The edges that move with the room

Every model has an edge is the argument this whole field exists for: a model’s range is a measurement rather than a matter of taste, and it is quoted with the model or the model is quoted without it. The edge that is a region sharpened it — a boundary is a band and not a line, and the width of the band is itself computable.

Both of those essays, and every boundary either of them measured, were computed at one temperature. So was every other edge in this collection. The condition is real, it was never stated, and this rung is about how much it is worth.

The one that moves exactly #

The cleanest case is the small-signal edge, and it is clean because the dependence is a theorem rather than a fit.

How small is small signal computed the amplitude at which a transconductance is one per cent optimistic: drive an exponential with a sinusoid of amplitude v̂ and the fundamental is $2 I_1(\hat v / V_T) / (\hat v / V_T)$ times the small-signal answer, and setting that to 1.01 gives the boundary. The whole expression depends on the amplitude only through $\hat v / V_T$.

So the edge is proportional to the thermal voltage, which is proportional to absolute temperature, and nothing else about the device enters at all.

At minus forty degrees the edge is 5.67 millivolts. At plus one hundred and twenty-five it is 9.69. The ratio is 1.709, and the ratio of the absolute temperatures is 398.15/233.15 = 1.708. They agree because they are the same statement.

That is a factor of 1.71 on a boundary this collection has been quoting as a single number, and the direction is the unhelpful one: a circuit qualified for its linearity on a cold bench has less margin than the measurement said, because the edge it was compared against moves up with the room.

The one that does not move at all #

The interesting cases are the ones where the dependence cancels, and there is a clean example in the filter field.

A Sallen–Key section’s designed Q is a ratio of capacitances. A tempco every capacitor on the board shares moves the pole frequency and leaves the ratio alone, so the section itself contributes nothing to a Q drift. What drifts is the amplifier: the realised Q exceeds the designed Q by an amount set by the gain-bandwidth product, and the gain-bandwidth product is a transconductance over a capacitance.

If the tail current is set by a resistor, the transconductance is that current over the thermal voltage, so it falls as one over absolute temperature. The Q error rises in proportion: 1.544 per cent at minus forty, 2.577 at plus one hundred and twenty-five, a factor of 1.669 against the 1.708 the temperatures alone give.

If the tail current is instead made proportional to absolute temperature — which is one of the two things a bandgap circuit is for — the two temperatures cancel exactly and the gain-bandwidth product is flat. The Q does not move.

That is the shape worth carrying away. A temperature dependence in a boundary is not a fact of nature to be tolerated; it is usually the ratio of two quantities, and making the two track is a design choice available at the cost of a bias circuit.

Why a factor of 1.71 on a boundary is not a factor of 1.71 on anything else #

It is worth separating two things that a reader could reasonably conflate, because the consequence depends entirely on which one is meant.

The small-signal edge moves by 1.71. The error at a fixed drive does not move by 1.71; it moves by rather more, because the departure is quadratic in the drive amplitude near the edge. Driving a device with a fixed five millivolts of signal, the linear model’s error at minus forty is about twice what it is at plus one hundred and twenty-five — a device gets more linear as it warms, at a fixed drive, which is the opposite of the intuition that heat makes things worse.

That is a real and slightly counter-intuitive result and it falls straight out of the definition. The nonlinearity of an exponential is a function of $\hat v / V_T$; raising the temperature raises $V_T$; so a fixed $\hat v$ is a smaller argument and a smaller distortion.

What does get worse with temperature on the same device is everything the bias controls — the transconductance falls, the gain falls, the gain-bandwidth falls — which is the mechanism the Q section above measures. The two effects are opposite in sign and belong to different quantities, and lumping them into “performance degrades with temperature” loses both.

The one that moves for two reasons at once #

The diode drop is the awkward case, because two mechanisms move it and they are separable only if somebody separates them.

Two millivolts a kelvin has the first: at a fixed current the forward voltage falls by about two millivolts per kelvin, arrived at as the difference between a rising thermal voltage and a much faster rising saturation current.

