The sensor inside its own answer

Two millivolts a kelvin, and the wrong sign built the number everybody knows out of two rising quantities. The thermal voltage climbs by a third of a per cent per kelvin and the saturation current climbs far faster, and the drop at a fixed current is nVₜ·ln(I/Iₛ), so the second wins and the forward voltage falls at about two millivolts per kelvin.

That is a statement about a junction at a temperature. This rung asks what happens when the junction is the thing deciding the temperature, which it always is, and which nothing in that essay allowed for.

The dissipation at a fixed sense current is I·V, and V is a function of T. So the temperature is a fixed point of the same equation the two rungs below solve for a ferrite core and for an inrush thermistor:

T = Tₐ + Rₜₕ · I · V(T).

The sign, and what it rules out #

dV/dT is negative, so dP/dT is negative at a fixed current, so the loop gain Rₜₕ·dP/dT is negative at every temperature and every current on the range that matters.

A junction driven from a current source is therefore unconditionally stable, however badly it is mounted. There is no thermal resistance that removes the root; there is no ignition temperature; the disappearing-root failure that the loss that depends on what it causes draws for a wound component has no counterpart here.

Driven from a voltage source it is the opposite, and the contrast is the whole reason the distinction is drawn. At a fixed voltage the current rises with temperature, because the saturation current is climbing exponentially and nothing is holding the current down. Then dP/dT is positive and large, the loop gain crosses one at a thermal resistance a bench supply provides without effort, and the root disappears. That is thermal runaway in a junction, it is why a bare diode across a supply fails and a diode fed from a resistor does not, and it is the same disappearing root drawn one field over.

Two errors, and only one of them calibrates out #

Because the loop gain is negative and small, the interesting output is not stability. It is the two errors that a small negative loop gain produces in a measurement.

The offset is the junction sitting above ambient by Rₜₕ·P. At a microamp and six hundred kelvin per watt it is nought point two nine millikelvin, which is nothing. At ten milliamperes it is four point two six kelvin, which is not. A calibration removes it: measure the diode’s voltage at a known ambient, write down the constant, subtract it forever.

The gain error does not calibrate out, and it is the finding of this rung.

Raise the ambient by a kelvin. The junction warms, its forward voltage falls, its dissipation falls with it, and it therefore rises by slightly less than a kelvin. The factor is exactly

dTⱼ/dTₐ = 1/(1 − Rₜₕ·dP/dT),

which is below one because the loop gain is negative. The instrument reads a junction and is calibrated in ambient, so it under-reports every change by that factor.

At ten milliamperes into six hundred kelvin per watt the factor is 9,584 parts per million — just under one per cent. Calibrate the thing at twenty-five degrees and read it at eighty-five and it is out by five hundred and eighty-three millikelvin, from a device whose own law is exact and whose calibration was correct.

Why a subtraction cannot fix it #

A calibration at one point removes an offset. This is a gain error, and removing a gain error needs two points — which means two known ambients, which means a second reference, which is the thing the diode was there to avoid.

Worse, the factor is a property of the mounting, not of the device. Six hundred kelvin per watt is a small-signal diode with its leads in air; fifty is the same die soldered to a plane. A calibration certificate can carry a device’s own constants and cannot carry the thermal resistance of a board somebody has not built yet.

So the practical rule falls out of the arithmetic rather than out of judgement: make the sense current small enough that Rₜₕ·I·V is below the resolution wanted, and the whole problem disappears. At a microamp the offset is a third of a millikelvin and the gain error one and a half parts per million, on the same mounting where ten milliamperes costs a per cent.

The measurement that would show it, and why nobody takes it #

The gain error is invisible to every check anybody actually runs, and it is worth walking through why, because the same reasoning applies to a lot of instrumentation.

Put the diode in an oven and sweep the ambient. The voltage against ambient is a beautiful straight line — because the fixed point is very nearly linear over any range an oven covers — with a slope that is the true coefficient multiplied by the gain factor. A fit to that sweep returns one number, its residual is tiny, and the number is wrong by nine and a half thousand parts per million.

