What matching does about temperature

The previous rung measured a diode’s drop falling at 1.828 mV/K and made the comparison that gives the number its weight: it is larger than every amplitude boundary the semiconductor field has measured, put together. A circuit that has to hold a millivolt cannot hold it across a room that changes temperature.

The answer is not a better junction. It is two of them.

What a pair cancels built the differential pair and measured what it removes: every even harmonic of the drive, exactly, because the two halves’ nonlinearities are odd-symmetric about the balance point. This essay measures what it does to the other problem, and the structure of the answer is the same — a first-order cancellation that is exact by symmetry, and a second-order residue that is the whole design problem.

The cancellation, and why it is exact #

Both halves of a pair are on the same die, at the same temperature, carrying the same current. The quantity that matters at the input is the difference of their two base-emitter drops, and the difference of two equal things does not move — whatever the drops individually do with temperature, they do it together.

That is the whole first-order argument, and its exactness is worth pausing on. The 1.828 mV/K of the previous rung is not reduced by the pair; it is common to both halves and therefore appears at the output as a common-mode shift rather than as a differential one. The cancellation does not depend on knowing the coefficient, on the coefficient being linear, or on the junctions obeying any particular law. It depends only on the two halves being the same and being at the same temperature.

Which is where it stops being exact, and the rest of the essay is about the two ways it stops.

The residue is proportional to itself #

The halves are never identical. The mismatch that matters here is in the saturation current — a difference in junction area, in doping, in surface condition — and its effect on the input is strikingly simple.

To carry the same current through a junction whose saturation current is m times larger, the drop must be smaller by exactly the amount the exponential needs:

$$V_{os} = V_T \ln m$$

That is the input offset. A five per cent mismatch gives $V_T \ln 1.05$ = 1.2613 mV at 300 K, and the figure’s slider covers half a per cent to twenty per cent, which is 0.129 mV to 4.713 mV.

The consequence is the essay. $V_T$ is proportional to absolute temperature, and so, therefore, is the offset. An offset that is proportional to absolute temperature has a drift that is the offset divided by the temperature:

$$\frac{dV_{os}}{dT} = \frac{V_{os}}{T}$$

and the right-hand side contains no device parameter, no mismatch, no material, no band gap. It is one over the absolute temperature.

At 300 K that is 3 333 parts per million per kelvin, which the figure asserts at every position of the slider and which comes out identical to six digits at every one:

3.33 µV per kelvin for every millivolt of offset. It is one of the few genuinely universal numbers in analogue design, and it is derived here in one line and then measured on thirty-one solved operating points that know nothing about the derivation.

What that buys, which is the useful half #

The proportionality has a consequence that is not obvious from the algebra and is enormously useful in practice: trimming the offset to zero at one temperature trims the drift to zero with it.

If $V_{os}$ is proportional to $T$, then a trim that makes it zero at 300 K has made the constant of proportionality zero, and a quantity proportional to $T$ with a proportionality constant of zero is zero at every temperature. No separate adjustment for drift exists, is needed, or would have anything to act on.

This is why offset trimming works as well as it does, and why a datasheet quotes offset and drift as two numbers that are not independent. It is also the reason the ratio in the table above is the important column rather than the drift column: a part with a large offset and a proportionally large drift is a part that trims well, and a part whose drift is not $V_{os}/T$ has a mechanism in it that this model does not contain.

Against a single junction #

The comparison that puts the size of this in context is with the thing the pair replaced.

One junction drifts at 1.828 mV/K. A five per cent mismatched pair drifts at 4.204 µV/K. The pair is 435 times better, and the factor is not a constant — it is the single junction’s slope divided by $V_{os}/T$, so it shrinks as the mismatch grows: 4 254 times at half a per cent, 435 at five per cent, 116 at twenty.

That is worth stating as a design statement rather than as a ratio. Matching buys a factor of hundreds, and how many hundreds is decided entirely by how well the two devices are matched. A pair laid out carelessly on a die is not a pair; it is two junctions that happen to be near each other, and the whole benefit is in the layout.

Two routes, and why the agreement is not circular #

The figure computes the drift twice and the two are asserted equal to a part in a million, which is tight enough to be worth saying what the two are.

The measured route solves the pair at thirty-one temperatures. At each one it finds the input voltage that balances the two collector currents, which is a root of the difference of two exponentials with different saturation currents and different thermal voltages — the same Newton machinery the bias-point essay established, applied to a two-device network. It then fits a straight line through the thirty-one offsets.

The closed route evaluates $V_{os}/T$ at the midpoint of the range, from the offset at that temperature and nothing else.

