The source resistance the flicker moves

The floor a circuit has gave an amplifier two noise generators — a voltage in series with its input and a current across it — and found the source resistance at which their combined effect is least, relative to the source’s own noise. It is the ratio of the two densities, 6.67 kΩ for a part of 4 $\text{nV}/\sqrt{\text{Hz}}$ and 0.6 $\text{pA}/\sqrt{\text{Hz}}$. The bowl, and the bottom of it measured how wide the minimum is and closed on a limitation it could name but not measure: every noise source in it was a number rather than a spectrum. Real voltage and current noise both rise at low frequency, as 1/f in power below a corner, and the two corners are rarely at the same frequency. The optimum is a ratio of the two, so if they rise at different frequencies the optimum moves.

This essay gives the same part a spectrum and measures where the bottom of the bowl goes.

A part with two corners #

The white densities are the earlier essays’ part unchanged: 4 $\text{nV}/\sqrt{\text{Hz}}$ and 0.6 $\text{pA}/\sqrt{\text{Hz}}$. To them are added the two flicker corners a bipolar input stage typically has — the voltage noise’s low, here 10 Hz, and the current noise’s much higher, here 1 kHz, a hundred times above it. That ordering is the usual one for bipolar parts. Their current noise is the base current’s shot noise plus a flicker component of the same current, which is large; their voltage noise is set mostly by the thermal noise of the base spreading resistance and the shot noise of the collector current, whose flicker is much weaker, so its corner sits far lower. Below its corner each density rises as the square root of the corner over the frequency:

$$e_n(f) = e_{n0}\sqrt{1 + f_e/f}, \qquad i_n(f) = i_{n0}\sqrt{1 + f_i/f}.$$

At each frequency the part is the earlier essays’ two-generator amplifier with these spot densities, and every bowl is drawn by the same closure that drew theirs; nothing about the noise figure is re-derived.

At 100 Hz the voltage noise has barely moved, to 4.20 $\text{nV}/\sqrt{\text{Hz}}$, while the current noise has more than tripled, to 1.99 $\text{pA}/\sqrt{\text{Hz}}$. The optimum is their ratio, 2.11 kΩ, and a search on the drawn curve finds its bottom there to six figures. The floor has risen from 1.14 dB to 3.10 dB. A source left at the white optimum of 6.67 kΩ reads 4.49 dB at this frequency — 1.39 dB above the floor a better-chosen source would have given.

Slide to a hertz and the picture is extreme: the optimum is 699 Ω, the floor 15.1 dB, and the white optimum reads 21.8 dB, nearly seven decibels worse than it need be. Slide to 10 kHz and the spot bowl and the white one are nearly indistinguishable.

Reading the corners off a data sheet #

A data sheet rarely states a corner frequency as a number. It draws two curves — voltage noise density and current noise density against frequency — and states each at one or two spot frequencies, usually 10 Hz and 1 kHz. The corner is where each curve’s flat part meets its rising part, and it has to be read off the plot.

The two numbers that matter here are both on that plot. The white densities give the high-frequency optimum and floor, which is what the earlier essays used. The ratio of the corners gives the drift, and it can be read without reading either corner precisely: take the ratio $e_n/i_n$ at the two spot frequencies the data sheet quotes, and the change between them is the drift across that decade. For the part here that ratio is 4.74 kΩ at a kilohertz and 938 Ω at 10 Hz, a factor of five across two decades, and a designer who has computed the optimum at both spot frequencies has already seen the whole of this essay’s first figure in two numbers.

The trap is to compute it at one frequency only. The single optimum a design note quotes is almost always the white one, computed from the densities at a kilohertz or above, because that is where the densities are flat and the answer is stable. Stable is not the same as relevant, and for a low band it is the wrong frequency to have read.

Where the optimum goes #

Following the optimum across the whole spectrum shows it has two limits and a transition between them.

Above both corners it is the white ratio, 6.67 kΩ: 6.63 kΩ at 100 kHz and 6.36 at 10 kHz. Far below both, each density is dominated by its flicker term, the $1/f$ factors cancel in the ratio, and what is left is the white ratio times $\sqrt{f_e/f_i}$ — here $\sqrt{10/1000}$, a tenth, so 667 Ω. The measured optimum is 699 Ω at a hertz and 670 at a tenth of one, approaching it. Between the corners the optimum moves most: 938 Ω at 10 Hz, 2.11 kΩ at 100, 4.74 kΩ at a kilohertz.

