Only the real part is warm

The floor a resistor sets established the first boundary in these essays that bounds a model from below. A kilohm at room temperature produces 4.00 nanovolts per root hertz, it does so because it is warm rather than because of anything about how it was made, and gain does not help because gain amplifies it too.

That essay took a resistance and returned a density. Nyquist’s own statement is about an impedance, and the generalisation is one symbol long:

$$\overline{v^2} = 4kT\,\mathrm{Re}\{Z(f)\}\ \text{per hertz}$$

A resistance is a special case in which $\mathrm{Re}\{Z\}$ happens to be a constant. The interesting half of the statement is the other one. A reactance is not warm. A capacitor of a farad has an enormous impedance at low frequency and contributes nothing at all to the noise across its own terminals, at any temperature, because its impedance has no real part.

Which sounds like a piece of tidying until it is pointed at a real part, because real parts have loss in them, and how that loss is written down turns out to decide what the noise is.

Two ways of writing one capacitor, and the frequency they agree at #

The same part written two ways measured the conversion this essay is about to price. A capacitor’s loss is quoted either as a resistance in series with it — the equivalent series resistance a data sheet prints — or as one across it, and the pair of expressions that converts between them is exact at one frequency and at no other. How wide the band is around that frequency is set entirely by the quality factor.

Here the two models are matched at 10 kHz at a loss tangent of 0.02. In series that is

$$R_s = \frac{\tan\delta}{2\pi f_0 C} = 3.183\ \Omega$$

and across it,

$$R_p = \frac{1}{\tan\delta \cdot 2\pi f_0 C} = 7.958\ \mathrm{k\Omega}$$

which differ by a factor of 2500, being $1/\tan^2\delta$. A designer picks whichever is convenient — the series form for a bypass capacitor where the resistance is in the current path, the parallel form for a hold capacitor where the resistance is a leak — and the choice is normally treated as a change of notation.

The real part of the impedance says otherwise.

For the series model $\mathrm{Re}\{Z\} = R_s$, a constant at every frequency. For the parallel model $\mathrm{Re}\{Z\} = R_p/(1 + (2\pi f R_p C)^2)$, which falls as $1/f^2$ once the capacitor’s reactance is below $R_p$. Two quite different functions that happen to cross at 10 kHz.

What that does to the noise, in decibels #

The noise density is the square root of $4kT\mathrm{Re}\{Z\}$, so the table above becomes a table of noise densities with the ratios square-rooted.

At 100 Hz the series model says 0.226 nV per root hertz and the parallel model says 10.10 — a factor of 45, which is 33.0 dB. At a megahertz the series model says 0.226 and the parallel model says 0.00226, and the disagreement is 40.0 dB the other way.

Thirty-three decibels of noise from a choice made for algebraic convenience is not a modelling nicety. A 100 nF capacitor in the feedback path of an electrometer amplifier is the kind of place where ten nanovolts per root hertz decides whether the instrument works, and the two ways of writing down the same measured loss tangent disagree by more than an order of magnitude about it.

There is a right answer and it is neither model. A real capacitor’s loss is a distributed thing, its loss tangent is roughly constant over decades rather than either form predicting, and both models are fits to one measurement at one frequency. What the two disagreeing curves actually show is how much of the answer was never measured — the data sheet gave one number at one frequency, and everything the models say away from that frequency is extrapolation.

That is a familiar shape here and it is worth naming as such. The capacitance that is not one number makes the same point about the capacitance itself; the constant that is a window makes it about an ideality factor. A parameter extracted at one point and used across a decade is a measurement plus a model, and the model is usually invisible.

Two routes to the same curve #

Every density drawn here is computed twice and the two computations have nothing in common but the element values.

The first injects a probe current into the one-port and solves for the voltage across it, giving the complex impedance; its real part is multiplied by $4kT$ and square-rooted. No noise appears anywhere in that calculation — it is an impedance measurement.

The second takes the same netlist, removes the resistor, puts a one-volt source in series with a resistor of the same value in its place, and re-solves the whole network at each frequency to find what fraction of that source’s voltage reaches the terminals. That gain multiplies $\sqrt{4kTR}$. No impedance appears anywhere in that calculation — it is a transfer measurement, and it is the construction the resistor the noise comes from uses to apportion a filter’s noise among its resistors.

They agree to three parts in $10^{16}$ across a range from ten hertz to ten megahertz on both models. That is the arithmetic’s floor rather than a tolerance, and it is the check that matters here because the essay’s whole claim is that a single quantity — the real part — determines the answer. If the two routes disagreed anywhere, the claim would be that they do not.

Where the disagreement stops mattering, and where it starts #

A forty-decibel spread between two models is only a problem where the larger of the two is large compared with something else in the circuit, so the useful form of the answer is a frequency.

