Circuit Response
every model has an edge, and the figure carries it
Concept

Flicker noise — where it appears

Noise whose power spectral density falls as one over frequency, so that every decade of bandwidth contributes the same amount. It appears wherever conduction is through something granular or through a surface, it is a property of how a device was made rather than of its resistance, and no amount of averaging escapes it.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The resistors own the floor between 2.91k Ω and 85.9k Ω, and the part owns it outside. computed by solving, not by drawing. The noise at the load of the two-path compensation against the impedance of its own feedback network, with the resistors scaled together and the compensation capacitor taken down in proportion so that Rf·Cf — the handover between the two feedback paths — does not move. Four contributions are integrated over 10 Hz to 100 MHz: the amplifier's 4 nV/√Hz, fitted as the 0.005 power of the impedance and so flat; the two resistors' √(4kTR), the 0.501 power; and the amplifier's 0.60 pA/√Hz flowing in the feedback resistor, the 0.997 power. Two different powers of one quantity cross twice. The resistors carry more than half the power only between 2.91k Ω and 85.9k Ω; outside that window, in both directions, the part does. The part's share is least at 15.8k Ω, which is not eₙ/iₙ — it is that ratio multiplied by the noise gain of 2.000 and again by 1.187, the square root of the ratio of the bandwidths the two generators actually see; there it carries 26.26 per cent. The model stops where the amplifier's output current does: at 100 Ω the feedback resistor alone draws 10 mA a volt.

The window the resistors own

Eight essays have priced one compensated stage, and the fourth of them left two of the amplifier's own generators named and uncounted. With the current generator put in the netlist the floor at ten kilohms goes from 28.88 microvolts to 30.13, and the resistors carry more than half the noise power only between 2.91 kΩ and 85.9 kΩ — outside that window, in both directions, the part does. The flicker corner turns out to be worth 1.00009 in this stage's own band, and 3.474 one band away.

feedback · Capacitive load The floor a biased resistor is not standing on. computed by solving, not by drawing. A 100 kΩ resistor with 10 V across it. Its Johnson noise is 40.02 nV/√Hz and does not depend on the voltage; its excess noise is a 1/f density proportional to that voltage, 0.1 µV per volt per decade here, and the two are equal at 271 Hz. Below that the resistor is dominated by a generator that is a property of how it was made rather than of its resistance, and the crossover moves with the SQUARE of the voltage. Splitting the same total resistance into eight parts in series leaves the Johnson noise exactly where it was and divides the excess by √8 = 2.828, taking the total at 1 Hz from 660.2 to 236.4 nV/√Hz. With no voltage across it the second generator is absent rather than small.

The floor that is only a floor while nothing flows

Johnson noise depends on a resistance and a temperature and on nothing else, which is what makes it a floor. A real resistor has a second generator that depends on how it was made and on the voltage across it: 0.1 µV per volt per decade for a metal film, rising as 1/√f. On 100 kΩ with 10 V across it the two are equal at 271 Hz, and the crossover moves with the SQUARE of the voltage — 0.678 Hz at half a volt, 2.44 kHz at thirty. Splitting the same total resistance into eight parts in series divides the excess by exactly √8 and leaves the Johnson noise where it was.

noise · Johnson noise At 100 Hz the bowl's bottom is at 2.11 kΩ and 3.10 dB, against 6.67 kΩ and 1.14 dB in the white region. computed by solving, not by drawing: the spot noise figure against source resistance of an amplifier with 4 nV/√Hz and 0.6 pA/√Hz of white noise, whose voltage noise has a flicker corner at 10 Hz and whose current noise has one at 1 kHz. At 100 Hz the spot densities are 4.20 nV/√Hz and 1.99 pA/√Hz, so the optimum is 2.11 kΩ — found by search on the drawn curve at 2.11 kΩ — with a floor of 3.102 dB. Dashed: the white-region bowl, 6.67 kΩ and 1.138 dB. A source left at the white optimum reads 4.492 dB at this frequency. The slider is the frequency.

The source resistance the flicker moves

An amplifier's optimum source resistance is the ratio of its voltage noise to its current noise, and both densities rise below their flicker corners — which are rarely the same frequency. With the voltage noise's corner at 10 Hz and the current noise's at 1 kHz, a part whose optimum is 6.67 kΩ at high frequency wants 2.11 kΩ at 100 Hz and 699 Ω at 1 Hz, tending to the white value times the square root of the ratio of the corners. Over the audio band the drift costs 0.012 dB; over a band from 1 to 100 Hz, choosing the white optimum costs 3.77 dB, and the spot optimum at the band's centre recovers all but a tenth of a decibel.

noise · Device noise With a correlation of −0.3 between the two generators the bowl's bottom stays at 6.67 kΩ and its floor moves from 1.14 dB to 0.83. computed by solving, not by drawing, with the optimum found twice — in closed form and by a direct search of the noise factor, which agree to a part in a million. The 4 nV/√Hz, 0.6 pA/√Hz amplifier of the earlier essays, its two generators given a real correlation coefficient γ = −0.3. The noise figure against source resistance: the cross term 2γ·e·i·R adds the same fraction of the source's own noise at every resistance, so the bowl moves vertically and its bottom stays at e/i = 6.67 kΩ. The floor's excess over one scales as 1 + γ: 0.827 dB against 1.138 for independent generators. The slider is the correlation.

