The ripple that is a temperature

The Q the amplifier decides measures a quantity that a schematic says belongs to two capacitors and finds that it belongs to the amplifier: a Sallen–Key section built with a part whose gain–bandwidth is a hundred times the corner has a quality factor 1.97 per cent above the designed one, and a pole 1.97 per cent below it, and the number is the designed Q divided by the ratio.

The work that produced it ended with a list of what it had not done, whose last item was:

And no temperature anywhere in the eleven. It moves the ceramic’s Vₖ, the core’s saturation flux density, the amplifier’s gain-bandwidth (and therefore the realised Q), the switch’s threshold (and therefore the injected charge), and the board’s surface resistance. Five of the eleven arguments have a temperature coefficient that is not in the figures.

This is that item, for the third of the five. The interesting part is not that the answer moves; it is where the movement comes from, which turns out to be one design decision inside the amplifier that no data sheet describes as a decision.

Nothing in the filter has a temperature coefficient #

Start with what does not move, because it is more than it looks.

A Sallen–Key section’s quality factor, for the unity-gain arrangement this collection uses, is set by a ratio of capacitances. A temperature coefficient shared by both capacitors — which is what two parts of the same dielectric from the same reel have, to first order — cancels exactly in the ratio. The pole frequency does move, because it is a product of an RC and both terms have coefficients, but the Q does not.

So a filter built entirely from film capacitors and metal-film resistors, all with coefficients, has a Q that is temperature-independent to the extent that its parts match — and what matching does about temperature is the essay about how far that extent goes in a different field, and about the fact that matching buys the coefficient and not the absolute value.

Every temperature coefficient in what follows is therefore the amplifier’s.

Where the amplifier’s comes from, and the decision inside it #

A gain–bandwidth product is a transconductance over a compensation capacitance: ωₜ = gₘ over that capacitance. The capacitor is on a die and its coefficient is small. The transconductance is not small at all.

For a bipolar input pair, gₘ is the tail current over twice the thermal voltage, and the thermal voltage Vₜ = kT/q is proportional to absolute temperature. So what happens to the gain–bandwidth depends entirely on what happens to the tail current, and there are two ordinary answers:

• a tail current set by a resistor from the supply is roughly constant, so gₘ falls as 1/T and the gain–bandwidth falls with temperature;
• a tail current proportional to absolute temperature — which is what a bandgap reference and a ΔVᵦₑ circuit give — makes gₘ constant, and the gain–bandwidth does not move at all.

The second is standard practice in any part designed to be predictable, and it is standard practice because of this measurement rather than in spite of it. It is also invisible from outside: two parts with the same gain–bandwidth at 25 °C and the same package can differ by nearly a factor of two at the ends of their range, and the row that would say which is usually not on the data sheet.

What it does to a section #

A section designed for Q = 2 with a part that is a hundred times the corner at 300 K comes out 1.544 per cent high at −40 °C and 2.577 per cent high at +125 °C with the fixed tail current. The ratio between the two is 1.669, against the 1.708 that the ratio of absolute temperatures gives.

The three per cent it falls short by is worth a sentence rather than a fudge factor. The rung below’s identity — the fractional Q error is the designed Q over the gain–bandwidth ratio — is an asymptote, and the cold end sits at a ratio of 129 while the hot end sits at 75, so the cold end is further into the asymptote and its error is proportionately a little smaller than the simple law predicts. The departure is in the right direction and of the right size, which is a check on both statements rather than a discrepancy in either.

With the proportional tail current the same section is flat to six decimal places across the whole range. Not approximately flat: the two temperatures cancel algebraically and what is left has no temperature in it.

And the rung below’s other result survives intact. The pole frequency falls by what the Q rises by, at every temperature, to within three per cent of the error itself — so the identity is temperature-independent even though both of its terms are not.

What it does to a filter #

A quality factor two per cent out is not a specification anybody writes. A passband ripple is.

