A floor and a ceiling

Every model has an edge puts four of this collection’s boundaries on one frequency axis: the ideal amplifier at 1.42 kHz, the slew limit at 7.96 kHz, the ideal capacitor at 4.69 MHz, and Kirchhoff’s laws on ten centimetres of track at 3.97 MHz. All four are ceilings, and the axis they share is frequency.

This page is the other axis. Amplitude, with a floor at one end and a ceiling at the other, and every number on it taken from a different module of this site.

The two ends #

The floor is the Johnson noise of the source resistance in the measurement’s noise bandwidth. For a kilohm in a ten-kilohertz measurement that is 4.00 nV/√Hz times √(15.7 kHz) — using the noise bandwidth, π/2 times the corner, not the corner — which is 501 nV.

The ceiling is the amplitude at which the stage’s distortion reaches one per cent, which for an exponential transconductor is 1.034 mV. That number comes from the distortion a linear model cannot have, where it is bisected on a transformed transfer curve and checked against a Bessel series.

Between them is a factor of 2,062, which is 66.28 dB.

Two other ceilings are marked on the same axis and are further up: one per cent of gain error at 7.30 mV, and the amplitude at which a unity-gain follower stops being linear at all, 79.6 mV. Which of the three counts as “the” ceiling depends entirely on what is being asked — a limiter cares about the third and an audio stage about the first — and putting all three on one axis is the point of the figure.

Why the range is the quantity #

Neither end means much alone.

A noise floor of 501 nV is a small number and says nothing about whether a circuit is any good; a ceiling of 1.03 mV is a small number too, and a stage with that ceiling and a floor of a nanovolt would be excellent. What decides is the ratio, and the ratio is what a specification means when it says a system has “sixteen bits” or “ninety decibels”.

The reason the ratio is the right quantity rather than either end is that both ends move together under most of the things a designer can do. Lower the source resistance and the floor falls — and the signal available from a lower-impedance source usually falls too. Add gain and the floor rises exactly as much as the signal does. Narrow the bandwidth and the floor falls as its square root, and the measurement takes proportionally longer.

The last of those is the one that genuinely buys range, and it is why the slider on the figure is the bandwidth.

What the bandwidth buys #

The floor is √(density × bandwidth), so it falls as the square root of the bandwidth, and the ceiling does not move at all — the amplitude at which a stage distorts has nothing to do with how long anyone watches it.

So the range grows as the square root of the narrowing. Ten hertz of measurement bandwidth rather than ten kilohertz is a factor of 31.6 in the floor, which is 30 dB of range, bought entirely with time.

How much time is arithmetic. A measurement that averages for t seconds has a noise bandwidth of 1/2t, so ten hertz is fifty milliseconds and a tenth of a hertz is five seconds. Resolving a microvolt against a ten nanovolt-per-root-hertz floor at a signal-to-noise ratio of ten needs 100 Hz and takes five milliseconds; a hundred times better resolution needs ten thousand times the bandwidth reduction and takes fifty seconds.

That square root is the least negotiable relation in this field. It is why an extra digit of resolution costs a hundred times the measuring time, and why instruments that are specified as “6½ digit” and “8½ digit” differ by four orders of magnitude in integration time rather than by any improvement in their front ends.

Sixty-six decibels, and what that is worth #

A number of that size deserves a comparison, because “sixty-six decibels” is meaningless without one.

It is eleven bits. A converter with more than eleven bits placed behind this stage in this bandwidth is describing the noise with the extra ones. It is also, coincidentally, about the range of a telephone line and about a third of the range of human hearing, which spans roughly 120 dB from the threshold to the point of pain.

The measurement bandwidth moves it a decade at a time in twenty-decibel steps — 96.3 dB at ten hertz, 86.3 at a hundred, 76.3 at a kilohertz, 66.3 at ten, 56.3 at a hundred, 46.3 at a megahertz. Sixteen bits is 96 dB, so this stage reaches sixteen bits’ worth of range in a ten-hertz bandwidth, which is to say in about fifty milliseconds per reading.

That is the arithmetic behind a fact anybody who has used a bench multimeter knows without having derived: the slow ranges are the accurate ones, and each further digit costs a hundredfold in time. The instrument is not doing anything clever on the slow ranges. It is narrowing a bandwidth, and the floor is falling as its square root.

