The drift a tee multiplies

The tee that charges for its own compensation found what a feedback tee costs a photodiode amplifier. The tee reaches a large transimpedance with small resistors, $R_1(1 + R_2/R_3) + R_2$, and the tap that makes it sets the stage’s noise gain at direct current to $1 + R_2/R_3$ — nine, for the essay’s tee, where a single feedback resistor of any value gives exactly one, because the photodiode is a capacitor and at direct current the source is an open circuit. The compensation that costs nothing then priced that noise gain against the alternative, a capacitor across a single resistor, and found the capacitor better at every phase margin.

Both essays priced the tee’s noise gain in noise. It is a gain, and it multiplies everything that enters where noise enters: the amplifier’s input offset among the rest. A gain at zero frequency does not care whether what it multiplies fluctuates or sits still. The second essay’s closing list put it plainly — the tee multiplies the amplifier’s offset, and so its offset drift, and over an industrial range that is a direct-current error nobody prices at the time the tee is chosen. This essay prices it, and finds the price depends on which other error it is standing beside.

The two direct-current errors #

A photodiode amplifier at direct current has two error sources that matter, and they enter at different places.

The offset is a voltage in series with the amplifier’s input. It reaches the output multiplied by the noise gain: one for a single resistor on a capacitive source, since nothing ties the summing node to ground but the diode’s shunt resistance — a gigaohm here — and the tee’s own factor for a tee.

The bias current is a current drawn from the summing node. It flows through the feedback network exactly as a signal current would, so it reaches the output multiplied by the transimpedance — and a tee and a single resistor of the same transimpedance multiply it identically. That is the definition of their being the same transimpedance.

Both are trimmed or calibrated out at one temperature, and what is left is how each moves with temperature. The offset of a good amplifier drifts by about a microvolt a kelvin. The bias current of a field-effect input is a junction leakage and doubles roughly every ten kelvin. The figures take those two numbers, a 10 MΩ transimpedance, and a calibration at 25 °C.

Ten times the offset, and the same bias current #

At −40 °C the offset has drifted by 65 µV. Through a single resistor that reaches the output once and reads as 6.56 pA of input current; through the tee of factor ten it reaches the output ten times and reads as 65.1 pA. The ratio is the tee’s factor to within the diode’s shunt resistance, which the figure requires. The bias current, meanwhile, has fallen to a ninetieth of its room value, and its contribution through the two networks is identical to the printed digit — required to a part in a million.

So the tee’s penalty is a pure multiplication of one of the two errors, and whether it matters depends on which error is larger. At the hot end the bias current has doubled ten times over by 125 °C — a nanoampere — and it dominates both networks: through the single resistor it overtakes the offset term at 40 °C, and through the tee, whose offset term is ten times larger, at 85 °C. Above 85 °C the tee and the single resistor are equally wrong, and the choice between them costs nothing. Below it the tee is wrong by its factor.

A tee of a hundred makes the offset term a hundred times larger, 650 pA at −40 °C, and the bias current never catches it inside the range drawn. For that tee the error at every temperature is the offset’s, multiplied, and the single resistor’s advantage is the full factor of a hundred at the cold end.

Where a microvolt a kelvin comes from #

The offset drift the figures use is a round number, and it is worth saying where such numbers come from, because the tee multiplies whatever the amplifier’s input stage has. What matching does about temperature derives it for a bipolar pair: a mismatch between the two transistors’ saturation currents appears as an offset proportional to absolute temperature, so its drift is the offset divided by the temperature — a millivolt of untrimmed offset at 300 K drifts 3.3 µV/K. Trimming the offset at room temperature removes the offset and not that drift, unless the trim is made in a way that tracks.

A field-effect input’s offset drifts for a different reason and by a comparable amount, and its bias current — the other error on this page — is a gate leakage, which is why it doubles every ten kelvin rather than following anything linear. The pair of numbers used here, a microvolt a kelvin and a picoampere doubling every ten, is the pair a precision field-effect amplifier is specified by; a bipolar input would have a much larger bias current, which hides the tee’s penalty at every temperature, and a correspondingly larger noise current, which the error a bigger resistor cannot help is about.

