The bowl, and the bottom of it

The floor a circuit has put two generators on an amplifier’s input — a voltage noise in series with it and a current noise across it — and found that their sum, compared with the source’s own noise, has a minimum at the ratio of the two.

That essay also carried a warning in prose, with two numbers in it and no picture: a source of 100 Ω shows the part 10.41 dB and a source of 6.67 kΩ shows it 1.138, while the total input-referred density at the first is 4.196 nV/√Hz and at the second 11.780. The configuration with the better noise figure is nearly three times noisier.

Two points are an assertion. This page draws the curves they are on, measures how wide the bottom of the bowl is, and finds that the bowl’s width is set by its depth in the direction nobody would guess.

Two curves that only rise #

A noise figure is a ratio, so it has two curves in it, and the minimum belongs to neither of them.

The total has three regimes and each is one of the three terms. Below about a kilohm the voltage generator dominates and the curve is flat at eₙ; through the optimum the source’s own 4kTR is comparable to both; above ten kilohms the current generator’s iₙR dominates and the curve rises in proportion to R rather than to its square root, which is why the ratio turns round.

So the two questions have different answers and the difference is not small.

If the source resistance is fixed — a transducer, an antenna, a sensor 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. Nothing about the source can be changed, so a ratio against it is a fair scoring of the amplifier.

If the source resistance is a free choice — a bias network, a divider, a feedback resistor, a transformer ratio — then the noise figure is minimising the wrong thing, because it improves when the reference gets worse. The quantity to minimise is the absolute density, and it wants the smallest source resistance the circuit can be built with.

That is the first of the two inversions on this page and the smaller one.

Why there are two generators and not one #

The two-generator model is the reason a minimum exists at all, and it is worth restating because everything above depends on the two being about different things rather than being two views of one thing.

A resistor’s noise is a single quantity: 4kTR, a voltage, and there is nothing to choose about it — the floor a resistor sets establishes that no property of the resistor beyond its value and its temperature appears in it. An amplifier is not like that. Its input stage has a voltage fluctuation referred to its input terminals, which exists whether or not anything is connected, and it has a current that must flow into or out of those terminals, which produces no voltage at all into a short and produces iₙR into a source of R ohms.

One matters when the source is small, the other when it is large, and they are properties of different mechanisms inside the device — a channel or a base region for the first, a gate or base current for the second. Nothing requires them to be related, which is what makes the product and the ratio independent, and that independence is the whole of the last section on this page.

A model with one generator would have no minimum: it would be either flat or monotone against source resistance, and the whole subject would be the instruction to use the smallest source available. The second generator is what turns that into a design decision, and the price of admitting it is that the answer now depends on which question was asked.

The bowl is one shape #

Substituting u = Rₛ/Rₒₚₜ into 1 + (eₙ² + (iₙRₛ)²)/4kTRₛ collapses both generators into the minimum they already produce:

$F(u) = 1 + (F_{\min} - 1)\,\dfrac{u + 1/u}{2}$

There is one curve. Every part in this field has the same bowl, centred on its own optimum, scaled vertically by its own Fₘᵢₙ − 1, and symmetric in the logarithm of u because u + 1/u is. The expression is not used to draw anything here — every curve on this page is the same closure the rung below plots — but it is checked against that closure at each part, and it says what the drawing is a picture of.

The consequence is the one that decides how much matching effort is worth spending, and it needs stating carefully because the algebra has a sign in it that intuition does not.

F is a factor and a noise figure is its logarithm. A part with Fₘᵢₙ close to one has a bowl whose absolute variation is small, so in decibels — which is a logarithm of F rather than of F − 1 — the same relative excursion of (u + 1/u)/2 produces a much smaller number. A part with a large Fₘᵢₙ has the opposite. So the depth of the bowl in decibels and the sharpness of the bowl in decibels rise together.

The width, measured #

The quietest part tolerates a factor of thirty and the noisiest a factor of two. All four have their optimum at exactly the same resistance, so the ordering is not about where the bottom is; it is about how flat the bottom is, and it runs opposite to every instinct about matching. The part that makes the optimum look important — a bowl 7.6 dB deep, a curve that plainly rises on both sides — is the part on which a factor of two of source resistance already costs most of a decibel. The part whose bowl is 0.081 dB deep, and which therefore looks as though the optimum barely exists, is the one that keeps its performance across a decade and a half either way.

