What a ratio does not see

The shape a tolerance does not print drew four resistors for a divider from four populations that all honour ±1 per cent, and found the divider’s spread varying by a factor of 2.26 and its yield at a tight specification by a factor of two, with the same number printed on every part. Every part in it was drawn independently of the others. That was the second assumption the earlier essays made without saying so, and the more consequential one.

Parts are rarely independent. Four resistors taken from one reel were made in one run, from one batch of material, in one furnace; their errors have a common part and a part of their own. A resistor array puts its elements on one substrate, a few hundred microns apart, and their errors are nearly all common. And every resistor on a board sees the same temperature, which moves all of them in the same direction by nearly the same amount — the most completely shared error there is. The tolerance that is not on any part pointed at this in its last section: a divider made of two halves of one part has a ratio whose temperature coefficient is the difference of two nearly equal numbers. This essay measures that, and what follows from it for ratios and for sums.

A shared part and a part of one’s own #

Each part’s error is modelled as the sum of two independent pieces: one common to all four parts in one copy, and one of its own, scaled so that each part still has the standard deviation of a uniform 1% part, 0.577 per cent. The fraction of the variance that is common is $\rho$, the correlation between any two parts. At $\rho = 0$ the parts are the earlier essays’ independent ones; at $\rho = 1$ they are four copies of one error.

The four are then built two ways from the same draws. As a divider tapped at the middle, the output is the lower pair over the whole — a ratio, and each part moves it by a quarter of its error, with the upper two pulling one way and the lower two the other. As a series chain, the total resistance is the sum, and each part moves it by a quarter of its error, all in the same direction. The first four hundred dividers are solved on the netlist as well as from the ratio; the two agree to rounding.

At $\rho = 0.9$, twenty thousand builds give the divider a spread of 0.0912 per cent and the chain a spread of 0.5587 per cent. With independent parts both were 0.289: the correlation has made one answer three times better and the other nearly twice as bad.

The two laws #

The arithmetic is short. In the divider the common piece enters the upper pair and the lower pair equally and cancels, so only the independent pieces remain, and they carry $1 - \rho$ of the variance: the ratio’s spread is its independent value, half the part’s, times $\sqrt{1-\rho}$. In the chain the common piece enters all four with the same sign and adds in full, while the independent pieces average down by the count: the total’s spread is the part’s times $\sqrt{\rho + (1-\rho)/4}$.

Across seven shared fractions the measured spreads follow both laws to within the scatter of twenty thousand builds. The ratio’s falls from 0.289 per cent at $\rho = 0$ through 0.204 at a half and 0.091 at nine-tenths to 0.029 at 0.99. The total’s rises from 0.290 through 0.454 and 0.559 to 0.573, which is the part’s own spread: four parts that share everything are one part four times over, and averaging them buys nothing.

At $\rho = 0$ the two are equal, and that is a coincidence of this circuit worth naming — the divider’s quarter sensitivities and the chain’s averaging over four happen to halve the part’s spread by the same amount. Any correlation at all separates them, and in opposite directions. The correlation that a designer of ratios wants is exactly the one a designer of absolute values fears, and a board holds both kinds of circuit.

A correlated part, as an equivalent tolerance #

The ratio law has a practical form: in a ratio, a correlated part behaves like an independent part of a smaller tolerance.

The divider’s spread from correlated parts, divided by its spread from independent ones, is $\sqrt{1-\rho}$ to within two per cent at every shared fraction measured: 0.706 at a half, 0.498 at three-quarters, 0.316 at nine-tenths, 0.100 at 0.99 and 0.070 at 0.995. So a 1% part sharing ninety per cent of its error with its neighbours is, in a ratio, a 0.32% part; sharing ninety-nine per cent, a 0.1% part.

The curve is steep only at its end, which is the useful fact about it. Sharing half the error buys 29 per cent of the spread; sharing nine-tenths buys 68; and the factor of ten that turns a 1% part into a 0.1% one needs 99 per cent shared. Correlation is valuable in proportion to how nearly complete it is, and parts from one reel — which share a batch but were made at different places on it — typically sit on the flat part of the curve, while parts on one substrate sit at the steep end.

A resistor array #

That steep end is what a matched array sells.

