The current the instrument draws

Every amplifier drawn on this site has had inputs that take no current. That is not an idealisation of a small quantity. It is an idealisation of a quantity whose size is decided by something outside the part — and that is a different kind of assumption, because the designer of the amplifier has no control over the thing that sets it.

Fifty nanoamps is nothing. Fifty nanoamps in a megohm is fifty millivolts, and an instrument reading fifty millivolts of its own input stage is not an instrument.

Two currents, and the design turns on the difference #

The bias current is what both inputs draw. It flows in whatever resistance each input looks back into, and it can be cancelled: make the two resistances equal, and the two voltages it produces are equal and subtract.

The offset current is what is left of it after the two inputs differ. Nothing symmetric removes it, because it is itself the asymmetry, and it is typically a tenth of the bias.

So the classical cure is worth the ratio of the two — a factor of ten on an ordinary part, not a factor of a thousand — and every number below follows from that.

The three curves are the whole picture and each is measured rather than assembled from an expression. The offset voltage is flat: fifty microvolts, wherever the source resistance is. The bias current times the imbalance between the two resistances is linear in the source and is what balancing removes. The offset current times the source is linear too and is what balancing leaves.

Unbalanced, the current overtakes the voltage at 1.77 kΩ. Balanced, at 10.0 kΩ, which is the offset voltage divided by the offset current exactly — the ratio of the two currents further along the axis, as promised.

Below a kilohm, balancing makes the reading worse, by a factor of about five at a hundred ohms. That half of the instruction is never stated: the feedback network is already the larger resistance there, so equalising means adding resistance to the source rather than removing it from the feedback, and the bias current’s contribution grows instead of cancelling. The instruction “add a resistor equal to the parallel combination of the feedback network” assumes the source is the larger of the two, and says nothing about the case where it is not.

The three slopes are what makes it diagnosable #

The figure’s real content is not the crossings but the slopes, because they are what lets a measurement say which error it is looking at.

The offset voltage is flat. The two current terms are linear in the source. So a bench measurement that reads the output offset at two source resistances a decade apart separates them completely: if the reading is unchanged the offset voltage dominates, if it goes up by ten the current does, and if it goes up by three the two are comparable and the crossing is nearby. No part number, no data sheet, no assumption about which of the three the part is bad at.

That is the same manoeuvre the two-wire ohmmeter essay uses to separate the lead resistance from the meter’s own loading, and the same one the common-impedance essay uses to separate a shared return’s contribution from a real signal: change one thing whose exponent differs between two candidate mechanisms, and read which curve the measurement is on. It works here because the three exponents are 0, 1 and 1 against source resistance and 0, 1 and 0 against the imbalance, so the two measurements together identify all three.

The second of those is worth having explicitly. Balancing the source changes only the middle term, so the difference between the balanced and unbalanced readings is the bias current times the imbalance, and what remains after balancing is the offset current times the resistance plus the offset voltage. Three unknowns, three measurements, no part number.

Balancing a large source means moving the feedback network #

There is a practical asymmetry hiding in that, and it is the reason the cure is more expensive than it sounds.

If the feedback network is the larger resistance, balancing is a resistor in series with the source. It costs a resistor and its own Johnson noise, and that is the version every textbook draws.

If the source is the larger — which is the case in which any of this mattered, because a bias current is only a problem in a large resistance — there is no resistor to add. The feedback network has to be scaled up to meet it, keeping its ratio and therefore the gain: for a gain of eleven with a hundred-kilohm source, the feedback network goes from a kilohm and ten to a hundred kilohms and a megohm.

That is a real cost and it is not the one the instruction mentions. A megohm in the feedback path is a noise source, it forms a pole with the amplifier’s own input capacitance and with the board’s stray, and it makes the circuit’s bandwidth depend on a capacitance nobody specified.

What the cure costs at the floor #

Balancing takes the density from 42.3 to 59.5 nanovolts per root hertz at a hundred-kilohm source — a factor of 1.41, which is √2, because both inputs now look back into the same large resistance instead of one of them.

