The floor a circuit has

The floor a resistor sets computed 4kTR exactly, checked it two ways that share no arithmetic, and then named what sits above it: the amplifier’s own noise, and the excess that rises as 1/f. It named them and measured neither. This essay measures the first.

The distinction is worth stating before anything is drawn, because it is the whole reason this is a separate essay rather than a paragraph. A resistor’s noise is not a property of the resistor. It is 4kTR — Boltzmann’s constant, the absolute temperature and the resistance — and every resistor of a given value at a given temperature has exactly the same noise, because nothing about how it was made appears in the expression. There is nothing to choose and no better resistor to buy.

An amplifier is not like that at all. Its noise is a property of the device, two devices of the same nominal function differ by twenty decibels, and — the part that makes it a design problem rather than a purchase — how much of that noise reaches the output depends on what is connected to the input.

Two generators, because one will not do #

The standard model of an amplifier’s noise is two sources at its input, and the reason there are two rather than one is that one cannot reproduce what is observed.

Short the input of a real amplifier and something still comes out. That is a voltage, it is independent of the source, and it is called eₙ: a voltage noise density in series with the input, quoted in nanovolts per root hertz. If a voltage in series were the whole story, then raising the source resistance would change nothing at all — and it does. So there is a second generator, a current iₙ across the input, which produces no voltage into a short and produces iₙRₛ into a source resistance Rₛ.

Those are the two numbers on the front page of a datasheet, and everything in this essay follows from them plus the resistor’s own 4kTR. The three add in power because they are uncorrelated, so the total density at the input is

$$e_\text{total} = \sqrt{4kTR_s + e_n^2 + (i_n R_s)^2}$$

and the noise figure — how much worse the amplifier makes a source that already had noise of its own — is that total divided by the source’s own contribution:

$$F = 1 + \frac{e_n^2 + (i_n R_s)^2}{4kTR_s}$$

The minimum, and what it depends on #

Both terms in the numerator are divided by Rₛ. The first therefore falls as 1/Rₛ and the second rises as Rₛ, which is the classic shape whose minimum is where the two are equal:

$$\frac{e_n^2}{R_s} = i_n^2 R_s \quad\Longrightarrow\quad R_\text{opt} = \frac{e_n}{i_n}$$

For the part drawn above that is 4 nV/√Hz over 0.6 pA/√Hz, which is 6.67 kΩ, and its noise figure there is 1.138 dB.

That the optimum is the ratio is the well-known half. The sharp half is what the value at the optimum turns out to be. Substituting back,

$$F_\text{min} = 1 + \frac{2 e_n i_n}{4kT}$$

which depends on the product of the two generators and on nothing else. Two amplifiers with the same eₙiₙ are exactly equally good; they differ only in which source they are good for. That is a statement with teeth, and the slider is what makes it visible: moving eₙ from 1 to 32 nV/√Hz moves Rₒₚₜ from 1.67 kΩ to 53.3 kΩ and the best achievable figure from 0.314 dB to 5.312 dB, because with iₙ held fixed the product moves with eₙ. Move both the other way and the floor returns while the resistance slides.

The figure does not take the minimum on trust. It is asserted three ways: that the located optimum equals eₙ/iₙ to twelve digits, that the noise factor there equals the closed form above, and — because a derivative can be right about a stationary point and wrong about which kind it is — that walking a factor of three either side makes things worse. At the default part that is 1.138 dB at the optimum against 1.760 dB at 2.22 kΩ and 1.760 dB at 20 kΩ — the same number in both directions, which is not a coincidence and is the strongest single check in the figure. The two terms enter as Rₛ and 1/Rₛ, so the curve is symmetric in log Rₛ about its minimum, and a stationary point that was not where the ratio says it is would break that symmetry immediately.

The other matching condition, which is a different number #

The lines field spent an essay on maximum power transfer and the instruments field spent one on loading, so this site has already established that a source and a load can be matched. What it has not said until now is that there are two matching conditions on the same circuit and they are different numbers.

Maximum power into the amplifier happens when the source resistance equals the amplifier’s input resistance. That is a property of the input stage’s bias network and its device geometry, and it has nothing whatever to do with either noise generator. Minimum noise happens at eₙ/iₙ. Nothing requires the two to be close and for most parts they are not.

For the amplifier in the figure the input resistance is 1 MΩ and the noise optimum is 6.67 kΩ: a factor of 150 apart. Matching for power — transforming the source up to 1 MΩ, which is what a reader trained on the lines field would reach for — gives a noise figure of 13.71 dB against the 1.138 dB available. The cost of matching the wrong thing is 12.57 dB, which is a factor of eighteen in noise power and is not recoverable anywhere downstream.