A fifth-order 0.5 dB Chebyshev built from three sections, with a part whose gain–bandwidth at 300 K is three hundred times the corner, has

• 0.824 dB of ripple at −40 °C,
• 0.916 dB at 27 °C,
• 1.049 dB at +125 °C,

against the 0.500 it was designed for. It crosses a one-decibel specification at 88.6 °C.

That is a boundary in temperature, and this collection has not drawn one before. Every other edge here is a frequency, an amplitude or a size — the three nouns in the site’s own rule — and the reason is not that temperature is unimportant but that nothing had been measured against it.

The design that fails is not a marginal one. It meets its specification at room temperature with eight per cent to spare, its passives are perfect, its amplifier is three hundred times faster than the filter it is building, and it fails at a temperature an industrial part is sold to work at. What takes it there is one factor of 1.33 in a transconductance.

With the proportional tail current it is 0.916 dB at both ends of the range and never crosses anything.

Which sections move, and by how much #

The rung below’s second result is that the error is proportional to the designed Q, so a filter’s sections move by different amounts — and a fifth-order Chebyshev’s sections have quality factors that differ by nearly an order of magnitude.

The consequence for the temperature sweep is that the passband does not shift; it tilts, and it tilts by a different amount at each temperature. That is why the number that moves is the ripple rather than the corner: trimming one frequency cannot recover it at any temperature, and trimming it at one temperature makes it worse at another.

That last observation is the practical one and it is worth stating on its own. The temperature sweep is a walk along the gain–bandwidth curve the rung below already drew. Nothing new happens; what is new is that the walk is not optional, its two ends are set by a specification the customer chose, and its length is a property of how the amplifier’s bias was designed.

Where this leaves the design rule #

The rule of thumb the rung below examined is use an amplifier a hundred times the corner. It established that the rule is out by a decade at room temperature: a hundred times gives two per cent of Q error and it takes a thousand to recover a half-decibel design.

Temperature makes the required margin a function of the range the part must work over, and the factor is the ratio of the extreme absolute temperatures — 1.71 for the industrial range, 1.83 for the automotive one, and 1.0 for a part whose bias is proportional to absolute temperature.

So the corrected rule has two numbers in it rather than one: use an amplifier a thousand times the corner at the hot end of the range, and the hot end is where the gain–bandwidth is smallest, which is the end nobody measures at.

Telling the two amplifiers apart from outside #

Neither bias is on a data sheet by name, and the parameter that distinguishes them is usually not tabulated either. It is, however, a two-point measurement.

Measure the gain–bandwidth product at two temperatures — the unity-gain frequency of the open loop, which is one sweep of a follower — and take the ratio. A part with a fixed tail current gives the ratio of the absolute temperatures; a part with a proportional one gives one. Between 0 °C and 100 °C that is 1.37 against 1.00, which no measurement is going to confuse.

The same measurement answers the question the rung below leaves open about margin. The ideal amplifier, and where it stops being one computes where a closed-loop answer departs from the ideal one, from a gain–bandwidth product; if that product is itself a function of temperature then so is the departure, and the number to design against is the one at the hot end. Every boundary in this collection that has a gain–bandwidth in it inherits this, which is a large fraction of the feedback field.

The other four on the list #

Nine consecutive phases named temperature as an omission, and the last of them listed five specific places it bites. This essay is one of them. The other four are worth naming here rather than left as a sentence in a plan, because each has an essay already and each would move by an amount that is computable from what those essays already build.

The ceramic. The capacitance that is not one number pins a class II dielectric’s model with two data-sheet numbers, and both of them have temperature coefficients — a X7R is specified to ±15 per cent over its range and a Y5V to +22/−82. The three capacitances that essay separates would each move, and not together.

The core. The inductance the current decides has three saturation currents in it and all three are set by a saturation flux density that falls by roughly a third between 25 °C and 100 °C in a manganese-zinc ferrite. A part with thirty per cent of nameplate margin at room temperature has none at all when it is hot, and a switching converter is hot because it is switching.

