Concept

Random walk — where it appears

A running sum of independent steps, whose spread grows as the square root of their number while its mean stays where it began. A clock's period noise and a core's per-cycle flux imbalance both walk, so the worst excursion is set by how long the record runs rather than by any one step.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

An amplifier gain of 1000 caps a switched-capacitor integration at 8000 cycles' worth of averaging. computed by solving, not by drawing. The noise, referred to the input, of the estimate a switched-capacitor integrator makes by integrating a steady input for N cycles, with 1 pF sampling and 4 pF feedback capacitors, against N, for amplifier gains of 100, 1,000, 10,000 and infinity. Each cycle adds 2kT/Cs — 89.5 µV rms — and an ideal integrator averages it as 1/√N without limit. A finite gain leaks, and the noise stops falling at √(kT/(A·Cf)) = 1.000 µV for A = 1000, reached after about 8000 cycles. The closed form is checked against a seeded march.

The average an integrator can hold

A switched-capacitor integrator is an averager whose samples are exactly independent, because every cycle resets the sampling capacitor — and it pays for that with two switch phases, 2kT/Cs a cycle, 89.5 µV rms with a picofarad. An ideal integrator averages that as 1/√N without end. A real one leaks through its amplifier's finite gain, and the leak caps the average at 2A·Cf/Cs cycles: 8,000 for a gain of a thousand and a 4 pF feedback capacitor, where the noise stops at √(kT/(A·Cf)) = 1.00 µV. That floor contains the feedback capacitor and the gain and not the sampling capacitor at all, which decides only how long the integration takes to reach it.

noise · kT/C noise

With a leak of a hundred periods the flux settles to a spread of 1.22% of the swing at β = 0.5 — 0.759 of what independent duty errors would give. Seeded: three runs of sixty thousand periods of the event map, 5 mV of threshold noise, β = 0.5, driving a magnetising branch whose flux leaks with τ = L/R. The stationary spread of the period-end flux in volt-nanoseconds against τ in periods, quoted here as a percentage of one half cycle's swing V·T/2 = 5493 V·ns: 1 → 0.146% (closed form 0.145%), 3 → 0.233% (closed form 0.232%), 10 → 0.401% (closed form 0.397%), 30 → 0.685% (closed form 0.673%), 100 → 1.25% (closed form 1.22%), 300 → 2.16% (closed form 2.10%). The closed form linearises one period, keeping the leak inside each half and the covariance −AB between consecutive periods' imbalances; dropping that covariance gives the upper curve, 1.61% at a hundred periods. For a slow leak the spread is V·σ·|B − A|·√(τ/T), the drawn asymptote.

The spread a winding resistance holds

A relaxation oscillator's threshold noise walks an ideal core's flux without limit. Put the winding's resistance back and the walk stops at a spread: 1.22% of a half cycle's swing when L/R is a hundred periods, growing as the square root of L/R, while a comparator skew's offset grows as L/R itself. The anticorrelation between consecutive duty errors is worth 24% of that spread at β = 0.5 and 62% at β = 0.2 — but only in a core slow enough to remember several periods. And the resistance that bounds the walk is a boundary in its own right: with a millihenry and a fifth of the swing to spare, an ohm tolerates 12.9 mV of noise, and a hundred picoseconds of skew needs at least 0.14 Ω.

applied · Hysteresis

With a 300 ps skew and 5 mV of noise a core with 100 ns of headroom saturates after 502 periods on average — the 500 the skew alone gives — and one drive in a hundred by 184. Seeded: 2,000 drives of an ideal core by the event map at β = 0.5, 5 mV of threshold noise and a comparator skew of 300 ps, each run until the summed high-minus-low time leaves ±100 ns. A histogram of the counts on logarithmic bins, against the closed-form first-passage distribution, between two barriers, of a walk with drift 0.200 ns a period and spread 1.89 ns per root period. Mean 501.9 against 499.9; quantiles 1% 184, 10% 275, 50% 463; the empirical and closed-form cumulative distributions differ by at most 0.014.

The count a core survives

A relaxation oscillator with a 300 ps comparator skew and 5 mV of threshold noise saturates an ideal core with 100 ns of volt-second headroom after 502 periods on average — the 500 the skew alone gives — and one drive in a hundred by 184. The mean does not see the noise at all until the noise is large enough to saturate the core the other way first; the early tail sees nothing else. Every skew and noise level falls on one curve in one ratio, the skew's count over the count at which its ramp overtakes the walk. Read as a burst, a thousand periods with one failure in a thousand allows 150 ps of skew with no noise, 93.5 ps at a millivolt, and none at all above 2.41 mV.

applied · Hysteresis

Forty thousand periods at β = 0.5: each period's error against the same period's duty error, correlated by 0.008. Seeded: forty thousand periods of the event map, β = 0.5, 5 mV of threshold noise; one in twenty is drawn, each scaled by its own standard deviation (3.4 ns for the period, 2.49 ns for the high-minus-low time). A period is three draws with coefficients A, A + B, B and its duty error the same three with A, B − A, −B, so their covariance is σ²[A² + (A + B)(B − A) − B²], which is zero at every β. Measured, the correlation is 0.0080, inside the ±0.0100 that forty thousand samples resolve. With normal draws, uncorrelated is independent: the cloud is round.

Every draw shared, and nothing in common

A relaxation oscillator's period error and its volt-second error are made of the same threshold draws, and they are exactly uncorrelated. A period weighs its three draws A, A + B and B; the high half less the low weighs the same three A, B − A and −B; the products sum to A² + B² − A² − B², which is zero at every hysteresis. Over forty thousand periods the measured correlation is 0.008 against a resolution of 0.010. What the two do share is one draw with the period on either side, ±0.157 at β = 0.5, and over any whole burst those two lags cancel term for term. Four thousand bursts judged against twenty-five pairs of limits fail both ways exactly as often as the product of the two rates says: the margins specified separately double-count nothing.

applied · Hysteresis

Named alongside it

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

Seeded generator Headroom Monte carlo Relaxation oscillator Volt-seconds Duty cycle Averaging Closed form Design tradeoff Integrator kT/C noise Loop gain

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