So the total drift is fixed by one number, the ratio of the two corners, and the ratio enters under a square root. Corners a decade apart move the optimum by a factor of 3.2; corners a hundred times apart by 10. A part whose two corners coincide has an optimum that does not move at all, however high the corners are — its floor rises at low frequency, but the source that minimises it stays put. That is the one condition under which the single-number optimum of the earlier essays is the whole story.

The floor tells the other half. It rises from 1.144 dB at 100 kHz through 1.54 at a kilohertz, 3.10 at 100 Hz and 7.21 at 10 Hz to 15.1 at a hertz and 24.8 at a tenth of one. The floor goes as the product of the two densities rather than their ratio, and the product rises below both corners, so it has no low-frequency limit.

Over a band, the band decides #

A spot noise figure is a statement about one frequency, and a real signal occupies a band. The noise figure that matters for it is the ratio of total noise to source noise integrated across the band, and its optimum is the square root of the ratio of the two integrated densities — each flicker term integrating to its corner times $\ln(f_2/f_1)$ and each white term to the band’s width.

Two routes produce each curve. One integrates the spot densities numerically on a four-thousand-point logarithmic grid, knowing nothing about their form; the other uses the closed-form integrals. They agree to 10⁻⁵ on every band.

Over the audio band, 20 Hz to 20 kHz, the optimum is 5.76 kΩ — below the white value, but only by fourteen per cent — and choosing the white optimum instead costs 0.012 dB. The band is three decades wide and almost all of its hertz are in the top one, where the part is white, so the flicker barely enters the integral. The earlier essays’ single number serves perfectly.

Over 10 Hz to 1 kHz the optimum is 2.87 kΩ and the white choice costs 0.64 dB. Over 1 Hz to 100 Hz — the band of a seismometer, a strain gauge read slowly, a biological signal — the optimum is 1.17 kΩ and the white choice costs 3.77 dB. On that band the part’s own data-sheet optimum is the wrong source resistance by a factor of 5.7, and choosing it throws away more than half the signal-to-noise ratio a correct choice would have kept.

The asymmetry is the same weighting the bandwidth noise sees rests on: noise power is counted per hertz, so an integral over a band is dominated by where the band has most hertz, and on a linear frequency axis that is always its top. Flicker lives at the bottom of every band, so it weighs heavily only when the band itself is down among the corners.

A rule that recovers it #

A designer rarely has the band integral to hand, but usually has the spot noise densities at a few frequencies. The last two figures ask how good the simple rules are.

For two-decade bands swept along the spectrum, the white optimum costs up to 6.41 dB on the lowest band, 3.77 on the band from a hertz, 0.64 on the band from 10 Hz and 0.021 on the band from 100 Hz. The spot optimum at the band’s geometric centre — the ratio of the two densities read at ten times the lower edge — does far better: it never costs more than 0.11 dB, and wherever the white choice costs anything the centre’s choice costs less. Its worst is on the band from about 3 Hz, where it sits a quarter below the band’s own optimum, because the integral is weighted towards the band’s upper decade and the centre is at the bottom of that decade.

The practical rule, then, is to read the voltage and current noise at the middle of the band on a logarithmic scale, take their ratio, and use that. It is the same rule the earlier essays gave with one change of where to read, and on a two-decade band it captures all but a tenth of a decibel of what the full integral would buy.

It has a limit, and the band figure’s own numbers show it. The logarithmic centre is the right place to read only because a two-decade band’s weight is not very lopsided. On the three-decade audio band the centre is 632 Hz, its spot optimum is 4.18 kΩ, and it costs 0.057 dB — more than the white optimum’s 0.012, because the band’s weight is so heavily in its top decade that the white value is closer. The sharper rule reads the ratio at the centre of the band’s top decade instead: that costs 0.003 dB on the audio band, 0.019 on the band from 10 Hz and 0.028 on the band from a hertz, the best of the three rules on all three bands. It is also the rule that follows from the argument rather than from a fit: the band’s noise is counted per hertz, most of the hertz are in the top decade, and so that is where the densities should be read.

This is the noise version of an argument the optimum a spectrum moves made about a transformer winding in magnetics: an optimum derived for a single frequency and quoted as a property of the component is a property of the spectrum it was derived at, and a different spectrum moves it. There it was the foil thickness and the harmonics of a square current; here it is a source resistance and the flicker of two noise generators. In both, the fix is to weight by where the power actually is.

Deeper, and narrower, where it drifts #

The bowl, and the bottom of it found that the bowl is one shape scaled by its own depth: a shallow bowl is wide, and a deep one narrow. It closed by saying that the width was what would decide whether a drifting optimum mattered — a shallow bowl could drift a decade and lose a decibel, a deep one could not. The spectrum puts the two on the same axis.

