The reading a data sheet does not take

The coefficient that is about one reading ends on an unpinned model. A class II ceramic’s temperature envelope is a coefficient of exactly one reading — what a bridge sees at zero bias with a one-volt test signal — and the charge model behind the part has two parameters. One equation, two unknowns. Every way of dividing a ±15 per cent envelope between them honours the envelope to six digits while putting the working capacitance anywhere across 29 points at the cold end and 79 across the whole envelope, against the 30 the specification bounds.

That is where the argument stopped, and it stopped one question early. An underdetermined system is underdetermined by the measurements taken, not in itself, and nothing there asked whether some other reading tells the candidates apart.

The five parts, and how far apart they really are #

It is worth having the candidates as objects before treating them as points on a curve, because the spread the first figure draws is a spread of one derived quantity and the parts differ in more than that.

The nominal part is ten microfarads at zero volts falling to two at its rated five, which pins the charge law’s two parameters: a saturating polarisation of nine microfarads’ worth of amplitude and a characteristic voltage of 2.836 volts. Cool it until the bridge reads 15 per cent low and the five candidates are five different parts. The one that puts the whole change into the amplitude keeps $V_k$ at 1.291 volts and holds ten microfarads at zero bias; the one that puts it all into the characteristic voltage keeps $V_k$ at 2.836 and holds 8.491. At the rated five volts they hold 1.0156 and 1.8323 microfarads — a factor of 1.804 between two parts that a bridge reads as the same number — and the charge-average capacitance a reservoir would obey runs from 3.3223 to 5.0065, a range of 50.7 per cent.

None of that is a tolerance in the ordinary sense, which is why it is not covered by one. The parts are not scattered about a nominal; each of them is exactly on the specification, at the only condition the specification mentions, and they diverge everywhere else. A tolerance stack that treats the temperature coefficient as ±15 per cent on the working value is not conservative and not optimistic; it is answering a different question, and the shape of that mistake is the one the tolerance that is not on any part collects.

What a bridge reading is a function of #

A bridge does not measure a capacitance. It applies a sinusoid, takes the fundamental of the charge waveform, divides by the fundamental of the voltage, and reports the quotient — which for a capacitance that varies with the volts across it is a quantity with two arguments: the direct voltage the part is sitting on, and the amplitude of the sinusoid used to interrogate it.

The specification names one point on that surface. A kilohertz, no direct bias, one volt of test signal is not a footnote to the temperature coefficient; it is a coordinate, and the coefficient is a statement about the value of the surface there and nowhere else. The rest of the surface has never been drawn on this site — the rung below reads it along the bias axis at one amplitude, and the rung below that along the amplitude axis at zero bias.

The two ends of that picture disagree about the sign, and that is the feature worth having. At zero bias the small-signal capacitance is at its maximum, so a sinusoid centred there spends all of its time below the peak and the average comes out low — lower the harder the drive. Near the rated voltage the curve is convex, so the swing spends more of its time above the local value than below it and the average comes out high. Between the two there is exactly one bias at which the two effects cancel, and at that bias a tenfold change of test level moves the reading by nothing at all.

For this part it is at 1.8848 volts, which is 0.6645 of the polarisation’s characteristic voltage. Across the slider the bias moves from 1.570 volts to 3.430 while that fraction stays between 0.6603 and 0.6671, which is the first sign that the turnover is a property of the charge law rather than of the part.

Where the turnover actually is #

It is worth knowing exactly, because a bisection that lands near a round number is the kind of result this collection distrusts on principle — the habit the digits the arithmetic did not have is about.

The small-signal capacitance is C∞ + A·sech²(v/$V_k$), and a bridge reading at small amplitude is that value plus a term in the square of the amplitude times the curve’s own curvature. The turnover is therefore where the curvature vanishes — the inflection of sech² — and differentiating twice gives 3·tanh²x = 1, so x = artanh(1/√3) = 0.658479. That is arrived at with no bridge in it at all.

The two routes share the charge law and nothing else. One is a derivative taken by hand; the other simulates a bridge, sums four thousand points of a charge waveform at each of two amplitudes, and bisects on the difference. That they agree to a part in seventy thousand is the calibration this essay is quoted against, and the fitted exponent of 1.998 is what says the remaining gap is the arithmetic of a finite swing rather than a fault in the search — the same distinction every model has an edge insists on between a model’s error and a method’s.