Take the capacitor’s own implied noise and compare it with the Johnson noise of a kilohm — 4.00 nV per root hertz, which is about the smallest source resistance an ordinary amplifier is used with. The series model implies 0.226 nV per root hertz at every frequency, which is twenty-five decibels below the kilohm and can be ignored anywhere. The parallel model implies 11.29 nV per root hertz at low frequency, falling as $1/f$ once the reactance is below $R_p$, and it crosses the kilohm’s floor at 528 Hz.

So the whole disagreement lives below about half a kilohertz for this part. Above it, both models say the capacitor is quiet compared with any resistor it will sit beside, and the choice between them does not change a design. Below it, one of them says the capacitor is the dominant noise source in the circuit and the other says it contributes nothing, and there is no measurement in the data sheet that distinguishes them.

Which is an uncomfortable place for the boundary to be, because it is exactly the band where the corner where averaging stops working operates and where an instrument’s own drift and flicker live. A capacitor’s loss model is a low-frequency question, and low frequency is where nobody measures loss tangent, because the bridges that measure it are built for a kilohertz.

The crossing moves with the part. At a loss tangent of 0.002 the parallel resistance is ten times larger, so its low-frequency noise is 35.7 nV per root hertz and it crosses the kilohm at a higher frequency, not a lower one — a better capacitor’s parallel model is noisier. That is not a paradox; it is what the model says, and the fact that it says something absurd for a good capacitor is the clearest evidence available that it is the wrong model for a part whose loss is dielectric rather than a leak.

The reactance that is refused #

The sharpest statement in the essay is a refusal, and it is the one the whole generalisation rests on.

A capacitor with no loss in it at all has an impedance of $1/j\omega C$: a large imaginary part and a real part of exactly zero. Asked for its noise, the expression returns zero — not a small number, not something below a floor, but zero at every temperature and every frequency. The figure states that by measuring the impedance of a lossless capacitor and checking that its real part is under $10^{-12}$ ohms while its reactance is over a kilohm.

That refusal is what makes the rest of the field coherent. It is why an inductor in front of an amplifier costs no noise figure, which is the loss in front, counted twice’s finding that a lossless reactive network in the same position as a cable costs nothing while the cable costs a decibel for a decibel. It is why the total that has no resistor in it can integrate a resistor’s noise through a capacitor and get $\sqrt{kT/C}$: the capacitor sets the bandwidth and the resistor makes all the noise, and the total contains no resistance because the two dependences cancel exactly.

And it is why the question “how much noise does this capacitor make” has no answer until the loss is specified. Not a small answer. No answer.

The same sentence for an inductor, where the loss is a material #

An inductor has the dual problem and it is worse, because its loss has two mechanisms and they have different shapes.

The winding’s own resistance is a series resistance and is nearly constant at low frequency, rising as the square root of frequency once skin effect takes hold. The core’s loss is not a resistance at all — it is hysteresis and eddy currents in a material, and what it contributes to the impedance is a real part that rises roughly in proportion to frequency, which is what a complex permeability is a way of writing down. So an inductor’s $\mathrm{Re}\{Z\}$ is the sum of a constant, a square root and a first power, and its noise is the square root of that sum.

None of that is a resistor anybody drew. It is measured as an inductor’s quality factor against frequency — the resistance that grows with frequency measures the first two mechanisms and one dissipation, two exponents separates the core’s — and turning that measured quality factor into a noise density is one line: $\mathrm{Re}\{Z\} = \omega L/Q$, so

$$e_n = \sqrt{4kT\,\omega L/Q}$$

which rises as the square root of frequency for a constant $Q$. A millihenry of $Q = 100$ at 100 kHz has a real part of 6.28 ohms and produces 0.32 nV per root hertz; at 10 MHz, if the quality factor held, it would be 628 ohms and 3.2 nV per root hertz. It does not hold, which is the point — the quality factor is a measurement with its own frequency dependence, and the noise inherits every bit of it.

The practical consequence is the mirror of the capacitor’s. A designer who wants a quiet reactance should ask for a quality factor rather than for a resistance, because a quality factor is what is measured and a resistance is what is inferred. And a gyrator, which makes an inductance out of an amplifier and resistors, has a real part that is a genuine resistor and a noise that the inductor that is an amplifier computes directly — which is the one case where the fiction is not a fiction.

Where this changes a design decision #

Three places, and in each of them the two models point in opposite directions.

A sample-and-hold capacitor. Its loss is a leak, so the parallel model is the physical one, and its real part falls as $1/f^2$. The noise on a held value is therefore dominated by low frequencies — which is exactly where a hold spends its time — and the model that makes the part look quiet at a megahertz makes it loud at a hertz. A design that took the series number from the data sheet and assumed it flat has understated the low-frequency noise by twenty decibels or more.