Four discrete 1% parts put the divider’s ratio inside ±0.1 per cent in 27.3 per cent of builds, inside ±0.05 per cent in 13.9, and inside ±0.2 in 50.8. Four parts of the same 1% absolute tolerance sharing 99.5 per cent of their error — a thin-film array on one substrate — put it inside ±0.1 per cent in every one of twenty thousand builds, and inside ±0.05 per cent in 98.7. In a ratio the array’s parts behave fourteen times better than their printed tolerance, and that is exactly what an array’s data sheet means when it quotes an absolute tolerance of a per cent beside a ratio tolerance of a few hundredths.

It also says where an array is wasted. Built into the series chain, the same parts give a total whose spread is the full 0.57 per cent — no better than one discrete part, and twice as wide as four independent ones averaged. An array bought for an absolute resistance buys nothing; bought for a ratio, it buys a decade. The same parts, the same price, fourteen times better or twice as bad depending on which question the circuit asks of them — which is why the first thing to establish about any resistor in a precision design is whether its value enters the answer alone or as part of a ratio.

Temperature, the shared error that never goes away #

Tolerance is fixed at manufacture. Temperature is applied every day, and it is almost entirely shared.

Four ordinary thick-film resistors with a coefficient of 100 ppm per kelvin, each within ±25 ppm of that, are taken from −40 to 85 °C. The chain’s total follows the shared coefficient: up to 0.759 per cent away from its 25 °C value, the whole of the temperature coefficient times the swing. The divider’s ratio follows only the difference between the coefficients, since the shared hundred ppm moves the upper and lower pairs alike: up to 0.135 per cent, 5.6 times less, and inside the 0.163 per cent that a ±25 ppm spread allows over sixty-five kelvin at worst.

That factor is the ratio of the coefficient to its mismatch, and the mismatch is the number a ratio circuit should buy against. A divider built from 25 ppm/K parts that track each other to 5 ppm/K drifts like one built from 5 ppm/K parts, at a fraction of the price; an array whose elements track to 2 ppm/K drifts by two parts per million per kelvin of swing, whatever its absolute coefficient. The absolute coefficient is what a current-setting resistor or a series chain sees; the tracking coefficient is what a gain, a ratio or a bridge sees. Two different numbers on one data sheet, for two different kinds of circuit.

What the two laws mean for a design #

The practical content is a sorting rule, applied to every place a resistor’s value enters the answer.

Where the answer is a ratio, buy correlation. A gain set by two resistors, a divider, a bridge’s arms, an attenuator’s steps: all are ratios, all cancel what their parts share, and all are improved by an array or by halves of one part far more than by tighter discrete parts. Their temperature coefficient is the parts’ mismatch, not their own.

Where the answer is an absolute value, correlation is no help and can hurt. A current set by a resistor against a reference voltage, a time constant against a crystal, a total resistance: all see the shared error in full, and parts from one reel do not average down against each other the way independent ones would.

And a circuit can be moved from the second list to the first. A current source whose current is set by the ratio of two resistors against a reference divided by the same ratio has turned an absolute into a ratio, and its temperature coefficient falls from the parts’ own to their mismatch. That is the sensitivity argument run backwards, as the earlier essay put it — a sensitivity removed by choosing a different network — and correlation is what makes the removal pay.

The three tolerances that do nothing found a ladder filter blind to whole directions of component variation, among them the direction in which every part moves together. The divider is the simplest case of the same fact: a ratio is blind to the common direction, and correlation concentrates the parts’ error in exactly the direction it cannot see.

A worked ratio: the difference amplifier #

The ratio circuit where correlation matters most is one whose whole purpose is a ratio match. A difference amplifier rejects a common-mode voltage only as well as its four resistors form two equal ratios, and the four resistors that decide measured one built from one-tenth-per-cent parts giving 54 decibels of rejection. Its rejection falls as the ratio mismatch grows, in proportion, so anything that shrinks the mismatch raises the rejection by the same factor in decibels’ terms: twenty times the logarithm of the shrinkage.