The sensor inside its own answer has the second: the junction is not at ambient. At ten milliamperes into a badly mounted part it is four and a quarter kelvin above it, which is eight and a half millivolts of drop that belong to the mounting rather than to the room.

So the boundary that the constant-drop model’s error is measured against — the one current a constant is right at — is a function of the ambient through the first mechanism and of the layout through the second, and a single quoted crossover current carries neither condition.

The ones that barely move, and why that is worth stating #

Not every edge in this collection is temperature-sensitive, and saying which are not is as much a part of the answer as saying which are.

The lumped-element boundary is set by a physical length against a wavelength, and the wavelength depends on the dielectric constant of a board material. Common laminates move by a few per cent over the industrial range, so the boundary moves by a per cent or two — real, and two orders below the small-signal edge’s factor of 1.71.

A capacitor’s self-resonance is set by its capacitance and its lead inductance. The inductance is a geometry and does not care; the capacitance does, and for a class II ceramic it cares enormously — which the capacitance that is not one number measured against bias and which moves against temperature by a comparable amount.

So the boundaries divide into three kinds, and the division is more useful than any individual number. Those set by kT/q move as absolute temperature, exactly, and can be predicted without measuring anything. Those set by a ratio of two like quantities move hardly at all, and can sometimes be made not to move by construction. Those set by a component value move by whatever that component does, which has to be looked up.

The audit, and what it costs to run #

There is a mechanical version of this that is worth stating because it is cheap.

Take any boundary in this collection and look at the expression it is computed from. If a thermal voltage appears in it and nothing else temperature-dependent does, the boundary is proportional to absolute temperature and the factor across the industrial range is 1.708 — no measurement needed, no rebuild, no argument. That covers the small-signal edge, the amplitude edges of a differential pair, the diode’s per-decade slope, every intercept point derived from an exponential expansion, and the noise voltage of a resistance, which goes as the square root and therefore moves by 1.31.

If the expression is a ratio of two quantities of the same kind — two capacitances, two resistances on one die, two matched transistors — the boundary is first-order independent of temperature and what is left is a mismatch, which what matching does about temperature measures directly.

And if the expression contains a component value that is not in a ratio, the answer has to be looked up, and the looking up is the work. That is the third category and it is the smallest of the three, which is the encouraging part of this rung: most of the audit is arithmetic on expressions rather than measurement on circuits.

What this asks of every other essay #

The specific numbers matter less than the audit. Nearly two hundred essays in this collection quote a boundary, and the condition on almost all of them is “at 300 kelvin”, stated nowhere.

That is the same failure the constant that is a window records about a fitted ideality factor and the exponent nobody put in records about a Steinmetz exponent: a number extracted under a condition, used without it, and correct the whole time. The fix is not to recompute everything. It is to know which of the three kinds each boundary is, which is usually decidable by looking at what is in the expression rather than by running anything.

The range itself is a choice, and the choices are not close together #

Everything above is quoted across the industrial range, minus forty to plus one hundred and twenty-five, because that is the widest range a general-purpose part is specified over. The factor is 1.708 there and it is a different number everywhere else, and the differences are large enough to change conclusions.

The commercial range, nought to seventy degrees, gives 343.15/273.15 = 1.256. A laboratory that never leaves twenty to thirty degrees gives 1.034 — three per cent, which is genuinely negligible and is why the condition was never noticed. The automotive range up to a hundred and fifty gives 1.815, and a part qualified to two hundred gives 2.03.

So “does the temperature dependence of this boundary matter” has an answer between three per cent and a factor of two depending on a decision made elsewhere in the project, and the honest form of any boundary in this collection is the number, the temperature it was computed at, and the multiplier — which is the same three-part specification the constant that is a window argues for on a fitted coefficient, with the window being a temperature range instead of a current range.