The two share the mismatch and the temperature range. They do not share the balance condition, the iteration, the fit, or any expression for how an offset depends on temperature — the second route never differentiates anything. That is what makes the agreement a check: a wrong sign in the temperature dependence of the saturation current, a factor in the thermal voltage, or a mistake in the balance condition would move one and not the other.

It is also the reason the figure’s dashed line is drawn through absolute zero rather than fitted through the data. A fitted line is a weak claim that almost any smooth curve satisfies over 150 K. A line through the origin is the actual assertion — proportionality to absolute temperature — and a reader can see the data sitting on it or not.

The second way the cancellation fails #

Everything above assumes the two halves are at the same temperature. On a die they very nearly are, and “very nearly” is doing real work.

A pair on a die is a few tens of microns apart, with silicon between them, and silicon conducts heat well — so a thermal gradient across a pair is small. But the offset it produces is not divided down by anything. A gradient of a hundredth of a kelvin across the pair puts 18 µV of differential drop across it directly, at the full 1.828 mV/K of a single junction, because a gradient is precisely the condition under which the two halves are not doing the same thing.

That is larger than the drift of a well-matched pair, and it has a different signature: it follows whatever is dissipating power on the die rather than following the ambient. An amplifier whose output stage heats one side of its own input pair has a thermal feedback path from output to input with a lag of milliseconds, and the resulting distortion is the sort that shows up at low frequencies and disappears when the signal is small.

The site’s usual construction cannot reach this. It would need a thermal model beside the electrical one — a heat flow, a thermal resistance, a temperature that is a solved variable rather than a parameter — and this collection’s solver takes temperature as an input at every point. So the boundary is stated rather than measured, and it is stated with the number that makes it matter: a hundredth of a kelvin of gradient outweighs the drift of a pair matched to half a per cent.

What five per cent of mismatch actually is #

The slider is labelled in saturation-current ratio, which is the parameter the model has, and it is worth translating into the things a fabricator controls — because the translation explains why the numbers on the slider are the numbers they are.

Saturation current is proportional to junction area. Two devices drawn identically and etched identically differ in area by whatever the lithography and etch leave behind, which for a device a few microns across is a fraction of a per cent. Two devices drawn differently — one twice the width of the other, say — differ by whatever the ratio was meant to be, plus that same fractional error. So half a per cent on the slider is roughly what a careful common-centroid layout of matched geometries achieves, one to two per cent is an ordinary layout, and twenty per cent is two devices that were never intended as a pair.

The logarithm is what makes this tolerable. A five per cent area error is a five per cent error in saturation current and $\ln 1.05 = 0.0488$ of a thermal voltage — 1.26 mV. A hundred per cent error, a factor of two, is $\ln 2 = 0.693$, or 17.9 mV. Doubling the area mismatch does not double the offset; it adds $V_T \ln 2$ to it. That compression is why a differential pair is usable at all with ordinary tolerances, and it is the same logarithm that made the bias-point essay’s decade slope a constant.

It also explains a fact that surprises people meeting the arithmetic for the first time: a pair deliberately built with a 10:1 area ratio — which is how a bandgap reference generates its proportional-to-absolute-temperature term — has an intentional “offset” of $V_T \ln 10$ = 59.5 mV at 300 K, drifting at 198 µV/K. That is the signal in that circuit, it is fifty times the drift of the diode drop it is cancelling, and it is exactly the number the previous rung measured as the decade slope. The same expression is a defect here and a reference there, distinguished only by whether the ratio was chosen.

Why this rung sits on the pair and not on the temperature #

A word about where this essay is filed, because the choice says something about how the two ladders in this field relate.

It could have been the second rung of the temperature anchor: a diode’s drift, then a pair’s. It is instead the third rung of the differential pair, and the reason is that the argument is about the pair. The temperature dependence is the thing being acted on, and it arrived complete from the rung below — 1.828 mV/K, measured, two routes, done. What this essay adds is not a better measurement of that quantity but a structure that removes it, and the structure is the differential pair, which by this rung has already had two essays written about what its symmetry does.

That is also why the closing comparison is with what a pair cancels rather than with the diode. The two essays are the same argument applied to different quantities, and reading them together is the point of a ladder rather than a bibliography.

Where the model gives out #

Three other things are outside what this figure computes, and each is a real term in a real part.

Base current mismatch. A bipolar pair’s input current is not zero, and a mismatch in current gain puts a differential current through whatever source resistances are present. That produces an offset which depends on the external circuit rather than on the pair, and whose temperature coefficient follows current gain rather than $V_T$ — so it is not proportional to absolute temperature and it does not trim away.