The first stage decides the floor #

A signal chain has several stages, and one of them decides the answer.

Friis’s expression for a cascade divides each stage’s noise contribution by all the gain in front of it. The first stage has no gain in front of it, so its contribution enters in full; the second is divided by the first’s gain; the third by the product of the first two. With any reasonable gain in the first stage, everything after it very nearly does not matter.

The measured contributions make it stark. With the amplifier first: 1.2589 from the amplifier, 0.0900 from the mixer, 0.0030 from the intermediate amplifier — so everything after the first stage is 6.9% of the total. With the mixer first: 10.0000, 0.0259, 0.0030, and the answer is the mixer’s noise figure to three decimal places.

The design rule that follows is the one every radio engineer knows and it is worth deriving rather than repeating: put the quietest stage with useful gain first. Not the highest gain, and not the lowest noise — the first stage needs both, because its noise enters undivided and its gain divides everything after it.

The exception that proves it #

The slider on that figure puts a lossy cable in front of the chain, and the result reverses the usual reading in a way worth understanding.

An attenuator’s noise figure equals its loss — a 3 dB attenuator has a 3 dB noise figure — and its gain is less than one. Friis divides every later stage by the gain in front of it, and dividing by a number less than one multiplies. So a cable in front of an amplifier does not merely add its own loss to the noise figure; it multiplies every subsequent stage’s contribution.

With eight decibels of cable in front, the stages after the first contribute 26% of the total rather than 6.9%. The rule “the first stage decides” is really “the first stage decides, and a lossy component counts as a stage” — which is why a low-noise amplifier is mounted at the antenna rather than at the receiver, and why the length of coaxial cable in front of it is a specification.

Every number here came from somewhere else #

The unusual thing about this page is that it computes almost nothing of its own. Each mark on the axis is a result from a different part of the site, and the argument is entirely in putting them on one axis.

The floor comes from lib/noise.js — 4kTR, and the noise bandwidth from an integral over a solved response. The distortion ceiling comes from lib/semi.js, bisected on a transformed transfer curve. The gain-error ceiling comes from lib/limits.js, which has carried it since the foundation phase. The slew boundary comes from the same module, measured by integrating a rate-limited amplifier forward in time.

Four modules, four methods, one axis. That is the shape the whole collection has been building towards: the boundaries are not a list of warnings but a set of measured numbers that can be compared because each was computed from the model it belongs to.

It is worth being explicit that comparing them is not automatic. Four numbers computed by four methods are only comparable if they are in the same units and refer to the same thing, and getting that right took some care. Every number on the axis is an amplitude in volts, referred to the same point in the circuit — the floor is a root-mean-square voltage at the input, the distortion ceiling is the amplitude of a sinusoid at the same input, the slew boundary is the size of a step at the same input. Mixing a root-mean-square with a peak, or an input-referred quantity with an output-referred one, would put marks on the axis that could not be read against each other, and neither the figure nor a reader would notice.

That is the quiet cost of an argument built from four sources, and it is the reason this page’s machinery is a page of unit conversions and a plot rather than anything cleverer.

What closes the range from both ends #

The three techniques that raise the ceiling all appear elsewhere in this collection, and it is worth noticing that none of them lowers the floor and one of them raises it.

A differential pair raises the distortion ceiling by a factor of about eighteen — 18.2 mV rather than 1.03 mV for one per cent of harmonic content — because it removes the even harmonics exactly. It also contributes two devices’ worth of noise where there was one, so the floor rises by √2. The net gain in range is a factor of about twelve rather than eighteen.

Degeneration raises the ceiling in proportion to how much of the drive falls across something linear, and lowers the gain by the same factor. The floor referred to the input rises too, because the degeneration resistor is itself a warm resistor and contributes its own thermal noise. The trade is close to neutral in range terms and strongly favourable in linearity terms, which is why it is used.

Feedback raises the ceiling by the loop gain and does not change the input-referred noise at all, because the noise of the first stage is inside the loop and is amplified along with the signal. This is the one technique that buys range outright, and what it costs is the loop gain — which is the feedback field’s subject and comes with its own boundaries in phase margin and bandwidth.