That is the same trap the constant that is a window records about a diode’s ideality factor, arriving from a completely different direction: the residual reports that the model describes the data, which it does, and says nothing about whether the coefficient the model returns is the quantity anybody wanted.

The measurement that would show it is a sweep at two sense currents. The true junction coefficient depends on current through the logarithm, in a known way; the gain factor depends on current through the dissipation, in a different known way. Two sweeps separate them, and the separation is the diagnostic — one sweep cannot, however carefully it is taken.

The coefficient moves too, for a different reason #

There is a second dependence on the sense current and it has no heat in it at all.

The drop is nVₜ·ln(I/Iₛ) and the closed-form derivative is (V − Eg/q − xₜᵢ·Vₜ)/T. Both V and the logarithm depend on the current, so dV/dT depends on the current: minus two point four zero three millivolts per kelvin at a microamp and minus one point six one three at ten milliamperes, a difference of nearly a third.

That is not self-heating. It is the same law read at two currents, and it means the celebrated “two millivolts a kelvin” is a number about a bias as well as about a junction. A thermometer calibrated at one sense current and run at another is out by the difference between two coefficients before any of the thermal argument above applies.

The two effects push the same way at high current — larger self-heating, smaller magnitude of coefficient — so they compound rather than cancel, and a measurement that fits a single line through a voltage-against-ambient sweep absorbs both into one apparent slope and reports it as the device’s.

What Rₜₕ actually is here, and why it is worse than it looks #

Every number above is multiplied by a thermal resistance, and there is a subtlety in which one.

For the ferrite and the thermistor the relevant resistance is from the part to the ambient, and it is what the fixed point is about. For a sensor the relevant resistance is from the sensing junction to the thing being measured, which is not the same path and is often much worse.

A diode measuring the temperature of a heatsink it is bolted to has a short path to the heatsink and a long one to the air, so its self-heating mostly escapes into the thing it is measuring and the gain error is small. A diode measuring the temperature of the air has the opposite arrangement: its self-heating has nowhere to go but into the air immediately around it, which is the quantity being measured, so the sensor warms its own sample.

That second case is the one where a small dissipation is expensive, and it is the case a free-standing sensor is always in. It is also the case where the thermal resistance is least predictable, because still air is not still and a millimetre per second of draught changes it by a factor.

What this shares with the two rungs below #

Three parts, three signs, one equation.

A ferrite core has a negative loop gain over a hundred and fifty kelvin and then a wall, because its flux ceiling closes and the magnetising current explodes. Its failure is a root that disappears.

An inrush thermistor has a negative loop gain everywhere and no wall, so it always settles — and the protection that is gone by the second time is about the fact that where it settles is somewhere it is no longer doing its job.

A junction at a fixed current also has a negative loop gain everywhere, and its loop gain is tiny — parts per million to parts per hundred rather than order one. It never misbehaves. What it does instead is quietly scale a measurement, which is the failure mode that survives longest, because nothing about it looks wrong.

The transistor version, which is better and for a stated reason #

Nobody sensitive to any of this uses a plain diode. The standard arrangement is a transistor’s base-emitter junction run at two currents in a known ratio, and the quantity measured is the difference between the two forward voltages:

ΔV(be) = n·Vₜ·ln®,

with r the current ratio. Every device constant has left the expression. The saturation current is gone, the bandgap is gone, and what is left is a thermal voltage times the logarithm of a ratio — which is proportional to absolute temperature, with a coefficient that is a pure number.

What has not left the expression is this rung. The two currents dissipate differently, so the junction is at two temperatures during the two measurements, and the difference measured is between a junction at T₁ and a junction at T₂ rather than between two currents at one temperature. The error is Rₜₕ·(I₂ − I₁)·V, and it enters ΔV(be) directly rather than being divided by anything.

The usual defence is to keep the ratio modest and both currents small, which is the same conclusion this whole page reaches: the fix is a smaller current, not a cleverer arrangement.

