The coefficient that is about one reading

The capacitance that is not one number takes the two numbers a class II ceramic’s data sheet prints — the nominal capacitance and the capacitance at the rated voltage — pins a charge model with them, and finds that the part has three capacitances: 2.000 µF as a slope at its rated voltage, 5.814 µF as a charge average over the range, and 9.483 µF as a bridge reads it at zero bias.

A data sheet prints a third number as well, and this essay is about what it is a number of.

The temperature coefficient of a class II dielectric is stated as an envelope: X7R holds ±15 per cent from −55 to +125 °C, X5R the same over a narrower range, Y5V +22 and −82. It is one number and it describes one reading — the bridge’s, at zero bias, with a one-volt test signal, at a kilohertz. The model has two parameters. One printed coefficient does not determine two, and what is left over is not small.

Which two parameters #

The model is a charge rather than a capacitance, which is the rung below’s whole construction. A class II ceramic’s charge is a linear backbone plus a saturating ferroelectric polarisation:

$$q(v) = C_{\infty} v + Q_s \tanh\!\left(\frac{v}{V_k}\right)$$

with three constants. The backbone C∞ is the ordinary permittivity and does not saturate — without it the capacitance goes to zero at high bias rather than to a floor, which is the fault the rung below found by leaving it out. The polarisation has an amplitude and a characteristic voltage Vk, and these are the two that temperature acts on.

They act differently and for different reasons. The amplitude is how much aligned polarisation there is, and it falls as thermal agitation competes with it. The characteristic voltage is how much field it takes to align, and it falls as the material softens toward its Curie point. A dielectric formulated to be flat over a range is one where those two nearly cancel across it, which is what a ±15 per cent X7R is and why a Y5V, formulated for capacitance per volume rather than for flatness, is +22/−82.

The data sheet says what their combined effect on one reading is. It does not say how much of it is which, and nothing in the specification requires the manufacturer to know.

Every split honours the specification #

So the honest thing is to parameterise the ignorance: let a fraction f of the stated change be the amplitude’s and the rest the characteristic voltage’s, and for each f solve for the scale that puts the measured capacitance exactly on the envelope. Every point below is a part that meets its data sheet to six digits.

At the cold end of a ±15 per cent envelope the charge-average capacitance — the number a reservoir, a hold capacitor or a filter’s step response obeys — is anywhere from −42.9 to −13.9 per cent. That is 29.0 points of ambiguity at a single temperature, where the quantity the specification is written about is pinned exactly, to six digits, at −15. Across the whole envelope the working capacitance runs over 79.0 points against the specification’s 30 — a factor of 2.63 — and the specification is met exactly at every point of it.

At the hot end it runs from +13.9 to +36.2 per cent, 22.3 points, and one of the five splits is impossible: with the polarisation’s amplitude held fixed, the measured capacitance at zero bias is bounded above by the zero-bias capacitance itself, so no widening of the characteristic voltage can produce a fifteen per cent rise. That asymmetry is the only thing a data sheet’s envelope says about the split, and it says it at one end only.

Two parts a bridge cannot tell apart #

The two extremes of the cold curve are worth drawing as parts rather than as percentages.

At zero bias with a one-volt test signal both read 8.061 µF — the same number, to six figures, which is the constraint the data sheet imposes and the whole of it. At the rated voltage one is 1.016 µF and the other is 1.832 µF, a factor of 1.80.

Both are honest parts meeting a published tolerance. A circuit using them at their rated voltage is using two different capacitors, and no measurement made under the specified conditions distinguishes them.

This is the same shape as the rung below’s finding and one level further out. That essay’s point was that the capacitance is three numbers and a data sheet reports one of them; this one’s is that the temperature coefficient is likewise a coefficient of that one, and the other two inherit it with an uncertainty that the specification never mentions.

Where the ambiguity is paid for #

Three circuits use three different capacitances, and the ambiguity reaches each of them differently.