The trade is worse than that ratio suggests, and the reason is what the two quantities are. The offset it removes is a constant. It can be measured once, at one temperature, and subtracted — by a trim, by a calibration cycle, by a digital correction — and what is left is its drift. The noise it adds cannot be subtracted at all.

So the cure is right when the drift of the uncorrected offset is the problem, and wrong when the resolution is. For a direct-current instrument that is calibrated at switch-on, balancing is usually the wrong move; for one that must be right without calibration, it is the only move.

There is a third source in the same figure and it is the one nobody adds. The bias current is a current, and a current has shot noise: √(2qI) per root hertz, flowing in the source resistance. Above 1.0 MΩ that shot noise exceeds the source resistor’s own thermal noise, and the boundary is 2kT/qI — the current and the temperature, with no resistance in it at all, which is the same shape as the universal crossing voltage the shot-noise essay computes between shot and Johnson noise in a junction. A picoamp-input part pushes that boundary out to fifty gigohms; a fifty-nanoamp part meets it at a megohm, which is squarely inside the range where the part is being used for its precision.

What a data sheet’s two numbers are worth #

The two currents are quoted together on every data sheet and the pair is more informative than either.

Their ratio is the whole value of the classical cure, so a part whose offset current is a tenth of its bias current offers a factor of ten and one whose offset current is half its bias current offers a factor of two — and the second kind exists. Bias-compensated parts, which cancel most of the bias current with an internal source, are exactly that: the bias figure falls by an order and the offset figure does not move, so the arrangement that was worth ten is now worth very little, and the balancing resistor is left contributing noise for almost nothing. A designer who reads only the first number sees an improved part and applies the same cure to it.

Their absolute size decides where the crossings are, and the crossings move together: doubling both currents halves both resistances. So the single most useful thing to compute from a data sheet is not either current but the two ratios of offset voltage to bias current and of offset voltage to offset current, which are the two resistances at which the part changes character. For a fifty-nanoamp part they are 1.8 kΩ and 10 kΩ; for a twenty-picoamp part they are 2.4 MΩ and 25 MΩ; and for a bias-compensated part they are close together, which is the signature of the cure being worthless.

The input that has no path, which is not a large error #

Ask the netlist what the offset is when the input is capacitively coupled and the answer is not a big number. It is no number.

A bias current has to flow somewhere. With a capacitor between the source and the input, or a transformer, or a photodiode with no shunt resistance, there is no direct-current path, so the node’s potential is not determined by the network at all and the nodal matrix is singular. The solver refuses by name rather than returning a confident answer for a circuit that has none, which is the same refusal the singular-network essay is built on.

That refusal is the correct engineering answer as well as the correct numerical one. In a real circuit the input charges until something stops it — the amplifier’s own protection diodes, the input stage saturating, the capacitor’s leakage coming into balance — and which of those it is decides the answer, so the question has no answer until the designer supplies one. The instruction that falls out is not “add a large resistor to be safe”; it is that a bias-current return path is a component with a value, and its value is set by the offset it is allowed to produce.

The part chosen for picoamps #

A bipolar input’s bias current is a base current — a collector current divided by β — and both move slowly with temperature and in opposite directions, so it is nearly flat. A field-effect input’s is a reverse-biased junction’s leakage, which doubles about every ten kelvin.

At 300 K the field-effect part is five hundred times better and the choice looks settled. They cross at 111 °C, which is inside the operating range of a great many things and well inside a self-heating die’s.

So the instruction “use a FET-input part for a high source impedance” is a statement about a temperature as much as about an impedance, and the temperature clause is the one that is never in it. A design that is right on the bench at 25 °C and wrong in an enclosure at 85 is not a marginal design; it is a design whose dominant error term has changed identity.

The same doubling is why a field-effect input’s specification is quoted at 25 °C and its curve is printed on a logarithmic axis: there is no single number, and the number that is given is the best one.