Five parts a data sheet cannot tell apart #

Now put the candidates on that surface. Five parts, each honouring a −15 per cent envelope exactly, differing only in how the stated change is divided between the polarisation’s amplitude and its characteristic voltage — the five points along the lower curve of the first figure.

The condition the specification names is a node. Move a fiftieth of the way along the amplitude axis, on the same instrument with no bias supply and one knob turned, and five parts that were indistinguishable are separated by a sixth of their own value. The information was never absent from the part; it was absent from the one place the specification looks.

There is a reason that place is the worst one available, and it is not an accident of this model. At zero bias and vanishing amplitude the small-signal capacitance is C∞ + A exactly — sech²(0) is one, and the characteristic voltage does not appear in the expression at all. $V_k$ enters a zero-bias reading only through the curvature explored by the test swing, which is a second-order effect on a first-order quantity. So the specified condition is very nearly blind to one of the two parameters its own coefficient is a coefficient of, and the standard one-volt test signal is the only thing that lets it see anything.

Two axes, two parameters #

The reason one reading resolves the split at all is worth writing down as algebra, because it says which reading to take before any of them is taken — and that is the transferable part.

The charge law has the polarisation entering as A·tanh(v/$V_k$), so the small-signal capacitance is C∞ + A·sech²(v/$V_k$). At v = 0 that is C∞ + A and $V_k$ has cancelled out completely. Move the bias to the neighbourhood of $V_k$ and the sech² is falling at its fastest, so the reading is dominated by where $V_k$ is and only weakly by how large A is. The two axes of the surface are therefore not two views of one thing; they are two nearly independent questions, one asking how much saturating polarisation there is and the other asking at what voltage it saturates.

That is why the amplitude axis works at the cold end and fails at the hot one, and why the bias axis works at both. At the cold end the five candidates differ by 17.8 per cent in C∞ + A, so the quiet zero-bias reading — which is very nearly that sum — separates them. At the hot end the envelope’s own ceiling removes the division that would have moved the amplitude furthest, the four survivors differ by only 2.6 per cent in that sum, and the quiet reading has almost nothing to work with. Their characteristic voltages still differ, from 2.836 volts up to 3.985, and a reading taken where that matters finds them at once.

The cold end is the same story with the numbers larger. Read at 2.5 volts of bias with a fiftieth of a volt of signal, the five cold candidates are 176.02 per cent apart — a spread three and a half times the whole ambiguity in the working capacitance, from an instrument reading a part the specification says is one part.

The end of the envelope decides which reading to take #

Everything above is at the cold end. The hot end behaves differently, and the difference is practical rather than a curiosity.

A quiet reading at zero bias is nearly a direct measurement of C∞ + A, and at the hot end the candidates barely differ in that sum: they are pushed apart in $V_k$ instead, because the envelope’s ceiling has removed the division that would have moved the amplitude furthest. So the axis that worked at one end has almost nothing to say at the other.

The axis that does have something to say is the bias.

The reason is the mirror of the one above. A bias of 2.5 volts is close to the characteristic voltage of 2.836, which is exactly where sech² is falling fastest — so a reading taken there is a reading of $V_k$, the parameter a zero-bias reading cannot see. Two readings, at two conditions, against two parameters: the same shape as one number from two measurements, where a mutual inductance is extracted from a series-aiding and a series-opposing reading because neither alone contains it.

What one further reading is worth #

The separations above are qualitative until they are put against how well a real bridge reads, so the last figure asks the question a buyer would: given a second reading known to some tolerance, how much of the working capacitance is left undetermined?

Two facts in that picture are worth separating. The first is that a second reading is worth an enormous amount at all: one per cent on a bridge, which is a poor bridge, takes an eighteen-point uncertainty to half a point. The second is that the two candidate readings differ by 8.41 in what they buy for identical effort, and nothing in a data sheet, a specification or a purchase order says which to take. The choice is decided by which parameter the reading is sensitive to, and that is a statement about the model rather than about the instrument.

