The bow a trim leaves

The voltage at which the slope is zero built a reference from two junctions and showed that its slope at the trim temperature is read off its voltage: every weighting of the sum leaves the band gap at absolute zero, so a reference flat at 300 K is the one that reads 1.1717 volts there, and a millivolt either side is 3.33 microvolts per kelvin of slope. That is the whole of the first-order problem, and a single trim at room temperature solves it.

What it leaves is the second-order problem. The sum it trimmed moved 2.53 millivolts over the automotive range, −40 to 125 °C, and the weight that made it flat at 300 K could not make it flat anywhere else. Two millivolts a kelvin had seen the same thing in one junction — the drop’s slope is itself a function of temperature, so a straight line through the middle of a sweep misses the ends — and said that in a reference it would be the dominant error once the slope was cancelled. This essay measures it: what shape it has, what sets its size, where the trim should go to make it least, what it costs in the units a datasheet uses, and how a third junction removes it.

Flat at one temperature and nowhere else #

A reference whose core current is proportional to absolute temperature, trimmed to zero slope at 300 K, reads 1.17170 volts there. Swept across the automotive range it falls on both sides.

The output is 1.39 millivolts low at −40 °C and 2.53 millivolts low at 125 °C. It never rises: the residue is negative on both sides of the trim, with zero slope at the trim itself, so the reference’s output over temperature is an arch with its top at 27 °C. Every member of the family of weights has an arch of the same depth; changing the weight only tilts it, which moves its top along the range.

The size depends on how the core’s current is made. Run at a constant current the same reference falls 3.79 millivolts by 125 °C; with a current rising as the square of temperature, 1.27; with one rising as the cube, 0.010 — two hundredths of the proportional core’s figure, and that hundredth is the diode law’s “+1” rather than anything structural. That last case is the clue to what the arch is.

A shape that belongs to nobody #

The drop of a junction carrying $I_0 (T/T_0)^\alpha$ through a saturation current that goes as $T^{x_{ti}} e^{-E_g/kT}$ can be written exactly. Everything in it is either constant, proportional to $T$, or one term:

$$V(T) = \frac{E_g}{q} + \frac{T}{T_0}\left[V(T_0) - \frac{E_g}{q}\right] - (x_{ti} - \alpha)\,V_T\ln\frac{T}{T_0}.$$

The difference term the reference adds is proportional to $T$, so the weight can cancel the second term’s slope and nothing else. The third term has a $T\ln T$ in it, which no multiple of $T$ matches. With $x = T/T_0$, what is left once the slope at $T_0$ is zero is

$$\Delta V_\text{ref} = -(x_{ti} - \alpha)\,V_T(T_0)\,\bigl[x\ln x - (x - 1)\bigr].$$

That has two numbers in front and one function behind. The function is zero with zero slope at $x = 1$, positive on both sides, and its second derivative is $1/x$: the arch is steeper on the hot side than on the cold, which the figure above shows as 2.53 millivolts against 1.39 for a range that extends 98 kelvin above the trim and 67 below it.

Six references built differently — three laws of bias current, three exponents, currents from a hundred microamps to a milliamp — have bows from 1.25 to 3.76 millivolts at 125 °C. Divided by $(x_{ti} - \alpha)$ thermal voltages, every one of them lands on the same curve: 0.0485 at 125 °C and 0.0269 at −40 °C, to within one and a half per cent at the hottest point. The residue that is left is the “+1” again, which at a hundred microamps and 125 °C adds a hundredth of the bow and at ten microamps adds a twelfth — the same leakage edge the difference term had, appearing in the second-order budget an order of magnitude earlier because the second-order budget is that much smaller. Two currents with one name is the warning that the real leakage is larger still: the current a junction actually leaks is generated in its depletion region and does not follow the forward law’s saturation current at all.

So the bow’s size is two numbers and its shape is no number at all. The exponent $x_{ti}$ belongs to the silicon, $\alpha$ to the circuit, and $x\ln x - (x - 1)$ to the logarithm. A designer who knows the bias law and the exponent knows the bow before building anything, and one who changes the bias law from constant to proportional has removed a third of it without touching the trim. A current proportional to absolute temperature is also what the ripple that is a temperature gave a filter’s tail to hold its transconductance still; the same bias law serves both, for different reasons.

Where the zero belongs #

The trim makes the slope zero at the temperature the trim is done at, which on a production line is whatever temperature the tester sits at. It need not be. The weight can be set from the voltage that would make the slope zero at any temperature, and since the arch is lopsided, room temperature is not where it is least.

