The cure that needs one current

The current the instrument draws and the cure that becomes a different circuit measured the classical cure for an amplifier’s input current — make the resistances its two inputs look into equal — and found it worth “a factor of ten, not a thousand”: the ratio of the bias current to the offset current on a good bipolar part. The current a capacitor adds up found the same factor of ten lengthening an integrator’s usable time.

All three took the two currents as a data sheet quotes them: a bias current that both inputs draw, and a smaller offset current that is their difference. That description contains an assumption that holds for some amplifiers and fails badly for others. It assumes the two currents are mostly one current — a base current flowing into two matched transistors — with a small difference between them. Many modern precision amplifiers cancel their input current internally, by injecting a matched current back into each input, and what remains at each pin is a residue of the cancellation: small, of either sign, and with no particular relation to the residue at the other pin.

For such a part the cure’s premise is false, and this essay measures what the cure then does — over populations of parts rather than one typical part, because the question is statistical: it is about how much two currents share.

Populations of parts, each solved #

The stage is the non-inverting amplifier of the earlier measurements, a gain of eleven, solved as a netlist with each input current a current source at its pin. What changes is that there is no longer one part. A population of four hundred parts is drawn from a seeded generator, so that the same population appears every time the figure is drawn, and each part’s two input currents are built from a shared component and two independent ones:

$$I_\pm = \mu + \sigma\left(\sqrt{\rho}\,z_0 + \sqrt{1-\rho}\,z_\pm\right),$$

with $z_0$ common to both pins and $z_+$, $z_-$ each pin’s own. The mean μ is what every part draws, the spread σ is how parts differ, and the correlation ρ is how much of that difference the two pins share. Each part is then solved twice at a one-megohm source, with the inputs unbalanced and balanced, and the error referred to the input is read. The figures quote root-mean-square errors over the population.

Over the population the netlist is doing something simple. Unbalanced, the error is the non-inverting pin’s current in the source resistance; balanced, it is the difference between the two pins’ currents. So what balancing is worth is the ratio of the spread of one pin’s current to the spread of the difference, and everything below follows from what the two pins share.

For a zero-mean current, the correlation is everything #

Take a part whose input current has been cancelled to zero on average, so μ is zero and only the part-to-part spread remains. The error unbalanced is σ times the source resistance; balanced, it is $\sigma\sqrt{2(1-\rho)}$ times the source resistance, because the difference of two equally spread currents with correlation ρ has that spread. Balancing is worth the ratio, $1/\sqrt{2(1-\rho)}$.

That expression crosses one at ρ = ½. Above it, balancing helps, and by a lot as the correlation approaches one: at 0.99 it is worth 6.5 over four hundred solved parts. Below it, balancing hurts: with independent residues it is worth 0.72, and the error is 38 per cent larger with the cure than without it. The cure subtracts one pin’s current from the other’s, and if the two currents are unrelated, the difference of two independent quantities has a larger spread than either.

That is the whole content of the figure, and it inverts the instruction as usually given. “Balance the input resistances to remove the bias current” is a statement about a part whose two currents are mostly one current. On a part whose two currents are two currents, the same instruction adds a resistor’s noise and a larger error.

What the two kinds of input look like #

The distinction is not hypothetical, and drawing two populations side by side shows how different the two kinds of input are.

On a plain bipolar input the two currents are base currents of two matched transistors carrying the same collector current, so they rise and fall together from part to part: a part with a low current gain has a high current at both pins. The cloud lies along the diagonal, and what distinguishes one pin from the other — the offset current — is the thin width of the cloud. On a cancelled input the cloud is round. The cancellation circuit trims each pin to near zero, and what is left at each pin is the error of its own trim, unrelated to the other’s.

The residue has either sign #

A second property of cancelled inputs follows from the same picture, and it matters to anyone who leaves an input unconnected or relies on the input current to bias a node.