What a phase of thermal essays leaves behind #

This rung sits at the end of three others that put a temperature into places that had none — a core that heats itself, an inrush limiter that stops limiting, and a junction inside its own measurement — and what those three have in common is that the temperature was a variable being solved for.

Here it is not. Nothing on this page is a fixed point; the ambient is handed in and the boundaries move with it. That is the easier and much commoner case, and separating the two is worth a sentence: a temperature is a fixed point when the circuit’s own dissipation moves it appreciably, and it is a parameter when it does not. A small-signal amplifier drawing a milliamp at a volt into a hundred kelvin per watt is a tenth of a kelvin above its surroundings, and every boundary in this essay can therefore be read straight off the ambient.

The test for which case one is in is the same arithmetic each time: form $R_{th}\,dP/dT$ and see whether it is small. Where it is, the temperature is an input. Where it is not, it is an unknown, and the essays above are about what happens then.

The number worth carrying #

1.708, which is 398.15 over 233.15, and which is the factor every kT/q boundary in this collection moves by across the industrial temperature range.

It is not an error and nothing here is wrong. It is a condition, it is the same condition on a large number of otherwise unrelated results, and it is exactly predictable — which makes it the cheapest kind of caveat to carry: one ratio, applied wherever the expression has a thermal voltage in it, and nowhere else.

The second number worth carrying is 1.669 against 1.708, from the Q section. Those two are close and they are not equal, and the gap is the part of the drift that is not the thermal voltage — a residue of a per cent or two which, on a quantity whose whole error budget is two and a half per cent, is not a rounding. A boundary that moves as absolute temperature to within a per cent is still a boundary that has to be measured before that last per cent can be claimed, and the difference between predicting a dependence and measuring one is exactly that per cent.

The boundaries that do not move, and why they are the interesting ones #

Almost every number in this collection has a temperature in it, and the exceptions are worth listing because each is exceptional for a reason rather than by luck.

A boundary in volt-seconds is the clearest: the quantity that belongs to a core is $N\cdot A_e\cdot B_{sat}$ and it has no frequency in it — but it has a temperature, since $B_{sat}$ falls as the core warms, so the exception is only to the frequency axis. The frequency a sample rate invents is a genuine one: half the sample rate is decided by a clock and by nothing else, so the boundary moves only as far as the clock does. And the staircase on the way out is the purest, since $20\log(2/\pi)$ at half the sample rate is arithmetic and could not depend on a temperature if anybody wanted it to.

What separates those three from everything else on this page is that none of them is a property of a material. A thermal voltage, a saturation current, a permeability, a resistivity and a transconductance all have temperatures in them because they are properties of matter at a temperature; a ratio of two clock edges does not. So the practical form of this essay’s rule is a question rather than a warning: is this boundary made of a material constant, or of a ratio? If the first, it moves with the room and the coefficient is worth computing. If the second, it moves only as far as whatever sets the ratio does — which for a capacitor ratio on one die, or a pair of matched transistors, is very much less than either quantity moves alone.

What matching does about temperature is where that second case is measured, and its answer is exact rather than approximate: a pair’s residual offset is the thermal voltage times the logarithm of the mismatch, proportional to absolute temperature and therefore drifting at 3 333 parts per million per kelvin at every mismatch, because the ratio contains nothing about the device at all.

And the ratio test has a failure mode of its own, which the ripple that is a temperature is the worked example of. A filter section’s quality factor is a capacitance ratio and its pole frequency a product of four passive values, so by the rule above it should barely move — and it moves, from 0.82 decibels of ripple at −40 °C to 1.05 at +125, on passives with no temperature coefficient at all. The temperature enters through the amplifier’s transconductance, which is not in either design expression, and it enters or does not depending on how the tail current is biased. So the question is not only whether a boundary is made of a ratio, but whether the ratio is the whole of what decides it — and the terms a design expression omits are exactly the terms nobody thinks to give a coefficient. Which is the practical form of this essay’s rule: the temperature coefficient of a quantity is not computed from the expression that defines it, but from every element that turned out to be in the answer.