The capacitance that is not one number

Two rungs of this anchor have already found that a capacitor is not a capacitance. The capacitor that is an inductor measured the frequency above which the part’s series inductance owns its behaviour, and the capacitor that remembers measured a value that depends on when it was looked at — 1.00000 µF at a kilohertz and 1.00593 µF at a tenth of a millihertz, from a distribution of relaxation times.

Both of those are small effects on a good part. This one is not small, it is not a parasitic, and it is not at the edges of the range: a class II ceramic loses most of its capacitance under the direct voltage it is being used at, and the loss is a factor rather than a per cent.

The model is a charge, and it has two parameters #

The dielectric in a class II ceramic is a ferroelectric, and a ferroelectric’s polarisation saturates. So the honest statement is not a capacitance that varies but a charge that is not proportional to voltage:

$$q(v) = C_\infty v + Q_s \tanh(v/V_k)$$

— a linear backbone the vacuum and the electrode geometry supply, plus a polarisation with a limit. Its slope is $C_\infty + (Q_s/V_k)\,\mathrm{sech}^2(v/V_k)$, which is the nameplate value at zero and falls towards the backbone.

There are two parameters and both are pinned by two numbers a data sheet actually prints: the capacitance at zero bias and the capacitance at the rated voltage. There is no third degree of freedom to tune an answer with, which is the property this collection asks of every model it draws.

The linear backbone is load-bearing rather than decorative. Written without it — a bare tanh — the capacitance would go to zero at high bias instead of to a floor, and a circuit marched with such a part behaves in ways no capacitor does. The same structural point appears in the magnetics field, where a core’s B–H curve written without its μ₀ term makes an inductance collapse to nothing rather than to its air-core value.

Three capacitances, and the question each answers #

At the rated five volts this part has three values, and every one of them is the capacitance.

The small-signal value is the slope at the bias: 2.000 µF. That is what a ripple sees, what sets a decoupling network’s impedance, and what belongs in any alternating-current calculation on top of a direct bias.

The charge-average value is the total charge moved from zero divided by the voltage: 5.814 µF. That is what a reservoir obeys, what a sample-and-hold’s droop is computed from, and what decides how long a rail holds up when a supply is removed. It is nearly three times the first.

And what a bridge reads is neither: 2.105 µF, because a bridge applies a test signal of its own and takes the fundamental of the resulting charge waveform. That is a measurement condition rather than a property.

Which of the three is wanted is decided entirely by the question being asked, and the failure mode is not using the wrong one — it is not knowing there were three.

The measurement condition, with the application removed #

The last of those three deserves its own experiment, because it is the one that makes a data sheet’s small print load-bearing.

At zero bias — no application, no rail, nothing but the part and the meter — a one-volt test reads 94.8 per cent of the small-signal value and a two-volt test reads 83.1. The saturating term is an even function of voltage, so a sinusoid centred on zero still spends most of its cycle away from the peak of the curve, and the fundamental it returns is smaller than the slope at the origin.

So a data sheet’s 1 kHz, 1 V rms is part of the specification and not a footnote to it. Two parts measured under different conditions are not comparable, and a part measured at one volt is being reported five per cent low before it has been fitted to anything.

That is the same class of statement how small is small signal makes about a transistor: there is an amplitude below which a linear description is right, the amplitude is computable, and above it the measurement is of something else.

How large the derating actually is, and on what #

The figure above uses a part that keeps a fifth of its value at rated voltage, which is an ordinary X5R in a small case. The derating is not a property of the dielectric alone: it is a property of the field, so a thinner dielectric derates faster at the same terminal voltage, and a smaller case at the same capacitance and voltage rating has a thinner dielectric.

That produces the result which surprises people most, and it is a consequence rather than an anecdote: a larger case size of the same nominal value and voltage rating is a larger capacitance in the application, sometimes by a factor of two, with both parts meeting their published specification exactly. There is nothing to appeal to, because both were measured at zero bias with a one-volt test.

The same argument runs the other way for voltage rating. A 25 V part used at 5 V is at a fifth of its rated field and has lost almost nothing; a 6.3 V part used at 5 V has lost most of it. Two parts of identical nominal value, in identical cases, differing only in a number that describes a limit neither is anywhere near.