The width of the 1 dB bottom is ±4.0× at high frequency. It narrows as the floor rises — ±3.0 at 309 Hz, ±2.35 at 31 Hz, ±2.03 at 0.3 Hz — because the same drop in frequency that moves the optimum also deepens the bowl. The drift of the optimum from its white value goes the other way: 1.02× at 31 kHz, 2.03 at 309 Hz, 5.04 at 31 Hz and 9.85 at 0.3 Hz.

They cross at 142 Hz. Above it a source set at the white optimum is inside the 1 dB bottom of the spot bowl; below it, outside, and by 30 Hz the cost is 3.2 dB. So the earlier essay’s prediction is right in its form and sharper than it could say: the bowl is narrowest at exactly the frequencies where its bottom has moved furthest, and the two effects compound rather than offset.

The figure and the volts #

The bowl, and the bottom of it warned that the noise figure’s optimum and the lowest absolute noise are different questions: the figure compares the amplifier against the source’s own noise, and a smaller source resistance always has less noise of its own, so the quietest circuit in volts is the one with the smallest source, while the best figure is at the optimum. The two come apart whenever the source can be chosen.

The flicker moves the optimum towards the smaller source. On the low band here the figure’s optimum is 1.17 kΩ rather than 6.67 kΩ, so the gap between the figure’s answer and the volts’ answer narrows exactly where the flicker is worst. A designer who chooses the band’s optimum is simultaneously lowering the absolute noise, because the current generator — now more than three times larger at 100 Hz than at high frequency — is what a large source resistance turns into voltage.

It also changes which of the given sources the earlier essays sorted are well placed. A shunt of tens of milliohms, like the one in the ammeter that is a resistor, is still four decades below any optimum here and still sees the voltage noise alone. A source of nine megohms, like the probe that takes a tenth, is further above the optimum at low frequency than at high, because the optimum has come down while the source has not, and it sees the current generator’s flicker in full. For that source the useful choice is a part with a lower current-noise corner, which on this model is the same as a part with a smaller corner ratio.

How the numbers were obtained #

At each frequency the amplifier is constructed from its two spot densities and handed to the same two-generator closure the earlier essays used: the noise factor $1 + (e_n^2 + i_n^2 R_s^2)/4kTR_s$ at 290 K, the optimum $e_n/i_n$ and the floor. The optimum is also found independently by a ternary search on the drawn curve and required to agree to a part in a million. Band figures integrate the spot densities over the band by the trapezoid rule on a logarithmic grid of four thousand points and compare against the closed-form integrals, requiring agreement to 10⁻⁵. The bowl’s width is the factor at which the noise figure crosses its floor plus a decibel, bisected on the same closure; the frequency at which the drift equals the width is bisected, and at it the white optimum’s cost is required to be one decibel to a ten-thousandth.

What it leaves out #

Correlation between the generators. In a real device the voltage and current noise are partly correlated, especially where both come from the same flicker process, and correlation moves the optimum off the real axis — the best source then has a reactance. The model here keeps them independent at every frequency.

A reactive source. A piezoelectric sensor or a capacitive microphone has an impedance that falls with frequency, so its effective source resistance moves across the band too. Only the real part is warm is the rule for its own noise; how its moving impedance combines with a moving optimum is a second drift on top of the first.

Averaging. Flicker noise does not average down, as the corner where averaging stops working measured, so a slow measurement cannot buy its way out of a low band’s flicker by integrating longer. That is why the band figures here are the right figures of merit for low-frequency work and not a pessimistic one.

Still open: correlation, the reactive source and the transformer #

Correlated generators. The two-generator model assumes the voltage and current noise independent. When they share a flicker source, a correlation coefficient enters the noise factor, the optimum becomes a complex impedance, and the floor falls. How much of the low-frequency floor measured here a correlated part would recover, and what reactance its best source would carry, is the natural next measurement on this part.

A source whose impedance moves with the band. A capacitive transducer’s impedance falls as $1/f$, and on a low band it sweeps through the optimum this essay found moving. Whether the two drifts can be made to track — a transducer chosen so that its impedance follows the part’s optimum across the band — is a design question with an integral behind it.

Matching through a transformer. A step-up transformer changes the source resistance the amplifier sees by the square of its turns ratio, which is the usual way to bring a low-impedance source up to an optimum. With a moving optimum, one turns ratio suits one frequency; which ratio suits a band, and how much of the band-optimum it recovers against the loss in front, counted twice that a real transformer’s winding resistance adds, is the question this one leaves.