A plain bipolar input always draws its base current in the same direction: into an NPN input stage, out of a PNP one. Its sign is a property of the part type, printed on the data sheet. A cancelled input’s residue has either sign, and in the seeded population a little over half the parts push current out while the rest draw it in. So the sign of the error a source resistance produces is not knowable from the type number, and a circuit that relies on the input current to pull a floating node one way — a disconnected-sensor detector, a bias path through a large resistor — will work on some parts and fail on others.

What a data sheet’s two numbers say #

A designer usually has no population to draw, only two numbers from a data sheet: an input bias current and an input offset current. Those two numbers already contain the answer.

Over every population drawn — zero-mean currents at nine correlations, and two with a shared mean — the benefit of balancing is the ratio of the input current to the offset current, to within three per cent. That is the factor of ten the earlier measurements found, now seen as a general law rather than a property of one part: a plain input with an offset current a tenth of its bias current gains ten; a cancelled input whose data sheet quotes an offset current about equal to its bias current gains nothing; and a part whose offset current exceeds its bias current loses.

Data sheets of cancelled-input amplifiers do quote offset currents comparable to, or larger than, their bias currents, and they are telling the designer exactly this. Read as a ratio, the two numbers are an instruction: balance if the offset current is much smaller, and do not if it is not.

A shared mean is removed whatever the spread #

The zero-mean case isolates the correlation. Most parts have a mean as well — a plain bipolar input’s 50 nanoamps is mostly mean — and a mean is shared by both pins by definition.

The benefit grows with the mean in proportion: $\sqrt{\mu^2 + \sigma^2}/(\sigma\sqrt{2(1-\rho)})$. At a correlation of a half, where the spread contributes nothing either way, a mean eight times the spread makes balancing worth eight. So the cure works on a plain bipolar input for two reasons at once — its two currents are correlated, and most of each is a mean — and fails on a cancelled input for both reasons at once: the cancellation has removed the mean, and with it the correlation the mean was carrying.

That gives the cleanest statement of what balancing does. It removes the common part of the two input currents and leaves the difference. How large the common part is, relative to the difference, is what a data sheet’s ratio of bias to offset current measures, and it is the only number that decides whether the cure is worth its resistor.

The same arithmetic as a ratio of resistors #

The law that balancing removes the shared part and keeps the difference is not special to input currents. It is the arithmetic of every circuit that works by subtraction, and the rest of this subject has met it under other names. What a ratio does not see found it in resistors: parts from one reel share part of their error, and a divider that uses them as a ratio cancels exactly the part they share — four one-per-cent resistors sharing ninety per cent of their error make a far better ratio than four independent ones, and four sharing none make a ratio no better than its parts. A balanced input pair is the same divider in the current domain. Its benefit is the ratio of what the two pins share to what they do not, and it is large only when the sharing is large.

The correlation that matters is also the one a population of parts carries rather than one part. A single amplifier has two fixed currents and a fixed difference between them; what the correlation describes is how those numbers vary across a production run. So a designer who measures one sample, finds its offset current small, and concludes the cure will work, has measured one point of the scatter and not its shape. The shape a tolerance does not print makes the same point for resistors: two populations that honour the same tolerance can behave entirely differently in a circuit, because the circuit responds to how the values are distributed and the tolerance does not say.

Why a matched pair correlates and a trim does not #

The plain input’s correlation of 0.991 is not an accident of the model; it is what matching produces. Two transistors made side by side on one die share their doping, their temperature and their geometry, so their current gains are nearly equal and their base currents rise and fall together from die to die. What a pair cancels, and what it only halves measured how much a matched pair removes from its own transfer characteristic by the same symmetry, and what matching does about temperature found that what a pair leaves behind is set by the part of the two devices that is not shared — a saturation-current mismatch whose offset is proportional to absolute temperature.

A cancellation circuit breaks that sharing on purpose. It measures each input’s current and injects an equal and opposite one, so the common part — the base current both transistors draw — is removed at each pin separately. What is left at each pin is the error of its own cancellation, which depends on that pin’s own injection circuit, and the two injection circuits’ errors are no more related than any two independent trims. The trim that takes the drift with it found that a trim acting inside the mechanism it corrects can carry its drift away with it, and one acting outside cannot; an input-current cancellation acts outside the base current it corrects, and leaves two unrelated residues behind.