The switch. The offset that knows the signal computes an injected charge from a gate area, an oxide capacitance and an overdrive, and the threshold voltage in that overdrive moves by a couple of millivolts per kelvin — so the offset that essay measures as 5.67 mV has a coefficient of its own, and the 0.186 per cent of gain error it decomposes into does too.

The board. The current that does not reach the input computes a guard’s worth from a surface resistance, and surface resistance is the most temperature-and-humidity-dependent quantity in this collection by a wide margin — orders of magnitude rather than per cent.

Four different mechanisms, four different sizes, and the common feature is that in each case a number the essay treats as a property of the part is a property of the part at a temperature.

What is not in this model #

No passive temperature coefficients at all, which is deliberate and is what makes the argument clean. A real film capacitor is a few hundred parts per million per kelvin and a metal-film resistor is tens, and both move the pole rather than the Q. Adding them would move the corner by a per cent or two over the range and would not change one number above.

No amplifier offset drift, which moves the direct-current output and not the response. The two millivolts a kelvin that a base–emitter voltage moves by is the same physics arriving in a different place, and it is the reason a bandgap reference exists at all — the same circuit that fixes this one’s tail current.

And no self-heating. The temperature above is ambient and uniform. An amplifier dissipating tens of milliwatts is warmer than its board by tens of kelvin, so the junction temperature is what sets the gain–bandwidth and the ambient is what a specification is written against. That is a fixed point between a dissipation and a temperature, of the kind the reverse-recovery ladder solves in a different field, and it is not solved here.

What it would take to remove #

Three cures exist and they are the usual three, with the usual costs.

Buy the bias. A part whose tail current is proportional to absolute temperature has none of this, and it is the cheapest cure by a wide margin because somebody else has already paid for it. What it costs is knowing which part that is, which the two-point measurement above settles in an afternoon.

Buy margin. A thousand times the corner instead of three hundred puts the ripple at 0.53 dB at the hot end. That is a factor of three in gain–bandwidth, which is roughly a factor of three in supply current for the same architecture, and on a three-section filter it is three amplifiers’ worth.

Or change the family. A Butterworth has no ripple to specify and its band-edge section has a Q of about 1.3 rather than 4.5, so the same amplifier moves it by a third as much. That is a real trade and what a steep skirt costs is where its price is measured: the skirt a Chebyshev buys is what is being given up, and it is being given up to a temperature range rather than to a filter requirement, which is not how a family is usually chosen.

The third of those is the one worth dwelling on. A family is chosen against a stopband requirement and a passband requirement, both stated at one temperature, and the measurement above says the choice also buys or sells a temperature range. Nothing in the design procedure asks about that, and the amount is large: the same amplifier that leaves a Butterworth inside its specification across the industrial range takes a Chebyshev outside it at eighty-nine degrees.

The habit this belongs to #

The site’s rule names three variables — frequency, amplitude and size — and a hundred and eighty essays have drawn boundaries in them. The fourth was in the list of omissions of nine consecutive phases and had never been an axis.

What it produces is not a new kind of result. It is the same identity the Q the amplifier decides measured, walked along, with its two ends chosen by a customer rather than by a designer — and the finding is that a half-decibel filter with perfect passives and an amplifier three hundred times faster than it needs fails a one-decibel specification at eighty-nine degrees, for a reason that is one line of a bias circuit.

The collection has a place for results of this shape and it is worth putting this one in it. The edges that move with the room gathers the boundaries that are functions of a temperature nobody stated, and its central observation applies here without change: the numbers are right, the condition attached to them was left off, and it is the same condition every time. What this essay adds to that collection is a case where the temperature reaches the answer through a bias arrangement rather than through a material property — the same transconductance, the same passives and the same design, with a coefficient of zero or of a factor of 1.4 across an industrial range depending on how the tail current is set. That is not a component choice a filter designer would think of as a filter decision, and it is the largest single term in the result.