The resistors. A pair is loaded, usually by two resistors, and a mismatch between them produces an offset with the resistors’ own temperature behaviour. Matched resistors on a die track well; resistors of different types do not track at all.

Stress. Packaging puts mechanical stress on a die, silicon is piezoresistive, and the stress changes with temperature because the package and the die have different expansion coefficients. This is the mechanism behind most of the drift that remains after everything above has been accounted for, and it is the reason a precision part’s drift specification depends on its package.

The common feature of all three is that they are not proportional to absolute temperature, and so they are exactly the part of the drift that a room-temperature trim leaves behind. The clean result in this essay is the reason the residue is identifiable at all.

What the ladder now says, from the bottom up #

This anchor has three rungs and they compose into one statement, which is worth assembling because no single essay makes it.

Rung one built the pair and measured what its symmetry removes: every even harmonic of the drive, exactly, leaving a leading distortion term that is third order rather than second. The consequence there was an ordering — a pair’s gain gives way before its distortion does, at 10.4 mV against 18.2 mV, and only 1.75 times apart, where an exponential’s two boundaries are 7.06 times apart in the other order.

Rung two, in the temperature ladder beside this one, measured what one junction does across 150 K: 1.828 mV/K, larger than every amplitude in the field.

Rung three is this essay: the pair removes the second of those by the same symmetry that removed the first, and what survives is again second order — $V_{os}/T$ rather than $dV/dT$, 4.2 µV/K rather than 1.8 mV/K, and proportional to a mismatch that layout controls rather than to a physical constant that nothing does.

The shape is the same both times, and it is the reason differential circuits are the default in precision analogue rather than a technique for particular problems. A symmetry that is exact removes a first-order term entirely; what is left is a product of two small things, and small things multiply down faster than they add.

The rule this model stops being true under #

3 333 parts per million per kelvin, and 0.01 K.

The first is what a matched pair leaves behind, exactly, at every mismatch: an offset proportional to absolute temperature, drifting in proportion to itself, and vanishing entirely when it is trimmed. The second is the gradient across the pair at which that clean result is swamped by a mechanism this solver has no way to see — which is a hundredth of a degree, and is why the last decade of analogue precision is a thermal problem rather than an electrical one.

What a hundredth of a degree is, in the circuits that have to have it #

A gradient of ten millikelvin across two devices a few tens of micrometres apart sounds like a requirement nothing could fail, and two measurements in this collection say otherwise.

The sensor inside its own answer is the small-scale case: a junction driven from a current source runs 4.26 kelvin above ambient at ten milliamperes on a badly mounted part, entirely from its own dissipation. Ten millikelvin is a four-hundredth of that, so a pair carrying different currents — which is what a mirror with a deliberate ratio in it is — has a gradient two orders above the threshold from nothing but its own operation.

The degrees a thermocouple cannot see is the large-scale one and adds the part that is easy to get backwards: cooling a heat source better does not reduce the gradient inside it. A core on a cold plate has a larger centre-to-surface difference than one in still air — 3.37 kelvin against 2.59 — because the heat is being extracted from one face rather than from all of them. A die bonded hard to a heatsink on one edge is the same arrangement.

Which is why the 3 333 parts per million per kelvin measured here is the good news and the hundredth of a degree is the design problem. The electrical residue after matching is exact, small and predictable; the thermal residue is none of those, and it is set by where the power is dissipated on the die rather than by anything on the schematic.

The coefficient being cancelled is worth naming for scale. Two millivolts a kelvin, and the wrong sign measures a single junction’s drift at 1.828 mV/K — larger than every amplitude boundary the semiconductors field has — and this essay’s 3 333 parts per million per kelvin on an offset of a few millivolts is four hundred times smaller. That factor is what a matched pair is bought for, and it is why the edges that move with the room puts a ratio in a different class from a material constant: what survives matching is exactly the part of the coefficient that is a ratio.

What the pair is used for, and what it does not fix #

A pair whose residual drift is its offset divided by the absolute temperature is the foundation of three later arrangements. What a pair cancels, and what it only halves is the same symmetry read as a harmonic cancellation rather than as a drift. The copy, and its two errors is the mirror that depends on the same matching in the same way. The rejection the parts have is where matching between two amplifiers rather than two junctions sets a floor. Two millivolts a kelvin, and the wrong sign is the drift a single junction has and this arrangement removes, and The distortion a linear model cannot have is the error the matching does nothing about at all.