The slider makes the cost move in a way worth watching: at 1 nV/√Hz the penalty is 13.39 dB, at 32 nV/√Hz it is 8.41 dB. A noisier part is less punished by the wrong match, because its own floor is already high enough that the mismatch adds proportionally less. That is not a reason to choose a noisy part; it is a reason to notice that the penalty is a ratio and not an absolute.

Why the noise figure is the wrong thing to minimise on its own #

There is a trap in the quantity, and this site’s habit of drawing what a model stops being true at makes it easy to walk into.

The noise figure is a ratio. It compares the total noise with the source’s own, so it can always be improved by making the source noisier — and it goes to infinity as the source resistance goes to zero, not because the circuit got worse but because the reference did. A source of 100 Ω shows this part a noise figure of 10.4 dB, and a source of 6.67 kΩ shows it 1.14 dB, but the total input-referred density at 100 Ω is 4.20 nV/√Hz and at 6.67 kΩ is 11.78 nV/√Hz. The quieter-looking configuration is nearly three times noisier in absolute terms.

Both statements are true and they answer different questions. If the source resistance is fixed by what is being measured — a sensor, an antenna, a transducer whose impedance is what it is — then the noise figure is exactly the right quantity, because it says how much worse the amplifier makes a situation that already exists. If the source resistance is a free choice, minimising the noise figure is minimising the wrong thing, and the quantity to minimise is the absolute density.

This is the same shape of error the divider essay found in a different field: a ratio that is correct about the comparison it makes and silent about whether that comparison is the one being asked for.

The transformer, and what it does and does not buy #

There is one move that changes the source resistance an amplifier sees without changing the source: a transformer, or any other lossless impedance transformation.

A transformer of turns ratio n presents n²Rₛ to the amplifier, and it multiplies the signal voltage by n — so the signal and the source’s own noise are both transformed by the same factor and their ratio is untouched. What changes is where the amplifier’s two generators land relative to them. A source of 100 Ω stepped up by a ratio of 8.2 arrives as 6.7 kΩ, which is the optimum, and the noise figure goes from 10.4 dB to 1.14 dB.

The gain in the last sentence is real and it is worth being precise about what produced it. Nothing about the source got quieter, and nothing about the amplifier got quieter. The transformation moved the operating point along a curve that was always there.

And there is a limit on it that this site is obliged to state, because it is the frequency at which the model stops applying: a transformer is an inductor pair, its own winding resistance is a resistor with 4kTR of its own, and it has a bandwidth. The idealisation used above — lossless, infinite bandwidth, no coupling capacitance — is exactly the kind of assumption the rest of this collection spends its time bounding. The magnetics field, which this site has not built yet, is where that belongs.

Where in the chain it matters, which is almost entirely the first stage #

Everything above concerns one amplifier. Real signal chains are several, and the field has already built the arithmetic that decides how their noise adds: Friis’s formula, which divides each stage’s noise contribution by the gain in front of it.

The consequence is one that this essay’s optimum has to be read against. A stage’s noise matters in proportion to how little gain precedes it, so the first stage’s eₙ and iₙ are nearly the whole answer and everything after it is a correction. In the cascade the noise field already measures, the whole of the chain after the first stage contributes 6.9% of the total noise factor, and reordering the same three stages — identical parts, identical total gain of 50.0 dB — moves the chain’s noise figure from 1.31 dB to 10.01 dB.

That is why the source-resistance optimum is a first-stage question and not a system question. It is also why the 12.57 dB penalty computed above is not something a later stage can be chosen to compensate: it enters at the point in the chain where there is no gain in front of it to divide it down.

Noise temperature, which is the same statement without the reference #

The noise figure’s dependence on the source it is measured against — worse for a quiet source, better for a noisy one — is awkward enough that a second quantity exists to avoid it.

An amplifier’s noise temperature is the temperature the source resistor would have to be at, with the amplifier made perfect, to produce the noise actually observed. It is Tₑ = (F − 1)·T₀, where T₀ is the 290 K this field’s arithmetic is referred to. The part above, at its optimum, has F = 1.300 and therefore a noise temperature of 87 K.

Nothing new is being computed. It is the same measurement with the reference divided out, and it is preferred wherever the source is not at 290 K — a radio telescope’s antenna looking at cold sky, a cryogenic front end — because there the noise figure’s reference temperature is a fiction and every number quoted in it has to be corrected before it can be used.

The pair is worth having because the two make different mistakes obvious. A noise figure invites the error of comparing two parts measured against different sources. A noise temperature invites the error of forgetting that the amplifier is not, in fact, cold.

A density is not a voltage until a bandwidth is chosen #

One more thing has been held fixed throughout and should be said plainly: every number above is a density, in volts per root hertz, and a density is not a noise voltage.