So the cure and the cancellation are two attempts at the same job, and they do not combine. The cancellation removes the common part at the source, inside the amplifier; the cure removes it outside, in the resistor network. Once the first has been done there is nothing common left for the second to remove, and the second only adds its resistor’s noise and the difference of two unrelated currents.

What the resistor costs #

The cure is never free, and on a cancelled input the price is paid for nothing. The balancing resistor adds its own Johnson noise at the non-inverting input — the cure that becomes a different circuit measured it settling at $\sqrt{2}$ of the source’s noise once the network is scaled to the source — and, at high source impedances, its own capacitance and a pole with the input capacitance. On a plain input those costs buy a tenfold reduction in error, a good trade. On a cancelled input they buy a 38 per cent increase.

The noise cost also has the same correlation structure. The amplifier’s own current noise is two noise currents at two pins, and whether a balancing resistor cancels part of their effect depends on how correlated they are, which for a bipolar input is set by its input-stage tail current and for a cancelled input is not the same thing at all. The current the instrument draws found the cure costing forty per cent of the noise density on a plain part; on a cancelled part the same forty per cent is spent with no offset benefit to set against it.

The practical recommendation that follows is unglamorous: check the data sheet’s ratio before adding the resistor. Most precision amplifiers sold for high-impedance work are of the cancelled kind, and their application notes say, in small print, that the balancing resistor should be omitted. The cure that every textbook teaches is correct for the amplifier the textbook was written about.

What the measurement leaves out #

The populations are Gaussian with a single correlation. Real parts are not; a production lot has a structure — wafers, trim steps, a temperature at final test — and a cancelled input’s residue may correlate with its own offset voltage or with temperature in ways a single ρ does not capture. The law found here, benefit equals current over offset current, is a statement about second moments and holds for any distribution with those moments; what the shape changes is how often a particular part falls far from the root-mean-square.

The currents are constant in temperature. A cancelled input’s residue is the difference of two currents that drift, and it can change sign across temperature; the current the instrument draws measured how differently bipolar and field-effect inputs drift, and a cancelled input is a third behaviour with its own data-sheet curve.

And the stage is one non-inverting amplifier at one source resistance. At source resistances small enough that the offset voltage dominates, none of this matters, and the crossing where it starts to matter is the one the earlier measurements located at $V_{os}/I_B$; for a cancelled input with a small bias current that crossing moves to a much higher resistance, which is the reason such parts exist.

The number to keep #

Balancing removes what the two input currents have in common and leaves their difference. It is worth exactly the input current over the offset current, read as root-mean-square over parts — tenfold on a plain bipolar input whose two pins are one current, correlated at 0.991. For a zero-mean, cancelled input it is worth $1/\sqrt{2(1-\rho)}$: nothing at ρ = ½, and at the measured correlation of 0.10 a 38 per cent increase in error. A cancelled input’s residue has either sign, 54 and 46 per cent here, so its error’s sign is not knowable from the part number. A data sheet’s ratio of bias to offset current is the instruction.

Still open: noise correlation, drift that changes sign, and the bias current as a function of voltage #

The noise currents’ correlation. A bipolar input’s two noise currents share part of the tail current’s shot noise; a cancelled input’s do not. The same population method applied to noise rather than offset would say when a balancing resistor cancels part of the amplifier’s current noise and when it only adds its own.

A residue that changes sign with temperature. A cancelled input’s residue is the difference of two drifting currents and can pass through zero. Over a temperature range the error then has a minimum that the room-temperature data does not predict, and its location varies part to part.

Current that depends on the input voltage. A cancelled input’s residue is often a strong function of the common-mode voltage, which turns the offset into a nonlinearity in a follower. Measured as a function of the input, it is a distortion rather than an offset.