Both statements are the same statement: a shallow bowl is shallow in both directions. There is nothing paradoxical in it once the shape is one curve, and it is worth having because the design effort — a transformer, a paralleled input stage, a resistance chosen for noise rather than for gain — is expensive and is usually spent on the part whose datasheet makes matching look most urgent.

At three decibels of allowed penalty the widths become ±110.25, ±30.52, ±10.54 and ±4.17, which is the same ordering four times over and the practical answer for a circuit that has to work with a range of sources rather than one.

What a width is worth, in turns #

There is one move that changes the source resistance an amplifier sees without changing the source, and it is the move the width measurement is really about: a transformer, or any other lossless impedance transformation. A ratio n presents n²Rₛ to the amplifier and multiplies the signal by n, so the source’s own noise and the signal are scaled together and only the amplifier’s two generators move relative to them.

Because the transformation goes as the square, a tolerance in resistance is a much tighter tolerance in turns. The ±4.00× of resistance the 1.138 dB part allows for a decibel is ±2.00 in turns ratio; the quiet part’s ±30.13× is ±5.49; the noisy one’s ±2.16× is ±1.47. On a three-decibel budget the same four become ±10.5, ±5.52, ±3.25 and ±2.04.

Those are large numbers in turns, and they say something the bowl’s depth alone does not: a transformer only has to be about right. Anything between a 1:2 and a 1:8 step-up in front of the 0.081 dB part costs it under a decibel, and a ratio chosen from a catalogue rather than wound to order is inside that. Where the effort has to go instead is into the transformer’s own losses — its winding resistance is a resistor with 4kTR of its own, in front of everything — and into its bandwidth, because the transformation stops being lossless well before the amplifier stops being useful.

Two routes to the bottom #

The minimum is asserted here rather than assumed, in two ways that share no arithmetic.

The first is the closed form: Rₒₚₜ = eₙ/iₙ exactly, and the noise factor evaluated there equals 1 + 2eₙiₙ/4kT to twelve figures. That is a derivative set to zero and it locates a stationary point without saying which kind it is.

The second is a walk. The curve is evaluated at a third of the optimum and at three times it, and both are required to be worse — 1.760 dB and 1.760 dB against the 1.138 for the part in the first figure. A stationary point that is a maximum, or an inflection, fails that immediately, and the check costs two evaluations of a closure that is already being called sixty times to draw the line. It is the same habit what a network answers applies to a solve: the derivation says where to look and the evaluation says what is there.

That the two are the same number to three decimals is not a coincidence and is the third check without being written as one. u + 1/u is unchanged by u → 1/u, so the bowl is exactly symmetric about its optimum on a logarithmic axis, and a factor of three below costs precisely what a factor of three above costs. The measured curve reproducing that symmetry is evidence that the closure being drawn is the expression the section is about, and a departure from it would have meant one of the two generators was entering the arithmetic in the wrong place.

Equally good, and good for different things #

The other half of the closed form is the half the rung below stated and did not draw: Fₘᵢₙ contains only the product eₙiₙ, and Rₒₚₜ contains only the ratio. The two are independent, so a product line is a family of parts that are exactly equally good and are good for completely different sources.

Three parts, one figure of merit, three different circuits. A single number on the front of a datasheet — “0.31 dB noise figure” — is true of all three and distinguishes none of them, and the one that suits a 1 kΩ source is four times quieter in volts than the one that suits 16 kΩ, at the resistance each of them is best at.

The reason the density ratio stays at exactly four while the floor moves by three decibels is that at its own optimum a part sits on 4kTRₒₚₜ multiplied by Fₘᵢₙ, and only the first factor differs across a product line. It is a small piece of algebra with a practical edge on it: comparing two parts by noise figure alone compares their products and discards the information about which source each is for.

Where a bowl is worth anything at all #

The bowl belongs to the first stage and very nearly to nothing else.

Six point nine per cent is the whole justification for spending anything on matching, and it is also the whole limit on it. Optimising the source resistance of the second stage of a chain buys at most a few hundredths of a decibel, because whatever it saves arrives divided by the first stage’s gain. The budget goes to the front, and the loss in front, counted twice is what happens when something with gain below one is put there instead.