A decoupling bank uses the small-signal value at the working bias, because the ripple it is carrying is small. The pair that is worse than either computes an anti-resonance from the ceramic’s capacitance against the bulk part’s inductance, and its frequency goes as one over the square root of that capacitance — so a factor of 1.80 in the ceramic moves the anti-resonance by a factor of 1.34, and the capacitor that is not where the load is shows what a moved anti-resonance does to a target-impedance band. Nothing in the bank’s design procedure has that factor in it.

A hold or reservoir capacitor uses the charge average, because it is being charged and discharged across a range rather than perturbed about a point. That is the number the envelope above puts across twenty-nine points, and the direction is the bad one: a droop specification computed from the data sheet’s own figure can be out by a factor of nearly two in the direction that makes the droop worse.

A timing capacitor uses whichever of the three matches how it is driven, which is usually the charge average again — and a timing error of forty per cent is not a tolerance, it is a different circuit.

What the distortion does, which is the other half #

The rung below’s second finding is that the same nonlinearity is a distortion, and second order without bias and first order with one. Both the amplitude and the characteristic voltage appear in that result, so the ambiguity reaches it too.

The distortion at a stated drive goes roughly as the drive over the characteristic voltage, so the part whose cold-end change is all in the characteristic voltage — the one with Vk down at 1.29 V against the nominal 2.84 — has more than twice the distortion of the part whose change is all in the amplitude, at the same measured capacitance and the same temperature.

So a designer who has measured the distortion of one part at one temperature has measured the distortion of that part at that temperature, and the specification does not carry the number forward. Neither does the tolerance.

What would settle it #

One extra printed number closes the whole thing, and it is a number a manufacturer already measures: the capacitance at the rated voltage, at the extremes of temperature. The rung below’s model is pinned by exactly two readings, so two readings at each temperature pin it at each temperature, and nothing above remains ambiguous.

That is not a request for a new measurement. Every manufacturer plots capacitance against bias at several temperatures in an application note; what is missing is the specification, which is stated at one temperature and one bias while the derating curves are drawn as typical performance. A typical curve is not a bound, and a bound on the wrong reading is not a bound on what a circuit uses.

The same distinction is the tolerance that is not on any part’s, in the other direction. There the quantity that matters is a ratio and every part is specified absolutely, so the specification is far looser than what the circuit needs and matched parts do much better than their tolerance. Here the quantity that matters is a different reading of the same part, and the specification is looser than it looks in a way nothing on the page discloses.

What a bench measurement would settle, and what one usually measures #

The two-reading construction says exactly what to measure and it is two sweeps rather than a programme.

Put the part in an oven, hold it at the cold extreme, and measure the small-signal capacitance at zero bias and at the rated voltage. Two numbers pin the model at that temperature. Repeat hot. The whole of the ambiguity above is gone, and what is left is a part whose behaviour is known at every bias and every temperature in the range from four measurements.

What is usually measured instead is the zero-bias value over temperature, because that is the number the specification is written in and the one a bridge gives without a bias supply in the way. It is the measurement that confirms the part meets its data sheet and tells a designer nothing about the quantity their circuit uses — the same relationship as the point the device is never at describes for a transistor’s specification point: a number measured under conditions chosen for reproducibility rather than for relevance.

The bias supply is the reason, and it is a real one. Measuring a 10 µF part at five volts of bias on a bridge needs a bias tee that is not a source of its own capacitance, and the part’s own series inductance puts an upper limit on the test frequency that a bias network makes worse. Nothing here is difficult and all of it is why the measurement is not routine.

Three specifications, one reading, three different consequences #

It is worth putting this essay beside the other two in this collection that are about the same structure, because three is a pattern.

The ripple that is a temperature finds a filter’s passband ripple crossing its specification at 89 °C, with the whole of the dependence in an amplifier’s transconductance and none of it in the filter, and with the deciding parameter — how the tail current is biased — not on the data sheet at all.

The three tolerances that do nothing finds that five component tolerances reach a response through two combinations, so three fifths of a tolerance budget buys nothing and no per-component specification says which.

And this one finds a coefficient that is exactly true about a reading nobody uses.