The single resistor’s own curve is the baseline, and it already has the shape the tee exaggerates: an offset term symmetric about the calibration temperature and a bias term that is negligible below it and dominant above it. The single resistor’s crossover is at 40 °C, only fifteen kelvin above the calibration; with a tee of ten it moves to 85 °C, and with a tee of a hundred off the top of the range. The tee does not change the shape of the problem, only where the two errors meet.

The transimpedance above which it hides #

The offset’s error, as an input current, is the offset times the noise gain divided by the transimpedance, so it falls as the transimpedance rises. The bias current’s does not fall at all. So for any tee there is a transimpedance above which the bias current’s error dominates at every temperature, and there the tee’s multiplication is hidden.

The worst error of a tee of ten comes within a factor of two of the single resistor’s from about 10 MΩ; a tee of a hundred, from about 100 MΩ. Below those transimpedances the tee’s drift penalty is the whole of its worst-case error; above them, the bias current has taken over and the tee costs nothing in drift.

That turns the choice into a comparison of two quantities a designer already has. The tee is used because a single resistor of the required value is impractically large — above a few hundred megohms resistors are expensive, slow and leaky. Those are exactly the transimpedances at which the bias current dominates the drift anyway. The tee’s drift penalty is largest where a tee is least needed and smallest where it is most needed, which is an unusually benign coincidence, and it holds only as long as the bias current is large enough to hide behind.

The reasons a large single resistor is avoided are worth keeping beside that. Its Johnson noise current falls as its value rises — the floor a resistor sets is the arithmetic — so on noise alone a large resistor is the better choice, and it is the practical defects that push designers to a tee: the stray capacitance across a resistor of hundreds of megohms, its voltage coefficient, and its surface leakage, which is its own drift. A tee trades those for the offset multiplication, and the trade is good exactly where the resistor’s defects are worst.

An amplifier whose bias current is small #

The coincidence depends on the bias current, and the direction of amplifier design is to make it smaller. An amplifier with a femtoampere of bias current — the electrometer grade that the current the instrument draws is concerned with — moves the hiding transimpedance up by the same thousandfold. For such an amplifier the offset’s drift, multiplied by the tee, is the dominant direct-current error at every transimpedance a tee would be used for, and the tee’s factor is its drift penalty in full.

That is why the choice between a tee and a large resistor is not settled by the noise argument alone. The noise argument said the tee costs its factor in amplifier noise; the drift argument says it costs its factor in offset drift, and the second cost is only excused by a bias current that a better amplifier removes. A designer who upgrades the amplifier to lower its bias current and keeps the tee has made the drift worse relative to everything else, not better.

Which drift a calibration leaves #

The figures calibrate at 25 °C, and that choice is worth making explicit because it decides which errors are drifts. An offset trimmed at room temperature is zero there and grows linearly either way; a bias current calibrated at room temperature is subtracted as a constant, and what is left is its growth — tiny below room temperature, since the current has fallen, and exponential above it. So the two drifts have opposite shapes: the offset’s is symmetric about the calibration point and the bias current’s is almost all on the hot side.

That asymmetry is why the tee’s penalty lives at the cold end. At −40 °C the offset has drifted as far as at 90 °C, and there is no bias current to hide behind. An instrument that operates only warm — inside a heated enclosure, say — would see a much smaller tee penalty than one that has to work outdoors in winter, and a drift specification written only at the hot extreme would miss the tee’s contribution entirely.

The earlier estimate, restated #

The earlier essay’s closing list said the tee multiplies the amplifier’s offset “by its factor less one”. The solved network says it multiplies the offset’s contribution by its factor: ten for a tee of ten, to within the diode’s shunt resistance. The two statements agree about the increase — a tee of ten adds nine times the single resistor’s offset error — and the multiplication is the cleaner way to put it, since it is the direct-current noise gain and the noise argument uses the same number.