Where the number belongs in practice #

A silicon junction is one of the two commonest ways to put a temperature into a circuit, and the one with the fewest parts — a transistor’s base-emitter junction on the same die as everything else, run at a known current, is a thermometer that costs nothing and needs no package.

That is exactly the configuration in which the sense current is chosen for convenience rather than for this, and in which the thermal resistance from the sensing junction to the thing being sensed is a layout question nobody wrote down.

The copy, and its two errors makes the neighbouring point about a current mirror: two junctions on one die at one temperature is the assumption the whole device rests on, and the temperature is one only if neither of them is doing enough work to have its own. What matching does about temperature prices the mismatch case.

The number worth carrying #

Nine thousand five hundred parts per million, which is the gain error at ten milliamperes on a badly mounted small-signal diode, and one and a half, which is the gain error at a microamp on the same part.

The ratio between them is the sense current, entering the arithmetic twice — once through the dissipation and once through the logarithm — and it is entirely under the designer’s control. That is the useful shape of the result: an error four orders of magnitude wide, decided by one number that is usually chosen for a reason having nothing to do with it.

The reason it gets chosen badly is worth stating, because it is not carelessness. A larger sense current gives a larger signal, a better signal-to-noise ratio against the amplifier reading it, and a forward voltage far enough above the leakage region that the ideality factor is well behaved — which is the argument the constant that is a window makes in detail. Every one of those pressures pushes the current up, and the one pressure pushing it down is invisible unless somebody has drawn this page.

The floor a circuit has is the shape of the trade: a quantity with an optimum in the middle rather than a monotone preference, where two mechanisms of opposite sign meet. Here the optimum exists, it is computable from the mounting and the amplifier’s own noise, and the usual practice of picking a round number and moving on lands somewhere on the curve rather than at its bottom.

An instrument inside its own measurement, in four places #

A thermometer warmer than what it is measuring is one instance of a shape this collection keeps finding, and the four together are worth setting beside each other because the remedies differ.

The probe is part of the circuit is the electrical version: a reading is two solves, the circuit and the circuit-with-the-instrument, and the quantity drawn is the difference — one per cent wrong at 6.8 kHz for an ordinary probe on a two-kilohm source. The remedy is a smaller instrument, and it is available.

Two terminals measure the leads as well is the version where the instrument’s own connection is the error: a reading that is the resistance plus the wire, one per cent high at ten ohms and a hundred per cent high at a tenth. The remedy is four wires, and it is exact.

The degrees a thermocouple cannot see is the version where the instrument is in the right place and the object has more than one temperature — 2.59 kelvin between a core’s centre and its surface in still air, growing to 3.37 on a cold plate. There is no remedy; there is a decision about which temperature the specification is about.

And this one is the version with a loop in it, which is why the error is a gain rather than an offset. The sense current’s dissipation falls as the reading rises, so a calibration at one temperature cannot remove it — 583 millikelvin at 85 degrees on a part calibrated at 25 — and the only remedies available are a smaller sense current, which costs signal, or a better mounting, which is the quantity the thermometer is embedded in.

The 9 584 parts per million is the number worth carrying rather than the 4.26 kelvin, and the reason is the distinction this collection keeps returning to. An offset is removable by measurement — read the sensor at a known temperature, subtract, and it is gone. A gain error is removable only by a second calibration point, and this one is a gain error: the self-heating falls as the ambient rises, so the sensor under-reports every change rather than every reading. A single-point calibration leaves it entirely, and a two-point calibration removes it only over the interval between the two points.

That is the same structure the offset that knows the signal finds in a switched capacitor, where 5.67 mV of what a data sheet would call an offset moves by 1.86 mV across a volt of signal — a gain error of 0.186 per cent and only 3.4 µV of anything else. In both cases the quantity is named as an offset because that is what it looks like at one operating point, and in both cases the operating point is the thing it depends on. The tell is the same in both: sweep the quantity the specification says the error does not depend on, and see whether it moves.