gives two curves, and they run in opposite directions for the reason the mechanism predicts.

A factor of thirteen on one knob and sixty-three on the other are not tolerances. Polypropylene at 0.02 per cent and polyester at 0.2 are a factor of ten apart and that gap is treated as a choice of material; the same modelled part, with nothing about it changed, spans more than that between a tenth-of-a-second short and a thousand-second one.

Nothing in either picture is a defect of the part. Both are correct measurements of the same dielectric, taken by the same procedure with one number in it changed, and the procedure has no way of saying which of them is the dielectric’s own.

The reading is a count of decades #

The two sweeps are one sweep. Plotting every test against the number of decades between its two durations, rather than against either of them, collapses three families of absolute time onto a single line.

The fit has no intercept because the physical statement has none: a test whose short and whose reading are the same duration collects nothing. Through the origin, over seventeen marches, the slope is the model’s own branch density to half a per cent — and the half a per cent is itself explicable rather than residual. The charge that comes back is divided by the whole capacitance the terminals see, which includes the slow branches it came off, so the recovered fraction runs slightly under the branch density and runs under it by more the more slow capacitance there is: 0.05 per cent low on a 0.02 per cent part, 0.53 on a 0.2 per cent part, 2.67 on a one per cent part. A first-order argument reproduced exactly would have been a first-order argument that was not being tested.

The residuals have a shape too, and it is a shape rather than scatter. At one decade of ratio every family sits about 4.5 per cent above the line; at two decades and beyond the worst is 1.23 per cent. The excess does not shrink when the model is given more branches per decade — one, two, four and eight branches a decade give 4.13, 3.76, 3.77 and 3.77 per cent — so it is not the model’s discreteness. It is the width of a single relaxation’s own transition: a branch does not switch from held to released at the instant the short reaches its time constant, it does so over roughly a decade around it, and a test only one decade wide is counting an edge rather than an interior.

One slope, measured four ways #

The claim that the answer counts decades is a claim about a single derivative, and the machinery already produces that derivative four separate ways, on curves that look nothing like one another.

The first is inside a single test. The recovery climbs a fixed amount per decade of waiting, and the specified test reports it directly: 0.09667 per cent of the charging voltage per decade, taken as the difference between the reading and the reading a decade earlier.

The second is across tests, moving the reading time. Fitting the recovery against the logarithm of the reading over four tests from a hundred to three thousand seconds gives 0.09829 per cent per decade.

The third is across tests, moving the short. The same fit over four shorts from 0.3 to 10 seconds gives −0.10128 per cent per decade — the same magnitude with the opposite sign, which is the statement that only the ratio matters, arrived at by differentiating rather than by collapsing.

The fourth is the collapse itself, 0.10288 per cent per decade over all seventeen tests, and behind all four sits the model’s own branch density at 0.10343.

Four readings inside seven per cent of one another, three of them measured on curves that look nothing like each other and one of them a parameter nobody was allowed to fit. What they share is the netlist; what they do not share is any part of the arithmetic that gets from it to a number, and the first and third do not even share the axis they are differentiated along. That is the standard the digits the arithmetic did not have asks of an agreement — two routes that share an assumption are one route — and the one thing all four do share is worth naming, because it is the subject of the section after next: every one of them assumes the distribution is broad enough that the test sits inside it.

What it means for a part rather than for a test #

If the quoted figure counts decades of the test, then a part whose relaxation times are spread over a different range answers differently everywhere except at the point where it was measured.

A factor of 3.31 at a short one decade longer than the specified one, between parts a purchasing specification would call interchangeable. And the narrowest of the four is the sharpest example of what the number conceals: it reads 0.296 per cent at a 300-millisecond short, half again its own specified figure, because its slow charge is concentrated exactly in the decades a fast test reads. A designer selecting on the data-sheet number, in an application whose own time scales are a decade off the test’s, has selected on a quantity that does not rank the candidates in the order that matters. This is the same failure the tolerance that is not on any part is about, arriving through a specification instead of through a distribution: a number that is true, and about something other than the thing being chosen.

Two routes to it, sharing no arithmetic #

A collapse this clean invites the suspicion that it is an artefact of the marching. The netlist has a second reading that never mentions time at all.

The two computations share the model and nothing else. One switches a resistor in and out and walks the charge forward with the trapezoidal rule; the other factorises a complex impedance twice and has no state, no time step and no switch in it. Their agreement in the interior is the kind one step, computed twice insists on, and it is what makes the disagreements at the ends readable: at one decade of ratio the march runs 4.39 per cent high, which is the same edge effect the residuals showed, and at a reading of nine thousand seconds it runs 3.23 per cent low, because the slowest relaxation the model has is ten thousand seconds and the test has run out of dielectric to collect.

That second departure is the honest statement of where the law stops. It is not a frequency and not an amplitude but a duration: the rule holds while both of the test’s times lie inside the distribution, and the disagreement between the two routes is what announces that one of them has left it. A model whose edge is visible as a divergence between two of its own readings is in better condition than one whose edge has to be asserted, which is the argument the edge that is a region makes for measuring edges rather than declaring them.