The step from one to the other is an integral over the bandwidth the measurement actually has, and the noise field has already established that this bandwidth is not the −3 dB point. For a single pole it is π/2 times it — 1.5706 measured against 1.5708 — so a noise voltage computed from a corner frequency comes out 20.2% low. The two essays compose exactly: this one gives the density at the input, that one gives the bandwidth to multiply it by, and neither is usable without the other.

What the two-generator model itself leaves out #

This site’s rule requires every model to be drawn with the condition under which it fails, and the two-generator model has one that has been quietly suppressed above.

eₙ and iₙ have been added in power, which assumes they are uncorrelated. They are not, in general: both arise in the same input transistor, and the full description carries a complex correlation coefficient between them. When it is significant, the optimum source is complex rather than real — there is a reactance as well as a resistance that minimises the noise — and the minimum achievable figure is lower than the expression above gives, because part of one generator can be cancelled against the other.

The figures here take the correlation as zero, which is what a datasheet’s two numbers assume and what is close to true for most parts across most of their band. It is stated rather than assumed away because it is a real boundary of the model, and because it is the second time in this essay that a “matching” turned out to be a different quantity than the one a reader was likely to reach for.

What this leaves for the next rung #

Everything above is a statement about noise density, and every density in it is flat: eₙ and iₙ have been treated as numbers rather than as functions of frequency, which is what a datasheet’s front page encourages.

They are not numbers. Below some frequency every real device’s noise density rises, and it rises as 1/f — which is a spectrum with no bottom, and which breaks the one thing everybody knows about noise. That is the next essay, and the reason it is a separate one is that it needs a different piece of machinery: a density that is not flat is not a constant in an integral, and it cannot be built by scaling a white sequence.

There is a second thing the flat-density treatment hides, and it is the reason the two essays have to be in this order. The optimum source resistance derived above is eₙ/iₙ at whatever frequency the two are quoted at. Both generators have 1/f regions, and they do not have the same corner — the voltage noise’s is typically well below the current noise’s — so the optimum source resistance is itself a function of frequency, and a design that sits at the optimum in one decade sits off it in another. Nothing in this essay can say by how much, because nothing in it has a spectrum.

The rule this model stops being true under #

6.67 kΩ, and 1.138 dB. The first is where this part is quietest and the second is how quiet it can be, and both are consequences of two numbers on a datasheet rather than of anything a designer does. What a designer decides is which side of the optimum to be on, and by how much — and the answer is worth 12.57 dB on this part, against a decision that looks from the lines field like the obviously correct one.

The optimum, and the two circuits that cannot sit at it #

An optimum source resistance is only useful where the source resistance is a design variable, and in two of this collection’s measurements it is not.

The ammeter that is a resistor works from a shunt — tens of milliohms, three or four decades below the 6.67 kΩ optimum — because the burden voltage is the quantity being minimised and it is a burden voltage that sets the source impedance. A current measurement is always made from a source far too low for the amplifier reading it, and the design question there is an offset rather than a noise.

Where the trouble is at the input works from a photodiode, whose impedance is enormous and whose capacitance is the design parameter — so the source is far above the optimum at direct current and sweeps through it as frequency rises. That essay’s finding, that the amplifier is still ninety-five per cent of the noise at a large diode, is this page’s arithmetic evaluated at a source nobody chose.

Which is the honest reading of the optimum. It is exact, it is a ratio of two device parameters, and it is available to a designer only in the middle range of circuits where the source impedance can be transformed — by a transformer, by a different sensor, by a choice of bias resistor. At either extreme the useful content of this essay is not the optimum but the two generators, since knowing which of them dominates says which specification on the part matters.

The two circuits that can move their source impedance are worth naming for balance. The loss in front, counted twice is where the cheapest way to do it is priced: a transformer is very nearly lossless, so it costs almost nothing in noise figure by that essay’s identity, and it transforms the source towards whatever this page’s optimum is. A floor and a ceiling is where the benefit is read — lowering the floor without touching the ceiling, which no linearity technique does — and it is the only move available that widens a dynamic range from below.

Where the optimum source resistance is met #

A noise figure with an optimum source resistance is the first place this field’s numbers stop being densities and start being design decisions. The floor a resistor sets is the density itself, measured two ways. The bandwidth noise sees is what turns it into a voltage. The loss in front, counted twice is where the optimum meets a cable and loses, because a matching network that is lossy adds its loss twice. The floor a current sets is the other mechanism entirely, crossing this one at a current that puts fifty millivolts across the source. A floor and a ceiling is the range this floor is one end of, and The divider, and the thing it does not know about is why the source resistance is never quite the one on the schematic.