The tee’s other bill, for comparison #

The two bills are one bill. The tee’s direct-current noise gain multiplies the amplifier’s noise across the band where the noise gain is high, and it multiplies the amplifier’s offset at the one frequency where offset lives. The noise essay found the tee buying nothing in signal-to-noise ratio; this one finds it buying nothing in drift either, except where the bias current’s own drift has already made the offset irrelevant.

An error budget, written with the tee in it #

It helps to write the direct-current budget for the 10 MΩ amplifier as a designer would, because the tee changes which line of the budget is the largest.

With a single resistor, over −40 to 85 °C, the budget’s largest lines are the offset drift at 6.6 pA at the cold end and the bias current’s growth, 63 pA at the hot end; the resistor’s own temperature coefficient scales the reading rather than adding to it; and the photodiode’s dark current, which is a signal the amplifier cannot distinguish from light, sits beside them. The bias current is the line to work on, and a designer choosing a lower-leakage amplifier is working on the right thing.

With a tee of ten the offset line becomes 65 pA at the cold end, as large as the bias current’s worst at the hot end, and the budget has two equal lines at opposite ends of the range. With a tee of a hundred the offset line is 650 pA and stands alone, ten times anything else; the bias current and the dark current — the quantities the rest of the design has been working to reduce — are invisible beneath it.

That is the practical form of the result. Where the trouble is at the input is the reminder that a photodiode amplifier’s difficulties are at its input, and the tee moves one of them — the offset, multiplied — to the head of the list. The factor the expression leaves out makes the same point about the tee’s noise gain in the frequency domain: the transimpedance expression has no term for it, and neither does a direct-current budget that lists the offset once.

How the errors were solved #

Each error is a direct-current solve of the netlist: an ideal amplifier forcing its inputs equal, the offset as a voltage source in series with the non-inverting input, the bias current as a current drawn from the summing node, the photodiode as its gigaohm of shunt resistance only, and the feedback network as a single resistor or a tee of 10 kΩ from the tap to the output with the tap resistor and the input resistor chosen for the stated factor and exactly the stated transimpedance.

Each drift is the difference between the error at a temperature and at 25 °C, with the offset drifting at 1 µV/K and the bias current doubling every 10 K from 1 pA. Checked: the tee’s offset contribution is its factor times the single resistor’s, to within the shunt resistance’s own effect; the bias current’s contribution is the same through both to a part in a million; and the transimpedance above which each tee’s worst error is within a factor of two of the single resistor’s rises with the factor.

A shunt resistance, resistors and a drift that stay put #

The shunt resistance is constant. A photodiode’s shunt resistance falls with temperature, roughly halving every ten kelvin like the bias current, and as it falls it raises the single resistor’s noise gain towards the tee’s — narrowing the tee’s penalty at the hot end, where it was already hidden.

The resistors do not drift. A tee’s ratio of $R_2$ to $R_3$ sets its factor and so its transimpedance, and resistors of different values drift differently; the tee’s transimpedance has a temperature coefficient of its own that a single resistor does not share.

And the offset drift is one number. An amplifier’s drift is specified as a maximum, and its sign varies from part to part, so a population of amplifiers has drifts of both signs; the tee multiplies each part’s by the same factor.

Still open: the tee with a capacitor across it, the stray that forces a tee, and the ratio’s own drift #

A tee with its own capacitor. A capacitor across the tee acts only at high frequency, so it should supply margin without touching the direct-current noise gain — and so without touching either the noise penalty at low frequency or the drift penalty here. Whether any combination approaches the single resistor’s noise curve is still unmeasured.

The stray that forces a tee. A single resistor’s own stray capacitance sets a floor on the feedback capacitance and so a ceiling on its band. Above some transimpedance that ceiling falls below what a tee could deliver, and there the tee’s noise and drift become the price of the band rather than costs for nothing — the transimpedance at which that happens, for stated strays, is the measurement that would complete the comparison.

The ratio’s own drift. A tee of factor ten built from a 10 kΩ and a 1.1 kΩ resistor has a transimpedance that drifts with the difference of their temperature coefficients, which is a gain error the single resistor has only through its own coefficient. For resistors of ordinary tracking, whether that gain drift is larger or smaller than the offset drift measured here is a number.