Reading those three together says something that is not obvious from any one of them: the floor is much harder to move than the ceiling. Every technique here raises the top by an order and the best of them leaves the bottom alone. That asymmetry is why low-noise design is a separate discipline from linear design, and why the first stage in a chain gets a disproportionate share of the attention.

The one thing on this axis that is not a model’s edge #

Three of the four marks on the amplitude axis are boundaries of a model: the drive at which distortion reaches one per cent, the drive at which the gain is one per cent wrong, the step at which a follower stops being linear. Each says “above here, the description in use stops applying”.

The floor is a different kind of object and the difference is worth stating. It is not a boundary of a model; it is a physical limit, and no better model reaches below it. A more careful analysis of a resistor’s noise does not produce a smaller number, because 4kTR is not an approximation to anything — it follows from thermodynamics, and its independence of every property of the resistor except resistance and temperature is the sign of that.

So the axis on this page carries three statements about what is known and one about what is possible, and only the last of the four cannot be improved by thinking harder. That asymmetry is what makes the noise field the right place for this collection’s expansion to end: every other boundary here is a limit of a description, and this one is a limit of the world.

What the range is not #

Two things this page does not claim, and they are worth stating because the phrase “dynamic range” is used loosely enough that both are sometimes implied.

It is not the number of bits. A converter’s bit count is a property of the converter, and the range measured here is a property of the analogue stage in front of it. A twenty-four-bit converter behind a stage with sixty-six decibels of range delivers eleven useful bits, and the other thirteen are describing the noise very precisely.

It is not a single number for a circuit. Every mark on the axis moved when the bandwidth slider moved, and the ceiling would move if the stage were degenerated or wrapped in feedback. The figure draws one operating condition of one stage, and the honest reading is the method rather than the number — a floor from the source resistance and the noise bandwidth, a ceiling from the distortion the topology produces, and the ratio between them.

Which end moves, and what moves it #

The 66.3 decibels is a difference of two numbers that respond to entirely different design decisions, and separating them is what makes the range a design object rather than a specification.

The ceiling is bought with linearity and every technique for raising it is measured in the semiconductors field. What a resistor in the emitter buys is the standard one and its exchange rate is not one to one: dividing the gain by six moves the amplitude at which distortion reaches one per cent by thirty-five times — the square of the factor — and the amplitude at which the gain is one per cent out by only twelve. So degeneration raises the ceiling faster than it raises the other edge, and the two close up.

The floor is not bought at all in the same sense. The floor a resistor sets is why: a kilohm at room temperature produces 4.00 nanovolts per root hertz because it is warm, gain does not help because gain amplifies it too, and the only variables are the resistance, the temperature and the bandwidth. The floor a circuit has adds the one genuine design choice available — a device’s two generators have a source resistance at which their sum is least, 6.67 kΩ, which is their ratio and not the resistance that transfers maximum power.

Which is why the arrangement of the stages decides which end moves. The loss in front, counted twice is the measurement of that: a decibel of cable before an amplifier costs a decibel of signal and a decibel of noise figure, so it lowers the ceiling and raises the floor at once, while the 8.70 dB that stage ordering is worth moves the floor alone. A designer who has spent everything available on linearity has moved one end of this range; a designer who has reordered the chain has moved the other; and only the second is free.

And there is one arrangement that moves neither end and enlarges the range anyway, which is worth naming because it is the only one. The floor a circuit has finds a source resistance at which a device’s two noise generators sum to a minimum — 6.67 kΩ, their ratio, and not the resistance that transfers maximum power — so transforming a source towards that value lowers the floor without touching the ceiling. It costs a transformer or a different sensor rather than headroom, which is why it is the first thing tried in a receiver and the last thing available in an instrument.

Where the two ends come from #

A dynamic range is two measurements and a subtraction, and both ends are established elsewhere. The floor a resistor sets and The floor a current sets are the two mechanisms at the bottom, crossing at a current that puts fifty millivolts across the source. The floor a circuit has is the amplifier’s own, with an optimum source resistance. What a resistor in the emitter buys is the ceiling, and The bandwidth noise sees is the factor without which neither end is a voltage.