The common structure is that a specification is a statement about a measurement, and a circuit is a statement about a quantity, and the map between the two has a kernel or an ambiguity in it that the specification does not describe. That is not a criticism of specifications: a measurement is what can be agreed on between a buyer and a seller, and a quantity is not. It is an argument for computing the map rather than assuming it is the identity.

What is not in this model #

No actual temperature coefficients. The envelope is an input, taken from a published class, and the split is swept because it is unknown rather than estimated. A measurement of two parts of one type over temperature would fix the split for that type in an afternoon, and this essay says what such a measurement would be worth — which is the point of computing the ambiguity rather than assuming a value for it.

No ageing. A class II ceramic loses capacitance logarithmically with time since it was last heated past its Curie point — a per cent or two per decade of hours, quoted from a stated reference time — and that is a third coefficient on the same one reading, with the same ambiguity about which parameter it acts on and the same consequences. The arithmetic here would carry over unchanged.

And no frequency. Everything above is at the kilohertz a bridge measures at. The polarisation that saturates is also the polarisation that relaxes, so the amplitude in this model is itself a function of frequency — the capacitor that remembers is the essay about that relaxation, and the two nonlinearities live in the same material and are not independent.

The direction the error goes #

One more thing falls out of the two curves and it is the one a designer can act on without measuring anything.

The ambiguity is not symmetric about the nominal. At the cold end every split gives a working capacitance below nominal — between 13.9 and 42.9 per cent below — and at the hot end every split gives one above. So the sign is known even though the size is not, and the two ends of the temperature range are not equally dangerous.

Which end is the bad one depends on the circuit, and the two ordinary cases point opposite ways. A decoupling bank wants capacitance and is worst when it is cold; a timing circuit or an anti-alias filter wants a known capacitance and is worst wherever the departure is largest, which is the cold end again because the range there is 29 points against 22. A reservoir sized for droop at room temperature and qualified at −40 °C has between fourteen and forty-three per cent less charge than the arithmetic assumed, and the arithmetic assumed the data sheet’s fifteen.

So the practical rule, with no measurement at all: derate a class II ceramic’s working capacitance by roughly twice its stated envelope at the cold end, and treat the hot end as a bound on nothing. That is a rule of thumb, which this collection does not usually offer, and it is offered here because the alternative is a rule of thumb that is wrong by the factor above.

The habit this belongs to #

The site’s rule is that no model is drawn without the frequency, amplitude or size at which it stops being true. A specification is a model too, and this one stops being true in a way that has no axis at all: it is exactly true, forever, about a quantity nobody uses.

What the measurement adds is a number for how much that costs. Twenty-nine points of working capacitance from a fifteen-point envelope, and a factor of 1.80 between two parts that a bridge, under the conditions the specification names, reports as identical.

Where a factor of 1.80 lands #

That number is abstract until it is put into a circuit, and three circuits in this collection are decided by a capacitance closely enough for it to matter.

The period a delay lengthens has a period directly proportional to the capacitance, so two parts a bridge cannot tell apart set periods 1.80 apart — on a circuit whose other error terms are measured to fractions of a per cent and whose whole essay is about a comparator delay worth two per cent.

Two requirements pulling one capacitor has a stability floor and a transient optimum ten per cent apart in the series resistance, computed at a stated capacitance — and its own sweep shows the floor moving from 939 milliohms at 10 µF to 657 at 2 µF. A factor of 1.80 in the capacitance moves the design across most of that table.

And the band that closes with the order has a tolerance of a tenth of a decibel bounding a band of impedance levels, on sections whose quality factor is a ratio of two capacitances. A ratio survives a common factor and does not survive two parts of the same nominal value sitting at different points of their own bias curves, which is what this essay’s two indistinguishable parts are.

The common thread is worth stating because it decides where the problem is worth solving. In all three the capacitance is not being measured — it is being relied upon, at a bias and an amplitude the specification never mentions. A part characterised the way this essay characterises it, with two parameters instead of one, is not more expensive to make; it is more expensive to print.