Where this sits beside the rest of the field #

This field has measured several ways in which an instrument disturbs what it is reading — the probe that loads the node, the ammeter that is a resistance in the circuit, the return conductor whose millivolts appear in somebody else’s reading. All of those are the instrument’s impedance acting on the circuit’s signal.

This one is different in kind: the instrument is a source, and what it injects is a current that the circuit’s own impedance turns into a voltage. That is why the error grows with source resistance rather than falling with it, and why every other cure in the field — buffering, shortening, guarding — makes it worse rather than better. A buffer with a high input impedance has, almost by definition, a bias current that is the problem’s whole content.

It also sets up the arrangement two essays along. A transimpedance amplifier’s input is a summing junction with a megohm of feedback around it, and the bias current flows in that megohm and appears at the output as fifty millivolts of offset on a signal that may be microvolts. That essay measures the noise and the stability of the arrangement and treats the bias current as a given; the number it is given is the one measured here.

What is not modelled #

The bias current is a constant. It is not: on a bipolar input it depends on the collector current and therefore on the supply, and on a field-effect input it depends on the voltage across the input junction and therefore on the common-mode level. Both dependences are of order tens of per cent over a part’s range, which does not move any boundary here and does mean that a bias current measured at one operating point is not the one a circuit will see at another.

And it has a sign. A bipolar NPN input draws current into the amplifier and a PNP input pushes it out, and a bias-compensated part — one with a current source arranged to cancel most of it — has a bias current whose sign is not specified at all, only its magnitude. That last case is the awkward one, because the compensation removes most of the bias and leaves an offset current that is now comparable to it, so the balancing resistor’s benefit collapses to nothing while its noise cost remains.

Nothing here drifts. The offset voltage has a temperature coefficient of a microvolt or two per kelvin, and the whole comparison between the flat curve and the two rising ones moves with it. What that changes is where the crossing sits over temperature, which is a band rather than a line, and the band is wider than the figure suggests.

And the amplifier is otherwise perfect. No finite gain, no common-mode rejection error, no input capacitance. The last of those matters most: the balancing resistor forms a pole with the input capacitance, so the cure that equalises the two direct-current resistances unbalances the two time constants, which is exactly the imbalance the instrumentation amplifier’s fourth rung measures a rejection corner from. The two cures are in direct opposition and this essay measures only one of them.

What the gate checks #

Each of the three contributions is measured by a solve with the other two set to zero, and the site’s gate asserts that they sum to the total — which is not free, because the circuit is a network and superposition of three independent sources through it is a claim about the network rather than about arithmetic.

The unbalanced crossing is asserted to be near the offset voltage divided by the bias current, and the balanced one to be exactly the offset voltage divided by the offset current, to two per cent. Two assertions rather than one, because the first has the feedback network’s own resistance in it and the second does not, and asserting only the loose one would hide the exact result.

Balancing is asserted to be worth close to the ratio of the two currents wherever the source is large, and to be worth less than nothing where it is small — the half of the instruction nobody states, kept as a live check rather than as a note.

The noise assertion requires that balancing never lowers the density at any source resistance drawn, and that it costs at least a fifth of it somewhere in the range, so a version whose cure were free would fail.

And the shot-noise crossing is asserted against 2kT/qI rather than against a resistance, which is what makes it a statement about the current rather than about this particular part.

That last sentence is where this essay hands over. The current that does not reach the input is the subject it becomes: a teraohm across a board from a fifteen-volt rail is fourteen millivolts of error through a gigohm source, and a humid morning takes that resistance down two decades. A ring held at the input’s own potential leaves a nanovolt — the error becoming proportional to the signal rather than to the rail, so an offset has turned into a gain error of one part in a billion — and the same wire multiplies the input resistance by the loop gain, making it $10^{18}$ Ω at direct current and $10^{12}$ Ω at a megahertz.

Which is the shape of the whole ladder. At fifty nanoamps the input current is a property of the part and the design question is what resistance it flows in. At twenty picoamps it is not the part’s at all: it is the board’s, it depends on the weather, and the repair is a copper ring rather than a better amplifier.