What temperature does, and does not do #

The rung below records that a polymer’s relaxation times move with temperature by roughly a decade per twenty kelvin, and that this is not modelled. The decade law makes the prediction without any new machinery: if the answer depends only on the interval between the two durations, sliding the whole distribution should change nothing at all.

It very nearly does not. Sliding the six-decade distribution up by one, two and three decades — with the branch fraction untouched — moves the specified test’s reading from 0.2000 per cent to 0.2011, 0.2014 and 0.2016. Three decades of relaxation time, some sixty kelvin, and the reading moves by eight parts in a thousand of itself.

Sliding it the other way is a different story. One decade down gives 0.1910 per cent and two decades down gives 0.1306, a fall of 35 per cent, because the slow end of the distribution has slid inside the fifteen-minute reading and there is nothing left arriving after it. The two-decade-wide part is worse in both directions: one decade down it reads 0.0578 per cent and one decade up it reads 0.2926.

So the temperature coefficient of a dielectric absorption figure is not a property of the dielectric’s temperature coefficient. It is a statement about where the edges of the distribution sit relative to ten seconds and fifteen minutes, and it is close to zero for a part whose distribution straddles the test and violent for a part whose distribution does not. That is the same shape as the edges that move with the room finds elsewhere — a coefficient that is really a boundary crossing — and it is why a part quoted at one temperature cannot be corrected to another with a percentage.

What this does not say #

It does not say the specified test is badly designed. Ten seconds and fifteen minutes sit in the middle of the relaxation range of the films the test was written for, which is exactly where a measurement should be taken, and the procedure is repeatable to the digit. What it does not carry is its own interval, and a number quoted without the two durations that produced it is a number missing half its content — the same omission the capacitance that is not one number finds in a ceramic’s bridge reading and the coefficient that is about one reading finds in its temperature envelope. In all three cases the specification is honest and incomplete in the same way: it reports a value and omits the condition that decides it.

It does not say the figure is useless for comparing parts. Two parts measured on the same test, whose distributions both straddle it, are compared correctly — that is what the collapse means. It says the comparison stops being valid when the application’s own time scales are elsewhere, and it gives the number for how far elsewhere is enough: about a decade, for a factor of three.

And the law itself has stated limits rather than implied ones. It is good to 1.23 per cent while the two durations are at least two decades apart and both inside the relaxation distribution; to about 4.5 per cent at one decade of ratio; and not at all once the reading passes the slowest relaxation there is. Those three regions are visible in one picture and each is announced by a different quantity — a residual, a residual with a systematic sign, and a disagreement between two independent routes.

The two figures together are the argument for the model having one degree of freedom and not two: the slope moves in proportion to the branch fraction and the shape does not move at all.

Where the part’s other numbers sit #

Dielectric absorption is the low-frequency end of a capacitor and it has a mirror image at the other end, where the quoted value is equally a condition rather than a property.

The two defects are opposite in shape and identical in kind. Above its self-resonance the part is an inductor and the specification says so with a frequency attached; at a hundredth of a hertz the same part is 0.3866 per cent more capacitance than the label says, and the specification says so with nothing attached at all. The difference is not physics. It is that one industry convention carries its condition and the other does not.

What a specification would have to say instead #

The repair is not a better dielectric model and it is not a longer test. It is two numbers already in the procedure and absent from the result.

A quoted absorption with its two durations attached — 0.2 per cent, 10 s / 900 s — converts to any other pair of durations by multiplying by the ratio of the decade counts, as long as both pairs lie inside the dielectric’s own range. A sample-and-hold acquiring for a millisecond and holding for ten spans one decade rather than the specification’s 1.954, so the same part contributes about half the quoted figure to the error the held sample inherits; a dual-slope integrator running a two-second cycle against a ten-millisecond autozero spans 2.3 decades and contributes about a sixth more. Neither of those is a correction a reader can make from 0.2 per cent alone, and both are one multiplication from 0.2 per cent over 1.954 decades.

What that conversion cannot do is tell a reader whether the second pair of durations is still inside the distribution, and that is exactly the case the four-part sweep makes dangerous. The honest version of the rule therefore has a condition in front of it rather than after it, and the condition is checkable with one extra reading: the same test run at a second pair of durations a decade away. If the two agree at the decade count, the distribution straddles both and the conversion is good; if they do not, the part has an edge inside the range being used and no single figure describes it. That is one more measurement on the same apparatus, and it is the difference between a number and a number with a range.

The number worth carrying #

1.0288 parts per thousand per decade, for the polyester modelled here, against a branch density of 1.0343 — which is to say that the recovery a soak test reads is the dielectric’s slow capacitance multiplied by the number of decades the test leaves in.

The habit that goes with it is a question to ask of any quoted figure with a procedure behind it. Which of the procedure’s parameters are sufficiency conditions, dropping out once they are large enough, and which are in the answer? The charging hour drops out and could be doubled without consequence. The other two are the answer, they are quoted in the same sentence as the number and read as though they were units, and moving either by a decade moves the result by half of it. A specification that names a condition is not thereby a specification that has accounted for it.