That also bounds this whole page. A bowl that is worth ±30× on a quiet part and sits behind twenty decibels of gain is worth nothing at all, and the honest reading of the width measurement is that it says how carefully to choose the source for one stage in a system.

The sources this collection cannot choose #

The distinction between a source that is given and a source that is chosen is not abstract here, because the collection contains several of each and they come out on opposite sides.

Given. The ammeter that is a resistor works from a shunt of tens of milliohms, chosen to keep the burden voltage down, and no amount of noise argument will raise it — the burden is the specification. That is four or five orders of magnitude below the optimum of every part on this page, so the noise figure there is enormous and entirely uninformative, and the quantity that matters is the absolute density, which is eₙ and nothing else.

Given, at the other end. The probe that takes a tenth presents nine megohms in series, which is three decades above the largest optimum here, so the current generator dominates completely and the useful move is the opposite one: reduce iₙ, or reduce the resistance the current has to flow through, and ignore eₙ.

Chosen. The four resistors that decide, and the two that do not is a bridge whose arm values are set by a gain and a rejection requirement, and there is a decade of freedom in the impedance level once those are fixed — the same filter a thousand times larger is the essay about what that freedom costs elsewhere. That is exactly the case where minimising the noise figure picks the wrong end of the range, because the ratio improves as the arms get larger and the volts get worse.

And chosen without anybody noticing. The current the instrument draws is about an input resistance that exists to bias the stage rather than to be a source, and it sits across the input in parallel with whatever is being measured. It sets a source resistance without appearing in any specification, and the bowl above says how far off the optimum that puts the part.

What it does not say #

It does not say the noise figure is a bad quantity. It is the right quantity whenever the source is given, which is most measurement problems, and the loss in front, counted twice shows it doing work that nothing else does: a passive loss has a noise figure exactly equal to its loss, and that statement has no absolute-density version.

It says the quantity answers a comparison and not an amount, and that the two come apart by a factor of 2.81 on the part in the first figure and 3.32 on a quieter one.

And every generator on this page is a number rather than a spectrum, which is the limit the rung below named and could not measure. Both eₙ and iₙ have regions where their density rises as 1/f — the corner where averaging stops working builds one of those as a circuit rather than asserting it as a shape — and the two corners are not the same frequency. The optimum source resistance is eₙ/iₙ at whatever frequency the two are quoted at, so a part that sits at its optimum in one decade sits off it in another, and the width measured here is what says whether that matters: a part with a shallow bowl can drift a decade and lose a decibel, and a part with a deep one cannot.

Nothing on this page can put a number on that drift, because nothing on it has a spectrum. What it can do is say what the drift would have to be before it mattered, which is the more useful half.

The other omissions are the ordinary ones. The two-generator model treats eₙ and iₙ as uncorrelated, which real parts are not exactly; the source is a pure resistance, where a real transducer has reactance and only its real part is warm; and the temperature is 290 K throughout, so every noise figure here is quoted against the reference the definition uses rather than against the source’s own temperature.

The one number a datasheet gives, and the three it does not #

The whole of this page can be read as an objection to a single figure of merit, so it is worth saying what would have to be printed instead.

A part’s noise is described by two numbers and one qualification: eₙ, iₙ, and the frequency at which each was taken. From those three the optimum resistance, the floor, the bowl’s depth and the bowl’s width all follow exactly, and every figure here was computed from nothing else. A noise figure is one number derived from those two plus a source resistance the manufacturer chose, and it discards the information about which source it was for.

That is why the equal-product family is the sharpest of the results rather than a curiosity. Three parts advertised as 0.3138 dB are three different components for three different circuits, and there is no arrangement of the advertised number that separates them.

The number worth carrying #

±30 times on the best part and ±2.16 on the worst, for the same decibel. The optimum source resistance is the same 6.67 kΩ for all four, so the number that decides how hard to chase it is the depth of the bowl and not its position — and the deeper the bowl, the less room there is around the bottom of it.

The habit that goes with it is to ask what a minimum is a minimum of before optimising towards it. A noise figure is a ratio and its minimum is a property of the comparison; the absolute density has no minimum anywhere on the same axis; and the two disagree by a factor of three on the resistance a design should aim for. Which of them is the right target is decided by whether the source is a given or a choice, and that is a question about the circuit rather than about the part.