A bypass capacitor. Its loss is in the plates and the leads, so the series model is the physical one, and the noise is flat and small. A designer who reached for the parallel form because it made an impedance calculation easier would conclude the part was noisy below a kilohertz, which it is not.

A capacitor in a feedback network. Here neither form is obviously right and the answer matters most, because the capacitor is in the signal path and its noise is multiplied by whatever the stage does. The honest procedure is to measure the loss tangent at the frequency the circuit works at and use it there — which is what the crossing point in every figure above is, and which is why the two models were ever declared equivalent in the first place.

The general form of the rule: write the loss as whichever resistance corresponds to the mechanism, and if the mechanism is not known, the noise is not known either, by up to forty decibels.

Where neither loss model is the truth #

That a capacitor’s real loss follows either model. It follows neither. A dielectric’s loss tangent is roughly constant over many decades, which means $\mathrm{Re}\{Z\}$ falls as $1/f$ — between the series model’s constant and the parallel model’s $1/f^2$, and equal to neither. Both models are two-parameter fits to a one-parameter measurement, and what this essay prices is the gap between them rather than the distance of either from the truth.

That 4kT·Re{Z} holds out of equilibrium. It is an equilibrium statement. A resistor carrying current has a second noise generator that is not thermal and is not in the expression at all, which is the subject of a companion essay here.

That the refusal makes an inductor free in a real circuit. A real inductor has a series resistance and a real capacitor has loss, so the noiseless reactance is a limit rather than a component. The statement is about the reactance itself, and it is why the resistance is where a designer should look.

That the two routes agreeing proves the model. They agree because they are two ways of computing one netlist’s behaviour, and the netlist is the model. What they establish is that the real part determines the noise — which is the theorem — and not that the netlist describes the capacitor.

One density by two routes, and the reactance that declines to be warm #

Every point is computed twice, once from the real part of a probe-measured impedance and once from a transfer function with the resistor turned into a source, over twelve frequencies and both models. Checked to a part in a billion and coming out at three parts in $10^{16}$.

The two models are checked to agree at the frequency they were matched at, to within $2\tan^2\delta$, which is the order of the conversion’s own error and is checked rather than assumed.

And to disagree by more than a hundredfold two decades either side of it, in opposite directions — which is the essay’s claim and cannot be made from a single point.

The lossless capacitor is refused by measurement, with a real part below $10^{-12}$ ohms beside a reactance above a kilohm, so that “a reactance is not warm” is a number rather than a statement.

A theorem about impedances, used as a theorem about resistors #

What this essay adds to the field is a change of the object the noise belongs to.

Three earlier essays here have treated noise as something a resistor does. That is the right picture for a network of resistors and it silently assumes that every dissipative element in a circuit has been drawn as one. The moment a part’s loss is a property of the part rather than a separate component — a capacitor’s dielectric, an inductor’s core, a cable’s dielectric, a resistor’s own skin effect — the noise is a property of the impedance and the resistor is a fiction introduced to carry it.

The fiction is a good one and it is the only reason a netlist can compute any of this: a solver handles resistors and does not handle “loss”. But the fiction has to be built, and building it is a modelling decision with a forty-decibel range in it. This essay’s contribution is the range.

Which suggests the discipline. Where the loss is a real component — a series resistor that exists, a bleed resistor that was fitted — there is no decision and no range. Where it is a property of a part, the model is a choice, and the right question is not “series or parallel” but “which mechanism, and measured where”. The crossing point in every figure above is the only frequency at which that question does not need answering.

Still open: the loss that is neither model, the parts whose loss is not warm, and the cable #

A constant loss tangent, which is what a dielectric actually has. $\mathrm{Re}\{Z\}$ then falls as $1/f$, so the noise density falls as $1/\sqrt{f}$ — a pink spectrum produced by a component with no resistor in it at all. Built as a network of parallel resistor–capacitor sections, the way a resistor made of a clock builds a resistance out of something that is not one, it would put a number on how many sections a decade of constant loss tangent needs and what the residual ripple in the noise is.

Whether every dissipation is warm. The theorem is an equilibrium statement, and some losses are not in equilibrium with the part: a ferrite’s hysteresis at large signal, a semiconductor junction’s conduction. Whether a lossy inductor’s core loss produces $4kT\mathrm{Re}\{Z\}$ at small signal, and where it stops, is a measurement on a magnetics part rather than an argument.

And a cable, which is the case where the real part is frequency-dependent and known. A transmission line’s series resistance rises as $\sqrt{f}$ from skin effect, so its input impedance has a real part with a known shape. The noise of a terminated cable ought to follow it, and it would be the first case in this field where the real part is neither a constant nor a fitted parasitic but a computed consequence of a geometry.