Flattened at 300 K the box — the height between the highest and lowest output over the range — is 2.53 millivolts. Flattened at the middle of the range it is 2.02. The least is 1.91 millivolts with the zero at 39 °C, a little below the middle, and it is the setting at which the outputs at −40 and 125 °C come out level: the arch is tilted until its two feet are at the same height, and its depth from there to the top is the whole box. Putting the zero at either end of the range costs 8.1 and 7.2 millivolts, four times the best.

The rule that falls out is about the trim, not the circuit. A reference meant for a range should be flattened where the ends of that range are level, which for an arch steeper on the hot side is below the middle; trimming to zero slope at room temperature, which is what a single-temperature trim to the zero-slope voltage does, costs a third more box than necessary over the automotive range. The cure needs no second temperature. The trim target is a voltage, and the voltage that flattens the reference at 39 °C, read at 27 °C, is a number the bow’s closed form supplies: the zero-slope voltage at 312 K, carried back along the tilted arch to 300 K.

The box per kelvin #

Datasheets quote this bow as a temperature coefficient by the box method: the height of the box, divided by the output, divided by the width of the range. The result is a number in parts per million per kelvin that is not the slope anywhere — the slope is zero at the trim and steepest at the ends — and that grows with the range quoted over, since the box grows as the square of the range and the divisor as its first power.

For a core biased proportional to absolute temperature, perfectly trimmed at its best temperature, the bow alone is 4.22 parts per million per kelvin over 0 to 70 °C, 7.90 over −40 to 85, 9.79 over −40 to 125 and 11.02 over −55 to 125. A constant-current core bows exactly half as much again in millivolts over every range — the ratio of $x_{ti} - \alpha$ for the two, three against two — and 1.47 times as much per million, because its output is 26 millivolts higher. Nothing about the trim, the area ratio or the resistors enters these numbers: they are the floor for this construction with an exponent of 3, and a part with a lower coefficient over a wide range has removed the bow by some further means.

That is worth having as a number because it reorders a common comparison. A reference specified at 10 ppm/K over the automotive range and one at 5 ppm/K over the commercial range are, on this arithmetic, the same circuit trimmed equally well; the first is not worse, it is quoted over a range two and a half times wider. Read per kelvin, a box coefficient compares parts sold into the same range and nothing else. The edges that move with the room made the same point about the site’s own boundaries: a coefficient quoted without its range has dropped the number that decides it.

What the bow is worth to a converter #

A reference exists to be divided into steps, so the bow is most honestly priced in the steps of the converter that uses it. Take the reference as a converter’s full scale and give it the best trim measured above, 1.91 millivolts of box over −40 to 125 °C on 1.1717 volts — 1,630 parts per million of full scale across the range.

A twelve-bit converter’s step is 1.1717 volts over 4,096, 286 microvolts — the step the floor a converter sets compares against the noise in front of it — so the bow is 6.7 steps: a reading taken at 27 °C and repeated at 125 °C, of the same input, comes back several codes apart with every other part of the converter perfect. At ten bits it is 1.7 steps and at sixteen it is 107. The ten-bit converter does not need to know the reference has a bow; the twelve-bit one does, and it is the reason a twelve-bit system with an internal reference and no calibration is specified for its absolute accuracy at a fraction of its resolution.

Over the commercial range the same reference bows about 0.35 millivolts, 1.2 twelve-bit steps, which is why a part that is accurate enough on a bench is not accurate enough under a bonnet. The two ranges differ by a factor of 2.4 in width and a factor of 5.5 in bow, because the bow grows as the square of the excursion from the trim, and the automotive range’s extra 55 kelvin are almost all on the steep hot side.

The first-order error measured in the previous essay sits in the same budget and is usually larger. A tenth of a unit of uncertainty in $x_{ti}$ costs 8.6 microvolts per kelvin of slope after a voltage trim, and over the 98 kelvin from 27 to 125 °C that is 0.84 millivolts — nearly half the bow, linear rather than curved, and opposite in sign on the two sides of the trim. A designer budgeting a twelve-bit system adds the two, and the order in which they are attacked follows from the arithmetic: characterising the exponent narrows both, while a curvature correction narrows only the second and inherits the first.

The comparison with the converter also says when none of this matters. A reference used ratiometrically — the same reference driving a bridge and setting the converter’s full scale — cancels its own bow, since the input and the step move together. The two leads nobody counts uses that arrangement for a bridge’s supply noise, and it is the reason a strain or temperature measurement made ratiometrically can use a reference an absolute measurement would reject.