The part in a circuit, where no capacitance is asked for #

The way to avoid choosing among three values is not to choose. Put the element in a netlist with its charge as the state, march it, and read what comes out.

The march is the one the transients field already had, extended with reactive elements defined by charge rather than by value. The trapezoidal rule is applied to $dq/dt = i$ rather than to $i = C\,dv/dt$, which is the same rule whenever C is constant and is the correct one when it is not: a charge-defined element conserves charge by construction, and a value-defined one marched with a value that moves does not.

The comparison beside it is the same netlist with an ordinary Cap at the value the caller says the part is, and it distorts by parts in 10¹³ — which is the check that the distortion being reported belongs to the nonlinearity rather than to the marching.

Second order without a bias, first order with one #

The two curves in that figure have different slopes, and the difference is the result.

With no bias, the distortion rises as the square of the drive: fitted exponent 2.01, so a doubling of amplitude is four times the distortion. The capacitance curve is even in voltage, so the leading term of the charge’s departure from linearity is cubic, and a cubic nonlinearity makes a third harmonic whose amplitude goes as the cube of the drive — a ratio going as the square.

With 2.5 volts of bias, it rises in proportion: fitted exponent 0.99. The bias has moved the operating point onto a part of the curve that is not symmetric, so there is a second-order term, and a second-order term makes a second harmonic whose ratio goes as the first power.

The same part is therefore a different kind of nonlinearity depending on where it is sitting, and the practical consequence is large: a one-volt signal on 2.5 volts of bias comes out 6.3 per cent distorted, against 0.20 per cent for the same signal about zero. Thirty times, from a direct voltage that carries no signal.

That first-order-against-second-order distinction is one this collection keeps meeting. The distortion a linear model cannot have found it in a transistor’s exponential, and what a pair cancels found a symmetry removing the even terms and leaving the odd ones. A ceramic capacitor at zero bias is a symmetric device; biased, it is not.

Where this lands in a real circuit #

Three places, and they are not the ones the effect is usually attributed to.

A coupling capacitor in an audio path normally has a direct voltage across it — that is what coupling means — and the signal rides on it. A 6.3 per cent distortion figure would be catastrophic and is what this model gives for a one-volt signal on 2.5 volts of bias in a source impedance comparable to the capacitor’s reactance. Real designs avoid it by using a film capacitor, and the reason is exactly this rather than anything about equivalent series resistance.

A decoupling bank is affected differently and less visibly: the pair that is worse than either computes an impedance curve out of nominal values, and the ceramics in it are at a fifth of those values at the rail voltage. That moves the self-resonance up by the square root of the derating and the anti-resonance with it, so a bank’s impedance curve is a function of the supply voltage.

And a timing network — a relaxation oscillator of the kind the period a delay lengthens measures — uses the charge-average value if it is charging from one voltage to another and the small-signal value if it is oscillating about a bias — two different numbers, differing by a factor of three, for the same part in the same circuit under two different modes of operation.

What the two rungs below are, seen from here #

Three defects of one component have now been measured and they are usefully different from each other.

The series inductance is a parasitic: an element that is not the capacitor, in series with it, dominating above a frequency. It makes the part behave like a different component.

The absorption is a distribution of relaxation times: the same dielectric responding on many timescales, so the value depends on the timescale of the question. It makes the part behave like many capacitors.

The voltage coefficient is neither. It is not a parasitic and not a distribution; it is the dielectric’s own polarisation being a bounded quantity. It makes the part behave like a capacitor whose value is a function of the state it is in — which is the only one of the three that makes it a nonlinear element, with harmonics, an operating point and an amplitude range.

Reading a data sheet backwards #

The model has two parameters and a data sheet prints two numbers, which is exactly enough — but the two numbers are not usually the two that are printed. What is printed is a curve, typically as a percentage change against direct bias, and the useful thing about having a two-parameter form is that the curve can be checked against it rather than interpolated.

