The capacitor that remembers

Every capacitor drawn on this site so far has been a number with two parasitics bolted to it. A series resistance and a series inductance, both of which make it worse at high frequency and neither of which changes what it is at low frequency: above its self-resonance the part is an inductor, and below it, it is the value on the label.

This is the defect at the other end, and it has the opposite shape. The part is more capacitance the longer it is looked at, and the error it makes in a circuit is a fraction of the last thing that happened rather than a constant.

The test that defines it #

Charge a capacitor to ten volts and leave it for an hour. Short it for ten seconds. Open the short and watch the terminals.

They do not stay at zero. They climb — to twenty millivolts on the part modelled here, two parts in a thousand of where they were, over the next fifteen minutes. Nothing leaked in and no energy was created: some fraction of the dielectric’s polarisation follows the field slowly, and the ten-second short discharged the fast part and not the slow part. When the short is removed the slow part is still polarised, and it puts its charge back into the plates.

The number a data sheet calls the dielectric absorption is that recovery divided by the charging voltage: 0.2 per cent here, which is a polyester film. Polypropylene manages 0.02, an X7R ceramic is around one, and an electrolytic is worse than either.

The shape is the finding, not the number #

The curve above is a straight line on a logarithmic time axis. That is not what a relaxation looks like. A relaxation is an exponential, which on this axis is flat and then falls off a cliff; this rises by a fixed amount per decade of time, gaining 0.097 per cent of the charging voltage in every factor of ten, for as many decades as the measurement runs.

So there is no time constant after which it is over. A decade before the reading it had reached 0.103 per cent — barely half the final figure — and a decade after it, it would be at 0.29. The fifteen minutes in the specification is not a settling time. It is a place on a slope, chosen by committee, and the number quoted is a property of that choice as much as of the part.

This is why the model here is a distribution of relaxation times rather than one. The nominal capacitance sits in parallel with seven series R–C branches whose time constants are spread a decade apart, from ten milliseconds to three hours. Equal capacitance per decade is what produces a response logarithmic in time: one branch comes to equilibrium per decade, each contributing the same step, for as long as the branches last.

That construction is not new here. The 1/f essay builds a noise density the same way — a chain of interleaved poles and zeros, one pair per decade — and gets a slope of −9.989 decibels per decade out of a network that was never told to have one. The same trick, applied to a polarisation instead of to a spectrum, gives a recovery that is logarithmic in time out of a network that was never told to be. In both cases the shape is a consequence of the spacing rather than of any single element, which is why neither has a characteristic frequency or a characteristic time.

A switch is a different netlist with the same charge in it #

The measurement is three phases and only two of them are marched.

The charging hour is not, because an hour is long against every time constant in the model, so the fully charged state is every capacitor at ten volts — exactly, not approximately. Marching an hour to arrive at a state that can be written down would be theatre.

The short and the open are marched, with the state handed from one netlist to the next. That is what a switch is in this collection: not an element with a law, but a different circuit with the same charge in it, which is the same treatment the sampled-noise essays give a sampling switch and the treatment the site’s marching machinery was extended to support.

The short phase has a trap in it worth recording. The nominal capacitance discharged through an ohm has a time constant of microseconds, and the phase lasts ten seconds, so a uniform step large enough to reach the end is ten thousand times the fastest constant in the netlist. The trapezoidal rule is stable there and it is not accurate: its amplification factor approaches −1, so the fast state alternates in sign and decays hardly at all. A first version did exactly that and reported an absorption of 2.9 per cent — four times the total slow charge in the model, which is impossible, and is what said the number belonged to the integrator rather than to the dielectric. Marching the phase by decades fixes it, and keeps the per-step error argument the transients field’s own gate rests on intact.

The same part is seven capacitors #

The second route to the same defect does not switch anything. It solves the same netlist as an admittance, over frequency, and reads C = Im(Y)/ω — which is what a bridge does.