The straightness of both lines matters too. What a reading is worth is proportional to how well it is known over the whole range tried, which means there is no threshold and no diminishing return: a reading twice as good is worth exactly twice as much, all the way down. That is a comfortable property to have and it is not the usual one — the trade in two requirements pulling one capacitor has an interior optimum and this does not.

What it does not say #

It does not say the part is now known. The model has three parameters and only two of them are ever discussed: C∞, the linear backbone the vacuum and the electrode geometry supply, is fixed here at a microfarad by assertion, and no data sheet prints it either. Two readings pin two parameters, and the third is still a choice. What a third reading would do is not pin anything — it would over-determine the model, and a third reading that disagreed with the two would be the first evidence that a tanh is the wrong charge law. That is the more valuable measurement of the two, and it is the one nobody takes, for the same reason nobody takes the second.

It does not say the specified condition is badly chosen. Zero bias and a one-volt test are reproducible on any instrument in any laboratory, need no bias supply, and are what makes two data sheets comparable at all. What the surface shows is that reproducibility and informativeness are different requirements, and that a condition optimised for the first can land on a node of the second — the same trade two terminals measure the leads as well makes about a resistance measurement, where the convenient connection and the honest one are not the same connection.

And it does not say a designer can act on any of this. The second reading is not on the data sheet, so obtaining it means having the part in hand and a bridge with a bias input — which is a bench measurement of a specific reel rather than a specification, and a different reel is a different answer. The result is about what a specification could determine and does not, which is why it belongs beside the tolerance that is not on any part rather than beside a design rule.

Why this is not a complaint about the part #

Nothing above is a defect in the ceramic. The part does exactly what its physics says: a ferroelectric polarisation saturates, and a saturating charge law has two parameters because it has a size and a scale. Nor is it a defect in the bridge, which reports the fundamental ratio it is asked for and reports it correctly at every point of the surface.

The defect is in what a specification is a specification of, and it has a shape that recurs wherever a model has more parameters than a document has lines. A published envelope constrains one scalar function of the parameters; the circuit obeys a different one; and the two agree only if the model has one parameter, which is precisely the case in which nobody would have written a model. The capacitance a reservoir obeys, the capacitance a small ripple sees and the capacitance a bridge reports are three different functionals of one part — that is the capacitance that is not one number’s finding — and adding a temperature coefficient adds one constraint to a model that has just been shown to support three distinct answers.

So the honest reading of a data sheet’s ±15 per cent is narrower than it looks and wider than it is usually taken to be. It is exactly true, at one coordinate, of one of the three capacitances, and it implies a range rather than a value everywhere else. Quoting it as a tolerance on the working value is the same category error as quoting a bridge’s zero-bias reading as the capacitance in a regulator’s output filter — a mistake the capacitor that was right once prices in a different setting, where the part’s value is exactly right at the one operating point it was chosen at.

What the surface opens #

The turnover has a use that this essay does not exploit and a later one might. It is the one bias at which a bridge reading is insensitive to the level it is taken with, so it is the condition at which two instruments with different test amplitudes would agree — and disagreement between instruments is a standing complaint about class II ceramics that the capacitance that is not one number traces to exactly this cause. A specification written at 0.658 of the characteristic voltage would be reproducible for a different and better reason than the one at zero bias is.

There is also the other defect of the same part. The capacitance that depends on when it is measured, in the capacitor that remembers, is a second surface with time on one axis, and its specified test — charge for an hour, short for ten seconds, read after fifteen minutes — is three coordinates rather than two. Nothing has yet asked whether that test, too, is taken at a point that happens to be blind to one of its model’s parameters. The machinery to ask is the same machinery, and the question is exactly the one this essay answered for the bias axis.

The number worth carrying #

±18.40 per cent from the specification, ±0.512 from the specification plus one reading known to one per cent — and only if the reading is taken at a bias. Taken at the condition the specification itself names, that same reading is worth nothing at all, because five parts that differ by 1.804 in what they do at their working voltage read 8.060799 µF there to six digits.

The habit that goes with it is about what underdetermination means. A model with two parameters and one measurement is not a model that cannot be known; it is a model that has been asked one question, and the useful work is finding the second question rather than widening the error bars on the first. The place to look for it is where the missing parameter has leverage, which is a property of the model and is knowable before any measurement is made — and in this case it is the one axis the specification holds at zero.