If the model is right, the whole curve is determined by any two points on it. Taking the value at zero and at the rated voltage and predicting the middle is the strongest available test of whether a tanh is the right shape, and it is a test a reader can run on any published curve in a minute.

The second thing to read backwards is the test amplitude, which is printed and is usually ignored. A part characterised at 0.5 V rms and one at 1.0 V rms differ by about three per cent before anything else is considered, and the difference has the same sign for every class II part, so it is not noise between manufacturers — it is a systematic offset between two measurement standards.

What is not in this model #

No frequency dependence. The charge here responds instantly, so this model has no absorption in it and the rung below has no voltage coefficient in it. A real class II part has both at once, and they interact: the slow polarisation is the same ferroelectric domain motion that saturates, so the absorption itself is a function of bias. Neither model says so.

No temperature. A class II dielectric is named for its temperature characteristic — X7R holds ±15 per cent from −55 to +125 °C, X5R over a narrower range — and the same domain physics that produces the voltage coefficient produces that curve. A part at 85 °C has a different $V_k$ and the whole of this essay moves.

No ageing. A ferroelectric’s permittivity falls logarithmically with time after its Curie point is crossed during manufacture, at a few per cent per decade of hours, and is reset by heating. So the nameplate value has a date on it, in exactly the way the capacitor that remembers’s recovery has a duration on it, and for a related physical reason.

And no hysteresis. The model’s q(v) is single-valued, so a cycle traverses the same curve in both directions and dissipates nothing. A real ferroelectric has a loop, the loop has an area, and the area is a loss per cycle that rises with amplitude — which is a distortion and a heating mechanism this model has neither of.

Two routes to the same charge #

The distortion measured by marching can be checked without marching, and the two routes share no arithmetic, which is the habit this collection runs on.

The march solves a netlist step by step with the charge as the state and never evaluates a capacitance. The other route takes q(v) directly, applies the drive waveform to it as a memoryless function, and takes the harmonics of the resulting charge — no netlist, no time stepping, no companion model. In the limit where the resistor’s impedance is small against the capacitor’s, the two must agree, and they do.

Where they disagree is informative rather than a failure: at drive levels where the resistor matters, the marched answer includes the phase shift the network adds between the voltage across the capacitor and the drive, so the harmonic amplitudes differ while the total distortion does not. The memoryless route is a statement about the dielectric; the march is a statement about the circuit.

The shape of the answer #

A number on a component, and three different measurements of it that disagree by a factor of three, none of them wrong. The disagreement is not a tolerance and it is not an error: it is what happens when a quantity defined as a derivative is quoted as though it were a property.

Everything above came from writing the charge instead of the capacitance and then never asking for a capacitance again. The three values are three questions put to one q(v); the distortion is that same q(v) marched in a circuit; and the measurement condition is a bridge’s own test signal put through the same function. One model, pinned by two numbers, and no fitting anywhere.

Which of the three a circuit is actually using #

Three correct answers are only a problem if it is unclear which one a given circuit asks for, and it usually is. Four circuits elsewhere in this collection ask four different questions of the same part.

A filter asks for the small-signal slope at the bias it is sitting at, because its corner is set by how much charge moves the voltage a little. That is the 2.000 µF column, the smallest of the three, and it is why the same filter a thousand times larger’s invariance — exact to a part in $10^{15}$ in the arithmetic — says nothing about whether two builds of one design have the same corner.

A relaxation oscillator asks the same question and is measured against it directly: the period a delay lengthens has a period proportional to the capacitance, so a part inside its printed tolerance and at the slope value sets a period two and a half times shorter than the printed number predicts.

A reservoir capacitor asks for the charge average, because what it does is hold a charge over a half cycle. That is the 5.814 µF column, the largest, and it is why the direct voltage that is a sawtooth can use a nominal value with less error than a filter can — its ripple depends on total charge delivered rather than on a local derivative.

And a regulator’s output capacitor asks for both at once, which is the worst case. Two requirements pulling one capacitor needs the slope for the loop’s pole and the stored charge for the load step’s droop, and its stability floor and its transient optimum sit ten per cent apart — a margin smaller than the spread between two of the columns on this page.