At a kilohertz the part reads 1.00000 µF — its nominal value to a part in ten thousand — because every slow branch is an open circuit at that frequency. That is why the defect is invisible to the instrument most people measure a capacitor with, and why a part can be within a per cent on a bridge and wrong by a per cent in a circuit.

At a tenth of a millihertz it reads 1.00593 µF. Same part, same terminals, 0.593 per cent more capacitance, and the difference is entirely which branches have had time to charge.

The loss and the dispersion are one quantity #

There is a relation between those two readings that neither of them was told about, and it comes out of the model exactly.

The dissipation factor over the band where the branches are — the ratio of the real to the imaginary part of the admittance — is 6.94×10⁻⁴, and it is flat: the same value across four decades, which is what a constant-phase element is. Meanwhile the capacitance falls with frequency along a straight line on a logarithmic axis, with a slope of one branch per decade.

Those two numbers are not independent. π/2 times the capacitance’s own logarithmic slope is 6.99×10⁻⁴, which is the measured loss angle to within a per cent — and nothing in the netlist was told to make it so. It is Kramers and Kronig arriving in a Foster network: a system that cannot respond instantly must dissipate, and the amount it dissipates is fixed by how much its response depends on frequency.

The practical consequence is that the two specifications a data sheet gives — dissipation factor and dielectric absorption — are not two facts about the part. They are one fact read twice, and a part with a low dissipation factor at a kilohertz has low absorption for the same reason rather than as a happy coincidence. Polypropylene’s 0.0002 loss angle and its 0.02 per cent absorption are the same number.

Why the number is a property of the test #

It is worth being exact about how much of the quoted figure belongs to the part.

Every one of the three durations in the specification decides the answer. The charging hour has to be long against the slowest branch or the slow branches never polarise; the ten-second short discharges every branch faster than ten seconds and leaves every branch slower than ten seconds alone; the fifteen-minute reading catches whatever has come back by then. 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.

The model makes that measurable rather than arguable. Adding branches outside the window the test exercises changes the reported absorption hardly at all: extending the distribution from six decades to ten — the same fraction per branch, the same shape, half again the total slow charge — moves the measured figure from 0.1934 per cent to 0.1940, four parts in a thousand of itself, because the extra branches are either far too fast to survive the short or far too slow to have returned by the reading.

So a data sheet’s absorption figure is a statement about the part inside a particular window of times, and a circuit whose own time scales sit outside that window is not described by it. A hold of ten microseconds is exercising branches the ten-second short never touched. A dual-slope integrator running at fifty conversions a second is exercising branches the fifteen-minute reading had long since finished with. Both are affected by the same dielectric and neither is predicted by the number that was measured, which is the practical reason this collection prefers a measurement over a specification wherever the two are available.

What it costs a circuit #

An offset that can be measured once and subtracted is not a problem. This one cannot, because it is not an offset.

Hold a sample-and-hold at ten volts for long enough that every branch has followed. Acquire zero volts through a switch. Open the switch and watch.

The held voltage creeps back towards where it was, by a fraction of the step. So the error is zero for a constant input, worst for a full-scale one, and proportional to the change since the last acquisition — which makes it signal-dependent, and a signal-dependent error appears as distortion rather than as a line in a specification. A converter with this in front of it has a total harmonic distortion that gets worse with signal amplitude and does not appear anywhere in its own error budget.

The numbers, for the 0.2 per cent part: ten microseconds of hold costs 1.0 part per million, which is one bit at 19.9 bits. A hundred microseconds costs 10 ppm — 16.5 bits. A millisecond costs 101 ppm, which is one bit at 13.3 bits.

So a sixteen-bit converter with a hundred-microsecond conversion time, built with a polyester hold capacitor, is at its last bit before anything else in the design has been considered. Not because the capacitor is out of tolerance — it is bang on value at a kilohertz — but because the quantity that matters is one the tolerance does not describe.

The curve is logarithmic again, for the same reason the recovery is: about a bit per three decades once the branches are in range. There is no hold time short enough to escape it, and none long enough to make it catastrophic.

Which capacitors this rules out #

The consequence for design is a short list and it is stricter than it looks.