Two generators with one cause

The two-generator model of an amplifier assumes its voltage and current noise independent. Give them a correlation γ and the source sees a cross term, 2·Re(γ·e·i·Z*), linear in its resistance and its reactance. A real correlation leaves the quietest source resistance at e/i and moves the floor in proportion to 1 + γ: 0.83 dB instead of 1.14 at γ = −0.3. An imaginary one moves the optimum off the resistance axis — 5.77 kΩ with 3.33 kΩ of capacitive reactance at γ = 0.5j — and the noise figure's contours become circles about it; a resistive source cannot use it at all. If the two flicker components share one cause, the sign of the sharing decides the low-frequency floor: at 1 Hz, 3.9 dB or 18.0 instead of 15.1. And the correlation is measurable with three resistors, best with the middle one next to the optimum.

noise · Device noise With a 100 kHz flicker corner the 0.1 ps oscillator's ratio stops falling ten decibels a decade of capture and starts falling twenty, past 2,277 samples — 1.42 periods of the corner. computed by solving, not by drawing, from the exact expected residual of a summed process, with seeded clocks as dots (24 at each length to 16,384). A free-running oscillator with a 0.1 ps period spread and a flicker of its frequency whose timing spectrum meets the walk's at 100 kHz, sampling a 10.01 MHz tone at 160 MHz. Signal-to-timing-error after removing each capture's best delay and frequency: the walk alone falls ten decibels a decade, 85.7 dB at 1024, 67.6 dB at 65536; with the flicker, 84.1 and 52.9 dB. The flicker's share equals the walk's at 2,277 samples, 14.2 µs, which is 1.42 periods of the corner frequency; beyond it the ratio falls 19.5 dB a decade. Removing a quadratic as well buys 3.80 dB at 65,536 samples and changes no slope.

A frequency that wanders

Below its flicker corner an oscillator's phase noise falls thirty decibels a decade instead of twenty, because the oscillator's frequency itself wanders. A clock with 0.1 ps of period spread and a 100 kHz corner samples a 10 MHz tone at 84.1 dB over 1,024 samples and 52.9 over 65,536 — 14.7 dB worse than the walk alone — because past 2,277 samples its error grows as the capture rather than its square root. That crossing is 1.42 periods of the corner frequency for every corner. Fitting a parabola as well as a line buys 3.88 dB at every length and changes no slope; each further term buys a fixed fraction. And a first-order locking loop cannot stop it: the locked clock's variance grows with the logarithm of the record, 1.92 dB lost between 4,096 and 262,144 samples, until a loop with an integrator holds it at 87.5.

digital · Aperture jitter A 50 Ω source over 20.0 Hz–20.0 kHz: the best turns ratio is 10.7, giving 1.30 dB; the white optimum's 11.5 gives 1.31, and no transformer at all 13.2. computed by solving, not by drawing, the band integrals checked by quadrature. The drifting part of the essay before — 4 nV/√Hz and 0.6 pA/√Hz white, flicker corners at 10 Hz and 1 kHz — fed from a 50 Ω source through an ideal step-up transformer of ratio n, so it sees n²·50 Ω. The noise figure over 20.0 Hz to 20.0 kHz against n: least at n = √(√(E/I)/Rₛ) = 10.73, where the amplifier sees 5.76 kΩ, at 1.298 dB. The ratio that matches the white optimum, 11.55, gives 1.310 dB; the source connected directly, 13.23 dB. The slider is the band.

One turns ratio for a band

A 50 Ω source and an amplifier whose optimum is 6.67 kΩ want a step-up transformer, and the ratio that matches the white optimum, 11.5, is the one a design note computes. Over the audio band it is nearly right: 1.31 dB against 1.30 at the band's own best ratio, 10.7. Over 1 Hz to 100 Hz it is badly wrong — 9.2 dB where a ratio of 4.8 gives 5.4 — because the band's noise lives in its bottom decade, where the flicker has moved the optimum down. One ratio for a two-decade band costs at most 0.24 dB against a ratio that could follow the optimum. The transformer's winding resistance multiplies the whole noise factor by one plus the winding's resistance over the source's, not just its own share: a tenth of the source costs 0.41 dB where its own noise alone would cost 0.31. And 115 amplifiers in parallel are an ideal 10.7:1 transformer, noise for noise.

noise · Device noise

Named alongside it

The objects these essays reach for when they reach for this one.

Design tradeoffOptimum source resistanceCurrent noiseNoise figureNoise floorVoltage noiseJohnson noiseAperture jitterCapacitive loadCorner frequencyCross termExcess noise

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