A third junction carries the bow’s shape #

The bow is $-(x_{ti} - \alpha)V_T\ln(T/T_0)$ with its slope removed. A quantity with exactly that shape is available from one junction: run it at the core’s current and at a constant current, and the difference between the two drops is

$$V(I_0 x^\alpha) - V(I_0) = \alpha\,V_T\ln x,$$

in which the saturation current has cancelled for the same reason it cancelled in the difference of two areas. Added to the reference with a weight of $(x_{ti} - \alpha)/\alpha$, and with the main weight re-solved for zero slope, it subtracts the bow term for term.

With the third term weighted 2 for a core whose current is proportional to absolute temperature, the bow falls from 2.53 millivolts to 18 microvolts — the “+1” once more — and the output settles at 1.12000 volts. That is the band gap the model was given. With the bow gone there is nothing left of the correction $(x_{ti} - \alpha)V_T$ that put the uncorrected reference at 1.1717: the reference’s output is the extrapolated band gap itself, and the circuit’s name, which was approximately true of the two-term version, is exact of the three-term one within this model.

Real curvature-corrected references use variants of this: a junction biased by a current with a different temperature law, a ratio of two such currents, or a resistor with a nonlinear temperature coefficient chosen to approximate the logarithm. Each supplies a term in $T\ln T$ or something close to it, and each has the same weakness, which the next figure measures.

A correction is only as good as the exponent #

The correction’s weight, $(x_{ti} - \alpha)/\alpha$, contains the silicon’s exponent, and the exponent is known to a few tenths at best.

Weighted for an exponent of 3 and built from junctions whose exponent is 3.5, the corrected reference bows 0.65 millivolts; at 2.5, 0.61. The uncorrected bow at those exponents would have been 3.16 and 1.90 millivolts. Half a unit of error in the exponent leaves a quarter of the uncorrected PTAT bow — the correction removes the bow in proportion to how well $x_{ti}$ is known, because the residue is $(x_{ti,\text{true}} - x_{ti,\text{assumed}})$ thermal voltages times the same shape. The correction never makes the reference worse anywhere in the range drawn, and it never makes it better than the designer’s knowledge of one device parameter.

The same parameter limited the first-order trim in the previous essay: a reference trimmed to the zero-slope voltage for an exponent of 3 drifts 86.2 microvolts per kelvin for every unit the exponent is wrong. So both the first-order and the second-order corrections rest on one number that is a property of the process and is measured by characterising parts over temperature. A design with curvature correction has not escaped the oven; it has moved its result one order of smallness further down.

What the model omits, and which way each omission pushes #

The junctions here are the diode law with a constant band gap, an exponent of 3 on the saturation current and an ideality of one, evaluated in logarithms so that it holds across the whole range. Every bow is a sweep at half-kelvin steps of a reference whose weight was solved by differencing at the trim temperature; the closed form is checked against the sweep to within the “+1”, and the six references of the collapse are checked against the shape to one and a half per cent.

Three omissions matter more than the others. The band gap narrows with temperature in real silicon, by about a quarter of a millielectronvolt per kelvin near room temperature and less at low temperature, which adds a curvature of its own with a sign that adds to this bow; measured references show bows of the same order but larger, and their curvature corrections are fitted rather than computed. The resistors drift, and since the weight is a ratio of two resistors the ratio’s own temperature coefficient enters the slope linearly — a voltage trim at one temperature absorbs its value but not its drift. And the transistors are not diodes: their base currents flow through the resistors and their Early effect moves the current ratio with the supply, both of which add terms with temperature laws of their own, as the trim that takes the drift with it found for a pair’s base currents.

None of those changes the structure measured here: a first-order cancellation that is exact at one temperature, a residue in $T\ln T$ whose size is the difference between two exponents, a best trim temperature below the middle of a range, and a correction that is as good as the exponent it assumes.

Still open: the band gap’s own bend, the supply, and the package #

The band gap that moves. Silicon’s band gap follows a law with a quadratic term near room temperature and a flattening below 150 K. Putting that law into the saturation current would make the bow’s shape a sum of two functions rather than one, and would say how much of a measured reference’s curvature is the exponent this essay used and how much is the gap itself — which decides whether a correction built on a third junction can ever be exact for real silicon.

The bow under a moving supply. A reference’s core current is set through transistors with a finite output resistance, so the current ratio and the weight both move a little with the supply voltage. A supply that is itself a function of temperature therefore adds a slope and a curvature the trim never saw, and the size of that term against the 1.9 millivolts measured here is the line-regulation specification read as a temperature coefficient.

Stress, which looks like temperature. Packaging stresses a die as the moulding compound cools and as it expands with heat, and junction drops move with mechanical stress by a fraction of a millivolt. The stress is a function of temperature with a hysteresis in it, so it adds a bow that depends on the direction the part was swept — the one term in a reference’s temperature behaviour that a single sweep cannot separate from the curvature measured here.