A hold capacitor cannot be ceramic above a few bits. An X7R at one per cent absorption costs a bit at seven, which is the resolution of nothing. NP0 is a different dielectric with a different mechanism and is fine; the point is that “a hundred nanofarads, five per cent” describes both.

An integrator’s capacitor decides its linearity. A dual-slope converter integrates up for a fixed time and down until it crosses zero, and dielectric absorption makes the down-slope depend on how far the up-slope went — which is a nonlinearity in the conversion, not an offset, and it is the reason the good ones use polypropylene or polystyrene.

And a sample-and-hold’s settling specification is about the wrong part of the settling. A specification of the form settles to 0.01% in 2 µs is a statement about the amplifier and the switch. The dielectric’s tail is at 0.01 per cent after ten microseconds and is still growing at a second, so the two quantities do not even overlap in time, and the specification that was met says nothing about the error that arrives afterwards.

That last one is the same shape as the doublet a pole-zero cancellation leaves behind: a fast settling that meets its number and a slow tail underneath it that no one measured, with the tail deciding the accuracy and the fast part deciding the specification. The two even have the same cure — a slower loop that does not need the fast settling — and the same reason for being missed.

What is not modelled #

The branch values are a construction, not a measurement. Seven branches a decade apart with equal capacitance is the simplest thing that produces the observed shape, and a real dielectric’s distribution is neither uniform nor exactly logarithmic. What that changes is the detail of the curve between decades; what it does not change is that the response is logarithmic rather than exponential, because that follows from the distribution being broad rather than from its being uniform.

Nothing here is nonlinear. A real dielectric’s absorption depends slightly on the field it was polarised in, which makes the error depend on the square of the step as well as on the step. That is a second-order distortion term and this model cannot have one: every element in it is linear, so the error is exactly proportional to the change and the harmonic structure it would produce is absent.

Nor temperature-dependent. The relaxation times of a polymer move with temperature by roughly a decade per twenty kelvin, which slides the whole curve sideways — so the absorption a part shows at 85 °C is the absorption it would have shown at room temperature had the test used a different set of durations. That is measurable with the same machinery and is not measured here.

And the leakage is separate. A real capacitor also has a parallel resistance, which produces a droop rather than a recovery and has the opposite sign. In a hold circuit the two are added and can partially cancel, which is a coincidence rather than a design, and would make the total error smaller at one hold time and larger at every other.

What the gate checks #

The two routes are required to agree in the one place they can. They are not the same measurement — one is a fifteen-minute reading after a ten-second short and the other is a pair of bridge readings — so what is asserted between them is the relation: the fraction of the capacitance that is slow, over the band of times the test exercises.

The recovery is asserted to be monotone over the whole open-circuit phase, and to be still rising at the end: the reading at a tenth of the time must be under half the final one, which is the claim that the fifteen minutes is a place on a slope. A version whose recovery had settled would fail.

The marched test is required to reproduce the absorption the model was built to, to two per cent, which is what makes alphaFor a bisection on a measurement rather than an algebraic inversion.

The loss angle is asserted against π/2 times the capacitance’s own logarithmic slope, to six per cent, across the flat part of the band — an assertion between two quantities the model was never told to relate.

And the hold error is asserted to grow with every hold time drawn and never to come back, and to cost more than four bits over four decades, so a model whose tail settled would fail rather than be quietly plotted.

Where this sits #

The frequency field measured a capacitor’s upper edge and found a self-resonance above which the part is an inductor. The noise field measured the floor a sampling capacitor sets and found kT/C, with no resistance in it. This is the third quantity a capacitor has that its label does not mention, and it is the only one of the three that depends on what the circuit did to the part before the measurement started — which puts it beside the tolerance that is not on any part as a quantity that is real, consequential, and absent from every number on the label.

That is what makes it hard to design against. A frequency limit can be checked against a spectrum and a noise floor against a bandwidth, but an error that is a fraction of the previous sample has to be checked against the signal — and the signal is what the circuit was built to be ignorant of.