The array result above says what shrinkage correlation buys. Parts that share 99.5 per cent of their error present, in a ratio, $\sqrt{0.005}$ of their tolerance — fourteen times less — so four such parts in a difference amplifier would add twenty-three decibels to the discrete parts’ rejection, taking one-tenth-per-cent-grade parts from 54 dB to about 77. That is why an integrated difference amplifier, whose four resistors are laser-trimmed on one substrate, reaches rejections that discrete parts of any tolerance a buyer can find do not: its resistors are not individually better, they are better together. The rejection the parts have found the same structure in an instrumentation amplifier’s active parts — two matched but individually mediocre amplifiers costing nothing — and this is the passive half of it.

Temperature makes the case stronger. A difference amplifier’s rejection over temperature depends on how its four resistors’ coefficients track, and four discrete parts that each drift by a hundred parts per million per kelvin, within ±25, lose their ratio match by up to the mismatch figure the figure above measured — 0.135 per cent over the industrial range, which is 1.4 times the whole tolerance of the tenth-per-cent parts. A difference amplifier trimmed at room temperature with discrete parts can lose most of its rejection at the ends of its range; one made from an array, whose coefficients track to a few parts per million, keeps it.

The same arithmetic in a chain of stages #

A multistage divider is a chain of ratios, and correlation between its stages behaves in two ways at once. The ladder built for the rest of itself found a chain of resistor dividers whose error sits in its front stage, and the stage that is wrong is the far one the case where it does not. Within a stage, parts that share their error cancel it from that stage’s ratio; between stages, a shared error moves every stage’s loading together, and whether it cancels there depends on whether the chain’s answer is a ratio of ratios or a product of absolute values. The two-law picture — one factor of $\sqrt{1-\rho}$ for what is a ratio and one of $\sqrt{\rho + (1-\rho)/n}$ for what is a sum — applies stage by stage, and a designer who knows which of the two each stage’s answer is knows which kind of part to buy for it.

How the numbers were obtained #

Each part’s fractional error is drawn as $\sigma(\sqrt\rho\,c + \sqrt{1-\rho}\,e)$ from a seeded generator, with $c$ one standard normal draw shared by the four parts in one copy and $e$ an independent standard normal draw for each, and $\sigma$ the standard deviation of a uniform 1% part, 0.577 per cent. The divider’s output is the lower two resistances over the four; the first three hundred builds are also solved on a netlist and must agree. The chain’s total is the mean of the four errors. Twenty thousand builds are drawn at each shared fraction; the laws $\tfrac12\sigma\sqrt{1-\rho}$ and $\sigma\sqrt{\rho + (1-\rho)/4}$ are checked against the measured spreads. The temperature runs draw each part’s initial value uniformly within 1% and its coefficient uniformly within 100 ± 25 ppm/K, and evaluate the ratio and the total at every five kelvin from −40 to 85 °C.

What it leaves out #

Correlation that is not uniform. Every pair of parts here shares the same fraction. On a real substrate neighbouring elements share more than distant ones, and a gradient across the array — hotter at one end — is a shared error with a direction, which a common-centroid layout cancels and a straight row does not.

Correlation of the shapes. The parts here are Gaussian-correlated uniform parts. A sorted population with correlation, or a shared error that is itself bimodal, would combine the earlier essay’s tails with this essay’s cancellation.

Self-heating. The temperature here is the ambient’s, shared by all four. A divider carrying current heats its parts by amounts that depend on their power, and the upper and lower pairs of a divider dissipate differently when their resistances differ — a temperature that is not shared, arriving through the circuit rather than the room.

Still open: gradients, self-heating, and the ratio that is not symmetric #

A gradient across an array. A temperature or process gradient moves each element by an amount proportional to its position, and a divider whose upper and lower halves are at different positions sees the gradient as a mismatch. Interleaving the halves — the common-centroid arrangement — cancels a linear gradient exactly and a quadratic one not at all, and how much of an array’s ratio tolerance a given gradient consumes is a question the correlated draw here could be extended to answer.

Self-heating in a ratio. The pulse that ends before the heat measured a shunt heating under current; in a divider the two halves heat unequally whenever their resistances or currents differ, and the ratio then drifts with the signal. That is a temperature error that correlation cannot cancel because the circuit itself makes it unshared.

Ratios that are not halves. A divider of ten to one has unequal sensitivities, and the shared error then cancels only in the ratio of the pair’s sums. Whether a correlated array still helps a highly unequal divider as much as this symmetric one is the next measurement, and it bears on every high-gain amplifier.